window.FINEPROOFS_DISTRIBUTION.problems.push(...[{"i":"f3f3302a365eba13","q":"In a acutangle triangle  $ABC, \\angle B>\\angle C$ . Let  $D$  the foot of the altitude from  $A$  to  $BC$  and  $E$  the foot of the perpendicular from  $D$  to  $AC$ . Let  $F$  a point in  $DE$ . Prove that  $AF$  and  $BF$  are perpendiculars if and only if  $EF\\cdot DC=BD\\cdot DE$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.08027,"x":0.27677,"p":[[0,19,0.0,0.16072,0.13243,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,13,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.12938,0.21237,0.0,0.07,0.14286,0.0,0.85714,16,0,0,16,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[8,19,0.4211,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.08027,0.09401,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.23652,0.14993,0.14286,0.14288,0.42857,0.0,0.4286,4,0,0,4,0,13,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.27677,0.14696,0.14286,0.28571,0.42857,0.0,0.571,3,0,0,3,0,8,0,0,10,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.4286,"k":"rising","v":0.14286,"x":0.433,"p":[[0,15,0.0,0.14286,0.17857,0.0,0.14286,0.14287,0.0,0.85714,13,0,0,13,0,12,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[4,15,0.2667,0.14724,0.14934,0.0,0.14286,0.14287,0.0,0.71429,9,0,0,9,0,18,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[8,15,0.5333,0.34371,0.23916,0.25,0.28571,0.4286,0.0,1.0,4,1,0,4,0,4,0,0,12,0,0,6,0,0,3,0,0,0,0,0,2,0,1],[12,15,0.8,0.433,0.24608,0.2857,0.42859,0.57111,0.0,1.0,2,1,0,2,0,4,0,0,7,0,0,8,0,0,4,0,0,4,0,0,2,0,1],[15,15,1.0,0.42852,0.20199,0.28571,0.42857,0.571,0.0,1.0,1,1,0,1,0,2,0,0,10,0,0,8,0,0,8,0,0,1,0,0,1,0,1]]}]},{"i":"37c3465db549e965","q":"The degrees of polynomials  $P$  and  $Q$  with real coefficients do not exceed  $n$ . These polynomials satisfy the identity\n\\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \\]\nDetermine all possible values of  $Q \\left( - \\frac{1}{2} \\right)$ .","t":[{"b":0,"e":0.57143,"k":"flat","v":0.44642,"x":0.60268,"p":[[0,54,0.0,0.53125,0.27019,0.28571,0.42857,0.57143,0.14286,1.0,0,7,0,0,0,1,0,0,9,0,0,9,0,0,6,0,0,0,0,0,0,0,7],[4,54,0.0741,0.59375,0.29904,0.39286,0.42857,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,7,0,0,10,0,0,3,0,0,0,0,0,1,0,10],[8,54,0.1481,0.60267,0.27137,0.42857,0.57121,1.0,0.0,1.0,1,9,0,1,0,0,0,0,2,0,0,12,0,0,8,0,0,0,0,0,0,0,9],[12,54,0.2222,0.60268,0.24152,0.42857,0.57143,0.67857,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,13,0,0,9,0,0,0,0,0,0,0,8],[16,54,0.2963,0.44642,0.12241,0.42857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,4,0,0,14,0,0,12,0,0,0,0,0,0,0,0],[20,54,0.3704,0.48214,0.10564,0.42857,0.42857,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,18,0,0,11,0,0,0,0,0,1,0,0],[24,54,0.4444,0.46429,0.12372,0.42857,0.42857,0.46429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,21,0,0,7,0,0,0,0,0,0,0,1],[28,54,0.5185,0.52231,0.18073,0.42857,0.49979,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,12,0,0,13,0,0,0,0,0,0,0,3],[32,54,0.5926,0.49107,0.12846,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,15,0,0,13,0,0,0,0,0,0,0,1],[36,54,0.6667,0.48214,0.08564,0.42857,0.42859,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,16,0,0,14,0,0,0,0,0,0,0,0],[40,54,0.7407,0.50893,0.1234,0.42857,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,13,0,0,16,0,0,0,0,0,0,0,1],[44,54,0.8148,0.49549,0.07969,0.42857,0.49979,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,15,0,0,16,0,0,0,0,0,0,0,0],[48,54,0.8889,0.4732,0.08326,0.42857,0.42857,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,18,0,0,12,0,0,0,0,0,0,0,0],[52,54,0.963,0.47768,0.07668,0.42857,0.42857,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,19,0,0,12,0,0,0,0,0,0,0,0],[54,54,1.0,0.49107,0.08702,0.42857,0.5,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,14,0,0,16,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.35714,"x":0.60714,"p":[[0,56,0.0,0.60714,0.26,0.42857,0.50001,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,12,0,0,6,0,0,0,0,0,2,0,8],[4,56,0.0714,0.45982,0.15458,0.39286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,11,0,0,12,0,0,0,0,0,0,0,1],[8,56,0.1429,0.45982,0.17029,0.39286,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,15,0,0,7,0,0,0,0,0,0,0,2],[12,56,0.2143,0.45982,0.16263,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,12,0,0,9,0,0,0,0,0,1,0,1],[16,56,0.2857,0.41964,0.18536,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,10,0,0,14,0,0,4,0,0,0,0,0,0,0,2],[20,56,0.3571,0.41963,0.12338,0.28571,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,2,0,0,7,0,0,14,0,0,9,0,0,0,0,0,0,0,0],[24,56,0.4286,0.35714,0.12877,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,16,0,0,12,0,0,1,0,0,0,0,0,1,0,0],[28,56,0.5,0.44196,0.10326,0.42857,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,15,0,0,10,0,0,0,0,0,0,0,0],[32,56,0.5714,0.41518,0.10326,0.28571,0.42857,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,15,0,0,7,0,0,0,0,0,0,0,0],[36,56,0.6429,0.43303,0.10999,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,13,0,0,10,0,0,0,0,0,0,0,0],[40,56,0.7143,0.39284,0.09447,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,12,0,0,16,0,0,4,0,0,0,0,0,0,0,0],[44,56,0.7857,0.40177,0.08326,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,20,0,0,3,0,0,0,0,0,0,0,0],[48,56,0.8571,0.41518,0.11495,0.28571,0.42857,0.46429,0.14286,0.57143,0,0,0,0,0,1,0,0,9,0,0,14,0,0,8,0,0,0,0,0,0,0,0],[52,56,0.9286,0.43307,0.10997,0.28571,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,13,0,0,10,0,0,0,0,0,0,0,0],[56,56,1.0,0.41964,0.10062,0.28571,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,16,0,0,7,0,0,0,0,0,0,0,0]]}]},{"i":"7b175bb9f02b9ed7","q":"Prove the following inequality:  $$  \\frac{1}{\\sqrt[3]{1^2}+\\sqrt[3]{1 \\cdot 2}+\\sqrt[3]{2^2} }+\\frac{1}{\\sqrt[3]{3^2}+\\sqrt[3]{3 \\cdot 4}+\\sqrt[3]{4^2} }+...+ \\frac{1}{\\sqrt[3]{999^2}+\\sqrt[3]{999 \\cdot 1000}+\\sqrt[3]{1000^2} }> \\frac{9}{2} $$  (The member on the left has 500 fractions.)","t":[{"b":5,"e":0.42857,"k":"falling","v":0.51786,"x":0.96875,"p":[[0,22,0.0,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[4,22,0.1818,0.90625,0.22477,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,26],[8,22,0.3636,0.87052,0.24055,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,4,0,22],[12,22,0.5455,0.83036,0.27302,0.57143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,0,0,0,1,0,22],[16,22,0.7273,0.58482,0.32214,0.28571,0.5,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,13,0,0,2,0,0,4,0,0,1,0,0,0,0,11],[20,22,0.9091,0.51786,0.29827,0.28571,0.28571,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,17,0,0,3,0,0,3,0,0,1,0,0,0,0,8],[22,22,1.0,0.52679,0.23538,0.28571,0.57141,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,9,0,0,5,0,0,10,0,0,1,0,0,3,0,3]]},{"b":6,"e":0.71429,"k":"flat","v":0.74107,"x":0.9375,"p":[[0,43,0.0,0.84374,0.23519,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,1,0,0,3,0,20],[4,43,0.093,0.91518,0.18851,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,24],[8,43,0.186,0.86161,0.26119,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,24],[12,43,0.2791,0.9375,0.18536,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,28],[16,43,0.3721,0.91518,0.18161,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,25],[20,43,0.4651,0.88393,0.21558,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,3,0,22],[24,43,0.5581,0.91071,0.18472,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,7,0,22],[28,43,0.6512,0.79909,0.12301,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,15,0,4],[32,43,0.7442,0.74107,0.15335,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,8,0,0,13,0,2],[36,43,0.8372,0.76784,0.09945,0.71429,0.78571,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,16,0,0],[40,43,0.9302,0.80357,0.09279,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,23,0,0],[43,43,1.0,0.76338,0.11073,0.71429,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,17,0,0]]}]},{"i":"42a95a1baa65037d","q":"Let  $n$  be a natural number. Find the least natural number  $k$  for which there exist  $k$  sequences of  $0$  and  $1$  of length  $2n+2$  with the following property: any sequence of  $0$  and  $1$  of length  $2n+2$  coincides with some of these  $k$  sequences in at least  $n+2$  positions.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0892,"x":0.31677,"p":[[0,67,0.0,0.12053,0.12428,0.0,0.14286,0.2857,0.0,0.28571,15,0,7,15,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.14286,0.26,0.0,0.0,0.2857,0.0,1.0,21,2,0,21,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[8,67,0.1194,0.14284,0.26724,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,1],[12,67,0.1791,0.29015,0.35441,0.0,0.07143,0.4642,0.0,1.0,16,4,0,16,0,2,0,0,1,0,0,5,0,0,2,0,0,2,0,0,0,0,4],[16,67,0.2388,0.1875,0.26108,0.0,0.14286,0.2857,0.0,1.0,15,1,0,15,0,7,0,0,5,0,0,1,0,0,1,0,0,1,0,0,1,0,1],[20,67,0.2985,0.24991,0.31137,0.0,0.14288,0.32143,0.0,1.0,14,3,0,14,0,3,0,0,7,0,0,4,0,0,0,0,0,0,0,0,1,0,3],[24,67,0.3582,0.17409,0.22509,0.0,0.0,0.2857,0.0,0.857,18,0,0,18,0,0,0,0,8,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[28,67,0.4179,0.10259,0.17213,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,1,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[32,67,0.4776,0.23212,0.27137,0.0,0.07143,0.42858,0.0,0.85714,16,0,0,16,0,1,0,0,5,0,0,4,0,0,2,0,0,3,0,0,1,0,0],[36,67,0.5373,0.28124,0.27076,0.14286,0.2143,0.32143,0.0,1.0,7,2,0,7,0,9,0,0,8,0,0,2,0,0,2,0,0,2,0,0,0,0,2],[40,67,0.597,0.31677,0.26911,0.14214,0.28571,0.42857,0.0,1.0,6,2,0,6,0,6,0,0,9,0,0,6,0,0,1,0,0,1,0,0,1,0,2],[44,67,0.6567,0.27232,0.27747,0.0,0.21428,0.32143,0.0,1.0,9,1,0,9,0,7,0,0,8,0,0,3,0,0,0,0,0,2,0,0,2,0,1],[48,67,0.7164,0.29464,0.31326,0.0,0.14286,0.42858,0.0,1.0,10,2,0,10,0,7,0,0,5,0,0,3,0,0,1,0,0,2,0,0,2,0,2],[52,67,0.7761,0.24553,0.22934,0.0,0.2857,0.32143,0.0,0.71429,10,0,0,10,0,5,0,0,9,0,0,4,0,0,0,0,0,4,0,0,0,0,0],[56,67,0.8358,0.2991,0.27514,0.14286,0.2143,0.42858,0.0,1.0,7,1,0,7,0,9,0,0,5,0,0,4,0,0,3,0,0,1,0,0,2,0,1],[60,67,0.8955,0.1875,0.2683,0.0,0.14286,0.2857,0.0,1.0,15,1,0,15,0,8,0,0,3,0,0,3,0,0,0,0,0,0,0,0,2,0,1],[64,67,0.9552,0.11606,0.20022,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,6,0,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[67,67,1.0,0.0892,0.14614,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,5,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.00893,"x":0.4821,"p":[[0,49,0.0,0.12054,0.12931,0.0,0.07143,0.28571,0.0,0.28571,16,0,8,16,0,5,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.09375,0.15407,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,3,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.20081,0.30065,0.0,0.0,0.28571,0.0,1.0,17,2,0,17,0,5,0,0,4,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[16,49,0.3265,0.2008,0.23654,0.0,0.14286,0.28571,0.0,0.857,14,0,0,14,0,5,0,0,6,0,0,4,0,0,0,0,0,2,0,0,1,0,0],[20,49,0.4082,0.29911,0.32996,0.0,0.2857,0.42857,0.0,1.0,12,3,0,12,0,3,0,0,7,0,0,4,0,0,0,0,0,1,0,0,2,0,3],[24,49,0.4898,0.22322,0.23674,0.0,0.14286,0.4286,0.0,0.71429,12,0,0,12,0,7,0,0,4,0,0,4,0,0,2,0,0,3,0,0,0,0,0],[28,49,0.5714,0.25887,0.26343,0.0,0.14286,0.4642,0.0,0.85714,11,0,0,11,0,7,0,0,3,0,0,3,0,0,6,0,0,0,0,0,2,0,0],[32,49,0.6531,0.05348,0.14611,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,49,0.7347,0.1875,0.21852,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,6,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[40,49,0.8163,0.42408,0.24085,0.2857,0.35714,0.57111,0.14286,1.0,0,2,0,0,0,6,0,0,10,0,0,6,0,0,4,0,0,3,0,0,1,0,2],[44,49,0.898,0.44194,0.25593,0.2857,0.42857,0.60682,0.14286,1.0,0,2,0,0,0,7,0,0,7,0,0,8,0,0,2,0,0,4,0,0,2,0,2],[48,49,0.9796,0.4821,0.26182,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,9,0,0,4,0,0,4,0,0,5,0,0,3,0,2],[49,49,1.0,0.4732,0.2586,0.2857,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,10,0,0,3,0,0,4,0,0,6,0,0,2,0,2]]}]},{"i":"063d708115c7bb63","q":"Let  $n$  be a positive integer and let  $(a_1,a_2,\\ldots ,a_{2n})$  be a permutation of  $1,2,\\ldots ,2n$  such that the numbers  $|a_{i+1}-a_i|$  are pairwise distinct for  $i=1,\\ldots ,2n-1$ .\nProve that  $\\{a_2,a_4,\\ldots ,a_{2n}\\}=\\{1,2,\\ldots ,n\\}$  if and only if  $a_1-a_{2n}=n$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.36604,"x":0.68303,"p":[[0,45,0.0,0.36604,0.18873,0.14286,0.42857,0.42857,0.0,0.857,2,0,0,2,0,7,0,0,1,0,0,18,0,0,2,0,0,1,0,0,1,0,0],[4,45,0.0889,0.60267,0.25688,0.42857,0.64286,0.85714,0.0,1.0,1,1,0,1,0,3,0,0,2,0,0,3,0,0,7,0,0,6,0,0,9,0,1],[8,45,0.1778,0.4508,0.29484,0.24999,0.42857,0.71429,0.0,1.0,2,2,0,2,0,6,0,0,7,0,0,5,0,0,3,0,0,2,0,0,5,0,2],[12,45,0.2667,0.47767,0.31259,0.24999,0.42857,0.75,0.0,1.0,3,2,0,3,0,5,0,0,6,0,0,4,0,0,2,0,0,4,0,0,6,0,2],[16,45,0.3556,0.59375,0.31564,0.39286,0.64286,0.85714,0.0,1.0,2,5,0,2,0,4,0,0,2,0,0,4,0,0,4,0,0,4,0,0,7,0,5],[20,45,0.4444,0.62052,0.31055,0.39286,0.71429,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,3,0,0,2,0,0,4,0,0,5,0,0,8,0,5],[24,45,0.5333,0.62054,0.27804,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,3,0,0,4,0,0,4,0,0,5,0,0,8,0,4],[28,45,0.6222,0.68303,0.29824,0.53571,0.857,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,3,0,0,1,0,0,2,0,0,5,0,0,11,0,6],[32,45,0.7111,0.5982,0.27765,0.42857,0.57143,0.85714,0.0,1.0,2,3,0,2,0,2,0,0,1,0,0,6,0,0,7,0,0,3,0,0,8,0,3],[36,45,0.8,0.65179,0.3008,0.42857,0.78571,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,1,0,0,4,0,0,3,0,0,3,0,0,11,0,5],[40,45,0.8889,0.56237,0.26243,0.39286,0.57121,0.85704,0.0,1.0,1,2,0,1,0,2,0,0,5,0,0,5,0,0,7,0,0,3,0,0,7,0,2],[44,45,0.9778,0.57142,0.24999,0.42857,0.57143,0.74996,0.0,1.0,1,2,0,1,0,2,0,0,3,0,0,6,0,0,8,0,0,4,0,0,6,0,2],[45,45,1.0,0.49549,0.24478,0.28571,0.571,0.60714,0.14286,0.85714,0,0,0,0,0,7,0,0,2,0,0,6,0,0,9,0,0,2,0,0,6,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.41507,"x":0.62052,"p":[[0,88,0.0,0.48214,0.19149,0.42857,0.42857,0.46431,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,20,0,0,3,0,0,1,0,0,3,0,1],[4,88,0.0455,0.54463,0.2976,0.39286,0.57143,0.85714,0.0,1.0,3,2,0,3,0,3,0,0,2,0,0,6,0,0,4,0,0,5,0,0,7,0,2],[8,88,0.0909,0.58482,0.29093,0.42857,0.57143,0.85714,0.0,1.0,1,3,0,1,0,5,0,0,1,0,0,5,0,0,5,0,0,4,0,0,8,0,3],[12,88,0.1364,0.57588,0.27312,0.42857,0.57143,0.85714,0.0,1.0,1,3,0,1,0,3,0,0,3,0,0,6,0,0,5,0,0,5,0,0,6,0,3],[16,88,0.1818,0.42856,0.28121,0.24999,0.42857,0.71429,0.0,1.0,3,1,0,3,0,5,0,0,7,0,0,6,0,0,2,0,0,4,0,0,4,0,1],[20,88,0.2273,0.41507,0.30176,0.14286,0.42857,0.60714,0.0,1.0,3,1,0,3,0,10,0,0,2,0,0,4,0,0,5,0,0,2,0,0,5,0,1],[24,88,0.2727,0.56245,0.32534,0.28571,0.57143,0.85714,0.0,1.0,3,4,0,3,0,3,0,0,4,0,0,4,0,0,3,0,0,3,0,0,8,0,4],[28,88,0.3182,0.62052,0.30223,0.42857,0.71429,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,2,0,0,2,0,0,6,0,0,5,0,0,7,0,5],[32,88,0.3636,0.5492,0.3391,0.28571,0.57143,0.85714,0.0,1.0,3,6,0,3,0,4,0,0,5,0,0,2,0,0,3,0,0,5,0,0,4,0,6],[36,88,0.4091,0.58917,0.30684,0.42857,0.57143,0.85714,0.0,1.0,3,5,0,3,0,1,0,0,3,0,0,6,0,0,4,0,0,4,0,0,6,0,5],[40,88,0.4545,0.58928,0.31083,0.39286,0.57143,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,3,0,0,5,0,0,4,0,0,3,0,0,7,0,5],[44,88,0.5,0.58482,0.32608,0.28571,0.71429,0.85714,0.0,1.0,1,5,0,1,0,6,0,0,3,0,0,4,0,0,1,0,0,4,0,0,8,0,5],[48,88,0.5455,0.49999,0.33502,0.14286,0.5,0.857,0.0,1.0,4,4,0,4,0,5,0,0,3,0,0,4,0,0,4,0,0,3,0,0,5,0,4],[52,88,0.5909,0.60713,0.31339,0.25,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,8,0,0,1,0,0,1,0,0,4,0,0,6,0,0,7,0,5],[56,88,0.6364,0.44183,0.38016,0.10714,0.42857,0.85714,0.0,1.0,8,6,0,8,0,5,0,0,2,0,0,4,0,0,2,0,0,2,0,0,3,0,6],[60,88,0.6818,0.5311,0.3354,0.24999,0.5712,0.85714,0.0,1.0,3,4,0,3,0,5,0,0,4,0,0,3,0,0,3,0,0,3,0,0,7,0,4],[64,88,0.7273,0.45085,0.24249,0.2857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,5,0,0,6,0,0,7,0,0,3,0,0,4,0,0],[68,88,0.7727,0.55355,0.31693,0.28571,0.57143,0.85714,0.0,1.0,1,5,0,1,0,5,0,0,6,0,0,2,0,0,5,0,0,2,0,0,6,0,5],[72,88,0.8182,0.58916,0.30684,0.28571,0.57143,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,5,0,0,1,0,0,7,0,0,2,0,0,7,0,5],[76,88,0.8636,0.54462,0.31831,0.28571,0.57121,0.74996,0.0,1.0,3,5,0,3,0,3,0,0,4,0,0,5,0,0,2,0,0,7,0,0,3,0,5],[80,88,0.9091,0.51339,0.28763,0.28571,0.4286,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,4,0,0,7,0,0,2,0,0,6,0,0,5,0,2],[84,88,0.9545,0.48212,0.24156,0.28571,0.42857,0.60714,0.0,1.0,1,1,0,1,0,3,0,0,6,0,0,9,0,0,5,0,0,3,0,0,4,0,1],[88,88,1.0,0.49998,0.21723,0.28571,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,7,0,0,9,0,0,7,0,0,2,0,0,4,0,1]]}]},{"i":"5923ce8bff8cd916","q":"24. N1 (BLR) Let $\\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation  $$ \\tau(a n)=n $$  does not have a positive integer solution $n$.","t":[{"b":1,"e":0.571,"k":"rising","v":0.17856,"x":0.5023,"p":[[0,51,0.0,0.17856,0.23417,0.0,0.07143,0.28571,0.0,0.71429,16,0,2,16,0,5,0,0,6,0,0,0,0,0,2,0,0,3,0,0,0,0,0],[4,51,0.0784,0.38834,0.23749,0.25001,0.42857,0.571,0.0,0.85714,5,0,0,5,0,3,0,0,3,0,0,12,0,0,5,0,0,2,0,0,2,0,0],[8,51,0.1569,0.29018,0.24086,0.0,0.2857,0.42857,0.0,0.71429,9,0,0,9,0,4,0,0,6,0,0,7,0,0,2,0,0,4,0,0,0,0,0],[12,51,0.2353,0.41062,0.19817,0.2857,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,6,0,0,7,0,0,10,0,0,3,0,0,0,0,0],[16,51,0.3137,0.36155,0.2341,0.14286,0.35714,0.57143,0.0,0.71429,5,0,0,5,0,5,0,0,6,0,0,3,0,0,10,0,0,3,0,0,0,0,0],[20,51,0.3922,0.38389,0.24853,0.2857,0.28571,0.57111,0.0,1.0,4,1,0,4,0,3,0,0,11,0,0,3,0,0,5,0,0,5,0,0,0,0,1],[24,51,0.4706,0.37941,0.19097,0.28571,0.42857,0.571,0.0,0.71429,3,0,0,3,0,3,0,0,7,0,0,10,0,0,7,0,0,2,0,0,0,0,0],[28,51,0.549,0.41512,0.20931,0.28571,0.42857,0.571,0.0,0.85714,4,0,0,4,0,0,0,0,6,0,0,12,0,0,6,0,0,3,0,0,1,0,0],[32,51,0.6275,0.38387,0.19372,0.2857,0.42857,0.571,0.0,0.71429,4,0,0,4,0,0,0,0,10,0,0,8,0,0,8,0,0,2,0,0,0,0,0],[36,51,0.7059,0.34815,0.19204,0.2857,0.28571,0.571,0.0,0.71429,3,0,0,3,0,4,0,0,12,0,0,3,0,0,9,0,0,1,0,0,0,0,0],[40,51,0.7843,0.4687,0.17936,0.42857,0.571,0.57143,0.14286,0.857,0,0,0,0,0,5,0,0,2,0,0,8,0,0,14,0,0,2,0,0,1,0,0],[44,51,0.8627,0.45084,0.18246,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,8,0,0,8,0,0,8,0,0,4,0,0,1,0,0],[48,51,0.9412,0.45977,0.18463,0.42857,0.4286,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,3,0,0,10,0,0,13,0,0,0,0,0,2,0,0],[51,51,1.0,0.5023,0.21106,0.39286,0.4998,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,4,0,0,8,0,0,6,0,0,7,1,0,2,0,0]]},{"b":3,"e":0.28571,"k":"rising","v":0.17188,"x":0.41516,"p":[[0,47,0.0,0.17188,0.20505,0.0,0.14286,0.2857,0.0,0.85714,13,0,6,13,1,7,0,0,7,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[4,47,0.0851,0.41069,0.21352,0.2857,0.42857,0.57143,0.0,0.85714,3,0,0,3,0,3,0,0,5,0,0,10,0,0,7,0,0,3,0,0,1,0,0],[8,47,0.1702,0.37941,0.22187,0.24999,0.42857,0.571,0.0,0.71429,5,0,0,5,0,3,0,0,4,0,0,9,0,0,8,0,0,3,0,0,0,0,0],[12,47,0.2553,0.31247,0.20955,0.14286,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,6,0,0,7,0,0,9,0,0,2,0,0,3,0,0,0,0,0],[16,47,0.3404,0.28105,0.22732,0.105,0.28571,0.4286,0.0,0.71429,8,0,0,8,0,5,0,0,8,0,0,5,0,0,3,0,0,3,0,0,0,0,0],[20,47,0.4255,0.32366,0.22159,0.14286,0.42857,0.42858,0.0,0.71429,7,0,0,7,0,3,0,1,4,0,0,10,0,0,5,0,0,2,0,0,0,0,0],[24,47,0.5106,0.39284,0.20198,0.28571,0.42857,0.571,0.0,0.71429,3,0,0,3,0,3,0,0,7,0,0,8,0,0,8,0,0,3,0,0,0,0,0],[28,47,0.5957,0.3772,0.2203,0.2857,0.42857,0.4642,0.0,1.0,4,1,0,4,0,1,0,1,9,0,0,9,0,0,5,0,0,2,0,0,0,0,1],[32,47,0.6809,0.35265,0.20509,0.24999,0.28571,0.4642,0.0,0.85714,3,0,0,3,0,5,0,0,9,0,0,7,0,0,6,0,0,1,0,0,1,0,0],[36,47,0.766,0.36602,0.24464,0.14286,0.42857,0.57143,0.0,0.71429,6,0,0,6,0,4,0,0,5,0,0,4,0,0,9,0,0,4,0,0,0,0,0],[40,47,0.8511,0.40618,0.20232,0.2857,0.42857,0.571,0.0,0.85714,2,0,0,2,0,4,0,0,6,0,0,9,0,0,8,0,0,2,0,0,1,0,0],[44,47,0.9362,0.40623,0.23447,0.2857,0.35714,0.57143,0.0,0.85714,2,0,0,2,0,5,0,0,9,0,0,4,0,0,6,0,0,4,0,0,2,0,0],[47,47,1.0,0.41516,0.21533,0.25002,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,7,0,0,4,0,0,8,0,0,7,0,0,4,0,0,1,0,0]]}]},{"i":"969a2ce921cd534e","q":"51. (USS 5) Several segments, which we shall call white, are given, and the sum of their lengths is 1 . Several other segments, which we shall call black, are given, and the sum of their lengths is 1 . Prove that every such system of segments can be distributed on the segment that is 1.51 long in the following way: Segments of the same color are disjoint, and segments of different colors are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is 1.49 long.","t":[{"b":0,"e":0.57143,"k":"rising","v":0.125,"x":0.73649,"p":[[0,56,0.0,0.125,0.23351,0.0,0.0,0.14286,0.0,0.85714,23,0,17,23,0,2,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[4,56,0.0714,0.56237,0.24483,0.42857,0.57143,0.71429,0.14,1.0,0,3,0,0,0,4,0,0,2,0,0,6,0,0,9,0,0,5,0,0,3,0,3],[8,56,0.1429,0.68747,0.22143,0.42859,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,7,0,0,6,0,6],[12,56,0.2143,0.64282,0.23146,0.53539,0.71429,0.857,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,4,0,0,6,0,0,9,0,0,6,0,3],[16,56,0.2857,0.62047,0.24383,0.4286,0.57143,0.857,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,5,0,0,7,0,0,6,0,0,5,0,4],[20,56,0.3571,0.6429,0.25997,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,5,0,0,6,0,0,2,0,0,6,0,0,7,0,5],[24,56,0.4286,0.56248,0.2257,0.39286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,7,0,0,5,0,0,6,0,0,8,0,0,3,0,2],[28,56,0.5,0.61594,0.20673,0.5354,0.57143,0.75,0.14,1.0,0,1,0,0,0,2,0,0,1,0,0,5,0,0,10,0,0,6,0,0,7,0,1],[32,56,0.5714,0.70534,0.22571,0.5354,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,7,0,0,1,0,0,9,0,0,10,0,4],[36,56,0.6429,0.62943,0.22264,0.42857,0.64286,0.74996,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,7,0,0,5,0,0,8,0,0,4,0,4],[40,56,0.7143,0.69194,0.24253,0.571,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,2,0,0,4,0,0,4,0,0,9,0,0,6,0,6],[44,56,0.7857,0.71427,0.22869,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,9,0,0,4,0,8],[48,56,0.8571,0.64274,0.22888,0.42859,0.71429,0.85714,0.14,1.0,0,3,0,0,0,1,0,0,3,0,0,6,0,0,4,0,0,8,0,0,7,0,3],[52,56,0.9286,0.73649,0.22924,0.57132,0.85714,0.85714,0.14,1.0,0,7,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,5,0,0,10,0,7],[56,56,1.0,0.62487,0.2391,0.42857,0.57143,0.75,0.14,1.0,0,5,0,0,0,2,0,0,2,0,0,5,0,0,9,0,0,6,0,0,3,0,5]]},{"b":2,"e":1.0,"k":"rising","v":0.08036,"x":0.70977,"p":[[0,48,0.0,0.08036,0.14698,0.0,0.0,0.14286,0.0,0.57143,22,0,20,22,0,6,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,48,0.0833,0.57585,0.24086,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,4,0,0,6,0,0,10,0,0,3,0,0,3,0,4],[8,48,0.1667,0.5982,0.26351,0.42857,0.64286,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,6,0,0,3,0,0,6,0,0,7,0,3],[12,48,0.25,0.70977,0.2328,0.57143,0.78564,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,0,0,0,2,0,0,8,0,0,4,0,0,12,0,4],[16,48,0.3333,0.65177,0.2141,0.4286,0.64286,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,8,0,0,7,0,0,8,0,0,3,0,5],[20,48,0.4167,0.58032,0.25985,0.39286,0.57141,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,6,0,0,4,0,0,9,0,0,3,0,0,3,0,5],[24,48,0.5,0.58929,0.28065,0.28571,0.64286,0.85714,0.14286,1.0,0,2,0,0,0,4,0,0,6,0,0,2,0,0,4,0,0,4,0,0,10,0,2],[28,48,0.5833,0.59364,0.2816,0.39286,0.57143,0.85714,0.14,1.0,0,6,0,0,0,3,0,0,5,0,0,5,0,0,6,0,0,3,0,0,4,0,6],[32,48,0.6667,0.64731,0.24739,0.4286,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,11,0,0,4,0,0,3,0,0,6,0,6],[36,48,0.75,0.67852,0.24487,0.571,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,4,0,0,1,0,0,9,0,0,4,0,0,7,0,6],[40,48,0.8333,0.62257,0.23854,0.42857,0.677,0.74996,0.14,1.0,0,4,0,0,0,1,0,0,5,0,0,4,0,0,5,1,0,8,0,0,4,0,4],[44,48,0.9167,0.58926,0.30252,0.28571,0.57143,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,5,0,0,4,0,0,5,0,0,3,0,0,5,0,6],[48,48,1.0,0.70087,0.23519,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,2,0,0,9,0,0,6,0,0,4,0,8]]}]},{"i":"9db3b1f2a3aa1cde","q":"Let  $a$  and  $b$  be positive integers with  $b$  odd, such that the number   $$ \\frac{(a+b)^2+4a}{ab} $$   is an integer. Prove that  $a$  is a perfect square.","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.43747,"p":[[0,11,0.0,0.41963,0.27649,0.24999,0.42859,0.57143,0.0,1.0,7,1,0,7,0,1,0,0,3,0,0,6,0,0,9,0,0,4,0,0,1,0,1],[4,11,0.3636,0.43747,0.28556,0.24999,0.571,0.60714,0.0,1.0,7,1,0,7,0,1,0,0,3,0,0,4,0,0,9,0,0,6,0,0,1,0,1],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"rising","v":0.38392,"x":0.54013,"p":[[0,38,0.0,0.38392,0.31831,0.0,0.42859,0.60714,0.0,0.85714,12,0,0,12,0,0,0,0,0,0,0,5,0,0,7,0,0,5,0,0,3,0,0],[4,38,0.1053,0.41515,0.20933,0.39286,0.42857,0.57143,0.0,0.71429,5,0,0,5,0,1,0,0,2,0,0,9,0,0,14,0,0,1,0,0,0,0,0],[8,38,0.2105,0.46874,0.30977,0.35714,0.42859,0.71429,0.0,1.0,7,3,0,7,0,1,0,0,0,0,0,9,0,0,5,0,0,6,0,0,1,0,3],[12,38,0.3158,0.45534,0.25614,0.42857,0.57121,0.57143,0.0,1.0,6,1,0,6,0,0,0,0,1,0,0,8,0,0,12,0,0,3,0,0,1,0,1],[16,38,0.4211,0.49998,0.1428,0.42857,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,10,0,0,14,0,0,4,0,0,0,0,0],[20,38,0.5263,0.48209,0.19147,0.42857,0.571,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,4,0,0,7,0,0,12,0,0,6,0,0,0,0,0],[24,38,0.6316,0.48658,0.1631,0.42857,0.571,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,1,0,0,11,0,0,13,0,0,4,0,0,0,0,0],[28,38,0.7368,0.51335,0.12805,0.42857,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,13,0,0,12,0,0,5,0,0,0,0,0],[32,38,0.8421,0.51783,0.14173,0.42857,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,9,0,0,15,0,0,5,0,0,0,0,0],[36,38,0.9474,0.51338,0.16697,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,9,0,0,12,0,0,7,0,0,0,0,0],[38,38,1.0,0.54013,0.09931,0.42857,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,9,0,0,18,0,0,4,0,0,0,0,0]]}]},{"i":"6a5781aa25d2b836","q":"26. (YUG 4) Prove that the functional equations  $$ \\begin{aligned} f(x+y) & =f(x)+f(y) \\\\ \\text { and } \\quad f(x+y+x y) & =f(x)+f(y)+f(x y) \\quad(x, y \\in \\mathbb{R}) \\end{aligned} $$  are equivalent.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.24991,"x":0.57139,"p":[[0,77,0.0,0.24991,0.09459,0.14286,0.2857,0.28571,0.0,0.4286,1,0,0,1,0,9,0,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.56696,0.2461,0.42857,0.64286,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,5,0,0,4,0,0,13,0,0,0,0,3],[8,77,0.1039,0.54465,0.22711,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,3,0,0,10,0,0,6,0,0,8,0,0,0,0,3],[12,77,0.1558,0.57139,0.20825,0.42859,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,3,0,0,11,0,0,9,0,0,1,0,2],[16,77,0.2078,0.51783,0.1847,0.42857,0.57143,0.71429,0.14286,0.857,0,0,0,0,0,3,0,0,3,0,0,7,0,0,10,0,0,8,0,0,1,0,0],[20,77,0.2597,0.51334,0.24963,0.28571,0.57121,0.60714,0.0,1.0,1,3,0,1,0,4,0,0,4,0,0,3,0,0,12,0,0,5,0,0,0,0,3],[24,77,0.3117,0.5044,0.19878,0.42857,0.571,0.71429,0.0,0.71429,1,0,0,1,0,4,0,0,0,0,0,8,0,0,10,0,0,9,0,0,0,0,0],[28,77,0.3636,0.46868,0.20274,0.28571,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,5,0,0,8,0,0,8,0,0,6,0,0,1,0,0],[32,77,0.4156,0.50885,0.234,0.28571,0.571,0.60714,0.0,1.0,1,2,0,1,0,2,0,0,7,0,0,3,0,0,11,0,0,5,0,0,1,0,2],[36,77,0.4675,0.4598,0.23617,0.25,0.57121,0.71429,0.0,0.71429,2,0,0,2,0,6,0,0,2,0,0,4,0,0,9,0,0,9,0,0,0,0,0],[40,77,0.5195,0.50444,0.23953,0.39286,0.571,0.57143,0.0,1.0,1,3,0,1,0,3,0,0,4,0,0,7,0,0,10,0,0,4,0,0,0,0,3],[44,77,0.5714,0.43747,0.18534,0.28571,0.42859,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,7,0,0,9,0,0,9,0,0,4,0,0,0,0,0],[48,77,0.6234,0.48658,0.20472,0.28571,0.42859,0.60714,0.14286,1.0,0,1,0,0,0,2,0,0,8,0,0,8,0,0,6,0,0,6,0,0,1,0,1],[52,77,0.6753,0.4464,0.19478,0.2857,0.4286,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,10,0,0,4,0,0,9,0,0,6,0,0,0,0,0],[56,77,0.7273,0.46424,0.25503,0.28571,0.49979,0.57143,0.0,1.0,2,3,0,2,0,3,0,0,7,0,0,4,0,0,11,0,0,2,0,0,0,0,3],[60,77,0.7792,0.49551,0.2201,0.28571,0.57121,0.60714,0.14286,1.0,0,1,0,0,0,5,0,0,4,0,0,5,0,0,10,0,0,6,0,0,1,0,1],[64,77,0.8312,0.42857,0.20516,0.28571,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,3,0,0,12,0,0,5,0,0,6,0,0,0,0,0],[68,77,0.8831,0.40612,0.16419,0.28571,0.42857,0.57111,0.14,0.71429,0,0,0,0,0,6,0,0,5,0,0,10,0,0,10,0,0,1,0,0,0,0,0],[72,77,0.9351,0.36594,0.19871,0.14286,0.42857,0.4642,0.0,0.71429,3,0,0,3,0,6,0,0,3,0,0,12,0,0,6,0,0,2,0,0,0,0,0],[76,77,0.987,0.25893,0.16917,0.14286,0.2857,0.32143,0.0,0.71429,3,0,0,3,0,12,0,0,9,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[77,77,1.0,0.25446,0.20741,0.14286,0.14295,0.42857,0.0,0.71429,6,0,0,6,0,11,0,0,6,0,0,4,0,0,3,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.30347,"x":0.5714,"p":[[0,78,0.0,0.30347,0.15054,0.24999,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,7,0,0,16,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[4,78,0.0513,0.55353,0.17405,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,7,0,0,9,0,0,11,0,0,1,0,0],[8,78,0.1026,0.53569,0.17857,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,6,0,0,11,0,0,8,0,0,0,0,1],[12,78,0.1538,0.49548,0.17488,0.42857,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,4,0,0,7,0,0,11,0,0,7,0,0,0,0,0],[16,78,0.2051,0.53125,0.21498,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,3,0,0,4,0,0,8,0,0,12,0,0,1,0,0],[20,78,0.2564,0.48213,0.22516,0.28571,0.4286,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,6,0,0,9,0,0,8,0,0,3,0,0,1,0,2],[24,78,0.3077,0.51334,0.19514,0.42857,0.57121,0.60714,0.0,0.85714,2,0,0,2,0,0,0,0,4,0,0,6,0,0,12,0,0,7,0,0,1,0,0],[28,78,0.359,0.47763,0.191,0.28571,0.571,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,7,0,0,4,0,0,14,0,0,3,0,0,0,0,1],[32,78,0.4103,0.40624,0.23174,0.25,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,5,0,0,6,0,0,4,0,0,8,0,0,6,0,0,0,0,0],[36,78,0.4615,0.4821,0.19478,0.39286,0.4286,0.71429,0.0,0.71429,1,0,0,1,0,2,0,0,5,0,0,9,0,0,6,0,0,9,0,0,0,0,0],[40,78,0.5128,0.49757,0.18147,0.39286,0.5355,0.60607,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,7,0,1,8,0,0,7,0,0,1,0,0],[44,78,0.5641,0.45982,0.16648,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,8,0,0,7,0,0,6,0,0,0,0,0],[48,78,0.6154,0.42853,0.19882,0.28571,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,3,0,0,5,0,0,10,0,0,7,0,0,5,0,0,0,0,0],[52,78,0.6667,0.4598,0.21348,0.28571,0.4998,0.60714,0.0,0.71429,2,0,0,2,0,2,0,0,7,0,0,5,0,0,8,0,0,8,0,0,0,0,0],[56,78,0.7179,0.4732,0.16917,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,7,0,0,8,0,0,9,0,0,6,0,0,0,0,0],[60,78,0.7692,0.49082,0.2111,0.39286,0.57121,0.71429,0.0,0.71429,1,0,0,1,0,4,0,0,3,0,0,6,0,0,8,0,0,10,0,0,0,0,0],[64,78,0.8205,0.5714,0.16366,0.571,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,3,0,0,12,0,0,13,0,0,0,0,0],[68,78,0.8718,0.50442,0.19555,0.42857,0.571,0.71429,0.14286,0.71429,0,0,0,0,0,4,0,0,3,0,0,8,0,0,6,0,0,11,0,0,0,0,0],[72,78,0.9231,0.55798,0.18336,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,4,0,0,7,0,0,15,0,0,0,0,0],[76,78,0.9744,0.43749,0.14257,0.28571,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,15,0,0,4,0,0,4,0,0,0,0,0],[78,78,1.0,0.41514,0.16111,0.28571,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,4,0,0,7,0,0,12,0,0,6,0,0,3,0,0,0,0,0]]}]},{"i":"b60d54b675ab6ff9","q":"Find all integers  $x,y$  such that  $x^3(y+1)+y^3(x+1)=19$ .\n\n*Proposed by Bulgaria*","t":[{"b":2,"e":0.71429,"k":"flat","v":0.77232,"x":0.96875,"p":[[0,78,0.0,0.91071,0.13244,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,15,0,16],[4,78,0.0513,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[8,78,0.1026,0.93301,0.09444,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,19],[12,78,0.1538,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[16,78,0.2051,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[20,78,0.2564,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[24,78,0.3077,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[28,78,0.359,0.88838,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[32,78,0.4103,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[36,78,0.4615,0.88839,0.07771,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,9],[40,78,0.5128,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[44,78,0.5641,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[48,78,0.6154,0.89285,0.08748,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,21,0,10],[52,78,0.6667,0.86605,0.04972,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[56,78,0.7179,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[60,78,0.7692,0.875,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[64,78,0.8205,0.88838,0.07771,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,9],[68,78,0.8718,0.80804,0.16602,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,17,0,5],[72,78,0.9231,0.84821,0.10677,0.71429,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,14,0,8],[76,78,0.9744,0.85268,0.10403,0.71429,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,15,0,8],[78,78,1.0,0.77232,0.10012,0.71429,0.71429,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,5,0,4]]},{"b":3,"e":0.85714,"k":"flat","v":0.93304,"x":1.0,"p":[[0,198,0.0,0.93304,0.09439,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,19],[4,198,0.0202,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[8,198,0.0404,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,198,0.0606,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,198,0.0808,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,198,0.101,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,198,0.1212,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,198,0.1414,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,198,0.1616,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,198,0.1818,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,198,0.202,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[44,198,0.2222,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[48,198,0.2424,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[52,198,0.2626,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[56,198,0.2828,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[60,198,0.303,0.9866,0.04167,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[64,198,0.3232,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,198,0.3434,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,198,0.3636,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[76,198,0.3838,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[80,198,0.404,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[84,198,0.4242,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[88,198,0.4444,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[92,198,0.4646,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[96,198,0.4848,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[100,198,0.5051,0.94197,0.18161,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[104,198,0.5253,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,198,0.5455,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[112,198,0.5657,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[116,198,0.5859,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[120,198,0.6061,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[124,198,0.6263,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[128,198,0.6465,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[132,198,0.6667,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[136,198,0.6869,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[140,198,0.7071,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[144,198,0.7273,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[148,198,0.7475,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[152,198,0.7677,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[156,198,0.7879,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[160,198,0.8081,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[164,198,0.8283,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[168,198,0.8485,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[172,198,0.8687,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[176,198,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[180,198,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[184,198,0.9293,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[188,198,0.9495,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[192,198,0.9697,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[196,198,0.9899,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[198,198,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"bfca2eacaf1bd975","q":"Let  $a,b,c\\in\\mathbb{Z^{+}}$  such that   $$ (a^2-1, b^2-1, c^2-1)=1 $$  Prove that  $$ (ab+c, bc+a, ca+b)=(a,b,c) $$  (As usual,  $(x,y,z)$  means the greatest common divisor of numbers  $x,y,z$ )\n*Proposed by A. Golovanov*","t":[{"b":2,"e":0.14286,"k":"flat","v":0.16063,"x":0.27678,"p":[[0,22,0.0,0.25884,0.14487,0.14286,0.14286,0.42857,0.0,0.4286,1,0,0,1,0,17,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.25,0.13832,0.14286,0.14286,0.42857,0.0,0.4286,1,0,0,1,0,17,0,0,3,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.27678,0.17103,0.14286,0.2143,0.42857,0.0,0.857,1,0,0,1,0,15,0,0,4,0,0,11,0,0,0,0,0,0,0,0,1,0,0],[12,22,0.5455,0.25893,0.13092,0.14286,0.14286,0.42857,0.14286,0.4286,0,0,0,0,0,17,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.19642,0.09937,0.14286,0.14286,0.2857,0.14286,0.571,0,0,0,0,0,23,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,22,0.9091,0.17411,0.09268,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[22,22,1.0,0.16063,0.09944,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.15179,"x":0.25893,"p":[[0,37,0.0,0.25893,0.12595,0.14286,0.2143,0.42857,0.14286,0.4286,0,0,0,0,0,16,0,0,6,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.20536,0.11258,0.14286,0.14286,0.1786,0.14286,0.42857,0,0,0,0,0,24,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.18304,0.09606,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.20071,0.16704,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,27,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[16,37,0.4324,0.19188,0.09186,0.14286,0.14286,0.1786,0.14,0.42857,0,0,0,0,0,24,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.17411,0.08553,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.22758,0.21684,0.14286,0.14286,0.14287,0.14,1.0,0,2,0,0,0,25,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[28,37,0.7568,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.15615,0.07451,0.14286,0.14286,0.14286,0.14,0.571,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"b4484d5d91fde1a7","q":"On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top face are added. Determine whether one can select the integers on the faces so that the possible sums are  $2,3,4,5,6,7,8,9,10,11,12,13$ , all equally likely?","t":[{"b":0,"e":1.0,"k":"flat","v":0.88393,"x":1.0,"p":[[0,66,0.0,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[4,66,0.0606,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,66,0.1212,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,66,0.1818,0.90625,0.22759,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,27],[16,66,0.2424,0.95982,0.16457,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[20,66,0.303,0.91964,0.21706,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,28],[24,66,0.3636,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[28,66,0.4242,0.89286,0.25,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,27],[32,66,0.4848,0.88393,0.27534,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,27],[36,66,0.5455,0.89286,0.26,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,27],[40,66,0.6061,0.88839,0.24415,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,26],[44,66,0.6667,0.89732,0.27947,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[48,66,0.7273,0.93304,0.22011,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[52,66,0.7879,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[60,66,0.9091,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,66,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":2,"e":1.0,"k":"flat","v":0.80793,"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],[4,55,0.0727,0.9375,0.20806,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[8,55,0.1455,0.88393,0.26831,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,26],[12,55,0.2182,0.94195,0.15513,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,28],[16,55,0.2909,0.80793,0.33447,0.67856,1.0,1.0,0.0,1.0,2,23,0,2,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,23],[20,55,0.3636,0.90179,0.26592,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[24,55,0.4364,0.8125,0.34151,0.89286,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,24],[28,55,0.5091,0.8125,0.37019,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,25],[32,55,0.5818,0.84821,0.29001,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,25],[36,55,0.6545,0.90179,0.23808,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,27],[40,55,0.7273,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[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],[48,55,0.8727,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[52,55,0.9455,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[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":"4ce340eaf636dc79","q":"**Problem 2**\nDetermine all pairs  $(n, m)$  of positive integers satisfying the equation  $$ 5^n = 6m^2 + 1\\ .  $$","t":[{"b":4,"e":0.28571,"k":"rising","v":0.42409,"x":0.7946,"p":[[0,91,0.0,0.42409,0.18028,0.28571,0.35714,0.57111,0.14286,0.85714,0,0,0,0,0,1,0,0,15,0,0,7,0,0,3,0,0,5,0,0,1,0,0],[4,91,0.044,0.70982,0.29771,0.39286,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,6,0,0,1,0,0,1,0,0,4,0,0,8,0,10],[8,91,0.0879,0.63839,0.27429,0.28571,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,7,0,0,1,0,0,3,0,0,6,0,0,9,0,4],[12,91,0.1319,0.62946,0.29201,0.28571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,7,0,0,1,0,0,2,0,0,6,0,0,8,0,5],[16,91,0.1758,0.73661,0.27689,0.64286,0.78571,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,5,0,0,2,0,0,0,0,0,8,0,0,4,0,12],[20,91,0.2198,0.70088,0.23245,0.57132,0.71429,0.85714,0.1429,1.0,0,5,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,10,0,0,8,0,5],[24,91,0.2637,0.62944,0.30064,0.28571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,8,0,0,0,0,0,2,0,0,6,0,0,7,0,6],[28,91,0.3077,0.6607,0.23892,0.57132,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,5,0,0,1,0,0,4,0,0,14,0,0,1,0,6],[32,91,0.3516,0.72768,0.24317,0.57143,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,5,0,0,12,0,6],[36,91,0.3956,0.62053,0.25904,0.28571,0.71429,0.857,0.14286,1.0,0,3,0,0,0,2,0,0,7,0,0,1,0,0,2,0,0,11,0,0,6,0,3],[40,91,0.4396,0.65623,0.26931,0.39286,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,7,0,0,2,0,0,2,0,0,8,0,0,6,0,6],[44,91,0.4835,0.72766,0.25345,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,4,0,0,0,0,0,4,0,0,11,0,0,1,0,11],[48,91,0.5275,0.72322,0.24468,0.57143,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,5,0,0,2,0,0,2,0,0,8,0,0,7,0,8],[52,91,0.5714,0.66963,0.27535,0.28571,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,9,0,0,2,0,0,1,0,0,3,0,0,12,0,5],[56,91,0.6154,0.62053,0.30222,0.28571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,8,0,0,1,0,0,2,0,0,5,0,0,7,0,6],[60,91,0.6593,0.62032,0.28738,0.39286,0.71429,0.85714,0.14,1.0,0,5,0,0,0,5,0,0,3,0,0,1,0,0,6,0,0,6,0,0,6,0,5],[64,91,0.7033,0.59374,0.28371,0.28571,0.71429,0.857,0.14286,1.0,0,3,0,0,0,4,0,0,7,0,0,1,0,0,0,0,0,11,0,0,6,0,3],[68,91,0.7473,0.67409,0.27947,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,5,0,0,3,0,0,1,0,0,7,0,0,7,0,7],[72,91,0.7912,0.73214,0.24679,0.71429,0.78571,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,0,2,0,0,1,0,0,9,0,0,9,0,7],[76,91,0.8352,0.63837,0.2575,0.28571,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,9,0,0,1,0,0,3,0,0,9,0,0,5,0,5],[80,91,0.8791,0.64732,0.26722,0.28571,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,9,0,0,2,0,0,2,0,0,6,0,0,8,0,5],[84,91,0.9231,0.74107,0.26591,0.67857,0.85714,1.0,0.0,1.0,1,9,1,1,0,0,0,0,4,0,0,1,0,0,2,0,0,6,0,0,9,0,9],[88,91,0.967,0.7946,0.1381,0.71429,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,14,0,5],[91,91,1.0,0.77231,0.21088,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,4,0,0,17,0,5]]},{"b":6,"e":1.0,"k":"rising","v":0.41964,"x":0.86159,"p":[[0,83,0.0,0.41964,0.25488,0.28571,0.28571,0.46418,0.14286,1.0,0,4,0,0,0,2,0,0,20,0,0,2,0,0,2,0,0,2,0,0,0,0,4],[4,83,0.0482,0.72766,0.27284,0.53539,0.85707,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,4,0,0,3,0,0,3,0,0,4,0,0,6,0,11],[8,83,0.0964,0.7232,0.25488,0.53572,0.857,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,3,0,0,1,0,0,4,0,0,13,0,6],[12,83,0.1446,0.70088,0.28428,0.4286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,4,0,0,3,0,0,2,0,0,6,0,0,5,0,10],[16,83,0.1928,0.76786,0.28739,0.53571,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,7,0,0,1,0,0,1,0,0,2,0,0,6,0,15],[20,83,0.241,0.75,0.20203,0.71429,0.85707,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,8,0,0,12,0,5],[24,83,0.2892,0.75892,0.28446,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,4,0,0,1,0,0,0,0,0,6,0,0,6,0,13],[28,83,0.3373,0.74999,0.24223,0.57143,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,5,0,0,9,0,9],[32,83,0.3855,0.79909,0.22265,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,7,0,0,6,0,13],[36,83,0.4337,0.81696,0.2321,0.85711,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,13,0,12],[40,83,0.4819,0.7723,0.28763,0.57132,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,3,0,0,2,0,0,1,0,0,8,0,14],[44,83,0.5301,0.76786,0.22517,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,5,0,0,12,0,8],[48,83,0.5783,0.73657,0.25029,0.57142,0.857,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,6,0,0,9,0,8],[52,83,0.6265,0.6607,0.28065,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,5,0,0,4,0,0,2,0,0,4,0,0,9,0,6],[56,83,0.6747,0.77678,0.25489,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,5,0,0,8,0,12],[60,83,0.7229,0.83481,0.1992,0.857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,0,14,0,11],[64,83,0.7711,0.76339,0.22477,0.67857,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,4,0,0,12,0,8],[68,83,0.8193,0.79464,0.22286,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,5,0,0,10,0,11],[72,83,0.8675,0.83481,0.1992,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,9,0,14],[76,83,0.9157,0.86159,0.16556,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,11,0,14],[80,83,0.9639,0.8125,0.26592,0.64286,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,2,0,0,3,0,19],[83,83,1.0,0.79017,0.25996,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,4,0,0,2,0,17]]}]},{"i":"e7804e802754f4a8","q":"In an $n \\times n$ square array of $1 \\times 1$ cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.","t":[{"b":1,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,21,0.0,0.86607,0.27879,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,24],[4,21,0.1905,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[8,21,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,21,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,21,0.7619,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,21,0.9524,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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":6,"e":1.0,"k":"rising","v":0.83482,"x":1.0,"p":[[0,29,0.0,0.83482,0.25532,0.57143,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,3,0,20],[4,29,0.1379,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,29,0.2759,0.91963,0.1673,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,26],[12,29,0.4138,0.94196,0.1551,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,28],[16,29,0.5517,0.90179,0.17655,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,1,0,24],[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],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"666596957f52e794","q":"Find all functions  $f:\\mathbb{R}\\to \\mathbb{R}$  so that for any reals  $x,y$  the following holds:\n\\[f(x\\cdot f(x+y))+f(f(y)\\cdot f(x+y))=(x+y)^2\\]","t":[{"b":5,"e":0.85714,"k":"rising","v":0.49104,"x":0.97768,"p":[[0,209,0.0,0.49104,0.27879,0.28571,0.42857,0.60714,0.0,1.0,1,4,0,1,0,3,0,0,8,0,0,8,0,0,4,0,0,1,0,0,3,0,4],[4,209,0.0191,0.86161,0.28456,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,24],[8,209,0.0383,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,209,0.0574,0.85267,0.26363,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,1,0,23],[16,209,0.0766,0.86609,0.21407,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,3,0,21],[20,209,0.0957,0.89286,0.18558,0.82143,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,1,0,23],[24,209,0.1148,0.87051,0.22123,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,1,0,0,2,0,22],[28,209,0.134,0.88393,0.24856,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,25],[32,209,0.1531,0.89286,0.18211,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,5,0,21],[36,209,0.1722,0.89286,0.20516,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,23],[40,209,0.1914,0.92857,0.18558,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[44,209,0.2105,0.87945,0.2024,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,20],[48,209,0.2297,0.93304,0.17122,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,27],[52,209,0.2488,0.79908,0.24188,0.57132,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,3,0,0,2,0,17],[56,209,0.2679,0.89285,0.18898,0.82132,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,1,0,23],[60,209,0.2871,0.83927,0.22518,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,3,0,0,3,0,19],[64,209,0.3062,0.87946,0.2055,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,5,0,0,1,0,22],[68,209,0.3254,0.86158,0.24614,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,4,0,21],[72,209,0.3445,0.85713,0.23147,0.78571,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,0,0,0,2,0,22],[76,209,0.3636,0.93749,0.14701,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,26],[80,209,0.3828,0.90179,0.19377,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,23],[84,209,0.4019,0.91518,0.17076,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,25],[88,209,0.4211,0.89286,0.21128,0.85714,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,23],[92,209,0.4402,0.85267,0.21574,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,4,0,19],[96,209,0.4593,0.86604,0.23132,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,21],[100,209,0.4785,0.88392,0.19047,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,1,0,22],[104,209,0.4976,0.84374,0.2609,0.78571,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,3,0,21],[108,209,0.5167,0.90625,0.17717,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,23],[112,209,0.5359,0.78125,0.31131,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,2,0,0,1,0,0,2,0,0,5,0,0,1,0,0,0,0,20],[116,209,0.555,0.83925,0.26186,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,4,0,20],[120,209,0.5742,0.88391,0.2354,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,23],[124,209,0.5933,0.78123,0.29664,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,2,0,0,3,0,0,2,0,0,2,0,0,4,0,17],[128,209,0.6124,0.89732,0.22084,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,23],[132,209,0.6316,0.77674,0.25241,0.571,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,2,0,0,4,0,15],[136,209,0.6507,0.83929,0.23623,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,1,0,0,3,0,20],[140,209,0.6699,0.78125,0.3009,0.4286,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,3,0,0,5,0,0,0,0,0,3,0,0,1,0,19],[144,209,0.689,0.75893,0.32623,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,4,0,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,17],[148,209,0.7081,0.89286,0.21429,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,23],[152,209,0.7273,0.86161,0.2382,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,0,2,0,22],[156,209,0.7464,0.8482,0.21707,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,2,0,20],[160,209,0.7656,0.84377,0.23239,0.67846,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,2,0,0,1,0,21],[164,209,0.7847,0.8079,0.27809,0.63965,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,6,0,0,0,0,0,2,0,0,3,0,19],[168,209,0.8038,0.83482,0.27225,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,22],[172,209,0.823,0.82143,0.30514,0.82143,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,21],[176,209,0.8421,0.87945,0.22049,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,24],[180,209,0.8612,0.93748,0.11816,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[184,209,0.8804,0.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0,19],[156,215,0.7256,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[160,215,0.7442,0.91517,0.18853,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,24],[164,215,0.7628,0.91518,0.20473,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,24],[168,215,0.7814,0.91071,0.17768,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,22],[172,215,0.8,0.9375,0.18189,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[176,215,0.8186,0.84822,0.2765,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,23],[180,215,0.8372,0.88839,0.25187,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,25],[184,215,0.8558,0.88839,0.23887,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,24],[188,215,0.8744,0.8973,0.20899,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,3,0,24],[192,215,0.893,0.91071,0.20124,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,24],[196,215,0.9116,0.85712,0.27202,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,24],[200,215,0.9302,0.94643,0.13243,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[204,215,0.9488,0.91069,0.17037,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,23],[208,215,0.9674,0.90625,0.21902,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,26],[212,215,0.986,0.79464,0.30501,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,3,0,0,6,0,17],[215,215,1.0,0.88391,0.25112,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,24]]}]},{"i":"0719802ff92490d8","q":"Let  $c$  be fixed natural number. Sequence  $(a_n)$  is defined by:\r\n\r $a_1=1$ ,  $a_{n+1}=d(a_n)+c$  for  $n=1,2,...$ .\r\n\r\nwhere  $d(m)$  is number of divisors of  $m$ . Prove that there exist  $k$  natural such that sequence  $a_k,a_{k+1},...$  is periodic.","t":[{"b":5,"e":0.85714,"k":"flat","v":0.88393,"x":0.94196,"p":[[0,35,0.0,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[4,35,0.1143,0.89286,0.08748,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,11],[8,35,0.2286,0.89285,0.08748,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,21,0,10],[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],[16,35,0.4571,0.9375,0.11258,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,21],[20,35,0.5714,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[24,35,0.6857,0.91517,0.07874,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[28,35,0.8,0.88393,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,9],[32,35,0.9143,0.92856,0.07144,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[35,35,1.0,0.88839,0.07771,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,9]]},{"b":6,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,31,0.0,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[4,31,0.129,0.94196,0.10012,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[8,31,0.2581,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,31,0.3871,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,31,0.5161,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,31,0.6452,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,31,0.7742,0.99106,0.03461,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,31,0.9032,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"f1785cb491440275","q":"For a positive integer  $n$ , let  $d(n)$  denote the number of positive divisors of  $n$ . Determine all positive integers  $n$  for which  $d(n)$  is the second largest divisor of  $n$ .","t":[{"b":5,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,67,0.0,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[4,67,0.0597,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[8,67,0.1194,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[12,67,0.1791,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,67,0.2388,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,67,0.2985,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,67,0.3582,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[28,67,0.4179,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[32,67,0.4776,0.93304,0.15146,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,25],[36,67,0.5373,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[40,67,0.597,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[44,67,0.6567,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[48,67,0.7164,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[52,67,0.7761,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[56,67,0.8358,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[60,67,0.8955,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[64,67,0.9552,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[67,67,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":6,"e":0.71429,"k":"flat","v":0.86157,"x":0.99554,"p":[[0,115,0.0,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,115,0.0348,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,115,0.0696,0.91964,0.15126,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,6,0,22],[12,115,0.1043,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[16,115,0.1391,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[20,115,0.1739,0.98884,0.05085,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,30],[24,115,0.2087,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,115,0.2435,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[32,115,0.2783,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,115,0.313,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,115,0.3478,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[44,115,0.3826,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[48,115,0.4174,0.92857,0.13363,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,23],[52,115,0.4522,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[56,115,0.487,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[60,115,0.5217,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[64,115,0.5565,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[68,115,0.5913,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,115,0.6261,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[76,115,0.6609,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[80,115,0.6957,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[84,115,0.7304,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[88,115,0.7652,0.94642,0.09943,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[92,115,0.8,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[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],[100,115,0.8696,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[104,115,0.9043,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[108,115,0.9391,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[112,115,0.9739,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[115,115,1.0,0.86157,0.14054,0.857,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,20,0,9]]}]},{"i":"9d9ea3c753331455","q":"Given an integer  $k \\geq 2$ , determine the largest number of divisors the binomial coefficient  $\\binom{n}{k}$  may have in the range  $n-k+1, \\ldots, n$  , as  $n$  runs through the integers greater than or equal to  $k$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.12054,"p":[[0,21,0.0,0.12054,0.16409,0.0,0.0,0.2857,0.0,0.57143,18,0,3,18,0,5,0,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,21,0.1905,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.05357,0.1171,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.04911,0.14555,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.2857,"p":[[0,23,0.0,0.09822,0.14914,0.0,0.0,0.14287,0.0,0.57143,20,0,5,20,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,23,0.1739,0.2857,0.17126,0.24999,0.28571,0.28571,0.0,1.0,2,1,0,2,0,6,0,0,19,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[8,23,0.3478,0.25,0.17128,0.14286,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,3,0,0,17,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[12,23,0.5217,0.08927,0.17764,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0357,0.11839,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,23,0.8696,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d7d22b4d43e015ac","q":"Let  $ABC$  be a triangle, the circle having  $BC$  as diameter cuts  $AB,AC$  at  $F,E$  respectively. Let  $P$  a point on this circle. Let  $C',B$ ' be the projections of  $P$  upon the sides  $AB,AC$  respectively. Let  $H$  be the orthocenter of the triangle  $AB'C'$ . Show that  $\\angle EHF = 90^o$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.66517,"x":0.85713,"p":[[0,83,0.0,0.74998,0.27665,0.67856,0.85707,1.0,0.0,1.0,1,12,0,1,0,1,0,0,2,0,0,2,0,0,2,0,0,7,0,0,5,0,12],[4,83,0.0482,0.85713,0.24484,0.857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,0,5,0,20],[8,83,0.0964,0.71429,0.27894,0.42857,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,9,0,0,1,0,0,5,0,0,3,0,12],[12,83,0.1446,0.78569,0.26488,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,5,0,0,4,0,15],[16,83,0.1928,0.66517,0.34554,0.42857,0.71429,1.0,0.0,1.0,3,12,0,3,0,2,0,0,1,0,0,6,0,0,0,0,0,5,0,0,3,0,12],[20,83,0.241,0.77677,0.28108,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,4,0,0,2,0,0,5,0,0,1,0,17],[24,83,0.2892,0.81696,0.22084,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,9,0,0,2,0,16],[28,83,0.3373,0.73212,0.3129,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,4,0,0,4,0,0,1,0,0,2,0,16],[32,83,0.3855,0.71875,0.28901,0.53569,0.78571,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,3,0,0,3,0,0,5,0,0,3,0,0,3,0,13],[36,83,0.4337,0.74553,0.28734,0.42857,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,1,0,0,6,0,0,0,0,0,5,0,0,5,0,13],[40,83,0.4819,0.72321,0.32915,0.42857,0.85707,1.0,0.0,1.0,2,15,0,2,0,2,0,0,1,0,0,4,0,0,1,0,0,5,0,0,2,0,15],[44,83,0.5301,0.73214,0.30671,0.42857,0.85707,1.0,0.0,1.0,1,15,0,1,0,0,0,0,5,0,0,4,0,0,0,0,0,5,0,0,2,0,15],[48,83,0.5783,0.77676,0.25987,0.57132,0.85714,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,5,0,0,3,0,15],[52,83,0.6265,0.73212,0.28516,0.4286,0.78564,1.0,0.0,1.0,1,13,0,1,0,0,0,0,3,0,0,5,0,0,1,0,0,6,0,0,3,0,13],[56,83,0.6747,0.78121,0.2422,0.57143,0.857,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,4,0,0,6,0,13],[60,83,0.7229,0.76338,0.30642,0.53539,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,3,0,0,3,0,0,1,0,0,4,0,0,2,0,17],[64,83,0.7711,0.74554,0.35667,0.53572,1.0,1.0,0.0,1.0,3,19,0,3,0,2,0,0,1,0,0,2,0,0,1,0,0,4,0,0,0,0,19],[68,83,0.8193,0.69195,0.29904,0.42859,0.71429,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,3,0,0,7,0,0,1,0,0,5,0,0,1,0,13],[72,83,0.8675,0.78124,0.28343,0.71421,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,6,0,0,3,0,16],[76,83,0.9157,0.81249,0.25365,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,0,0,18],[80,83,0.9639,0.74105,0.25365,0.67857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,2,0,0,3,0,0,2,0,0,9,0,0,5,0,10],[83,83,1.0,0.77678,0.24206,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,6,0,0,5,0,13]]},{"b":7,"e":0.0,"k":"falling","v":0.26339,"x":0.90625,"p":[[0,91,0.0,0.67409,0.33166,0.42859,0.71429,1.0,0.0,1.0,3,12,0,3,0,1,0,0,2,0,0,4,0,0,1,0,0,8,0,0,1,0,12],[4,91,0.044,0.77229,0.30694,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,1,0,0,3,0,0,3,0,0,1,0,0,4,0,17],[8,91,0.0879,0.79017,0.27661,0.67857,0.92857,1.0,0.1429,1.0,0,16,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,3,0,0,5,0,16],[12,91,0.1319,0.76338,0.26393,0.57132,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,2,0,0,3,0,15],[16,91,0.1758,0.78572,0.30093,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,2,0,0,3,0,0,2,0,0,3,0,0,1,0,19],[20,91,0.2198,0.82141,0.26002,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,5,0,0,3,0,18],[24,91,0.2637,0.75893,0.30396,0.53572,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,4,0,0,2,0,0,2,0,0,3,0,0,2,0,17],[28,91,0.3077,0.77677,0.2549,0.57143,0.85714,1.0,0.1429,1.0,0,15,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,4,0,0,3,0,15],[32,91,0.3516,0.70981,0.36332,0.42859,0.85714,1.0,0.0,1.0,5,15,0,5,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,4,0,15],[36,91,0.3956,0.81695,0.27488,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,4,0,0,4,0,18],[40,91,0.4396,0.683,0.30038,0.42859,0.71429,1.0,0.0,1.0,1,13,0,1,0,1,0,0,2,0,0,7,0,0,4,0,0,4,0,0,0,0,13],[44,91,0.4835,0.7991,0.274,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,0,0,0,1,0,0,3,0,0,2,0,0,9,0,14],[48,91,0.5275,0.90625,0.16982,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,20],[52,91,0.5714,0.76783,0.25693,0.53539,0.85707,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,6,0,0,3,0,0,3,0,0,3,0,15],[56,91,0.6154,0.78125,0.25996,0.64286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,6,0,0,0,0,0,5,0,0,4,0,15],[60,91,0.6593,0.68749,0.37532,0.28571,0.85714,1.0,0.0,1.0,4,15,0,4,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,0,4,0,15],[64,91,0.7033,0.68749,0.32427,0.39286,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,5,0,0,1,0,0,3,0,0,2,0,0,8,0,10],[68,91,0.7473,0.62498,0.34022,0.28571,0.71429,1.0,0.0,1.0,2,11,0,2,0,2,0,0,5,0,0,5,0,0,1,0,0,4,0,0,2,0,11],[72,91,0.7912,0.65175,0.30917,0.42857,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,5,0,0,4,0,0,4,0,0,3,0,0,5,0,9],[76,91,0.8352,0.67854,0.29881,0.42857,0.71429,1.0,0.0,1.0,2,9,0,2,0,1,0,0,1,0,0,6,0,0,2,0,0,6,0,0,5,0,9],[80,91,0.8791,0.54911,0.3409,0.28571,0.42857,1.0,0.0,1.0,3,9,0,3,0,2,0,0,6,0,0,7,0,0,0,0,0,5,0,0,0,0,9],[84,91,0.9231,0.59812,0.34534,0.28571,0.57144,1.0,0.0,1.0,2,10,0,2,0,2,0,0,7,0,0,5,0,0,0,0,0,3,0,0,3,0,10],[88,91,0.967,0.47317,0.30395,0.2857,0.42859,0.71429,0.0,1.0,3,3,0,3,0,4,0,0,7,0,0,4,0,0,3,0,0,5,0,0,3,0,3],[91,91,1.0,0.26339,0.31361,0.0,0.14286,0.32143,0.0,1.0,9,4,0,9,0,12,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,4]]}]},{"i":"d39c7ffe58401055","q":"Let  $n$  be a natural number. We define sequences  $\\langle a_i\\rangle$  and  $\\langle b_i\\rangle$  of integers as follows. We let  $a_0=1$  and  $b_0=n$ . For  $i>0$ , we let   $$ \\left( a_i,b_i\\right)=\\begin{cases} \\left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\\right) & \\text{if } a_{i-1}<b_{i-1},\n\\left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\\right) & \\text{if } a_{i-1}>b_{i-1},\n\\left(a_{i-1},b_{i-1}\\right) & \\text{if } a_{i-1}=b_{i-1}.\\end{cases} $$  Given that  $a_k=b_k$  for some natural number  $k$ , prove that  $n+3$  is a power of two.","t":[{"b":4,"e":0.71429,"k":"flat","v":0.49107,"x":0.65177,"p":[[0,18,0.0,0.5982,0.31831,0.28571,0.71429,0.85714,0.0,1.0,3,4,0,3,0,3,0,0,3,0,0,1,0,0,3,0,0,8,0,0,7,0,4],[4,18,0.2222,0.62052,0.30849,0.39286,0.71429,0.85714,0.0,1.0,2,6,0,2,0,3,0,0,3,0,0,1,0,0,5,0,0,7,0,0,5,0,6],[8,18,0.4444,0.49107,0.37446,0.14286,0.71429,0.74996,0.0,1.0,7,5,0,7,0,4,0,0,4,0,0,0,0,0,0,0,0,9,0,0,3,0,5],[12,18,0.6667,0.52679,0.32818,0.2857,0.57143,0.71429,0.0,1.0,5,4,0,5,0,2,0,0,4,0,0,2,0,0,4,0,0,8,0,0,3,0,4],[16,18,0.8889,0.63389,0.13805,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,14,0,0,3,0,0],[18,18,1.0,0.65177,0.17105,0.57143,0.71429,0.75,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,5,0,0,8,0,0,10,0,0,8,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.41069,"x":0.72745,"p":[[0,22,0.0,0.63391,0.33869,0.42857,0.71429,1.0,0.0,1.0,4,9,0,4,0,1,0,0,2,0,0,3,0,0,4,0,0,5,0,0,4,0,9],[4,22,0.1818,0.72745,0.27995,0.57143,0.85707,1.0,0.0,1.0,1,10,0,1,0,2,0,0,0,0,0,4,0,0,3,0,0,5,0,0,7,0,10],[8,22,0.3636,0.50892,0.32328,0.14286,0.64286,0.71429,0.0,1.0,5,2,0,5,0,4,0,0,2,0,0,2,0,0,3,0,0,10,0,0,4,0,2],[12,22,0.5455,0.41069,0.29611,0.14286,0.4998,0.57143,0.0,0.85714,7,0,0,7,0,4,0,0,2,0,0,3,0,0,9,0,0,3,0,0,4,0,0],[16,22,0.7273,0.51334,0.25718,0.42857,0.57143,0.71429,0.0,0.85714,4,0,0,4,0,1,0,0,2,0,0,4,0,0,12,0,0,4,0,0,5,0,0],[20,22,0.9091,0.51786,0.28738,0.2857,0.57143,0.71429,0.0,0.85714,4,0,0,4,0,3,0,0,2,0,0,3,0,0,6,0,0,8,0,0,6,0,0],[22,22,1.0,0.48649,0.27872,0.2857,0.57121,0.71429,0.0,0.85714,4,0,0,4,0,3,0,0,3,0,0,4,0,0,7,0,0,6,0,0,5,0,0]]}]},{"i":"c009b5e1cb87bc84","q":"Let  $n\\geq 4$  be a positive integer.Out of  $n$  people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of  $n$ ,such that for any possible outcome of the game,there always exist an ordered four people group  $(a_{1},a_{2},a_{3},a_{4})$ ,such that the person  $a_{i}$  wins against  $a_{j}$  for any  $1\\leq i<j\\leq 4$","t":[{"b":4,"e":0.85714,"k":"flat","v":0.83036,"x":0.9375,"p":[[0,16,0.0,0.84375,0.33189,0.96429,1.0,1.0,0.0,1.0,4,24,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[4,16,0.25,0.9375,0.13803,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[8,16,0.5,0.90625,0.12169,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,13,0,16],[12,16,0.75,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[16,16,1.0,0.83036,0.18708,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,10,0,12]]},{"b":7,"e":0.71429,"k":"falling","v":0.63838,"x":0.95536,"p":[[0,20,0.0,0.87944,0.2399,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,0,23],[4,20,0.2,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[8,20,0.4,0.9107,0.10565,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,16],[12,20,0.6,0.90625,0.14555,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,10,0,18],[16,20,0.8,0.91518,0.17445,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,5,0,23],[20,20,1.0,0.63838,0.24481,0.42857,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,7,0,0,3,0,0,7,0,0,8,0,3]]}]},{"i":"bdaa09c337c6f345","q":"**a)** Prove that the function  $ f:\\mathbb{Z}_{\\ge 0}\\longrightarrow\\mathbb{Z}_{\\ge 0} , $  given as  $ f(n)=n+(-1)^n $  is bijective.**b)** Find all surjective functions  $ g:\\mathbb{Z}_{\\ge 0}\\longrightarrow\\mathbb{Z}_{\\ge 0} $  that have the property that  $ g(n)\\ge n+(-1)^n , $  for any nonnegative integer.","t":[{"b":3,"e":1.0,"k":"flat","v":0.78125,"x":0.90625,"p":[[0,22,0.0,0.90625,0.13175,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[4,22,0.1818,0.82589,0.1504,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,3,0,12],[8,22,0.3636,0.80803,0.13176,0.71429,0.71429,1.0,0.714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,1,0,10],[12,22,0.5455,0.80357,0.12753,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,2,0,9],[16,22,0.7273,0.78125,0.11837,0.71429,0.71429,0.75,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,1,0,7],[20,22,0.9091,0.78125,0.11837,0.71429,0.71429,0.75,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,1,0,7],[22,22,1.0,0.84821,0.14258,0.71429,0.71429,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,15]]},{"b":4,"e":1.0,"k":"flat","v":0.87486,"x":0.98214,"p":[[0,22,0.0,0.90179,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,21],[4,22,0.1818,0.87486,0.14633,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,1,0,18],[8,22,0.3636,0.88837,0.15046,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,1,0,20],[12,22,0.5455,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[16,22,0.7273,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,22,0.9091,0.92411,0.13825,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,24],[22,22,1.0,0.9732,0.06624,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]}]},{"i":"21b87a5db1c5fe88","q":"Let  $x<105$  and  $y<120$  be positive integers. Determine the number of ordered pairs  $(x,y)$  for which  $\\frac{x}{105}$  and  $\\frac{y}{90}$  are in lowest terms, but their sum  $\\frac{x}{105} + \\frac{y}{90} = \\frac{6x+7y}{630}$  is not.","t":[{"b":2,"e":1.0,"k":"flat","v":0.95094,"x":1.0,"p":[[0,94,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,94,0.0426,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,94,0.0851,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,94,0.1277,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,94,0.1702,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,94,0.2128,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,94,0.2553,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,94,0.2979,0.95094,0.14094,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,27],[32,94,0.3404,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[36,94,0.383,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,94,0.4255,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[44,94,0.4681,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,94,0.5106,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[52,94,0.5532,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,94,0.5957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,94,0.6383,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,94,0.6809,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,94,0.7234,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,94,0.766,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,94,0.8085,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,94,0.8511,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,94,0.8936,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,94,0.9362,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,94,0.9787,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[94,94,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,82,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,82,0.0488,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,82,0.0976,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,82,0.1463,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,82,0.1951,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,82,0.2439,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,82,0.2927,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,82,0.3415,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,82,0.3902,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[36,82,0.439,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,82,0.4878,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,82,0.5366,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[48,82,0.5854,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,82,0.6341,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[56,82,0.6829,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,82,0.7317,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[68,82,0.8293,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[76,82,0.9268,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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],[82,82,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"9aaf22b21e88f27b","q":"12. G6 (USA) Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron, $G$ its centroid, and $A_{1}^{\\prime}, A_{2}^{\\prime}, A_{3}^{\\prime}$, and $A_{4}^{\\prime}$ the points where the circumsphere of $A_{1} A_{2} A_{3} A_{4}$ intersects $G A_{1}, G A_{2}, G A_{3}$, and $G A_{4}$, respectively. Prove that  $$ G A_{1} \\cdot G A_{2} \\cdot G A_{3} \\cdot G A_{4} \\leq G A_{1}^{\\prime} \\cdot G A_{2}^{\\prime} \\cdot G A_{3}^{\\prime} \\cdot G A_{4}^{\\prime} $$  and  $$ \\frac{1}{G A_{1}^{\\prime}}+\\frac{1}{G A_{2}^{\\prime}}+\\frac{1}{G A_{3}^{\\prime}}+\\frac{1}{G A_{4}^{\\prime}} \\leq \\frac{1}{G A_{1}}+\\frac{1}{G A_{2}}+\\frac{1}{G A_{3}}+\\frac{1}{G A_{4}} $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.9241,"x":1.0,"p":[[0,61,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,61,0.0656,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,61,0.1311,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,61,0.1967,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,61,0.2623,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,61,0.3279,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,61,0.3934,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,61,0.459,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,61,0.5246,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,61,0.5902,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,61,0.6557,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[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],[48,61,0.7869,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[52,61,0.8525,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,61,0.918,0.9241,0.07974,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[60,61,0.9836,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[61,61,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]},{"b":3,"e":1.0,"k":"flat","v":0.98214,"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],[4,28,0.1429,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,28,0.4286,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,28,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,28,0.7143,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,28,0.8571,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,28,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"2e2d62930fe0aa82","q":"16. G4 (ARM) Let $\\Gamma_{1}, \\Gamma_{2}, \\Gamma_{3}, \\Gamma_{4}$ be distinct circles such that $\\Gamma_{1}, \\Gamma_{3}$ are externally tangent at $P$, and $\\Gamma_{2}, \\Gamma_{4}$ are externally tangent at the same point $P$. Suppose that $\\Gamma_{1}$ and $\\Gamma_{2} ; \\Gamma_{2}$ and $\\Gamma_{3} ; \\Gamma_{3}$ and $\\Gamma_{4} ; \\Gamma_{4}$ and $\\Gamma_{1}$ meet at $A, B, C, D$, respectively, and that all these points are different from $P$. Prove that  $$ \\frac{A B \\cdot B C}{A D \\cdot D C}=\\frac{P B^{2}}{P D^{2}} $$","t":[{"b":2,"e":1.0,"k":"flat","v":0.95534,"x":0.99554,"p":[[0,27,0.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[4,27,0.1481,0.95534,0.1871,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[8,27,0.2963,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[12,27,0.4444,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,27,0.5926,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,27,0.7407,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,27,0.8889,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[27,27,1.0,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28]]},{"b":5,"e":0.571,"k":"flat","v":0.83929,"x":0.98214,"p":[[0,91,0.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[4,91,0.044,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,91,0.0879,0.91518,0.24707,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[12,91,0.1319,0.95534,0.16539,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[16,91,0.1758,0.93304,0.18552,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[20,91,0.2198,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[24,91,0.2637,0.9375,0.19212,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[28,91,0.3077,0.83929,0.34209,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,25],[32,91,0.3516,0.93304,0.16746,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,27],[36,91,0.3956,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[40,91,0.4396,0.93303,0.13825,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[44,91,0.4835,0.95536,0.11538,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[48,91,0.5275,0.92409,0.1923,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,24],[52,91,0.5714,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[56,91,0.6154,0.94641,0.11714,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[60,91,0.6593,0.94196,0.15916,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[64,91,0.7033,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[68,91,0.7473,0.88838,0.22513,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,19],[72,91,0.7912,0.92408,0.1129,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[76,91,0.8352,0.91964,0.14698,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,21],[80,91,0.8791,0.89272,0.12896,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16],[84,91,0.9231,0.91072,0.12242,0.85714,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,17],[88,91,0.967,0.87945,0.13885,0.85711,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,10,0,15],[91,91,1.0,0.87053,0.12037,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,15,0,11]]}]},{"i":"bf4f35990c7d8043","q":"Let  $x_i\\  (i = 1, 2, \\cdots 22)$  be reals such that  $x_i \\in [2^{i-1},2^i]$ . Find the maximum possible value of  $$ (x_1+x_2+\\cdots +x_{22})(\\frac{1}{x_1}+\\frac{1}{x_2}+\\cdots+\\frac{1}{x_{22}}) $$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.23214,"x":0.98661,"p":[[0,62,0.0,0.54018,0.27137,0.28571,0.57143,0.64286,0.14286,1.0,0,4,0,0,0,5,0,0,4,0,0,4,0,0,11,0,0,0,0,0,4,0,4],[4,62,0.0645,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25],[8,62,0.129,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,62,0.1935,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[16,62,0.2581,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[20,62,0.3226,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[24,62,0.3871,0.92411,0.16746,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,26],[28,62,0.4516,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[32,62,0.5161,0.92857,0.11845,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[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],[40,62,0.6452,0.94196,0.11769,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,24],[44,62,0.7097,0.82589,0.31488,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,22],[48,62,0.7742,0.875,0.21943,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,21],[52,62,0.8387,0.86607,0.25238,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,21],[56,62,0.9032,0.80357,0.31084,0.78571,1.0,1.0,0.14286,1.0,0,19,0,0,0,5,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,19],[60,62,0.9677,0.6875,0.34523,0.46429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,8,0,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,13],[62,62,1.0,0.23214,0.18123,0.14286,0.14286,0.42857,0.0,0.85714,2,0,0,2,0,21,0,0,0,0,0,7,0,0,1,0,0,0,0,0,1,0,0]]},{"b":5,"e":0.57143,"k":"rising","v":0.48661,"x":0.99107,"p":[[0,167,0.0,0.48661,0.2547,0.28571,0.5,0.57143,0.0,1.0,1,4,0,1,0,3,0,0,7,0,0,5,0,0,11,0,0,1,0,0,0,0,4],[4,167,0.024,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,167,0.0479,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[12,167,0.0719,0.9375,0.15542,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,27],[16,167,0.0958,0.94643,0.13243,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,27],[20,167,0.1198,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[24,167,0.1437,0.92857,0.18898,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,26],[28,167,0.1677,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,167,0.1916,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,167,0.2156,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,167,0.2395,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,167,0.2635,0.95089,0.13175,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[48,167,0.2874,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[52,167,0.3114,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[56,167,0.3353,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[60,167,0.3593,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[64,167,0.3832,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[68,167,0.4072,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,167,0.4311,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[76,167,0.4551,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[80,167,0.479,0.94643,0.16656,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[84,167,0.503,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[88,167,0.5269,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[92,167,0.5509,0.91964,0.15542,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,24],[96,167,0.5749,0.95089,0.16602,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[100,167,0.5988,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[104,167,0.6228,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[108,167,0.6467,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[112,167,0.6707,0.94196,0.11769,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,24],[116,167,0.6946,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[120,167,0.7186,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[124,167,0.7425,0.93304,0.14279,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,25],[128,167,0.7665,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[132,167,0.7904,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[136,167,0.8144,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[140,167,0.8383,0.93304,0.13356,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[144,167,0.8623,0.93304,0.18893,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,28],[148,167,0.8862,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[152,167,0.9102,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[156,167,0.9341,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[160,167,0.9581,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,25],[164,167,0.982,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[167,167,1.0,0.87946,0.19597,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,19]]}]},{"i":"0bb8493492cacd71","q":"Let  $ \\displaystyle{a,b,c}$  be positive real numbers such that  $\\displaystyle{a+b+c=1.}$  Prove that   $$  \\displaystyle{\\frac{1+ab}{a+b}+\\frac{1+bc}{b+c}+\\frac{1+ca}{c+a}\\geq 5.} $$","t":[{"b":2,"e":1.0,"k":"flat","v":0.22321,"x":0.81249,"p":[[0,72,0.0,0.81249,0.32032,0.71429,1.0,1.0,0.0,1.0,3,21,0,3,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,21],[4,72,0.0556,0.50893,0.37616,0.25001,0.35714,1.0,0.0,1.0,5,9,0,5,0,3,0,0,8,0,0,1,0,0,2,0,0,3,0,0,1,0,9],[8,72,0.1111,0.59374,0.37476,0.14286,0.71429,1.0,0.0,1.0,4,9,0,4,0,5,0,0,2,0,0,0,0,0,4,0,0,3,0,0,5,0,9],[12,72,0.1667,0.4375,0.37786,0.10714,0.28571,0.75,0.0,1.0,8,6,0,8,0,4,0,0,5,0,0,2,0,0,1,0,0,4,0,0,2,0,6],[16,72,0.2222,0.47321,0.40317,0.0,0.42857,1.0,0.0,1.0,9,9,0,9,0,3,0,0,3,0,0,2,0,0,2,0,0,4,0,0,0,0,9],[20,72,0.2778,0.39286,0.3977,0.0,0.14286,0.75,0.0,1.0,11,6,0,11,0,6,0,0,1,0,0,0,0,0,4,0,0,2,0,0,2,0,6],[24,72,0.3333,0.3482,0.37785,0.0,0.21428,0.60714,0.0,1.0,12,6,0,12,0,4,0,0,4,0,0,2,0,0,2,0,0,2,0,0,0,0,6],[28,72,0.3889,0.5,0.38631,0.14286,0.49999,0.89286,0.0,1.0,6,8,0,6,0,5,0,0,4,0,0,1,0,0,2,0,0,4,0,0,2,0,8],[32,72,0.4444,0.48659,0.38772,0.10714,0.4998,0.78571,0.0,1.0,8,8,0,8,0,3,0,0,3,0,0,2,0,0,2,0,0,6,0,0,0,0,8],[36,72,0.5,0.38392,0.36497,0.0,0.28571,0.71429,0.0,1.0,9,5,0,9,0,5,0,0,4,0,0,4,0,0,1,0,0,2,0,0,2,0,5],[40,72,0.5556,0.4732,0.42624,0.0,0.42857,1.0,0.0,1.0,11,10,0,11,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,1,0,10],[44,72,0.6111,0.22321,0.29867,0.0,0.07143,0.32143,0.0,1.0,16,2,0,16,0,4,0,0,4,0,0,2,0,0,2,0,0,2,0,0,0,0,2],[48,72,0.6667,0.37945,0.35284,0.0,0.28571,0.71429,0.0,1.0,10,3,0,10,0,3,0,0,6,0,0,0,0,0,4,0,0,3,0,0,3,0,3],[52,72,0.7222,0.40625,0.42724,0.0,0.14286,1.0,0.0,1.0,12,9,0,12,0,5,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,9],[56,72,0.7778,0.42856,0.40247,0.0,0.28571,0.75,0.0,1.0,11,7,0,11,0,3,0,0,3,0,0,0,0,0,3,0,0,4,0,0,1,0,7],[60,72,0.8333,0.54464,0.43805,0.0,0.71429,1.0,0.0,1.0,11,12,0,11,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,12],[64,72,0.8889,0.49551,0.3912,0.0,0.57121,0.85714,0.0,1.0,9,6,0,9,0,2,0,0,3,0,0,0,0,0,3,0,0,5,0,0,4,0,6],[68,72,0.9444,0.43302,0.37026,0.10714,0.28571,0.71429,0.0,1.0,8,6,0,8,0,4,0,0,5,0,0,0,0,0,5,0,0,3,0,0,1,0,6],[72,72,1.0,0.73213,0.32685,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,4,0,0,1,0,0,0,0,0,5,0,0,4,0,0,1,0,16]]},{"b":5,"e":0.28571,"k":"flat","v":0.45534,"x":1.0,"p":[[0,111,0.0,0.79017,0.2879,0.67836,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,3,0,0,1,0,0,2,0,0,4,0,0,2,0,18],[4,111,0.036,0.46875,0.33547,0.25,0.35714,0.71429,0.0,1.0,4,5,0,4,0,4,0,0,8,0,0,3,0,0,1,0,0,5,0,0,2,0,5],[8,111,0.0721,0.54016,0.41914,0.10714,0.64286,1.0,0.0,1.0,8,11,0,8,0,3,0,0,3,0,0,0,0,0,2,0,0,3,0,0,2,0,11],[12,111,0.1081,0.65625,0.39586,0.28571,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,4,0,0,2,0,0,1,0,0,2,0,0,1,0,16],[16,111,0.1441,0.50892,0.38949,0.14286,0.4998,1.0,0.0,1.0,7,9,0,7,0,3,0,0,4,0,0,2,0,0,1,0,0,6,0,0,0,0,9],[20,111,0.1802,0.57589,0.39039,0.28571,0.71429,1.0,0.0,1.0,6,12,0,6,0,0,0,0,7,0,0,2,0,0,0,0,0,5,0,0,0,0,12],[24,111,0.2162,0.45534,0.3489,0.14286,0.35714,0.71429,0.0,1.0,6,5,0,6,0,4,0,0,6,0,0,1,0,0,3,0,0,6,0,0,1,0,5],[28,111,0.2523,0.53125,0.36463,0.24999,0.64286,0.85714,0.0,1.0,6,7,0,6,0,2,0,0,4,0,0,3,0,0,1,0,0,7,0,0,2,0,7],[32,111,0.2883,0.7232,0.2695,0.67857,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,2,0,0,0,0,0,4,0,0,12,0,0,2,0,10],[36,111,0.3243,0.45962,0.35682,0.14286,0.28571,0.75,0.0,1.0,5,5,0,5,0,6,0,0,6,0,0,0,0,0,3,0,0,4,0,0,3,0,5],[40,111,0.3604,0.56697,0.30823,0.28571,0.64286,0.75,0.0,1.0,2,5,0,2,0,3,0,0,6,0,0,1,0,0,4,0,0,8,0,0,3,0,5],[44,111,0.3964,0.55803,0.32016,0.28571,0.64286,0.75,0.0,1.0,2,6,0,2,0,3,0,0,8,0,0,0,0,0,3,0,0,8,0,0,2,0,6],[48,111,0.4324,0.55357,0.35129,0.28571,0.57143,1.0,0.0,1.0,4,9,0,4,0,1,0,0,8,0,0,2,0,0,2,0,0,6,0,0,0,0,9],[52,111,0.4685,0.52229,0.33428,0.28571,0.571,0.75,0.0,1.0,4,6,0,4,0,2,0,0,7,0,0,2,0,0,4,0,0,5,0,0,2,0,6],[56,111,0.5045,0.54911,0.35013,0.28571,0.57143,1.0,0.0,1.0,3,9,0,3,0,2,0,0,10,0,0,0,0,0,2,0,0,6,0,0,0,0,9],[60,111,0.5405,0.62054,0.34183,0.28571,0.71429,1.0,0.0,1.0,3,11,0,3,0,0,0,0,8,0,0,1,0,0,2,0,0,7,0,0,0,0,11],[64,111,0.5766,0.65178,0.32525,0.5,0.71429,0.89286,0.0,1.0,4,8,0,4,0,0,0,0,4,0,0,0,0,0,1,0,0,12,0,0,3,0,8],[68,111,0.6126,0.53571,0.3607,0.28571,0.5,0.89286,0.0,1.0,4,8,0,4,0,3,0,0,7,0,0,2,0,0,1,0,0,5,0,0,2,0,8],[72,111,0.6486,0.62499,0.35129,0.28571,0.71429,1.0,0.0,1.0,4,9,0,4,0,1,0,0,5,0,0,1,0,0,1,0,0,7,0,0,4,0,9],[76,111,0.6847,0.53124,0.32583,0.28571,0.49979,0.78571,0.0,1.0,1,8,0,1,0,4,0,0,10,0,0,1,0,0,4,0,0,4,0,0,0,0,8],[80,111,0.7207,0.51337,0.36397,0.14286,0.571,0.85714,0.0,1.0,5,7,0,5,0,4,0,0,5,0,0,1,0,0,3,0,0,5,0,0,2,0,7],[84,111,0.7568,0.56247,0.35344,0.28571,0.71429,0.85714,0.0,1.0,5,7,0,5,0,1,0,0,6,0,0,1,0,0,2,0,0,7,0,0,3,0,7],[88,111,0.7928,0.53116,0.35944,0.24999,0.57143,0.85714,0.0,1.0,5,7,0,5,0,3,0,0,4,0,0,3,0,0,3,0,0,4,0,0,3,0,7],[92,111,0.8288,0.75,0.29451,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,5,0,0,1,0,0,2,0,0,6,0,0,2,0,15],[96,111,0.8649,0.70981,0.33973,0.28571,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,7,0,0,1,0,0,1,0,0,3,0,0,3,0,15],[100,111,0.9009,0.64286,0.40248,0.28571,0.71429,1.0,0.0,1.0,6,15,0,6,0,1,0,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,15],[104,111,0.9369,0.87499,0.25444,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,1,0,24],[108,111,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[111,111,1.0,0.91518,0.22548,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,27]]}]},{"i":"cf851e5192573ca6","q":"Let  $x_1,x_2,\\dots ,x_6$  be nonnegative real numbers such that  $x_1+x_2+x_3+x_4+x_5+x_6=1$ , and  $x_1x_3x_5+x_2x_4x_6 \\geq \\frac{1}{540}$ . Let  $p$  and  $q$  be positive relatively prime integers such that  $\\frac{p}{q}$  is the maximum possible value of  $x_1x_2x_3+x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$ . Find  $p+q$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.32143,"x":0.52679,"p":[[0,104,0.0,0.32143,0.13832,0.28571,0.35714,0.42857,0.0,0.57143,2,0,2,2,0,5,0,0,9,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[4,104,0.0385,0.48214,0.17405,0.42857,0.57143,0.57143,0.0,0.85714,1,0,1,1,0,1,0,0,5,0,0,8,0,0,13,0,0,3,0,0,1,0,0],[8,104,0.0769,0.46429,0.20516,0.28571,0.5,0.60714,0.14286,0.85714,0,0,0,0,0,3,0,0,11,0,0,2,0,0,8,0,0,7,0,0,1,0,0],[12,104,0.1154,0.50893,0.15947,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,10,0,0,14,0,0,2,0,0,2,0,0],[16,104,0.1538,0.45089,0.18595,0.28571,0.5,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,6,0,0,6,0,0,12,0,0,4,0,0,0,0,0],[20,104,0.1923,0.46428,0.19233,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,13,0,0,3,0,0,9,0,0,4,0,0,2,0,0],[24,104,0.2308,0.50893,0.17835,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,9,0,0,10,0,0,5,0,0,2,0,0],[28,104,0.2692,0.48214,0.17405,0.28571,0.42857,0.60714,0.14286,0.71429,0,0,0,0,0,1,0,0,9,0,0,7,0,0,7,0,0,8,0,0,0,0,0],[32,104,0.3077,0.48214,0.21354,0.28571,0.42857,0.71429,0.0,0.85714,1,0,1,1,0,0,0,0,12,0,0,4,0,0,5,0,0,8,0,0,2,0,0],[36,104,0.3462,0.52679,0.22428,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,4,0,0,8,0,0,8,0,0,5,0,0,2,0,2],[40,104,0.3846,0.49553,0.22583,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,11,0,0,6,0,0,7,0,0,1,0,0,5,0,1],[44,104,0.4231,0.50893,0.20497,0.39286,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,7,0,0,11,0,0,2,0,0,3,0,1],[48,104,0.4615,0.47768,0.19103,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,11,0,0,5,0,0,8,0,0,5,0,0,2,0,0],[52,104,0.5,0.49554,0.22864,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,7,0,0,7,0,0,8,0,0,3,0,0,2,0,2],[56,104,0.5385,0.41964,0.18536,0.28571,0.42857,0.46431,0.14286,1.0,0,1,0,0,0,2,0,0,12,0,0,10,0,0,5,0,0,1,0,0,1,0,1],[60,104,0.5769,0.41518,0.16115,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,4,0,0,14,0,0,8,0,0,0,0,0,1,0,0],[64,104,0.6154,0.43303,0.145,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,8,0,0,11,0,0,9,0,0,2,0,0,0,0,0],[68,104,0.6538,0.50893,0.18536,0.39286,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,7,0,0,10,0,0,4,0,0,3,0,0],[72,104,0.6923,0.50891,0.20183,0.28571,0.57121,0.60714,0.14286,0.85714,0,0,0,0,0,1,0,0,9,0,0,5,0,0,9,0,0,4,0,0,4,0,0],[76,104,0.7308,0.41964,0.16727,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,9,0,0,6,0,0,11,0,0,2,0,0,0,0,0],[80,104,0.7692,0.50891,0.17104,0.42857,0.57121,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,8,0,0,12,0,0,3,0,0,1,0,1],[84,104,0.8077,0.4375,0.17105,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,12,0,0,8,0,0,8,0,0,1,0,0,2,0,0],[88,104,0.8462,0.45089,0.17169,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,10,0,0,10,0,0,7,0,0,2,0,0,2,0,0],[92,104,0.8846,0.41518,0.15714,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,9,0,0,11,0,0,6,0,0,3,0,0,0,0,0],[96,104,0.9231,0.40179,0.1729,0.28571,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,15,0,0,6,0,0,7,0,0,0,0,0,2,0,0],[100,104,0.9615,0.47768,0.21011,0.28571,0.42857,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,11,0,0,8,0,0,4,0,0,5,0,0,2,0,1],[104,104,1.0,0.38393,0.18363,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,4,0,0,14,0,0,7,0,0,4,0,0,1,0,0,2,0,0]]},{"b":2,"e":0.42857,"k":"falling","v":0.1517,"x":0.49554,"p":[[0,76,0.0,0.33482,0.12169,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,11,0,0,13,0,0,2,0,0,0,0,0,0,0,0],[4,76,0.0526,0.40178,0.14914,0.28571,0.42857,0.46429,0.14286,0.71429,0,0,0,0,0,3,0,0,10,0,0,11,0,0,6,0,0,2,0,0,0,0,0],[8,76,0.1053,0.4375,0.21706,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,3,0,0,7,0,0,6,0,0,8,0,0,5,0,0,1,0,0],[12,76,0.1579,0.49554,0.18205,0.39286,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,11,0,0,7,0,0,2,0,0,4,0,0],[16,76,0.2105,0.4375,0.16728,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,10,0,0,8,0,0,9,0,0,2,0,0,1,0,0],[20,76,0.2632,0.45089,0.25281,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,8,0,0,4,0,0,7,0,0,4,0,0,1,0,2],[24,76,0.3158,0.33928,0.18123,0.2857,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,4,0,0,14,0,0,7,0,0,3,0,0,1,0,0,1,0,0],[28,76,0.3684,0.37946,0.18766,0.28571,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,16,0,0,3,0,0,8,0,0,1,0,0,1,0,0],[32,76,0.4211,0.40625,0.21461,0.28571,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,12,0,0,4,0,0,6,0,0,3,0,0,2,0,0],[36,76,0.4737,0.30803,0.19597,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,9,0,0,7,0,0,8,0,0,4,0,0,0,0,0,1,0,0],[40,76,0.5263,0.33036,0.21558,0.2857,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,2,0,0,14,0,0,4,0,0,4,0,0,2,0,0,1,0,0],[44,76,0.5789,0.28125,0.21572,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,7,0,0,9,0,0,5,0,0,3,0,0,1,0,0,1,0,0],[48,76,0.6316,0.30357,0.25692,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,4,0,0,12,0,0,3,0,0,3,0,0,0,0,0,2,0,1],[52,76,0.6842,0.3125,0.21558,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,4,0,0,8,0,0,9,0,0,3,0,0,1,0,0,1,0,0],[56,76,0.7368,0.16518,0.17169,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,5,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[60,76,0.7895,0.26786,0.21943,0.10714,0.2857,0.42857,0.0,0.71429,8,0,0,8,0,6,0,0,8,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[64,76,0.8421,0.26339,0.19597,0.14286,0.2857,0.42857,0.0,0.71429,7,0,0,7,0,6,0,0,9,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[68,76,0.8947,0.2366,0.19757,0.0,0.28571,0.42857,0.0,0.71429,10,0,0,10,0,4,0,0,8,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[72,76,0.9474,0.24107,0.16917,0.10714,0.28571,0.42857,0.0,0.4286,8,0,0,8,0,5,0,0,8,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[76,76,1.0,0.1517,0.14699,0.0,0.14286,0.28571,0.0,0.42857,13,0,0,13,0,7,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"50e0d1967ce98c68","q":"Let  $ABC$  a triangle and let  $I$  be the center of its inscribed circle. Let  $D$  be the symmetric point of  $I$  with respect to  $AB$  and  $E$  be the symmetric point of  $I$  with respect to  $AC$ .\nShow that the circumcircles of the triangles  $BID$  and  $CIE$  are eachother tangent.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,49,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.04464,0.12596,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.06696,0.18205,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[20,49,0.4082,0.05804,0.17076,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[24,49,0.4898,0.04018,0.12492,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,49,0.5714,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.02232,0.08827,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.04018,0.1439,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[44,49,0.898,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,78,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,78,0.0513,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,78,0.1026,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,78,0.1538,0.08036,0.19865,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[16,78,0.2051,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,78,0.2564,0.09375,0.23585,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[24,78,0.3077,0.03125,0.10556,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,78,0.359,0.02679,0.10374,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,78,0.4103,0.05804,0.1984,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[36,78,0.4615,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,78,0.5128,0.0625,0.15542,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[44,78,0.5641,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,78,0.6154,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,78,0.6667,0.04911,0.12682,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[56,78,0.7179,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,78,0.7692,0.04911,0.18766,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[64,78,0.8205,0.03571,0.10714,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,78,0.8718,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,78,0.9231,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,78,0.9744,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[78,78,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d6324aba602ee4e6","q":"Let  $G$  be a \ufb01nite simple graph and let  $k$  be the largest number of vertices of any clique in  $G$ . Suppose that we label each vertex of\u00a0 $G$  with a non-negative real number, so that the sum of all such labels is  $1$ . De\ufb01ne the\u00a0*value of an edge* to be the product of the labels of the two vertices at its ends. De\ufb01ne the *value of a labelling* to be the sum of values of the edges.\nProve that the maximum possible value of a labelling of\u00a0 $G$  is  $\\frac{k-1}{2k}$ .\n(A *\ufb01nite simple graph*\u00a0is a graph with \ufb01nitely many vertices, in which each edge connects two distinct vertices and no two edges connect the same two vertices. A *clique* in a graph is a set of vertices in which any two are connected by an edge.)","t":[{"b":0,"e":1.0,"k":"rising","v":0.54911,"x":1.0,"p":[[0,9,0.0,0.54911,0.30327,0.28571,0.5,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,5,0,0,6,0,0,4,0,0,2,0,0,5,0,5],[4,9,0.4444,0.9375,0.15126,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,27],[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],[9,9,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":1.0,"k":"rising","v":0.72768,"x":1.0,"p":[[0,11,0.0,0.72768,0.26332,0.57143,0.85714,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,2,0,0,9,0,9],[4,11,0.3636,0.83481,0.19599,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,6,0,15],[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":"e2ce67e822973f0e","q":"Let  $ABC$  be a triangle with incenter  $I$ , and suppose the incircle is tangent to  $CA$  and  $AB$  at  $E$  and  $F$ . Denote by  $G$  and  $H$  the reflections of  $E$  and  $F$  over  $I$ . Let  $Q$  be the intersection of  $BC$  with  $GH$ , and let  $M$  be the midpoint of  $BC$ . Prove that  $IQ$  and  $IM$  are perpendicular.","t":[{"b":2,"e":1.0,"k":"flat","v":0.7991,"x":0.98661,"p":[[0,134,0.0,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[4,134,0.0299,0.91071,0.1948,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,25],[8,134,0.0597,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[12,134,0.0896,0.92857,0.16366,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,26],[16,134,0.1194,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[20,134,0.1493,0.92857,0.18898,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,28],[24,134,0.1791,0.875,0.23623,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,24],[28,134,0.209,0.88838,0.20121,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,1,0,23],[32,134,0.2388,0.87946,0.22335,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,2,0,23],[36,134,0.2687,0.89286,0.21724,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,2,0,24],[40,134,0.2985,0.90625,0.19103,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,24],[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],[48,134,0.3582,0.92856,0.16754,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,26],[52,134,0.3881,0.9375,0.16728,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,27],[56,134,0.4179,0.88839,0.21939,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,24],[60,134,0.4478,0.84821,0.23402,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,5,0,0,2,0,20],[64,134,0.4776,0.9241,0.17491,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,26],[68,134,0.5075,0.91072,0.22517,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[72,134,0.5373,0.7991,0.30485,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,4,0,0,2,0,0,3,0,0,0,0,0,1,0,21],[76,134,0.5672,0.88838,0.24154,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,24],[80,134,0.597,0.88393,0.19704,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,4,0,21],[84,134,0.6269,0.90625,0.19102,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,24],[88,134,0.6567,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[92,134,0.6866,0.94643,0.13716,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[96,134,0.7164,0.90625,0.18423,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,24],[100,134,0.7463,0.87054,0.20316,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,2,0,21],[104,134,0.7761,0.87942,0.18602,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,2,0,21],[108,134,0.806,0.89286,0.20516,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,23],[112,134,0.8358,0.8482,0.24985,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,3,0,20],[116,134,0.8657,0.875,0.21943,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,2,0,22],[120,134,0.8955,0.87946,0.19597,0.85711,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,17],[124,134,0.9254,0.86161,0.23003,0.82143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,4,0,0,4,0,20],[128,134,0.9552,0.85714,0.20516,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,5,0,0,5,0,18],[132,134,0.9851,0.84375,0.25091,0.85711,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,7,0,18],[134,134,1.0,0.87053,0.18336,0.71429,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,17]]},{"b":5,"e":1.0,"k":"flat","v":0.82143,"x":0.96874,"p":[[0,146,0.0,0.92411,0.21124,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,27],[4,146,0.0274,0.92857,0.17496,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,26],[8,146,0.0548,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[12,146,0.0822,0.94643,0.16269,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[16,146,0.1096,0.96427,0.12376,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[20,146,0.137,0.93302,0.17125,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,26],[24,146,0.1644,0.92857,0.19885,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,27],[28,146,0.1918,0.9375,0.17474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[32,146,0.2192,0.94642,0.18814,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[36,146,0.2466,0.91964,0.15542,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,24],[40,146,0.274,0.82143,0.26964,0.42857,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,7,0,0,0,0,0,0,0,0,2,0,21],[44,146,0.3014,0.90179,0.21852,1.0,1.0,1.0,0.2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 $C_1$  and  $C_2$  be given circles. Let  $A_1$  on  $C_1$  and  $A_2$  on  $C_2$  be fixed points. If chord  $A_1P_1$  of  $C_1$  is parallel to chord  $A_2P_2$  of  $C_2$ , find the locus of the midpoint of  $P_1P_2$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.88839,"x":1.0,"p":[[0,43,0.0,0.88839,0.20119,0.85714,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,20],[4,43,0.093,0.97322,0.10971,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[8,43,0.186,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,43,0.2791,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,43,0.3721,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,43,0.4651,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,43,0.5581,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[28,43,0.6512,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,43,0.7442,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[36,43,0.8372,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[40,43,0.9302,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[43,43,1.0,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,25,0.0,0.91518,0.21683,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,25],[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],[8,25,0.32,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,25,0.48,0.93304,0.2082,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[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],[20,25,0.8,0.96875,0.10555,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[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],[25,25,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"ac211ff9ead51ed4","q":"Let  $C_0$  be the midpoint of hypotenuse  $AB$  of triangle  $ABC$ ;  $AA_1, BB_1$  the bisectors of this triangle;  $I$  its incenter. Prove that the lines  $C_0I$  and  $A_1B_1$  meet on the altitude from  $C$ .\n*Proposed by A.Zaslavsky*","t":[{"b":4,"e":0.0,"k":"falling","v":0.27677,"x":0.96428,"p":[[0,104,0.0,0.96428,0.11294,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[4,104,0.0385,0.91071,0.22517,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[8,104,0.0769,0.95982,0.12993,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,29],[12,104,0.1154,0.92409,0.1923,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,27],[16,104,0.1538,0.875,0.28291,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[20,104,0.1923,0.91072,0.19804,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,26],[24,104,0.2308,0.88839,0.269,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,26],[28,104,0.2692,0.83705,0.30369,0.85711,1.0,1.0,0.07143,1.0,0,23,0,0,1,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,23],[32,104,0.3077,0.87499,0.23353,0.85714,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,22],[36,104,0.3462,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[40,104,0.3846,0.90177,0.1938,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,3,0,23],[44,104,0.4231,0.88839,0.23887,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,3,0,24],[48,104,0.4615,0.94643,0.17678,1.0,1.0,1.0,0.07143,1.0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,28],[52,104,0.5,0.87054,0.25843,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,24],[56,104,0.5385,0.89954,0.26177,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[60,104,0.5769,0.74775,0.38507,0.53539,1.0,1.0,0.0,1.0,4,21,0,4,1,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,21],[64,104,0.6154,0.76113,0.3252,0.571,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,4,0,0,1,0,0,3,1,0,2,0,0,0,0,19],[68,104,0.6538,0.84374,0.28874,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,23],[72,104,0.6923,0.89062,0.24023,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,2,0,24],[76,104,0.7308,0.86161,0.26119,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,22],[80,104,0.7692,0.8125,0.31224,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,22],[84,104,0.8077,0.80357,0.33645,0.71429,1.0,1.0,0.0,1.0,3,21,0,3,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,21],[88,104,0.8462,0.83259,0.29807,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,1,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,2,0,22],[92,104,0.8846,0.75222,0.36596,0.49968,1.0,1.0,0.0,1.0,2,20,0,2,0,3,0,1,2,0,0,0,0,0,2,0,0,0,0,0,2,0,20],[96,104,0.9231,0.85714,0.25,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,21],[100,104,0.9615,0.62277,0.43088,0.14289,0.92857,1.0,0.0,1.0,6,16,0,6,1,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,16],[104,104,1.0,0.27677,0.37232,0.0,0.14286,0.32143,0.0,1.0,14,6,0,14,0,6,0,2,2,0,0,1,0,0,1,0,0,0,0,0,0,0,6]]},{"b":7,"e":1.0,"k":"flat","v":0.87499,"x":0.98661,"p":[[0,48,0.0,0.96428,0.12877,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[4,48,0.0833,0.87946,0.2969,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,26],[8,48,0.1667,0.92856,0.1856,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[12,48,0.25,0.92857,0.19885,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[16,48,0.3333,0.88837,0.23349,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,4,0,23],[20,48,0.4167,0.92411,0.18893,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,26],[24,48,0.5,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,48,0.5833,0.88839,0.23072,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,3,0,24],[32,48,0.6667,0.8817,0.25791,0.96429,1.0,1.0,0.07143,1.0,0,24,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,24],[36,48,0.75,0.88392,0.21852,0.857,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,4,0,0,3,0,22],[40,48,0.8333,0.87499,0.24158,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,24],[44,48,0.9167,0.89955,0.19793,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,1,0,0,0,3,0,0,3,0,23],[48,48,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"0e815905c83bb51c","q":"Let  $S$  be the set of all ordered pairs  $(a,b)$  of integers with  $0<2a<2b<2017$  such that  $a^2+b^2$  is a multiple of  $2017$ . Prove that \\[\\sum_{(a,b)\\in S}a=\\frac{1}{2}\\sum_{(a,b)\\in S}b.\\]\n\nProposed by Uwe Leck, Germany","t":[{"b":6,"e":0.28571,"k":"flat","v":0.16071,"x":0.53125,"p":[[0,45,0.0,0.16071,0.12242,0.14286,0.14286,0.17857,0.0,0.42857,7,0,2,7,0,17,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.46429,0.31744,0.25,0.42857,0.85714,0.0,1.0,4,1,0,4,0,4,0,0,7,0,0,4,0,0,1,0,0,3,0,0,8,0,1],[8,45,0.1778,0.53125,0.31791,0.28571,0.42857,0.85714,0.0,1.0,1,5,0,1,0,5,0,0,7,0,0,4,0,0,2,0,0,3,0,0,5,0,5],[12,45,0.2667,0.35268,0.24219,0.24999,0.28571,0.32143,0.0,1.0,1,1,0,1,0,7,0,0,16,0,0,2,0,0,1,0,0,1,0,0,3,0,1],[16,45,0.3556,0.29909,0.18334,0.14286,0.28571,0.32143,0.0,0.85714,2,0,0,2,0,8,0,0,14,0,0,4,0,0,2,0,0,1,0,0,1,0,0],[20,45,0.4444,0.30357,0.24419,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,11,0,0,9,0,0,5,0,0,0,0,0,1,0,0,2,0,1],[24,45,0.5333,0.34821,0.29001,0.14286,0.28571,0.42857,0.0,1.0,4,3,1,4,0,7,0,0,11,0,0,4,0,0,1,0,0,0,0,0,2,0,3],[28,45,0.6222,0.40179,0.27765,0.2857,0.28571,0.42857,0.0,1.0,1,3,0,1,0,6,0,0,13,0,0,5,0,0,1,0,0,0,0,0,3,0,3],[32,45,0.7111,0.20089,0.21387,0.0,0.14286,0.28571,0.0,0.85714,12,0,1,12,0,6,0,0,8,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[36,45,0.8,0.31696,0.26901,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,10,0,0,9,0,0,3,0,0,1,0,0,1,0,0,3,0,1],[40,45,0.8889,0.22768,0.17445,0.14286,0.14286,0.28571,0.0,0.85714,5,0,0,5,0,12,0,0,9,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[44,45,0.9778,0.24553,0.11425,0.14286,0.28571,0.28571,0.0,0.42857,3,0,1,3,0,7,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.22321,0.11811,0.14286,0.2857,0.28571,0.0,0.42857,3,0,1,3,0,12,0,0,13,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.18741,"x":0.51339,"p":[[0,58,0.0,0.18741,0.20654,0.14214,0.14286,0.14286,0.0,0.85714,7,0,0,7,0,19,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,0],[4,58,0.069,0.51339,0.30484,0.2857,0.42857,0.85714,0.0,1.0,1,2,0,1,0,5,0,0,8,0,0,3,0,0,3,0,0,1,0,0,9,0,2],[8,58,0.1379,0.36161,0.28568,0.14286,0.28571,0.46429,0.0,1.0,3,1,0,3,0,9,0,0,9,0,0,3,0,0,2,0,0,0,0,0,5,0,1],[12,58,0.2069,0.39732,0.30037,0.14286,0.28571,0.60714,0.0,1.0,2,1,0,2,0,11,0,0,5,0,0,4,0,0,2,0,0,1,0,0,6,0,1],[16,58,0.2759,0.26785,0.13716,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,8,0,0,21,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[20,58,0.3448,0.33027,0.24344,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,11,0,0,10,0,0,6,0,0,0,0,0,1,0,0,1,0,2],[24,58,0.4138,0.34375,0.24448,0.2857,0.28571,0.42857,0.0,1.0,3,3,0,3,0,3,0,0,17,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[28,58,0.4828,0.34374,0.15091,0.28571,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,2,0,0,14,0,0,10,0,0,3,0,0,1,0,0,0,0,0],[32,58,0.5517,0.31696,0.13234,0.28571,0.28571,0.28571,0.14286,0.857,0,0,0,0,0,4,0,0,21,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[36,58,0.6207,0.375,0.20124,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,19,0,0,8,0,0,0,0,0,0,0,0,1,0,2],[40,58,0.6897,0.36161,0.21124,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,2,0,0,19,0,0,7,0,0,0,0,0,0,0,0,1,0,2],[44,58,0.7586,0.29911,0.11495,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,3,0,0,22,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[48,58,0.8276,0.3125,0.14914,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[52,58,0.8966,0.28571,0.12372,0.2857,0.28571,0.32143,0.0,0.57143,2,0,1,2,0,5,0,0,17,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[56,58,0.9655,0.29464,0.11259,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,6,0,0,20,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[58,58,1.0,0.26786,0.09279,0.14289,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,9,0,0,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"689162af88065ebd","q":"Let $A B C D$ be a convex quadrilateral with area $S$. We denote $a=A B, b=B C, c=C D$ and $d=D A$. For any permutation $x, y, z, t$ of $a, b, c, d$, show that\n\n$$\nS \\leqslant \\frac{1}{2}(x y+z t)\n$$","t":[{"b":0,"e":0.2857,"k":"flat","v":0.28125,"x":0.30804,"p":[[0,32,0.0,0.28125,0.05629,0.28571,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.28571,1e-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],[8,32,0.25,0.30803,0.1017,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[12,32,0.375,0.28571,1e-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],[16,32,0.5,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.30804,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,32,0.75,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,32,0.875,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.28571,1e-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]]},{"b":6,"e":0.28571,"k":"flat","v":0.27678,"x":0.31697,"p":[[0,35,0.0,0.28572,5e-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],[4,35,0.1143,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,1,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.31697,0.12745,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,35,0.3429,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,35,0.4571,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.28571,1e-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],[28,35,0.8,0.28572,5e-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],[32,35,0.9143,0.28571,1e-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],[35,35,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b29061adb7ca20a6","q":"Let  $V$  be the set of all continuous functions  $f\\colon [0,1]\\to \\mathbb{R}$ , differentiable on  $(0,1)$ , with the property that  $f(0)=0$  and  $f(1)=1$ . Determine all  $\\alpha \\in \\mathbb{R}$  such that for every  $f\\in V$ , there exists some  $\\xi \\in (0,1)$  such that\n\\[f(\\xi)+\\alpha = f'(\\xi)\\]","t":[{"b":6,"e":1.0,"k":"flat","v":0.98661,"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],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,32,0.5,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,32,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,25,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,25,0.16,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,25,0.32,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,25,0.48,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,25,0.64,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,25,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,25,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"648e65ba66a664cb","q":"For each integer  $k\\ge2$ , the decimal expansions of the numbers  $1024,1024^2,\\dots,1024^k$  are concatenated, in that order, to obtain a number  $X_k$ .  (For example,  $X_2 = 10241048576$ .)  If \\[ \\frac{X_n}{1024^n} \\] is an odd integer, find the smallest possible value of  $n$ , where  $n\\ge2$  is an integer.\n\n*Proposed by Evan Chen*","t":[{"b":6,"e":0.85714,"k":"flat","v":0.81248,"x":0.95536,"p":[[0,93,0.0,0.8125,0.2299,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,6,0,0,8,0,13],[4,93,0.043,0.92411,0.1988,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,24],[8,93,0.086,0.87946,0.21756,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,19],[12,93,0.129,0.94196,0.12299,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[16,93,0.172,0.91964,0.17104,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,24],[20,93,0.2151,0.87277,0.20339,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,0,8,1,17],[24,93,0.2581,0.91071,0.16656,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,7,0,21],[28,93,0.3011,0.90178,0.15746,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,5,0,21],[32,93,0.3441,0.87946,0.21756,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,6,0,21],[36,93,0.3871,0.92857,0.14286,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,22],[40,93,0.4301,0.90625,0.21609,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,24],[44,93,0.4731,0.85268,0.25626,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,0,4,0,21],[48,93,0.5161,0.88393,0.1357,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,13,0,14],[52,93,0.5591,0.89286,0.17857,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,19],[56,93,0.6022,0.89732,0.11425,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,16],[60,93,0.6452,0.82589,0.22512,0.85711,0.85714,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,0,12,0,13],[64,93,0.6882,0.91518,0.13296,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,19],[68,93,0.7312,0.93304,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,10,0,20],[72,93,0.7742,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[76,93,0.8172,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[80,93,0.8602,0.88838,0.09268,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,20,0,10],[84,93,0.9032,0.89286,0.07986,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,10],[88,93,0.9462,0.89732,0.11425,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,15],[92,93,0.9892,0.81248,0.18014,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,0,8,0,11],[93,93,1.0,0.82589,0.19799,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,7,0,0,8,0,13]]},{"b":7,"e":0.71429,"k":"flat","v":0.62944,"x":0.97321,"p":[[0,81,0.0,0.83929,0.19805,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,0,5,0,16],[4,81,0.0494,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[8,81,0.0988,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[12,81,0.1481,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,81,0.1975,0.91518,0.13296,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,10,0,19],[20,81,0.2469,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[24,81,0.2963,0.87945,0.21463,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,8,0,19],[28,81,0.3457,0.85267,0.17307,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,9,0,14],[32,81,0.3951,0.80357,0.23077,0.85714,0.85714,0.89286,0.0,1.0,1,8,1,1,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,17,0,8],[36,81,0.4444,0.82589,0.23072,0.82143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,9,0,15],[40,81,0.4938,0.71429,0.24223,0.67857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,10,0,0,8,0,6],[44,81,0.5432,0.66964,0.25614,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,2,0,0,2,0,0,2,0,0,3,0,0,8,0,0,12,0,2],[48,81,0.5926,0.71874,0.16936,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,10,0,0,7,0,4],[52,81,0.642,0.65624,0.22548,0.42857,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,5,0,0,9,0,3],[56,81,0.6914,0.62944,0.24708,0.53539,0.71429,0.85714,0.0,1.0,1,1,1,1,0,1,0,0,4,0,0,2,0,0,7,0,0,5,0,0,11,0,1],[60,81,0.7407,0.77677,0.16729,0.71429,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,13,0,5],[64,81,0.7901,0.71874,0.25377,0.71429,0.85714,0.85714,0.0,1.0,1,2,0,1,0,2,0,0,1,0,0,2,0,0,1,0,0,5,0,0,18,0,2],[68,81,0.8395,0.72768,0.19352,0.67857,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,7,0,0,15,0,2],[72,81,0.8889,0.75892,0.18012,0.71429,0.78564,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,13,0,3],[76,81,0.9383,0.70982,0.20666,0.71429,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,11,0,0,12,0,2],[80,81,0.9877,0.71429,0.17857,0.71429,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,12,0,0,12,0,1],[81,81,1.0,0.73436,0.19663,0.71429,0.78571,0.85714,0.0,1.0,1,2,1,1,0,0,0,0,0,0,0,3,0,0,2,1,0,9,0,0,14,0,2]]}]},{"i":"3448760998597021","q":"We have, in the plane, 16 distinct points, which we denote by $A_{i, j}$ for $i, j \\in\\{1,2,3,4\\}$. These points satisfy the following alignment and cocyclicity relations:\n$\\triangleright$ for all $i \\in\\{1,2,3,4\\}$, the points $A_{i, 1}, A_{i, 2}, A_{i, 3}$, and $A_{i, 4}$ are collinear;\n$\\triangleright$ for all $j \\in\\{1,2,3,4\\}$, the points $A_{1, j}, A_{2, j}, A_{3, j}$, and $A_{4, j}$ are collinear;\n$\\triangleright$ the quadrilaterals $A_{1,1} A_{1,2} A_{2,2} A_{2,1}, A_{2,1} A_{2,2} A_{3,2} A_{3,1}, A_{3,1} A_{3,2} A_{4,2} A_{4,1}, A_{1,2} A_{1,3} A_{2,3} A_{2,2}$, $A_{1,3} A_{1,4} A_{2,4} A_{2,3}, A_{1,1} A_{2,2} A_{2,3} A_{1,4}$, and $A_{1,1} A_{2,2} A_{3,2} A_{4,1}$ are cyclic.\nShow that the quadrilateral $A_{4,1} A_{3,2} A_{3,3} A_{4,4}$ is also cyclic.\nNote: a quadrilateral is said to be cyclic if its four vertices lie on the same circle.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.00893,"x":0.16517,"p":[[0,54,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.04,0.06395,0.0,0.0,0.14,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.0758,0.07966,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.12447,0.05907,0.14,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.14715,0.11,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[32,54,0.5926,0.13813,0.02482,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.16063,0.09944,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,54,0.7407,0.14688,0.05634,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.16063,0.09944,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,54,0.8889,0.14724,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.16517,0.08821,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[54,54,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.08027,"p":[[0,32,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.08027,0.13329,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,13,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,32,0.25,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.04464,0.0974,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"c4d0b9c9452d7a0d","q":"A cyclic pentagon is given. Prove that the ratio of its area to the sum of the diagonals is not greater than the quarter of the circumradius.","t":[{"b":4,"e":1.0,"k":"flat","v":0.86606,"x":1.0,"p":[[0,26,0.0,0.86606,0.17476,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,1,0,19],[4,26,0.1538,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,26,0.3077,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[26,26,1.0,0.95089,0.19759,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30]]},{"b":5,"e":1.0,"k":"rising","v":0.83482,"x":1.0,"p":[[0,32,0.0,0.83482,0.17896,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,0,4,0,15],[4,32,0.125,0.94643,0.1915,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[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],[24,32,0.75,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"bf1f5d0105214758","q":"Find all pairs of polynomials  $P(x),Q(x)$  with integer coefficients such that  $P(Q(x)) = (x - 1)(x - 2)...(x - 9)$  for all real numbers  $x$","t":[{"b":1,"e":0.71429,"k":"falling","v":0.61157,"x":0.97321,"p":[[0,43,0.0,0.94195,0.11773,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,24],[4,43,0.093,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[8,43,0.186,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[12,43,0.2791,0.91964,0.15542,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,24],[16,43,0.3721,0.93304,0.14719,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26],[20,43,0.4651,0.88393,0.19045,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,0,0,23],[24,43,0.5581,0.92411,0.15966,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,26],[28,43,0.6512,0.69643,0.18814,0.57143,0.57143,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,6,0,0,0,0,8],[32,43,0.7442,0.63392,0.15541,0.57143,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,10,0,0,2,0,2],[36,43,0.8372,0.61607,0.12078,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,17,0,0,9,0,0,1,0,1],[40,43,0.9302,0.63391,0.15542,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,5,0,0,0,0,4],[43,43,1.0,0.61157,0.09608,0.57143,0.57143,0.60714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,7,0,0,0,0,1]]},{"b":6,"e":0.57143,"k":"falling","v":0.60714,"x":0.95535,"p":[[0,41,0.0,0.95535,0.13571,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[4,41,0.0976,0.92857,0.17496,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,27],[8,41,0.1951,0.9375,0.12846,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[12,41,0.2927,0.90624,0.18076,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,2,0,24],[16,41,0.3902,0.91295,0.14367,0.85714,1.0,1.0,0.5,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,1,2,0,0,2,0,0,6,0,21],[20,41,0.4878,0.85268,0.21572,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,3,0,20],[24,41,0.5854,0.71875,0.22724,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,7,0,0,2,0,10],[28,41,0.6829,0.63393,0.15126,0.57143,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,17,0,0,7,0,0,1,0,3],[32,41,0.7805,0.70088,0.20629,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,7,0,0,2,0,8],[36,41,0.878,0.63838,0.2082,0.57132,0.57143,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,5,0,0,14,0,0,4,0,0,1,0,6],[40,41,0.9756,0.65179,0.1234,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,11,0,0,1,0,2],[41,41,1.0,0.60714,0.16366,0.53572,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,15,0,0,5,0,0,1,0,3]]}]},{"i":"bddccf4b375f0c34","q":"Find all pairs of  $p,q$  prime numbers that satisfy the equation   $$ p(p^4+p^2+10q)=q(q^2+3) $$","t":[{"b":0,"e":0.28571,"k":"flat","v":0.18304,"x":0.33928,"p":[[0,111,0.0,0.22321,0.13803,0.10714,0.2857,0.28571,0.0,0.42857,8,0,0,8,0,1,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,111,0.036,0.22321,0.15126,0.0,0.28571,0.28571,0.0,0.42857,9,0,0,9,0,1,0,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,111,0.0721,0.26339,0.2372,0.0,0.28571,0.28571,0.0,1.0,9,2,0,9,0,0,0,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[12,111,0.1081,0.26786,0.23891,0.0,0.28571,0.28571,0.0,1.0,9,2,0,9,0,0,0,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[16,111,0.1441,0.27678,0.23941,0.10714,0.28571,0.32143,0.0,1.0,8,2,0,8,0,2,0,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,2],[20,111,0.1802,0.29018,0.25626,0.10714,0.28571,0.28571,0.0,1.0,8,2,0,8,0,1,0,0,16,0,0,4,0,0,0,0,0,0,0,0,1,0,2],[24,111,0.2162,0.33928,0.31491,0.21427,0.28571,0.32143,0.0,1.0,8,5,0,8,0,0,0,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,5],[28,111,0.2523,0.21428,0.24484,0.0,0.2857,0.28571,0.0,1.0,14,1,0,14,0,1,0,0,11,0,0,4,0,0,0,0,0,0,0,0,1,0,1],[32,111,0.2883,0.32143,0.31542,0.0,0.28571,0.42857,0.0,1.0,9,4,0,9,0,2,0,0,12,0,0,4,0,0,0,0,0,0,0,0,1,0,4],[36,111,0.3243,0.2009,0.22829,0.0,0.2143,0.32143,0.0,1.0,15,1,0,15,0,1,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[40,111,0.3604,0.31696,0.24415,0.2857,0.28571,0.32143,0.0,1.0,6,2,0,6,0,0,0,0,18,0,0,5,0,0,0,0,0,0,0,0,1,0,2],[44,111,0.3964,0.27232,0.25843,0.0,0.28571,0.32143,0.0,1.0,10,2,0,10,0,1,0,0,13,0,0,5,0,0,0,0,0,1,0,0,0,0,2],[48,111,0.4324,0.1875,0.14914,0.0,0.28571,0.28571,0.0,0.42857,12,0,0,12,0,0,0,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,111,0.4685,0.26786,0.13243,0.28571,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,1,0,0,19,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[56,111,0.5045,0.21875,0.16745,0.0,0.28571,0.28571,0.0,0.71429,10,0,0,10,0,0,0,0,19,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[60,111,0.5405,0.19643,0.21354,0.0,0.2857,0.28571,0.0,0.85714,15,0,0,15,0,0,0,0,11,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[64,111,0.5766,0.19643,0.16269,0.0,0.2857,0.28571,0.0,0.4286,12,0,0,12,0,1,0,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[68,111,0.6126,0.21875,0.1636,0.0,0.28571,0.28571,0.0,0.4286,10,0,0,10,0,2,0,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[72,111,0.6486,0.26339,0.12931,0.2857,0.28571,0.28571,0.0,0.4286,5,0,0,5,0,1,0,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[76,111,0.6847,0.23214,0.15465,0.0,0.2857,0.28571,0.0,0.42857,9,0,0,9,0,0,0,0,17,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[80,111,0.7207,0.23215,0.14173,0.14286,0.2857,0.28571,0.0,0.4286,7,0,0,7,0,3,0,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[84,111,0.7568,0.19196,0.15815,0.0,0.2857,0.28571,0.0,0.42857,12,0,0,12,0,1,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[88,111,0.7928,0.24553,0.21793,0.0,0.28571,0.28571,0.0,1.0,9,1,0,9,0,3,0,0,14,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[92,111,0.8288,0.21429,0.27433,0.0,0.07145,0.28571,0.0,1.0,16,1,0,16,0,1,0,0,9,0,0,2,0,0,0,0,0,2,0,0,1,0,1],[96,111,0.8649,0.22321,0.13803,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,1,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[100,111,0.9009,0.23661,0.14555,0.21427,0.28571,0.28571,0.0,0.4286,8,0,0,8,0,0,0,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[104,111,0.9369,0.19196,0.1411,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,3,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[108,111,0.973,0.18304,0.17215,0.0,0.2857,0.28571,0.0,0.4286,14,0,0,14,0,1,0,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[111,111,1.0,0.24107,0.0974,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,2,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.28571,"k":"flat","v":0.16071,"x":0.375,"p":[[0,245,0.0,0.26785,0.11152,0.24999,0.2857,0.28571,0.0,0.4286,2,0,0,2,0,6,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,245,0.0163,0.29018,0.18724,0.2857,0.28571,0.42857,0.0,1.0,5,1,0,5,0,2,0,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[8,245,0.0327,0.25,0.12877,0.2857,0.28571,0.28571,0.0,0.42857,6,0,0,6,0,0,0,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,245,0.049,0.26785,0.14617,0.2857,0.28571,0.42857,0.0,0.42857,6,0,0,6,0,1,0,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[16,245,0.0653,0.3125,0.12078,0.28571,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,0,0,0,17,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[20,245,0.0816,0.30357,0.16656,0.2857,0.28571,0.28571,0.0,1.0,3,1,1,3,0,1,0,0,21,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[24,245,0.098,0.24553,0.1439,0.24999,0.28571,0.28571,0.0,0.4286,7,0,0,7,0,1,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[28,245,0.1143,0.27678,0.1234,0.2857,0.28571,0.28571,0.0,0.4286,4,0,0,4,0,1,0,0,20,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[32,245,0.1306,0.29464,0.19212,0.2857,0.28571,0.42857,0.0,1.0,6,1,0,6,0,0,0,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,1],[36,245,0.1469,0.31696,0.17762,0.2857,0.28571,0.42857,0.0,1.0,4,1,0,4,0,0,0,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,1],[40,245,0.1633,0.22768,0.16311,0.0,0.28571,0.28571,0.0,0.42857,10,0,0,10,0,0,0,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[44,245,0.1796,0.25446,0.11701,0.2857,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,0,0,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,245,0.1959,0.25,0.14725,0.24999,0.28571,0.28571,0.0,0.4286,7,0,0,7,0,1,0,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[52,245,0.2122,0.27232,0.14446,0.2857,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,0,0,0,17,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[56,245,0.2286,0.24554,0.18977,0.0,0.28571,0.28571,0.0,0.85714,9,0,0,9,0,1,0,0,15,0,0,6,0,0,0,0,0,0,0,0,1,0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integral solutions to the equation \\[(m^{2}-n^{2})^{2}=16n+1.\\]","t":[{"b":3,"e":1.0,"k":"rising","v":0.6875,"x":0.98661,"p":[[0,50,0.0,0.6875,0.32229,0.39286,0.78564,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,7,0,0,6,0,0,0,0,0,2,0,0,1,0,15],[4,50,0.08,0.81696,0.26782,0.64286,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,2,0,0,2,0,20],[8,50,0.16,0.875,0.21651,0.85711,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,3,0,0,4,0,21],[12,50,0.24,0.74107,0.2976,0.42857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,6,0,0,5,0,0,0,0,0,3,0,0,2,0,16],[16,50,0.32,0.74549,0.27836,0.42859,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,6,0,0,1,0,0,4,0,0,2,0,15],[20,50,0.4,0.78571,0.29667,0.42857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,6,0,0,3,0,0,0,0,0,3,0,0,0,0,20],[24,50,0.48,0.82589,0.28735,0.64286,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,23],[28,50,0.56,0.85714,0.23145,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,2,0,0,3,0,21],[32,50,0.64,0.87054,0.23244,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,0,2,0,23],[36,50,0.72,0.85267,0.22724,0.82132,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,5,0,19],[40,50,0.8,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[44,50,0.88,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,50,0.96,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[50,50,1.0,0.93302,0.16363,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,26]]},{"b":4,"e":1.0,"k":"rising","v":0.75891,"x":0.98214,"p":[[0,111,0.0,0.75891,0.27766,0.57132,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,7,0,0,0,0,16],[4,111,0.036,0.84375,0.22406,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,7,0,0,0,0,20],[8,111,0.0721,0.76786,0.28739,0.42857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,10,0,0,0,0,0,1,0,0,0,0,19],[12,111,0.1081,0.85714,0.24222,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,2,0,0,2,0,22],[16,111,0.1441,0.81249,0.26593,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,4,0,0,0,0,20],[20,111,0.1802,0.78129,0.26954,0.42964,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,6,0,0,0,0,0,4,0,0,2,0,17],[24,111,0.2162,0.81696,0.27254,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0,21],[28,111,0.2523,0.82143,0.27664,0.75001,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,5,0,0,0,0,0,0,0,0,4,0,20],[32,111,0.2883,0.87946,0.25028,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,24],[36,111,0.3243,0.91964,0.15126,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,1,0,24],[40,111,0.3604,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[44,111,0.3964,0.88393,0.23266,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,24],[48,111,0.4324,0.88837,0.21941,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,5,0,22],[52,111,0.4685,0.92409,0.14722,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,23],[56,111,0.5045,0.89732,0.19638,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,23],[60,111,0.5405,0.92856,0.14725,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,6,0,23],[64,111,0.5766,0.9241,0.13356,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,22],[68,111,0.6126,0.9375,0.15542,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,26],[72,111,0.6486,0.88393,0.20652,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,3,0,22],[76,111,0.6847,0.92857,0.16366,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,24],[80,111,0.7207,0.9107,0.14618,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,9,0,19],[84,111,0.7568,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[88,111,0.7928,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[92,111,0.8288,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[96,111,0.8649,0.90399,0.16135,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,1,20],[100,111,0.9009,0.94641,0.12757,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,25],[104,111,0.9369,0.95982,0.1439,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[108,111,0.973,0.8771,0.22825,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,2,1,22],[111,111,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"7d8ec79eaa370ad5","q":"Find the largest possible value of  $a+b$  less than or equal to  $2007$ , for which  $a$  and  $b$  are relatively prime, and such that there is some positive integer  $n$  for which \\[\\frac{2^3-1}{2^3+1}\\cdot\\frac{3^3-1}{3^3+1}\\cdot\\frac{4^3-1}{4^3+1}\\cdots\\frac{n^3-1}{n^3+1} = \\frac ab.\\]","t":[{"b":1,"e":1.0,"k":"flat","v":0.98214,"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],[8,40,0.2,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,40,0.3,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,40,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,40,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,40,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,40,0.7,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,40,0.8,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":2,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,78,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,78,0.0513,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,78,0.1026,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,78,0.1538,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,78,0.2051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,78,0.2564,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,78,0.3077,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,78,0.359,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,78,0.4103,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,78,0.4615,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,78,0.5128,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,78,0.5641,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,78,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,78,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,78,0.7179,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,78,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,78,0.8205,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,78,0.8718,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,78,0.9231,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[76,78,0.9744,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[78,78,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"e483a2fa3600a5c1","q":"Find all positive integer  $N$  which has not less than  $4$  positive divisors, such that the sum of squares of the  $4$  smallest positive divisors of  $N$  is equal to  $N$ .","t":[{"b":3,"e":0.2857,"k":"falling","v":0.58036,"x":0.93304,"p":[[0,133,0.0,0.79464,0.21997,0.71429,0.85707,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,9,0,0,4,0,13],[4,133,0.0301,0.87946,0.19597,0.82143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,3,0,21],[8,133,0.0602,0.91071,0.16656,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,22],[12,133,0.0902,0.91071,0.15465,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,22],[16,133,0.1203,0.89284,0.1557,0.82132,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,20],[20,133,0.1504,0.93304,0.15561,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,25],[24,133,0.1805,0.90625,0.15815,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,22],[28,133,0.2105,0.86161,0.22156,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,4,0,0,3,0,20],[32,133,0.2406,0.91964,0.14258,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,22],[36,133,0.2707,0.89286,0.18211,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,20],[40,133,0.3008,0.91964,0.13803,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,4,0,22],[44,133,0.3308,0.90178,0.15335,0.82132,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,21],[48,133,0.3609,0.87052,0.21537,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,21],[52,133,0.391,0.89731,0.15253,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,5,0,20],[56,133,0.4211,0.87945,0.14775,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,6,0,17],[60,133,0.4511,0.87053,0.23517,0.82132,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,22],[64,133,0.4812,0.875,0.22517,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,2,0,21],[68,133,0.5113,0.8973,0.215,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,23],[72,133,0.5414,0.92856,0.12881,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,24],[76,133,0.5714,0.85266,0.21276,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,4,0,19],[80,133,0.6015,0.85267,0.24086,0.82132,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,18],[84,133,0.6316,0.83929,0.26184,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,5,0,0,4,0,19],[88,133,0.6617,0.77677,0.30711,0.71429,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,3,0,16],[92,133,0.6917,0.86159,0.22443,0.82143,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,6,0,18],[96,133,0.7218,0.875,0.2165,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,0,4,0,21],[100,133,0.7519,0.74106,0.30187,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,0,0,0,4,0,0,2,0,0,5,0,0,2,0,15],[104,133,0.782,0.63839,0.36067,0.2857,0.71429,1.0,0.0,1.0,2,13,0,2,0,4,0,0,3,0,0,4,0,0,2,0,0,2,0,0,2,0,13],[108,133,0.812,0.74554,0.30249,0.71429,0.78571,1.0,0.0,1.0,1,14,0,1,0,3,0,0,1,0,0,1,0,0,1,0,0,9,0,0,2,0,14],[112,133,0.8421,0.88838,0.1881,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,4,0,21],[116,133,0.8722,0.82588,0.27138,0.57143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,1,0,0,0,0,22],[120,133,0.9023,0.78125,0.30511,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0,5,0,0,2,0,18],[124,133,0.9323,0.74551,0.24155,0.57132,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,6,0,0,1,0,13],[128,133,0.9624,0.74558,0.24669,0.53607,0.78571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,6,0,0,3,0,0,5,0,0,4,0,12],[132,133,0.9925,0.58928,0.27836,0.42857,0.4286,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,4,0,0,11,0,0,3,0,0,2,0,0,3,0,7],[133,133,1.0,0.58036,0.27879,0.28571,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,7,0,0,6,0,0,4,0,0,4,0,0,3,0,6]]},{"b":6,"e":0.0,"k":"falling","v":0.51784,"x":0.91963,"p":[[0,195,0.0,0.79016,0.20513,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,7,0,0,7,0,11],[4,195,0.0205,0.88839,0.17762,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,0,4,0,20],[8,195,0.041,0.91963,0.1285,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,3,0,22],[12,195,0.0615,0.91071,0.17405,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,23],[16,195,0.0821,0.89741,0.16063,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,7,0,19],[20,195,0.1026,0.83036,0.26351,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,3,0,0,2,0,20],[24,195,0.1231,0.88391,0.21263,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,21],[28,195,0.1436,0.79455,0.25761,0.57143,1.0,1.0,0.14,1.0,0,17,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,4,0,0,2,0,17],[32,195,0.1641,0.85714,0.23145,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,18],[36,195,0.1846,0.87052,0.18339,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,2,0,20],[40,195,0.2051,0.8125,0.30813,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,0,2,0,21],[44,195,0.2256,0.82588,0.24933,0.71429,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,0,4,0,18],[48,195,0.2462,0.88391,0.1938,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,4,0,21],[52,195,0.2667,0.78571,0.26726,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,4,0,0,3,0,0,4,0,0,3,0,16],[56,195,0.2872,0.89284,0.18213,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,20],[60,195,0.3077,0.89732,0.18977,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,2,0,23],[64,195,0.3282,0.81696,0.23753,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,5,0,0,3,0,17],[68,195,0.3487,0.89732,0.19638,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,22],[72,195,0.3692,0.74552,0.30038,0.67857,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,3,0,0,0,0,0,2,0,0,6,0,0,5,0,13],[76,195,0.3897,0.80802,0.2276,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,4,0,0,5,0,15],[80,195,0.4103,0.83036,0.28446,0.82143,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,4,0,20],[84,195,0.4308,0.80802,0.25658,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,2,0,0,3,0,18],[88,195,0.4513,0.69196,0.30745,0.42857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,6,0,0,5,0,0,2,0,0,3,0,0,1,0,14],[92,195,0.4718,0.78571,0.30304,0.64286,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,4,0,0,2,0,0,0,0,0,2,0,0,4,0,18],[96,195,0.4923,0.7723,0.26211,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,3,0,0,7,0,13],[100,195,0.5128,0.84373,0.20936,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,6,0,17],[104,195,0.5333,0.76339,0.29797,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,2,0,0,1,0,0,6,0,0,2,0,16],[108,195,0.5538,0.77678,0.29437,0.71421,0.92857,1.0,0.0,1.0,1,16,0,1,0,2,0,0,1,0,0,2,0,0,1,0,0,6,0,0,3,0,16],[112,195,0.5744,0.81249,0.27766,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,1,0,0,2,0,20],[116,195,0.5949,0.73212,0.29398,0.42859,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,2,0,0,5,0,0,2,0,0,2,0,0,7,0,12],[120,195,0.6154,0.82588,0.28736,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,1,0,21],[124,195,0.6359,0.7098,0.28457,0.5354,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,2,0,0,4,0,0,3,0,0,8,0,0,1,0,12],[128,195,0.6564,0.80357,0.28291,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,0,5,0,17],[132,195,0.6769,0.79017,0.27196,0.57132,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,5,0,0,2,0,0,3,0,0,3,0,17],[136,195,0.6974,0.79463,0.23943,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,8,0,0,4,0,14],[140,195,0.7179,0.85268,0.24086,0.82143,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,3,0,0,4,0,20],[144,195,0.7385,0.81696,0.28846,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,3,0,0,4,0,19],[148,195,0.759,0.77679,0.3008,0.64286,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,4,0,0,2,0,0,0,0,0,3,0,0,4,0,17],[152,195,0.7795,0.81249,0.25365,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,4,0,0,2,0,18],[156,195,0.8,0.77232,0.30274,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,2,0,0,2,0,0,3,0,0,1,0,0,4,0,17],[160,195,0.8205,0.80804,0.23038,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,6,0,0,3,0,16],[164,195,0.841,0.84375,0.20934,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,3,0,18],[168,195,0.8615,0.66517,0.37048,0.39286,0.78571,1.0,0.0,1.0,3,14,0,3,0,4,0,0,1,0,0,3,0,0,1,0,0,4,0,0,2,0,14],[172,195,0.8821,0.77006,0.24922,0.57142,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,1,0,1,5,0,0,4,0,0,5,0,13],[176,195,0.9026,0.8482,0.21708,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,0,2,0,19],[180,195,0.9231,0.74107,0.33204,0.42857,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,2,0,0,5,0,0,1,0,0,0,0,0,5,0,16],[184,195,0.9436,0.85267,0.21868,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,4,0,19],[188,195,0.9641,0.57142,0.26,0.42857,0.4998,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,5,0,0,9,0,0,3,0,0,6,0,0,2,0,5],[192,195,0.9846,0.54014,0.24415,0.42857,0.571,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,3,0,0,9,0,0,7,0,0,5,0,0,2,0,3],[195,195,1.0,0.51784,0.2714,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,4,0,0,7,0,0,3,0,0,5,0,0,5,0,2]]}]},{"i":"01665634af18be39","q":"Let  $P,Q,R,S$  be the midpoints of the sides  $BC,CD,DA,AB$  of a convex quadrilateral, respectively. Prove that\n\\[4(AP^2+BQ^2+CR^2+DS^2)\\le 5(AB^2+BC^2+CD^2+DA^2)\\]","t":[{"b":0,"e":1.0,"k":"flat","v":0.8125,"x":1.0,"p":[[0,25,0.0,0.87946,0.22619,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,0,1,0,24],[4,25,0.16,0.8125,0.32818,0.78571,1.0,1.0,0.14286,1.0,0,23,0,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,23],[8,25,0.32,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[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],[20,25,0.8,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,25,0.96,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":2,"e":0.28571,"k":"volatile","v":0.31248,"x":0.93747,"p":[[0,28,0.0,0.93747,0.17111,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,28],[4,28,0.1429,0.46427,0.27893,0.2857,0.35714,0.57143,0.14286,1.0,0,5,0,0,0,6,0,0,10,0,0,1,0,0,10,0,0,0,0,0,0,0,5],[8,28,0.2857,0.63393,0.32328,0.28571,0.57143,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,4,0,0,0,0,0,10,0,0,1,0,0,0,0,12],[12,28,0.4286,0.37052,0.18508,0.14289,0.28571,0.57143,0.14286,0.57143,0,0,0,0,0,9,0,0,9,0,0,0,0,0,14,0,0,0,0,0,0,0,0],[16,28,0.5714,0.4151,0.18349,0.2857,0.57143,0.57143,0.14,0.57143,0,0,0,0,0,7,0,0,7,0,0,0,0,0,18,0,0,0,0,0,0,0,0],[20,28,0.7143,0.4598,0.18117,0.2857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,11,0,0,0,0,0,18,0,0,0,0,0,0,0,1],[24,28,0.8571,0.31248,0.17288,0.14286,0.2857,0.35704,0.14286,0.71429,0,0,0,0,0,11,0,0,13,0,0,0,0,0,7,0,0,1,0,0,0,0,0],[28,28,1.0,0.40165,0.18716,0.24999,0.571,0.57143,0.14,0.57143,0,0,0,0,0,8,0,0,7,0,0,0,0,0,17,0,0,0,0,0,0,0,0]]}]},{"i":"efaf58537c6b10c4","q":"Let  $ABC$  be a triangle with  $\\angle{BAC} > 90$ . Let  $D$  be a point on the segment  $BC$  and  $E$  be a point on line  $AD$  such that  $AB$  is tangent to the circumcircle of triangle  $ACD$  at  $A$  and  $BE$  is perpendicular to  $AD$ . Given that  $CA=CD$  and  $AE=CE$ . Determine  $\\angle{BCA}$  in degrees.","t":[{"b":5,"e":0.0,"k":"falling","v":0.3571,"x":0.55804,"p":[[0,25,0.0,0.55804,0.32213,0.39286,0.50001,0.85714,0.0,1.0,4,5,0,4,0,1,0,0,3,0,0,8,0,0,1,0,0,5,0,0,5,0,5],[4,25,0.16,0.48202,0.35855,0.105,0.57121,0.71429,0.0,1.0,8,5,0,8,0,2,0,0,2,0,0,2,0,0,6,0,0,5,0,0,2,0,5],[8,25,0.32,0.46428,0.28121,0.39285,0.50001,0.71429,0.0,0.85714,7,0,0,7,0,0,0,0,1,0,0,8,0,0,4,0,0,10,0,0,2,0,0],[12,25,0.48,0.49099,0.28568,0.2857,0.71429,0.71429,0.0,0.85714,5,0,0,5,0,2,0,0,4,0,0,3,0,0,0,0,0,17,0,0,1,0,0],[16,25,0.64,0.45981,0.2806,0.25001,0.4998,0.71429,0.0,0.85714,6,0,0,6,0,2,0,0,1,0,0,7,0,0,4,0,0,10,0,0,2,0,0],[20,25,0.8,0.5089,0.23673,0.42857,0.57143,0.71429,0.0,0.71429,4,0,0,4,0,0,0,0,3,0,0,5,0,0,7,0,0,13,0,0,0,0,0],[24,25,0.96,0.44642,0.26666,0.28571,0.4286,0.71429,0.0,0.85714,6,0,0,6,0,0,0,0,5,0,0,6,0,0,4,0,0,10,0,0,1,0,0],[25,25,1.0,0.3571,0.30511,0.0,0.28571,0.60714,0.0,0.85714,10,0,0,10,0,2,0,0,5,0,0,3,0,0,4,0,0,5,0,0,3,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.5357,"x":0.94643,"p":[[0,10,0.0,0.5357,0.37115,0.10714,0.57143,0.85714,0.0,1.0,8,6,2,8,0,1,0,0,1,0,0,2,0,0,5,0,0,5,0,0,4,0,6],[4,10,0.4,0.67409,0.31791,0.57132,0.71429,0.89286,0.0,1.0,4,8,0,4,0,0,0,0,2,0,0,1,0,0,2,0,0,10,0,0,5,0,8],[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],[10,10,1.0,0.94196,0.16698,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,26]]}]},{"i":"f15cd3f5a4be0c6a","q":"Let  $a$  and  $b$  be integers. Is it possible to find integers  $p$  and  $q$  such that the integers  $p+na$  and  $q +nb$  have no common prime factor no matter how the integer  $n$  is chosen ?","t":[{"b":5,"e":0.57143,"k":"falling","v":0.57588,"x":0.90177,"p":[[0,25,0.0,0.8214,0.24487,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,3,0,0,1,0,19],[4,25,0.16,0.85713,0.1713,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,6,0,16],[8,25,0.32,0.85264,0.19725,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,3,0,0,4,0,18],[12,25,0.48,0.88839,0.17029,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,5,0,20],[16,25,0.64,0.90177,0.17658,0.85711,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,23],[20,25,0.8,0.61602,0.13094,0.57132,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,16,0,0,10,0,0,1,0,1],[24,25,0.96,0.58466,0.12023,0.5354,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,12,0,0,0,0,0],[25,25,1.0,0.57588,0.16935,0.57143,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,3,0,0,17,0,0,7,0,0,1,0,1]]},{"b":6,"e":0.57143,"k":"flat","v":0.73213,"x":0.84821,"p":[[0,35,0.0,0.80802,0.23586,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,0,5,0,15],[4,35,0.1143,0.80355,0.21945,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,4,0,0,5,0,14],[8,35,0.2286,0.77677,0.23675,0.57143,0.78571,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,0,0,0,7,0,0,7,0,0,3,0,13],[12,35,0.3429,0.79914,0.22542,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,6,0,0,1,0,16],[16,35,0.4571,0.84821,0.18536,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,6,0,16],[20,35,0.5714,0.73213,0.22231,0.57143,0.78564,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,4,0,0,8,0,8],[24,35,0.6857,0.7589,0.20961,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,5,0,0,4,0,11],[28,35,0.8,0.81695,0.20896,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,7,0,14],[32,35,0.9143,0.82589,0.17399,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,7,0,12],[35,35,1.0,0.73657,0.19923,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,7,0,0,5,0,8]]}]},{"i":"33aae68729f44792","q":"Let  $a_1,a_2,...a_n$  be positive numbers. And let   $s=a_1+a_2+...+a_n$ ,and  $p=a_1*a_2*...*a_n$ .Prove that  $2^{n}*\\sqrt{p} \\leq 1+\\frac{s}{1!}+\\frac{s^2}{2!}+...+\\frac{s^n}{n!}$","t":[{"b":4,"e":0.0,"k":"volatile","v":0.0,"x":0.78125,"p":[[0,25,0.0,0.29018,0.43811,0.0,0.0,0.89286,0.0,1.0,21,8,0,21,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,8],[4,25,0.16,0.78125,0.37625,0.71429,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[8,25,0.32,0.66071,0.44999,0.10714,1.0,1.0,0.0,1.0,8,20,0,8,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,20],[12,25,0.48,0.75223,0.3985,0.62496,1.0,1.0,0.0,1.0,6,21,0,6,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,2,0,21],[16,25,0.64,0.47768,0.47997,0.0,0.28571,1.0,0.0,1.0,15,13,0,15,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,13],[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":6,"e":1.0,"k":"rising","v":0.35938,"x":0.99777,"p":[[0,22,0.0,0.35938,0.46617,0.0,0.0,1.0,0.0,1.0,20,9,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,9],[4,22,0.1818,0.58929,0.46941,0.0,1.0,1.0,0.0,1.0,11,17,0,11,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,17],[8,22,0.3636,0.74554,0.42218,0.67857,1.0,1.0,0.0,1.0,7,22,0,7,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,22],[12,22,0.5455,0.76786,0.40681,0.82143,1.0,1.0,0.0,1.0,6,24,0,6,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[16,22,0.7273,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[22,22,1.0,0.99777,0.01243,1.0,1.0,1.0,0.92857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31]]}]},{"i":"9dc3f7ed7f0687e0","q":"Let  $f:\\mathbb C\\to\\mathbb C$  be a holomorphic function with the property that  $|f(z)|=1$  for all  $z\\in\\mathbb C$  such that  $|z|=1$ . Prove that there exists a  $\\theta\\in\\mathbb R$  and a  $k\\in\\{0,1,2,\\ldots\\}$  such that  $$ f(z)=e^{i\\theta}z^k $$  for all  $z\\in\\mathbb C$ .","t":[{"b":4,"e":0.28571,"k":"falling","v":0.41518,"x":0.74105,"p":[[0,13,0.0,0.62499,0.27606,0.28571,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,7,0,0,1,0,0,6,0,0,4,0,0,7,0,5],[4,13,0.3077,0.74105,0.32819,0.571,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,1,0,0,3,0,0,5,0,0,2,0,0,1,0,17],[8,13,0.6154,0.49552,0.28343,0.28571,0.57121,0.71429,0.0,1.0,4,3,0,4,0,0,0,0,7,0,0,4,0,0,8,0,0,4,0,0,2,0,3],[12,13,0.9231,0.53122,0.34113,0.28571,0.571,0.85714,0.0,1.0,5,7,0,5,0,0,0,0,7,0,0,3,0,0,5,0,0,3,0,0,2,0,7],[13,13,1.0,0.41518,0.22968,0.28571,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,15,0,0,4,0,0,3,0,0,4,0,0,3,0,0]]},{"b":5,"e":0.0,"k":"volatile","v":0.01339,"x":0.77232,"p":[[0,21,0.0,0.62499,0.26667,0.28571,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,9,0,0,1,0,0,9,0,0,1,0,0,6,0,6],[4,21,0.1905,0.77232,0.33855,0.57143,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0,0,0,20],[8,21,0.381,0.66518,0.34183,0.28571,0.78571,1.0,0.0,1.0,3,12,0,3,0,0,0,0,6,0,0,0,0,0,6,0,0,1,0,0,4,0,12],[12,21,0.5714,0.70089,0.31004,0.28571,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,9,0,0,2,0,0,3,0,0,1,0,0,3,0,14],[16,21,0.7619,0.39731,0.44711,0.0,0.0,0.89286,0.0,1.0,17,8,0,17,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,8],[20,21,0.9524,0.10714,0.27433,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[21,21,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fe5125fbd0e09623","q":"Let  $n \\geq 2$  be a positive integer and define  $k$  to be the number of primes  $\\leq n$ . Let  $A$  be a subset of  $S = \\{2,...,n\\}$  such that  $|A| \\leq k$  and no two elements in  $A$  divide each other. Show that one can find a set  $B$  such that  $|B| = k$ ,  $A \\subseteq B \\subseteq S$  and no two elements in  $B$  divide each other.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.15179,"p":[[0,43,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.14732,0.25626,0.0,0.0,0.17857,0.0,1.0,21,1,0,21,0,3,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,1],[8,43,0.186,0.08036,0.19212,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[12,43,0.2791,0.07143,0.14286,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.07143,0.14286,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.07589,0.13825,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.05804,0.12807,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.15179,0.18877,0.0,0.0,0.32143,0.0,0.4286,19,0,0,19,0,0,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.05357,0.13243,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.04464,0.12078,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.13392,0.20494,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,1,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[43,43,1.0,0.12946,0.16115,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,0,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.17412,"p":[[0,95,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,95,0.0421,0.10714,0.18898,0.0,0.0,0.17857,0.0,0.71429,23,0,0,23,0,1,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[8,95,0.0842,0.12054,0.19269,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,2,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[12,95,0.1263,0.09375,0.15815,0.0,0.0,0.17857,0.0,0.42857,23,0,0,23,0,1,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,95,0.1684,0.11607,0.20341,0.0,0.0,0.17857,0.0,0.71429,22,0,0,22,0,2,0,0,4,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[20,95,0.2105,0.0625,0.13333,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,95,0.2526,0.11607,0.16917,0.0,0.0,0.17857,0.0,0.42857,20,0,0,20,0,4,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[28,95,0.2947,0.12945,0.20625,0.0,0.0,0.17857,0.0,0.71429,21,0,0,21,0,3,0,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[32,95,0.3368,0.125,0.23351,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,0],[36,95,0.3789,0.08036,0.15542,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,3,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,95,0.4211,0.17412,0.23619,0.0,0.0,0.32164,0.0,0.71429,18,0,0,18,0,3,0,0,3,0,0,5,0,0,0,0,0,3,0,0,0,0,0],[44,95,0.4632,0.16964,0.20341,0.0,0.0,0.32143,0.0,0.71429,17,0,0,17,0,2,0,0,5,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[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],[52,95,0.5474,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,95,0.5895,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,95,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,95,0.6737,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,95,0.7158,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,95,0.7579,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,95,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[84,95,0.8842,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,95,0.9263,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,95,0.9684,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"245e0dc1e02e64a3","q":"Let  $m$  and  $n$  be positive integers. A sequence of points  $(A_0,A_1,\\ldots,A_n)$  on the Cartesian plane is called *interesting* if  $A_i$  are all lattice points, the slopes of  $OA_0,OA_1,\\cdots,OA_n$  are strictly increasing ( $O$  is the origin) and the area of triangle  $OA_iA_{i+1}$  is equal to  $\\frac{1}{2}$  for  $i=0,1,\\ldots,n-1$ . \nLet  $(B_0,B_1,\\cdots,B_n)$  be a sequence of points. We may insert a point  $B$  between  $B_i$  and  $B_{i+1}$  if  $\\overrightarrow{OB}=\\overrightarrow{OB_i}+\\overrightarrow{OB_{i+1}}$ , and the resulting sequence  $(B_0,B_1,\\ldots,B_i,B,B_{i+1},\\ldots,B_n)$  is called an *extension* of the original sequence. Given two *interesting* sequences  $(C_0,C_1,\\ldots,C_n)$  and  $(D_0,D_1,\\ldots,D_m)$ , prove that if  $C_0=D_0$  and  $C_n=D_m$ , then we may perform finitely many *extensions* on each sequence until the resulting two sequences become identical.","t":[{"b":1,"e":0.57143,"k":"rising","v":0.24553,"x":0.6339,"p":[[0,52,0.0,0.24553,0.10248,0.24999,0.28571,0.28571,0.0,0.42857,3,0,2,3,0,5,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.49557,0.21422,0.28571,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,7,0,0,4,0,0,5,0,0,3,0,1],[8,52,0.1538,0.49552,0.21423,0.28571,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,9,0,0,3,0,0,6,0,0,4,0,0],[12,52,0.2308,0.42855,0.16364,0.28571,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,1,0,0,10,0,0,15,0,0,2,0,0,2,0,0,2,0,0],[16,52,0.3077,0.44192,0.17624,0.2857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,12,0,0,7,0,0,9,0,0,2,0,0,0,0,1],[20,52,0.3846,0.6339,0.22286,0.42857,0.71429,0.74996,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,7,0,0,4,0,0,9,0,0,4,0,4],[24,52,0.4615,0.55353,0.22517,0.42857,0.571,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,3,0,0,9,0,0,6,0,0,8,0,0,2,0,2],[28,52,0.5385,0.60261,0.18809,0.42857,0.57121,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,9,0,0,9,0,0,6,0,0,4,0,2],[32,52,0.6154,0.54467,0.22426,0.39286,0.4293,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,9,0,0,4,0,0,5,0,0,4,0,2],[36,52,0.6923,0.54457,0.19045,0.42857,0.571,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,9,0,0,6,0,0,8,0,0,2,0,1],[40,52,0.7692,0.59821,0.22428,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,8,0,0,1,0,0,10,0,0,5,0,2],[44,52,0.8462,0.56244,0.2285,0.42857,0.571,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,2,0,0,8,0,0,7,0,0,8,0,0,1,0,3],[48,52,0.9231,0.6071,0.22869,0.42857,0.57143,0.85704,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,5,0,0,8,0,0,4,0,0,6,0,3],[52,52,1.0,0.47317,0.20023,0.2857,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,10,0,0,9,0,0,6,0,0,3,0,0,2,0,1]]},{"b":5,"e":0.42857,"k":"flat","v":0.25446,"x":0.58015,"p":[[0,34,0.0,0.25446,0.11701,0.2857,0.28571,0.28571,0.0,0.4286,4,0,3,4,0,3,0,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.54898,0.22351,0.42857,0.4998,0.71429,0.14,1.0,0,3,0,0,0,1,0,0,5,0,0,10,0,0,6,0,0,5,0,0,2,0,3],[8,34,0.2353,0.58015,0.22277,0.39286,0.57121,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,3,0,0,9,0,0,5,0,0,5,0,2],[12,34,0.3529,0.5357,0.22868,0.39286,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,7,0,0,9,0,0,5,0,0,4,0,0,4,0,2],[16,34,0.4706,0.44194,0.16886,0.28571,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,1,0,0,9,0,0,14,0,0,4,0,0,3,0,0,0,0,1],[20,34,0.5882,0.46871,0.17579,0.28571,0.4998,0.57143,0.1429,0.85714,0,0,0,0,0,1,0,0,11,0,0,4,0,0,11,0,0,4,0,0,1,0,0],[24,34,0.7059,0.53567,0.17855,0.42857,0.4286,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,13,0,0,6,0,0,6,0,0,2,0,1],[28,34,0.8235,0.4732,0.2271,0.28571,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,1,0,0,14,0,0,4,0,0,5,0,0,5,0,0,1,0,2],[32,34,0.9412,0.45973,0.20415,0.28571,0.42857,0.57143,0.1429,1.0,0,1,0,0,0,1,0,0,11,0,0,11,0,0,2,0,0,4,0,0,2,0,1],[34,34,1.0,0.36159,0.15964,0.2857,0.28571,0.4286,0.14286,0.85714,0,0,0,0,0,4,0,0,15,0,0,8,0,0,3,0,0,1,0,0,1,0,0]]}]},{"i":"806ac7fe3fb6795f","q":"23. G4 (RUS) Let $A_{1} A_{2} \\ldots A_{n}$ be a convex polygon, $n \\geq 4$. Prove that $A_{1} A_{2} \\ldots A_{n}$ is cyclic if and only if to each vertex $A_{j}$ one can assign a pair $\\left(b_{j}, c_{j}\\right)$ of real numbers, $j=1,2, \\ldots n$, such that  $$ A_{i} A_{j}=b_{j} c_{i}-b_{i} c_{j} \\quad \\text { for all } i, j \\text { with } 1 \\leq i \\leq j \\leq n $$","t":[{"b":1,"e":1.0,"k":"flat","v":0.81683,"x":0.96427,"p":[[0,59,0.0,0.88393,0.23266,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,23],[4,59,0.0678,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[8,59,0.1356,0.89731,0.17584,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,4,0,22],[12,59,0.2034,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[16,59,0.2712,0.88393,0.16917,0.82143,1.0,1.0,0.5714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,4,0,20],[20,59,0.339,0.87499,0.228,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,3,0,22],[24,59,0.4068,0.92409,0.18896,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,26],[28,59,0.4746,0.91963,0.17837,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,26],[32,59,0.5424,0.83929,0.21354,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,1,0,0,4,0,18],[36,59,0.6102,0.81683,0.22376,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,3,0,0,1,0,18],[40,59,0.678,0.86161,0.21866,0.57143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,0,1,0,22],[44,59,0.7458,0.85711,0.19566,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,5,0,18],[48,59,0.8136,0.84372,0.19355,0.57143,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,3,0,18],[52,59,0.8814,0.87508,0.19151,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,2,0,21],[56,59,0.9492,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[59,59,1.0,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.85714,"x":0.95088,"p":[[0,40,0.0,0.85714,0.26486,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,7,0,20],[4,40,0.1,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[8,40,0.2,0.93303,0.15966,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[12,40,0.3,0.95087,0.11638,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[16,40,0.4,0.90625,0.16602,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,3,0,23],[20,40,0.5,0.89732,0.21793,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,24],[24,40,0.6,0.87498,0.17407,0.82132,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,5,0,19],[28,40,0.7,0.87945,0.18251,0.71429,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,0,0,22],[32,40,0.8,0.93311,0.13354,0.965,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[36,40,0.9,0.95088,0.07669,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[40,40,1.0,0.94196,0.08646,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21]]}]},{"i":"87261983e0ab7131","q":"3. N3 (MON) Let $p_{1}, p_{2}, \\ldots, p_{n}$ be distinct primes greater than 3 . Show that $2^{p_{1} p_{2} \\cdots p_{n}}+1$ has at least $4^{n}$ divisors.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.08929,"p":[[0,22,0.0,0.08929,0.16269,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,7,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,22,0.1818,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.06696,0.09439,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.05357,0.10565,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.0,"k":"flat","v":0.0625,"x":0.18304,"p":[[0,27,0.0,0.0625,0.12339,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,27,0.1481,0.11598,0.08325,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.11161,0.09932,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.08027,0.09401,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.18304,0.16065,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,15,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[24,27,0.8889,0.13393,0.15126,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,13,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[27,27,1.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9e61fd2f04b1d756","q":"$a_{n}$  is a sequence of positive integers such that, for every  $n\\geq 1$ ,  $0<a_{n+1}-a_{n}<\\sqrt{a_{n}}$ . Prove that for every  $x,y\\in{R}$  such that  $0<x<y<1$   $x< \\frac{a_{k}}{a_{m}}<y$  we can find such  $k,m\\in{Z^{+}}$ .","t":[{"b":5,"e":1.0,"k":"rising","v":0.57588,"x":0.96874,"p":[[0,18,0.0,0.57588,0.27775,0.28571,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,6,0,0,5,0,0,4,0,0,7,0,0,1,0,6],[4,18,0.2222,0.69193,0.27226,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,6,0,0,4,0,0,5,0,0,2,0,11],[8,18,0.4444,0.96874,0.06903,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[12,18,0.6667,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[16,18,0.8889,0.87946,0.17536,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,9,0,17],[18,18,1.0,0.88392,0.13092,0.85711,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,15]]},{"b":7,"e":1.0,"k":"flat","v":0.64731,"x":0.87051,"p":[[0,49,0.0,0.66067,0.27838,0.42857,0.64286,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,5,0,0,6,0,0,5,0,0,1,0,10],[4,49,0.0816,0.73213,0.24937,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,3,0,0,7,0,0,5,0,0,5,0,10],[8,49,0.1633,0.82587,0.21648,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,1,0,0,6,0,16],[12,49,0.2449,0.87051,0.20004,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,1,0,21],[16,49,0.3265,0.83034,0.19705,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,4,0,16],[20,49,0.4082,0.77676,0.26231,0.57132,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,5,0,0,4,0,0,5,0,0,1,0,16],[24,49,0.4898,0.66071,0.21354,0.57143,0.64286,0.74996,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,2,0,0,11,0,0,8,0,0,3,0,5],[28,49,0.5714,0.69643,0.23351,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,2,0,0,10,0,0,8,0,0,2,0,8],[32,49,0.6531,0.72768,0.2212,0.57143,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,10,0,0,1,0,10],[36,49,0.7347,0.79908,0.24188,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,4,0,0,3,0,16],[40,49,0.8163,0.79909,0.2016,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,0,4,0,13],[44,49,0.898,0.77677,0.24207,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,3,0,0,6,0,13],[48,49,0.9796,0.7455,0.25441,0.571,0.78571,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,3,0,0,3,0,13],[49,49,1.0,0.64731,0.24996,0.42857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,3,0,0,5,0,0,6,0,0,8,0,0,3,0,6]]}]},{"i":"faeeb136233fd686","q":"A  $8 \\times 8$  board is given, with sides directed north-south and east-west.\nIt is divided into  $1 \\times 1$  cells in the usual manner. In each cell, there is most one *house*. A house occupies only one cell.\n\nA house is *in the shade* if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade.\n\nFind the maximal number of houses that can be placed on the board such that no house is in the shade.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.20534,"p":[[0,137,0.0,0.13393,0.23128,0.0,0.0,0.14286,0.0,1.0,17,1,7,17,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[4,137,0.0292,0.20534,0.25981,0.0,0.14286,0.42857,0.0,0.857,15,0,0,15,0,7,0,0,0,0,0,5,0,0,3,0,0,0,0,0,2,0,0],[8,137,0.0584,0.16963,0.22426,0.0,0.14286,0.1786,0.0,0.85714,15,0,0,15,0,9,0,0,1,0,0,3,0,0,3,0,0,0,0,0,1,0,0],[12,137,0.0876,0.10267,0.17937,0.0,0.0,0.14287,0.0,0.57143,22,0,0,22,0,4,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,137,0.1168,0.04902,0.10471,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,137,0.146,0.10705,0.21427,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[24,137,0.1752,0.07142,0.15563,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,5,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[28,137,0.2044,0.14282,0.24477,0.0,0.0,0.1786,0.0,1.0,21,1,0,21,0,3,0,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,1],[32,137,0.2336,0.04902,0.11064,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,137,0.2628,0.07588,0.15557,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[40,137,0.292,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,137,0.3212,0.03572,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[48,137,0.3504,0.04017,0.10846,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,137,0.3796,0.0625,0.11259,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,137,0.4088,0.04911,0.12682,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[60,137,0.438,0.08027,0.1333,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,9,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,137,0.4672,0.05804,0.13296,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,137,0.4964,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,137,0.5255,0.09813,0.14477,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,10,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[76,137,0.5547,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,137,0.5839,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,137,0.6131,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,137,0.6423,0.03125,0.08552,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,137,0.6715,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[96,137,0.7007,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,137,0.7299,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,137,0.7591,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,137,0.7883,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,137,0.8175,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,137,0.8467,0.03125,0.07771,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,137,0.8759,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,137,0.9051,0.04454,0.11525,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[128,137,0.9343,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,137,0.9635,0.03571,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[136,137,0.9927,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[137,137,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.00446,"x":0.41959,"p":[[0,174,0.0,0.125,0.1915,0.0,0.0,0.17857,0.0,0.85714,18,0,8,18,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[4,174,0.023,0.21426,0.29011,0.0,0.0,0.42857,0.0,1.0,17,1,0,17,0,4,0,0,2,0,0,2,0,0,4,0,0,1,0,0,1,0,1],[8,174,0.046,0.23651,0.27806,0.0,0.14286,0.4286,0.0,1.0,13,1,0,13,0,7,0,0,2,0,0,4,0,0,3,0,0,1,0,0,1,0,1],[12,174,0.069,0.28121,0.27771,0.0,0.14286,0.571,0.0,0.85714,10,0,0,10,0,9,0,0,0,0,0,3,0,0,7,0,0,1,0,0,2,0,0],[16,174,0.092,0.23661,0.26871,0.0,0.14286,0.42857,0.0,1.0,13,1,0,13,0,6,0,0,3,0,0,4,0,0,3,0,0,2,0,0,0,0,1],[20,174,0.1149,0.14729,0.2215,0.0,0.0,0.17857,0.0,0.85714,18,0,0,18,0,6,0,0,3,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[24,174,0.1379,0.11152,0.15864,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,8,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[28,174,0.1609,0.1116,0.21047,0.0,0.0,0.14286,0.0,0.857,21,0,0,21,0,6,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[32,174,0.1839,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,174,0.2069,0.11607,0.20652,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[40,174,0.2299,0.08481,0.14218,0.0,0.0,0.14286,0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a finite non empty set of primes  $P$ , let  $m(P)$  denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of  $P$ .\n\n(i) Show that  $|P|\\le m(P)$ , with equality if and only if  $\\min(P)>|P|$ .\n\n(ii) Show that  $m(P)<(|P|+1)(2^{|P|}-1)$ .\n\n(The number  $|P|$  is the size of set  $P$ )\n\n*Dan Schwarz, Romania*","t":[{"b":0,"e":0.42857,"k":"flat","v":0.28125,"x":0.40179,"p":[[0,30,0.0,0.40179,0.05576,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.36607,0.10677,0.28571,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,15,0,0,14,0,0,1,0,0,1,0,0,0,0,0],[8,30,0.2667,0.29678,0.11322,0.2857,0.28571,0.28571,0.14,0.71429,0,0,0,0,0,6,0,0,19,0,1,5,0,0,0,0,0,1,0,0,0,0,0],[12,30,0.4,0.28125,0.09771,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,2,0,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.28126,0.09771,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,5,0,0,25,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,30,0.6667,0.28125,0.07562,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,5,0,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.29469,0.0871,0.2857,0.28571,0.28571,0.14286,0.43,0,0,0,0,0,5,0,0,20,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.30357,0.06916,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.31696,0.05906,0.2857,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.42857,"k":"flat","v":0.34373,"x":0.39732,"p":[[0,56,0.0,0.375,0.07784,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,10,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.35714,0.12372,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,15,0,0,14,0,0,0,0,0,0,0,0,1,0,0],[8,56,0.1429,0.37947,0.14111,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,14,0,0,15,0,0,1,0,0,0,0,0,0,0,1],[12,56,0.2143,0.35937,0.09697,0.2857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,13,0,0,15,0,1,1,0,0,0,0,0,0,0,0],[16,56,0.2857,0.39732,0.12234,0.28571,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,18,0,0,0,0,0,1,0,0,1,0,0],[20,56,0.3571,0.38393,0.06622,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.37054,0.10631,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,11,0,0,18,0,0,0,0,0,1,0,0,0,0,0],[28,56,0.5,0.35719,0.08752,0.2857,0.42857,0.42857,0.14286,0.43,0,0,0,0,0,2,0,0,12,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.37504,0.07787,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,13,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[36,56,0.6429,0.34821,0.07936,0.2857,0.28571,0.42857,0.2857,0.5714,0,0,0,0,0,0,0,0,19,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[40,56,0.7143,0.37946,0.09182,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,18,0,0,0,0,0,1,0,0,0,0,0],[44,56,0.7857,0.37054,0.09354,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,16,0,0,0,0,0,1,0,0,0,0,0],[48,56,0.8571,0.38394,0.06621,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.34373,0.10627,0.28571,0.35714,0.42857,0.14286,0.571,0,0,0,0,0,4,0,0,12,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[56,56,1.0,0.35482,0.08676,0.28571,0.42857,0.42857,0.14,0.4286,0,0,0,0,0,2,0,0,12,0,1,17,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"850025ad4463abc8","q":"Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:\nThe last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. \nLisa repeats this until she gets a number for the first time was on the board.\n(a) Show that the sequence of moves always ends.\n(b) If Lisa begins with the number  $53^{2022} - 1$ , what is the last number on the board?\n\nExample: If Lisa starts with the number  $2022$ , she gets  $202 + 4\\cdot 2 = 210$  in the first move and overall the result   $$ 2022 \\to 210 \\to 21 \\to 6 \\to 24 \\to 18 \\to 33 \\to 15 \\to 21 $$  .\nSince Lisa gets  $21$  for the second time, the turn order ends.\n\n*(Stephan Pfannerer)*","t":[{"b":1,"e":0.71429,"k":"flat","v":0.72322,"x":0.96875,"p":[[0,74,0.0,0.78571,0.32927,0.57143,1.0,1.0,0.0,1.0,3,21,3,3,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,0,0,0,21],[4,74,0.0541,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[8,74,0.1081,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[12,74,0.1622,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[16,74,0.2162,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[20,74,0.2703,0.94643,0.13716,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[24,74,0.3243,0.94196,0.1551,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,27],[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],[32,74,0.4324,0.91518,0.16698,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,24],[36,74,0.4865,0.875,0.18472,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,3,0,20],[40,74,0.5405,0.91518,0.16698,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,24],[44,74,0.5946,0.8973,0.1864,0.96425,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,1,0,24],[48,74,0.6486,0.92411,0.13825,0.85714,1.0,1.0,0.5714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,23],[52,74,0.7027,0.90179,0.17655,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,23],[56,74,0.7568,0.91517,0.17448,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,25],[60,74,0.8108,0.91964,0.1234,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,21],[64,74,0.8649,0.89285,0.14286,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,4,0,19],[68,74,0.9189,0.76339,0.13651,0.71429,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,17,0,0,4,0,6],[72,74,0.973,0.73213,0.07786,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,0,4,0,1],[74,74,1.0,0.72322,0.09407,0.71429,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,21,0,0,7,0,0]]},{"b":3,"e":0.71429,"k":"flat","v":0.5803,"x":0.98661,"p":[[0,79,0.0,0.69196,0.40738,0.5,1.0,1.0,0.0,1.0,7,18,7,7,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,18],[4,79,0.0506,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[8,79,0.1013,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[12,79,0.1519,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[16,79,0.2025,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,79,0.2532,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[24,79,0.3038,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[28,79,0.3544,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[32,79,0.4051,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,79,0.4557,0.9375,0.12846,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,24],[40,79,0.5063,0.94642,0.13247,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[44,79,0.557,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[48,79,0.6076,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,79,0.6582,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[56,79,0.7089,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[60,79,0.7595,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[64,79,0.8101,0.9375,0.15126,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,27],[68,79,0.8608,0.85268,0.21572,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,3,0,20],[72,79,0.9114,0.91964,0.15542,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,24],[76,79,0.962,0.83482,0.17536,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,4,0,15],[79,79,1.0,0.5803,0.07087,0.57142,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,24,0,0,5,0,0,0,0,0]]}]},{"i":"68bacecbca05f477","q":"For any real number  $p\\geq1$  consider the set of all real numbers  $x$  with\n\\[p<x<\\left(2+\\sqrt{p+\\frac{1}{4}}\\right)^2.\\]\nProve that from any such set one can select four mutually distinct natural numbers  $a, b, c, d$  with  $ab=cd.$","t":[{"b":4,"e":0.0,"k":"falling","v":0.04018,"x":0.90177,"p":[[0,89,0.0,0.57588,0.38875,0.14286,0.57143,1.0,0.0,1.0,6,11,0,6,0,3,0,0,2,0,0,0,0,0,6,0,0,3,0,0,1,0,11],[4,89,0.0449,0.90177,0.14483,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,21],[8,89,0.0899,0.87945,0.22337,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,1,0,0,2,0,23],[12,89,0.1348,0.8125,0.24856,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,7,0,0,5,0,15],[16,89,0.1798,0.74991,0.26984,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,2,0,0,1,0,0,2,0,0,10,0,0,3,0,12],[20,89,0.2247,0.85268,0.14934,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,5,0,14],[24,89,0.2697,0.75446,0.27254,0.71429,0.85707,1.0,0.0,1.0,2,11,0,2,0,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,6,0,11],[28,89,0.3146,0.74999,0.31945,0.67857,0.85714,1.0,0.0,1.0,2,13,0,2,0,2,0,0,2,0,0,0,0,0,2,0,0,3,0,0,8,0,13],[32,89,0.3596,0.70534,0.32131,0.57132,0.78571,1.0,0.0,1.0,1,13,0,1,0,4,0,0,1,0,0,1,0,0,5,0,0,4,0,0,3,0,13],[36,89,0.4045,0.65173,0.26231,0.53539,0.71429,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,3,0,0,3,0,0,6,0,0,8,0,0,4,0,6],[40,89,0.4494,0.76786,0.21943,0.57143,0.78571,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,7,0,0,5,0,11],[44,89,0.4944,0.76784,0.21944,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,9,0,0,2,0,12],[48,89,0.5393,0.65178,0.31529,0.42857,0.71429,0.89286,0.0,1.0,2,8,0,2,0,3,0,0,2,0,0,2,0,0,2,0,0,9,0,0,4,0,8],[52,89,0.5843,0.71429,0.29233,0.71429,0.71429,1.0,0.0,1.0,3,9,0,3,0,0,0,0,1,0,0,2,0,0,1,0,0,11,0,0,5,0,9],[56,89,0.6292,0.66964,0.34337,0.42857,0.71429,1.0,0.0,1.0,3,11,0,3,0,3,0,0,1,0,0,2,0,0,1,0,0,8,0,0,3,0,11],[60,89,0.6742,0.70088,0.27748,0.67857,0.71429,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,1,0,0,2,0,0,2,0,0,12,0,0,4,0,8],[64,89,0.7191,0.70536,0.28333,0.67857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,1,0,0,1,0,0,2,0,0,10,0,0,6,0,8],[68,89,0.764,0.58482,0.38359,0.21429,0.71429,0.89286,0.0,1.0,8,8,0,8,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,5,0,8],[72,89,0.809,0.42853,0.3444,0.14286,0.42857,0.60714,0.0,1.0,7,5,0,7,0,4,0,0,4,0,0,3,0,0,6,0,0,2,0,0,1,0,5],[76,89,0.8539,0.18304,0.2402,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,8,0,0,5,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[80,89,0.8989,0.23657,0.30637,0.0,0.07143,0.46418,0.0,1.0,16,1,0,16,0,4,0,0,3,0,0,1,0,0,4,0,0,1,0,0,2,0,1],[84,89,0.9438,0.16961,0.21255,0.0,0.0,0.2857,0.0,0.57143,17,0,0,17,0,3,0,0,6,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[88,89,0.9888,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"falling","v":0.38838,"x":0.89731,"p":[[0,98,0.0,0.55802,0.37005,0.24999,0.57143,1.0,0.0,1.0,4,10,0,4,0,4,0,0,4,0,0,2,0,0,4,0,0,3,0,0,1,0,10],[4,98,0.0408,0.79908,0.22551,0.67857,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,8,0,12],[8,98,0.0816,0.84375,0.20935,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,4,0,16],[12,98,0.1224,0.78125,0.2575,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,7,0,0,4,0,14],[16,98,0.1633,0.77677,0.23403,0.57143,0.85707,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,5,0,12],[20,98,0.2041,0.83481,0.23176,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,3,0,0,6,0,17],[24,98,0.2449,0.89731,0.16843,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,4,0,21],[28,98,0.2857,0.808,0.18769,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,8,0,11],[32,98,0.3265,0.85267,0.1731,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,6,0,15],[36,98,0.3673,0.85268,0.16554,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,6,0,15],[40,98,0.4082,0.8482,0.20807,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,8,0,0,0,0,19],[44,98,0.449,0.82142,0.17128,0.71429,0.857,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,0,8,0,10],[48,98,0.4898,0.79911,0.23652,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,3,0,14],[52,98,0.5306,0.7946,0.22855,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,8,0,0,6,0,12],[56,98,0.5714,0.75446,0.24284,0.57143,0.85714,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,4,0,0,8,0,10],[60,98,0.6122,0.75892,0.28221,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,2,0,0,1,0,0,1,0,0,2,0,0,8,0,0,4,0,13],[64,98,0.6531,0.70979,0.25627,0.57143,0.71429,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,1,0,0,1,0,0,6,0,0,9,0,0,6,0,7],[68,98,0.6939,0.73659,0.26513,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,2,0,0,0,0,0,1,0,0,5,0,0,8,0,0,5,0,10],[72,98,0.7347,0.61159,0.30354,0.42857,0.71429,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,1,0,0,5,0,0,4,0,0,8,0,0,2,0,7],[76,98,0.7755,0.62945,0.29637,0.39286,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,5,0,0,3,0,0,3,0,0,7,0,0,4,0,7],[80,98,0.8163,0.62052,0.23854,0.42857,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,2,0,0,6,0,0,4,0,0,11,0,0,4,0,3],[84,98,0.8571,0.69195,0.28819,0.57143,0.71429,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,2,0,0,1,0,0,5,0,0,7,0,0,6,0,8],[88,98,0.898,0.58036,0.31122,0.39286,0.57143,0.85714,0.0,1.0,3,6,0,3,0,1,0,0,4,0,0,5,0,0,5,0,0,4,0,0,4,0,6],[92,98,0.9388,0.50891,0.2922,0.2857,0.57143,0.71429,0.0,1.0,3,4,0,3,0,4,0,0,3,0,0,2,0,0,10,0,0,6,0,0,0,0,4],[96,98,0.9796,0.5,0.26486,0.28571,0.4286,0.71429,0.0,1.0,1,3,0,1,0,4,0,0,5,0,0,7,0,0,6,0,0,4,0,0,2,0,3],[98,98,1.0,0.38838,0.26542,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,3,0,0,6,0,0,3,0,0,9,0,0,4,0,0,0,0,1]]}]},{"i":"71916a4c7fa7a8b3","q":"$\\mathrm{f}(\\mathrm{x})$ is a polynomial of degree 3 with rational coefficients. If its graph touches the $\\mathrm{x}$-axis, show that it has three rational roots.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.5625,"x":0.67857,"p":[[0,22,0.0,0.5625,0.28333,0.28571,0.71429,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,5],[4,22,0.1818,0.61607,0.26351,0.28571,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,10,0,0,2,0,0,1,0,0,12,0,0,1,0,6],[8,22,0.3636,0.64731,0.27891,0.28571,0.71429,0.78571,0.14286,1.0,0,8,0,0,0,2,0,0,7,0,0,0,0,0,2,0,0,13,0,0,0,0,8],[12,22,0.5455,0.67411,0.10853,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,28,0,0,0,0,0],[16,22,0.7273,0.67857,0.13363,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,27,0,0,0,0,1],[20,22,0.9091,0.65179,0.20497,0.42857,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,5,0,0,0,0,0,19,0,0,0,0,4],[22,22,1.0,0.63839,0.14719,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,0,25,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.5,"x":0.64732,"p":[[0,31,0.0,0.62499,0.27141,0.28571,0.71429,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,13,0,0,0,0,7],[4,31,0.129,0.63839,0.25997,0.28571,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,15,0,0,1,0,6],[8,31,0.2581,0.64732,0.24996,0.28571,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,17,0,0,0,0,6],[12,31,0.3871,0.53571,0.22304,0.28571,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,11,0,0,2,0,0,1,0,0,16,0,0,0,0,1],[16,31,0.5161,0.62054,0.19103,0.64286,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,23,0,0,0,0,1],[20,31,0.6452,0.57589,0.18723,0.39286,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,3,0,0,1,0,0,20,0,0,0,0,0],[24,31,0.7742,0.56237,0.211,0.28571,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,10,0,0,3,0,0,0,0,0,18,0,0,0,0,1],[28,31,0.9032,0.58928,0.20439,0.28571,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,0,0,0,1,0,0,21,0,0,1,0,0],[31,31,1.0,0.5,0.20516,0.28571,0.42859,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,3,0,0,0,0,0,15,0,0,0,0,0]]}]},{"i":"cb868424f81d5c98","q":"11. (SWE 2) A function $f: I \\rightarrow \\mathbb{R}$, defined on an interval $I$, is called concave if $f(\\theta x+(1-\\theta) y) \\geq \\theta f(x)+(1-\\theta) f(y)$ for all $x, y \\in I$ and $0 \\leq \\theta \\leq 1$. Assume that the functions $f_{1}, \\ldots, f_{n}$, having all nonnegative values, are concave. Prove that the function $\\left(f_{1} f_{2} \\ldots f_{n}\\right)^{1 / n}$ is concave.","t":[{"b":1,"e":1.0,"k":"rising","v":0.17411,"x":1.0,"p":[[0,31,0.0,0.17411,0.24675,0.0,0.14286,0.14286,0.0,1.0,9,2,0,9,0,20,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[4,31,0.129,0.24554,0.32387,0.10714,0.14286,0.1429,0.0,1.0,8,4,0,8,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,4],[8,31,0.2581,0.19866,0.29706,0.0,0.14286,0.14286,0.0,1.0,10,3,0,10,1,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[12,31,0.3871,0.35259,0.39125,0.14286,0.14286,0.78571,0.0,1.0,5,8,0,5,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[16,31,0.5161,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,31,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,31,0.7742,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":0.14286,"k":"flat","v":0.07589,"x":0.20536,"p":[[0,10,0.0,0.20536,0.26471,0.14286,0.14286,0.14286,0.0,1.0,6,3,0,6,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,10,0.4,0.12938,0.23517,0.0,0.14143,0.14286,0.0,1.0,15,2,0,15,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,10,0.8,0.07589,0.07973,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.09812,0.06616,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b1866ca743c606b0","q":"Let  $a_i=min\\{ k+\\dfrac{i}{k}|k \\in N^*\\}$ , determine the value of  $S_{n^2}=[a_1]+[a_2]+\\cdots +[a_{n^2}]$ , where  $n\\ge 2$  . ( $[x]$  denotes the greatest integer not exceeding x)","t":[{"b":6,"e":1.0,"k":"rising","v":0.78571,"x":1.0,"p":[[0,19,0.0,0.78571,0.30094,0.53571,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,5,0,0,2,0,0,2,0,0,0,0,0,3,0,19],[4,19,0.2105,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,19,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":7,"e":1.0,"k":"flat","v":0.90625,"x":1.0,"p":[[0,3,0.0,0.90625,0.19103,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,24],[3,3,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"37b9e01793f41d8a","q":"(a) Show that, for each positive integer  $n$ , the number of monic polynomials of degree  $n$  with integer coefficients having all its roots on the unit circle is finite.\n(b) Let  $P(x)$  be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer  $m$  such that  $y^m=1$  for each root  $y$  of  $P(x)$ .","t":[{"b":0,"e":0.2857,"k":"falling","v":0.39732,"x":0.83929,"p":[[0,12,0.0,0.83483,0.23447,0.82143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,2,0,0,7,0,17],[4,12,0.3333,0.83929,0.22799,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,5,0,0,4,0,18],[8,12,0.6667,0.66518,0.30223,0.39286,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,8,0,0,6,0,0,1,0,0,3,0,0,2,0,12],[12,12,1.0,0.39732,0.05906,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.89732,"x":0.97767,"p":[[0,24,0.0,0.89732,0.19638,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,7,0,21],[4,24,0.1667,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,24,0.3333,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[12,24,0.5,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,24,0.6667,0.94195,0.07874,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[20,24,0.8333,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[24,24,1.0,0.95981,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]}]},{"i":"451de4f60106f2e3","q":"For a positive integer \\( n \\), let \\( P_n \\) denote the product of all prime numbers not exceeding \\( n \\). \n\nProve that there exists a constant \\( c > 0 \\) such that for any sufficiently large integer \\( n \\), if all integers not exceeding \\( P_n \\) and coprime with \\( P_n \\) are arranged in a sequence, then there exist two adjacent numbers in this sequence whose difference is at least \\( c \\cdot \\frac{n \\cdot \\ln n}{(\\ln \\ln n)^2} \\). \n\nFurthermore, consider whether this bound can be strengthened to \\( c \\cdot \\frac{n \\cdot \\ln n \\cdot \\ln \\ln \\ln n}{(\\ln \\ln n)^2} \\). \n\n*Created by Mucong Sun, Tianjin Experimental Binhai School*","t":[{"b":5,"e":1.0,"k":"rising","v":0.15179,"x":0.58472,"p":[[0,27,0.0,0.15179,0.12846,0.0,0.14286,0.28571,0.0,0.4286,11,0,0,11,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.48212,0.21351,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,9,0,0,8,0,0,6,0,0,2,0,0,5,0,0],[8,27,0.2963,0.56245,0.21997,0.39286,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,4,0,0,8,0,0,6,0,0,5,0,1],[12,27,0.4444,0.54008,0.14163,0.42859,0.571,0.60714,0.2857,0.857,0,0,0,0,0,0,0,0,3,0,0,10,0,0,11,0,0,7,0,0,1,0,0],[16,27,0.5926,0.45967,0.2643,0.2857,0.49979,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,8,0,0,2,0,0,9,0,0,2,0,0,4,0,1],[20,27,0.7407,0.53792,0.21502,0.33918,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,7,0,1,5,0,0,7,0,0,8,0,0,1,0,2],[24,27,0.8889,0.56022,0.2436,0.42857,0.57141,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,4,0,0,5,0,0,7,1,0,5,0,0,6,0,1],[27,27,1.0,0.58472,0.2536,0.42857,0.64286,0.75,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,6,0,0,3,0,0,8,0,0,6,0,2]]},{"b":7,"e":0.71429,"k":"rising","v":0.14732,"x":0.52675,"p":[[0,30,0.0,0.14732,0.15765,0.0,0.14286,0.28571,0.0,0.57143,14,0,1,14,0,7,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,30,0.1333,0.49552,0.19227,0.28571,0.57121,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,8,0,0,6,0,0,10,0,0,5,0,0,2,0,0],[8,30,0.2667,0.48204,0.23089,0.2857,0.5712,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,8,0,0,3,0,0,8,0,0,6,0,0,3,0,0],[12,30,0.4,0.51775,0.23074,0.39286,0.571,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,4,0,0,11,0,0,6,0,0,2,0,1],[16,30,0.5333,0.47319,0.18705,0.28571,0.4286,0.60707,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,8,0,0,4,0,0,6,0,0,2,0,0],[20,30,0.6667,0.45537,0.16532,0.28571,0.4286,0.57143,0.14286,0.857,0,0,0,0,0,1,0,0,9,0,0,11,0,0,6,0,0,4,0,0,1,0,0],[24,30,0.8,0.49998,0.21723,0.28571,0.571,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,6,0,0,6,0,0,7,0,0,9,0,0,0,0,1],[28,30,0.9333,0.49093,0.22863,0.28571,0.42857,0.60714,0.14,1.0,0,2,0,0,0,2,0,0,9,0,0,8,0,0,5,0,0,4,0,0,2,0,2],[30,30,1.0,0.52675,0.16916,0.42857,0.57121,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,7,0,0,8,0,0,9,0,0,1,0,0]]}]},{"i":"ca7d8324b4c4900e","q":"Find all integer solutions  $(p, q, r)$  of the equation  $r + p ^ 4 = q ^ 4$  with the following conditions:  $\\bullet$   $r$  is a positive integer with exactly  $8$  positive divisors.  $\\bullet$    $p$  and  $q$  are prime numbers.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.73213,"x":0.85267,"p":[[0,15,0.0,0.73213,0.09278,0.71429,0.71429,0.857,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,18,0,0,9,0,0],[4,15,0.2667,0.76785,0.12752,0.71429,0.71429,0.85714,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,11,0,3],[8,15,0.5333,0.7857,0.09448,0.71429,0.857,0.85714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,19,0,0],[12,15,0.8,0.81025,0.16714,0.85714,0.85714,0.85714,0.0,0.92857,1,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,27,1,0],[15,15,1.0,0.85267,0.02485,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0]]},{"b":1,"e":0.71429,"k":"flat","v":0.69642,"x":0.75893,"p":[[0,60,0.0,0.71428,0.10714,0.71429,0.71429,0.74996,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,17,0,0,8,0,0],[4,60,0.0667,0.73217,0.11142,0.71429,0.71429,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,21,0,0,7,0,1],[8,60,0.1333,0.75893,0.0974,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,17,0,0,13,0,0],[12,60,0.2,0.70534,0.07088,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,24,0,0,3,0,0],[16,60,0.2667,0.71875,0.05628,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0],[20,60,0.3333,0.70969,0.05629,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,27,0,0,2,0,0],[24,60,0.4,0.69643,0.04725,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[28,60,0.4667,0.69642,0.04728,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[32,60,0.5333,0.70076,0.07455,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,26,0,0,2,0,0],[36,60,0.6,0.72753,0.06552,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,0,5,0,0],[40,60,0.6667,0.71875,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,2,0,0],[44,60,0.7333,0.70536,0.04971,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0],[48,60,0.8,0.71429,0.05051,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,2,0,0],[52,60,0.8667,0.71875,0.05629,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0],[56,60,0.9333,0.70534,0.04975,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,1,0,0],[60,60,1.0,0.70771,0.06298,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,28,1,0,1,0,0]]}]},{"i":"db4ef2b79d111b23","q":"Find all permutations  $(a_1, a_2, \\cdots, a_{2024})$  of  $(1, 2, \\cdots, 2024)$  such that there exists a polynomial  $P$  with integer coefficients satisfying  $P(i) = a_i$  for each  $i = 1, 2, \\cdots, 2024$ 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all positive integers  $N$  having only prime divisors  $2,5$  such that  $N+25$  is a perfect square.","t":[{"b":2,"e":0.2857,"k":"falling","v":0.42415,"x":0.86606,"p":[[0,40,0.0,0.80804,0.16602,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,8,0,10],[4,40,0.1,0.8616,0.15765,0.82132,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,10,0,14],[8,40,0.2,0.85712,0.14288,0.82132,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,12,0,12],[12,40,0.3,0.85713,0.18898,0.85711,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,12,0,14],[16,40,0.4,0.86606,0.15544,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,13,0,13],[20,40,0.5,0.76786,0.23623,0.67857,0.85714,1.0,0.0,1.0,1,9,1,1,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,0,10,0,9],[24,40,0.6,0.79018,0.17122,0.71429,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,17,0,5],[28,40,0.7,0.83929,0.21651,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,0,10,0,15],[32,40,0.8,0.80357,0.21354,0.85711,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,2,0,0,17,0,8],[36,40,0.9,0.67857,0.29451,0.39286,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,6,0,0,3,0,0,1,0,0,2,0,0,11,0,7],[40,40,1.0,0.42415,0.28004,0.2857,0.28571,0.46536,0.0,1.0,2,2,0,2,0,4,0,0,11,0,0,7,0,0,1,0,0,0,0,0,5,0,2]]},{"b":5,"e":0.57143,"k":"falling","v":0.52673,"x":0.91072,"p":[[0,81,0.0,0.80356,0.16657,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,7,0,10],[4,81,0.0494,0.90624,0.13654,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,20],[8,81,0.0988,0.91072,0.13716,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,19],[12,81,0.1481,0.91072,0.14616,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,10,0,19],[16,81,0.1975,0.8125,0.20025,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,6,0,0,8,0,12],[20,81,0.2469,0.82589,0.16263,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,0,7,0,12],[24,81,0.2963,0.79017,0.22584,0.67857,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,4,0,0,8,0,12],[28,81,0.3457,0.83927,0.1777,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,0,12,0,11],[32,81,0.3951,0.76335,0.1877,0.57143,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,4,0,0,14,0,5],[36,81,0.4444,0.78576,0.21121,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,5,0,0,13,0,8],[40,81,0.4938,0.74552,0.20434,0.57143,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,5,0,0,11,0,6],[44,81,0.5432,0.80357,0.18814,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,8,0,0,11,0,9],[48,81,0.5926,0.70533,0.26949,0.57132,0.857,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,4,0,0,1,0,0,4,0,0,4,0,0,10,0,7],[52,81,0.642,0.79014,0.20515,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,0,11,0,9],[56,81,0.6914,0.72316,0.17478,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,10,0,0,8,0,4],[60,81,0.7407,0.67844,0.22611,0.571,0.71429,0.85714,0.14,1.0,0,6,0,0,0,2,0,0,0,0,0,4,0,0,7,0,0,10,0,0,3,0,6],[64,81,0.7901,0.70085,0.18683,0.57132,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,10,0,0,10,0,2],[68,81,0.8395,0.59815,0.18013,0.571,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,3,0,0,12,0,0,8,0,0,5,0,0],[72,81,0.8889,0.52673,0.18362,0.42857,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,7,0,0,10,0,0,7,0,0,2,0,0],[76,81,0.9383,0.6205,0.17355,0.571,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,2,0,0,7,0,0,16,0,0,3,0,0],[80,81,0.9877,0.57139,0.15971,0.4286,0.57143,0.71429,0.1429,1.0,0,1,0,0,0,1,0,0,1,0,0,8,0,0,11,0,0,10,0,0,0,0,1],[81,81,1.0,0.60253,0.15856,0.5713,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,5,0,0,11,0,0,11,0,0,3,0,0]]}]},{"i":"716743df3baef2a4","q":"Consider 51 strictly positive integers with a sum of 100 on a line. Show that for any integer $1 \\leqslant k<100$, there exist consecutive integers with a sum of $k$ or $100-k$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.27668,"x":1.0,"p":[[0,25,0.0,0.27668,0.12346,0.14286,0.28571,0.32143,0.0,0.571,1,0,1,1,0,9,0,0,14,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[4,25,0.16,0.47321,0.30813,0.24999,0.42857,0.71429,0.0,1.0,1,5,1,1,0,7,0,0,6,0,0,6,0,0,3,0,0,2,0,0,2,0,5],[8,25,0.32,0.83479,0.22051,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,3,0,0,5,0,17],[12,25,0.48,0.83036,0.20652,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,0,5,0,16],[16,25,0.64,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":2,"e":0.71429,"k":"rising","v":0.24554,"x":0.94643,"p":[[0,41,0.0,0.24554,0.12992,0.14286,0.2857,0.28571,0.0,0.71429,1,0,1,1,0,13,0,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,41,0.0976,0.56696,0.32826,0.28571,0.42857,0.89286,0.0,1.0,2,8,0,2,0,2,0,0,5,0,0,10,0,0,0,0,0,1,0,0,4,0,8],[8,41,0.1951,0.81249,0.31429,0.71429,1.0,1.0,0.0,1.0,2,21,1,2,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,21],[12,41,0.2927,0.81696,0.2402,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,2,0,0,5,0,17],[16,41,0.3902,0.85713,0.24486,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,1,0,0,2,0,22],[20,41,0.4878,0.74553,0.30459,0.42859,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,4,0,0,3,0,0,2,0,0,1,0,0,5,0,15],[24,41,0.5854,0.875,0.20124,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,20],[28,41,0.6829,0.61604,0.27302,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,4,0,0,4,0,0,6,0,0,4,0,0,6,0,5],[32,41,0.7805,0.875,0.20124,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,3,0,21],[36,41,0.878,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[40,41,0.9756,0.84821,0.19865,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,5,0,0,5,0,17],[41,41,1.0,0.84374,0.17629,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,8,0,14]]}]},{"i":"560f86e9ef737496","q":"For any positive integer  $n$ , let  $s(n) = 1 + 2 + \\cdots + n.$  Define a strictly increasing sequence of positive integers  $\\{a_n\\}_{n \\geq 1}$  such that  $a_1 = 1$  and $\\[\na_{n+1} = \\min \\left\\{ m \\mid s(m) - s(a_n) \\text{ is a perfect square} \\right\\}\n\\]$ for all positive integers  $n.$  Find the value of  $a_{2024}.$","t":[{"b":1,"e":0.57143,"k":"falling","v":0.74554,"x":1.0,"p":[[0,100,0.0,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[4,100,0.04,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,100,0.08,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,100,0.12,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,100,0.16,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[20,100,0.2,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,100,0.24,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,100,0.28,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,100,0.32,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[36,100,0.36,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,100,0.4,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[44,100,0.44,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[48,100,0.48,0.95088,0.11076,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[52,100,0.52,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,100,0.56,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[60,100,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,100,0.64,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[68,100,0.68,0.95982,0.1197,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[72,100,0.72,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[76,100,0.76,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[80,100,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[84,100,0.84,0.90179,0.16536,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,4,0,21],[88,100,0.88,0.89732,0.18293,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,1,0,23],[92,100,0.92,0.91071,0.13716,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,7,0,20],[96,100,0.96,0.83034,0.23809,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,4,0,0,2,0,19],[100,100,1.0,0.74554,0.24415,0.57143,0.78571,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,7,0,0,6,0,10]]},{"b":7,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,61,0.0,0.93302,0.1923,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,26],[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],[8,61,0.1311,0.91964,0.24468,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[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],[16,61,0.2623,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,61,0.3279,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[24,61,0.3934,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,61,0.459,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,61,0.5246,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[36,61,0.5902,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[40,61,0.6557,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,61,0.7213,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[48,61,0.7869,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[52,61,0.8525,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[56,61,0.918,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[60,61,0.9836,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[61,61,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]}]},{"i":"52541fe2ba61933a","q":"An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if  $x$  likes  $y$  then  $y$  likes  $x$ ).\nThe race wants to colonize a planet and sends  $n$  males,  $n$  females and  $n$  emales. Every expedition member likes at least  $k$  persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow.\n\na) Prove that if  $n$  is even and  $k\\geq 1/2$  then there might be no married triple.\nb) Prove that if  $k \\geq 3n/4$  then there can be formed  $n$  married triple ( i.e. everybody is in a triple).","t":[{"b":3,"e":0.28571,"k":"flat","v":0.47314,"x":0.67408,"p":[[0,23,0.0,0.56694,0.24611,0.28571,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,11,0,0,2,0,0,6,0,0,6,0,0,4,0,3],[4,23,0.1739,0.59373,0.29904,0.28571,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,12,0,0,1,0,0,3,0,0,4,0,0,4,0,7],[8,23,0.3478,0.58915,0.27369,0.28571,0.64071,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,12,0,0,2,0,0,2,0,0,7,0,0,4,0,5],[12,23,0.5217,0.64732,0.28344,0.39286,0.64286,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,8,0,0,5,0,0,3,0,0,2,0,0,6,0,8],[16,23,0.6957,0.67408,0.28846,0.28571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,9,0,0,2,0,0,2,0,0,4,0,0,6,0,9],[20,23,0.8696,0.53112,0.23197,0.28571,0.4293,0.71107,0.2857,1.0,0,2,0,0,0,0,0,0,10,0,0,7,0,0,6,0,0,2,0,0,5,0,2],[23,23,1.0,0.47314,0.16912,0.28571,0.42859,0.5711,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,10,0,0,10,0,0,1,0,0,1,0,1]]},{"b":6,"e":1.0,"k":"falling","v":0.31696,"x":0.74554,"p":[[0,21,0.0,0.50433,0.28334,0.28571,0.5,0.71429,0.0,1.0,3,4,0,3,0,1,0,0,7,0,0,5,0,0,5,0,0,7,0,0,0,0,4],[4,21,0.1905,0.74554,0.26663,0.53572,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,4,0,0,6,0,12],[8,21,0.381,0.66961,0.27302,0.28571,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,9,0,0,0,0,0,4,0,0,5,0,0,7,0,7],[12,21,0.5714,0.50891,0.28557,0.28571,0.28571,0.74996,0.14286,1.0,0,4,0,0,0,2,0,0,15,0,0,1,0,0,3,0,0,3,0,0,4,0,4],[16,21,0.7619,0.38838,0.15249,0.2857,0.28571,0.4286,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,8,0,0,5,0,0,0,0,0,0,0,1],[20,21,0.9524,0.35266,0.10086,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,21,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[21,21,1.0,0.31696,0.12745,0.2857,0.2857,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"e8558400271fb3e4","q":"A quadrilateral  $ABCD$  without parallel sides is inscribed in a circle  $\\omega$ . We draw a line  $\\ell_a \\parallel BC$  through the point  $A$ , a line  $\\ell_b \\parallel CD$  through the point  $B$ ,  a line  $\\ell_c \\parallel DA$  through the point  $C$ , and a line  $\\ell_d \\parallel AB$  through the point  $D$ . Suppose that the quadrilateral whose successive sides lie on these four straight lines is inscribed in a circle  $\\gamma$  and that  $\\omega$  and  $\\gamma$  intersect in points  $E$  and  $F$ . Show that the lines  $AC, BD$  and  $EF$  intersect in one point.\n*Proposed by A. Kuznetsov*","t":[{"b":2,"e":0.14286,"k":"flat","v":0.07143,"x":0.22322,"p":[[0,31,0.0,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.22322,0.17835,0.14286,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,14,0,0,2,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[8,31,0.2581,0.15179,0.15126,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,14,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.18304,0.1394,0.14286,0.14286,0.14287,0.0,0.4286,5,0,0,5,0,20,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.17857,0.14725,0.14286,0.14286,0.1429,0.0,0.57143,6,0,0,6,0,19,0,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[20,31,0.6452,0.14723,0.14054,0.0,0.14286,0.1786,0.0,0.42857,11,0,0,11,0,13,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.1875,0.16536,0.0,0.14286,0.42857,0.0,0.4286,9,0,0,9,0,13,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.15179,0.13333,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,21,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[31,31,1.0,0.16964,0.14032,0.14286,0.14286,0.14287,0.0,0.4286,7,0,0,7,0,18,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.09366,"x":0.25446,"p":[[0,71,0.0,0.09366,0.09177,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.16071,0.15047,0.0,0.14286,0.17857,0.0,0.42857,10,0,0,10,0,14,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[8,71,0.1127,0.20517,0.15546,0.14286,0.14286,0.32143,0.0,0.571,5,0,0,5,0,17,0,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[12,71,0.169,0.19187,0.14113,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,18,0,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[16,71,0.2254,0.22768,0.16698,0.14286,0.14286,0.42857,0.0,0.4286,7,0,0,7,0,10,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[20,71,0.2817,0.21429,0.18211,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,14,0,0,1,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[24,71,0.338,0.22312,0.15131,0.14286,0.14286,0.42857,0.0,0.42857,4,0,0,4,0,16,0,0,2,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[28,71,0.3944,0.20088,0.16308,0.14286,0.14286,0.32143,0.0,0.571,7,0,0,7,0,14,0,0,3,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[32,71,0.4507,0.20527,0.16345,0.14214,0.14286,0.42857,0.0,0.42857,7,0,0,7,0,14,0,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[36,71,0.507,0.25446,0.12745,0.14286,0.14288,0.42857,0.14286,0.4286,0,0,0,0,0,17,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[40,71,0.5634,0.21875,0.15966,0.14286,0.14286,0.28571,0.0,0.71429,4,0,0,4,0,16,0,0,5,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[44,71,0.6197,0.16518,0.14773,0.14286,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,21,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[48,71,0.6761,0.21429,0.16366,0.14286,0.14286,0.42857,0.0,0.4286,7,0,0,7,0,12,0,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[52,71,0.7324,0.21429,0.15568,0.14286,0.14286,0.42857,0.0,0.4286,6,0,0,6,0,13,0,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[56,71,0.7887,0.22768,0.14223,0.14286,0.14286,0.42857,0.0,0.42857,2,0,0,2,0,19,0,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.23214,0.15047,0.14286,0.14286,0.42857,0.0,0.4286,4,0,0,4,0,14,0,0,4,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.16518,0.13883,0.10714,0.14286,0.1786,0.0,0.4286,8,0,0,8,0,16,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[68,71,0.9577,0.22322,0.14258,0.14286,0.14286,0.42857,0.0,0.42857,3,0,0,3,0,17,0,0,3,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.23661,0.14555,0.14286,0.14286,0.42857,0.0,0.4286,3,0,0,3,0,15,0,0,4,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"df11dc0a366005e4","q":"6. (GDR 3) Show that for any $n \\not \\equiv 0(\\bmod 10)$ there exists a multiple of $n$ not containing the digit 0 in its decimal expansion.","t":[{"b":2,"e":0.57143,"k":"rising","v":0.21873,"x":0.37052,"p":[[0,15,0.0,0.21873,0.27887,0.0,0.14286,0.2857,0.0,1.0,14,1,1,14,0,6,0,0,6,0,0,0,0,0,2,0,0,2,0,0,1,0,1],[4,15,0.2667,0.34374,0.21385,0.25,0.42857,0.42858,0.0,0.85714,6,0,0,6,0,2,0,0,6,0,0,11,0,0,6,0,0,0,0,0,1,0,0],[8,15,0.5333,0.34375,0.22829,0.14286,0.42857,0.4286,0.0,1.0,6,1,1,6,0,3,0,0,5,0,0,11,0,0,6,0,0,0,0,0,0,0,1],[12,15,0.8,0.34821,0.22569,0.24999,0.42857,0.4286,0.0,1.0,6,1,1,6,0,2,0,0,6,0,0,11,0,0,6,0,0,0,0,0,0,0,1],[15,15,1.0,0.37052,0.15508,0.2857,0.42857,0.4286,0.0,0.57143,2,0,0,2,0,3,0,0,7,0,0,14,0,0,6,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.08929,"x":0.4241,"p":[[0,28,0.0,0.18303,0.27717,0.0,0.0,0.32143,0.0,1.0,19,1,2,19,0,3,0,0,2,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[4,28,0.1429,0.34822,0.26711,0.14286,0.42857,0.57143,0.0,1.0,7,1,0,7,0,6,0,0,2,0,0,5,0,0,9,0,0,2,0,0,0,0,1],[8,28,0.2857,0.30358,0.26905,0.10714,0.28571,0.4286,0.0,1.0,8,2,0,8,0,5,0,0,6,0,0,9,0,0,0,0,0,2,0,0,0,0,2],[12,28,0.4286,0.4241,0.25375,0.24999,0.42857,0.57143,0.0,1.0,4,1,0,4,0,4,0,0,2,0,0,10,0,0,6,0,0,4,0,0,1,0,1],[16,28,0.5714,0.37944,0.25655,0.14289,0.42857,0.57143,0.0,1.0,5,1,0,5,0,4,0,0,6,0,0,6,0,0,7,0,0,2,0,0,1,0,1],[20,28,0.7143,0.2098,0.2395,0.0,0.14286,0.4286,0.0,0.71429,14,0,0,14,0,7,0,0,1,0,0,3,0,0,6,0,0,1,0,0,0,0,0],[24,28,0.8571,0.23204,0.20126,0.0,0.2857,0.28571,0.0,0.71429,10,0,0,10,0,4,0,0,11,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[28,28,1.0,0.08929,0.12752,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"de4f5672e8f66887","q":"Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the numbers of divisors of $s n$ and of $s k$ are equal.  (Ukraine)","t":[{"b":0,"e":1.0,"k":"rising","v":0.65624,"x":0.95982,"p":[[0,48,0.0,0.65624,0.14224,0.57143,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,10,0,0,1,0,3],[4,48,0.0833,0.88839,0.15865,0.82143,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,19],[8,48,0.1667,0.90178,0.16917,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,3,0,22],[12,48,0.25,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[16,48,0.3333,0.89732,0.1439,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,3,0,20],[20,48,0.4167,0.90625,0.15407,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,22],[24,48,0.5,0.89286,0.19885,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,21],[28,48,0.5833,0.89285,0.15152,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,3,0,20],[32,48,0.6667,0.90624,0.14111,0.857,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,20],[36,48,0.75,0.94195,0.12811,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[40,48,0.8333,0.875,0.17035,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,3,0,19],[44,48,0.9167,0.94432,0.11383,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,1,0,5,0,24],[48,48,1.0,0.95982,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26]]},{"b":2,"e":0.1429,"k":"flat","v":0.48652,"x":0.94196,"p":[[0,71,0.0,0.62493,0.09946,0.57143,0.57143,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,7,0,0,1,0,1],[4,71,0.0563,0.91517,0.12807,0.85711,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,21],[8,71,0.1127,0.85714,0.15152,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,2,0,16],[12,71,0.169,0.84374,0.20316,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,0,2,0,18],[16,71,0.2254,0.88391,0.17293,0.82132,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,5,0,19],[20,71,0.2817,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[24,71,0.338,0.90625,0.18423,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,23],[28,71,0.3944,0.83479,0.16797,0.71429,0.85707,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,2,0,15],[32,71,0.4507,0.91518,0.18509,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,24],[36,71,0.507,0.70089,0.34323,0.53572,0.71429,1.0,0.0,1.0,2,14,0,2,0,4,0,0,1,0,0,1,0,0,1,0,0,8,0,0,1,0,14],[40,71,0.5634,0.72766,0.31209,0.57132,0.85714,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,4,0,0,0,0,0,5,0,0,4,0,0,0,0,16],[44,71,0.6197,0.79464,0.30501,0.67857,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,2,0,0,0,0,0,3,0,0,3,0,0,2,0,19],[48,71,0.6761,0.84375,0.25344,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,0,3,0,20],[52,71,0.7324,0.67411,0.35035,0.28571,0.71429,1.0,0.0,1.0,1,14,0,1,0,5,0,0,3,0,0,1,0,0,2,0,0,5,0,0,1,0,14],[56,71,0.7887,0.7232,0.31731,0.49968,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,5,0,0,0,0,0,4,0,0,1,0,0,5,0,14],[60,71,0.8451,0.71874,0.29339,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,3,0,0,0,0,0,6,0,0,5,0,0,2,0,13],[64,71,0.9014,0.64286,0.34256,0.42857,0.57143,1.0,0.0,1.0,2,13,0,2,0,4,0,0,0,0,0,5,0,0,6,0,0,2,0,0,0,0,13],[68,71,0.9577,0.80357,0.26666,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,9,0,0,1,0,17],[71,71,1.0,0.48652,0.33864,0.14286,0.5,0.71429,0.0,1.0,2,6,0,2,0,9,0,0,4,0,0,1,0,0,3,0,0,7,0,0,0,0,6]]}]},{"i":"b769e52d1f17ef45","q":"Assume that the polynomial  $\\left(x+1\\right)^n-1$  is divisible by some polynomial  $P\\left(x\\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0$ , whose degree  $k$  is even and whose coefficients  $c_{k-1}$ ,  $c_{k-2}$ , ...,  $c_1$ ,  $c_0$  all are odd integers. Show that  $k+1\\mid n$ .","t":[{"b":0,"e":0.571,"k":"flat","v":0.49996,"x":0.65173,"p":[[0,38,0.0,0.49996,0.17495,0.42857,0.5712,0.57143,0.0,1.0,2,1,2,2,0,0,0,0,1,0,0,10,0,0,17,0,0,1,0,0,0,0,1],[4,38,0.1053,0.558,0.23517,0.42857,0.57121,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,2,0,0,8,0,0,7,0,0,7,0,0,3,0,2],[8,38,0.2105,0.54908,0.21756,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,3,0,0,5,0,0,8,0,0,11,0,0,1,0,1],[12,38,0.3158,0.5781,0.17529,0.42859,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,7,0,1,8,0,0,11,0,0,1,0,1],[16,38,0.4211,0.56249,0.16342,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,8,0,0,8,0,0,10,0,0,2,0,0],[20,38,0.5263,0.5982,0.16917,0.42859,0.64286,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,7,0,0,6,0,0,14,0,0,1,0,1],[24,38,0.6316,0.65173,0.1673,0.571,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,14,0,0,1,0,3],[28,38,0.7368,0.59151,0.15895,0.42859,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,8,0,0,8,1,0,11,0,0,1,0,1],[32,38,0.8421,0.57588,0.13592,0.42857,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,9,0,0,13,0,0,0,0,0],[36,38,0.9474,0.56692,0.15355,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,6,0,0,10,0,0,11,0,0,1,0,0],[38,38,1.0,0.5357,0.16366,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,10,0,0,8,0,0,9,0,0,1,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.52229,"x":0.70085,"p":[[0,22,0.0,0.52229,0.16213,0.42857,0.571,0.57143,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,14,0,0,13,0,0,2,0,0,1,0,1],[4,22,0.1818,0.55802,0.24577,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,1,0,0,6,0,0,8,0,0,6,0,0,4,0,2],[8,22,0.3636,0.53124,0.14826,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,7,0,0,13,0,0,6,0,0,1,0,0],[12,22,0.5455,0.60266,0.17762,0.5354,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,5,0,0,9,0,0,12,0,0,2,0,1],[16,22,0.7273,0.70085,0.1763,0.57143,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,12,0,0,1,0,6],[20,22,0.9091,0.67409,0.13941,0.57143,0.64286,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,11,0,0,2,0,3],[22,22,1.0,0.65179,0.15947,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,7,0,0,1,0,4]]}]},{"i":"46704ce669dda46e","q":"A 0-1 sequence of length  $2^k$  is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number  $s$  for which Bob can always pick  $s$  members of the sequence, and guess all their values correctly.\n\nAlice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the member of the sequence picked by Alice.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.16071,"x":0.39284,"p":[[0,59,0.0,0.16071,0.14174,0.0,0.28571,0.28571,0.0,0.28571,14,0,3,14,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.37053,0.2078,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,4,0,0,15,0,0,6,0,0,2,0,0,1,0,0,3,0,0],[8,59,0.1356,0.38839,0.21792,0.28571,0.28571,0.42858,0.0,0.85714,1,0,0,1,0,3,0,0,15,0,0,7,0,0,1,0,0,1,0,0,4,0,0],[12,59,0.2034,0.35265,0.19877,0.2857,0.28571,0.42858,0.0,0.85714,2,0,0,2,0,2,0,0,19,0,0,2,0,0,4,0,0,1,0,0,2,0,0],[16,59,0.2712,0.37053,0.26693,0.2857,0.28571,0.32143,0.0,1.0,3,1,0,3,0,3,0,0,18,0,0,1,0,0,0,0,0,2,0,0,4,0,1],[20,59,0.339,0.29464,0.14698,0.2857,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,1,0,0,22,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[24,59,0.4068,0.33481,0.21901,0.2857,0.28571,0.32143,0.0,0.85714,3,0,0,3,0,3,0,0,18,0,0,3,0,0,1,0,0,1,0,0,3,0,0],[28,59,0.4746,0.39284,0.19884,0.2857,0.28571,0.42858,0.0,0.85714,1,0,0,1,0,2,0,0,15,0,0,7,0,0,2,0,0,3,0,0,2,0,0],[32,59,0.5424,0.33927,0.15462,0.28571,0.28571,0.28571,0.14286,0.857,0,0,0,0,0,2,0,0,23,0,0,4,0,0,1,0,0,0,0,0,2,0,0],[36,59,0.6102,0.38391,0.18706,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,20,0,0,3,0,0,1,0,0,5,0,0,1,0,0],[40,59,0.678,0.35268,0.17491,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,1,0,0,21,0,0,5,0,0,1,0,0,1,0,0,2,0,0],[44,59,0.7458,0.3616,0.22582,0.2857,0.28571,0.42858,0.0,0.85714,3,0,0,3,0,2,0,0,16,0,0,5,0,0,1,0,0,2,0,0,3,0,0],[48,59,0.8136,0.36161,0.13356,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,22,0,0,6,0,0,1,0,0,3,0,0,0,0,0],[52,59,0.8814,0.36606,0.14697,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,20,0,0,6,0,0,3,0,0,1,0,0,1,0,0],[56,59,0.9492,0.34821,0.22851,0.2857,0.28571,0.42857,0.0,1.0,3,2,0,3,0,1,0,0,19,0,0,5,0,0,1,0,0,0,0,0,1,0,2],[59,59,1.0,0.25893,0.11538,0.2857,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,2,0,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.10696,"x":0.17402,"p":[[0,58,0.0,0.11607,0.14032,0.0,0.0,0.28571,0.0,0.28571,19,0,2,19,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.12946,0.12556,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,7,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,58,0.1379,0.17402,0.17401,0.0,0.14286,0.28571,0.0,0.85714,10,0,0,10,0,10,0,0,10,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[12,58,0.2069,0.125,0.13716,0.0,0.14286,0.14287,0.0,0.4286,14,0,0,14,0,11,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.11143,0.1055,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.12947,0.09689,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,20,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.14268,0.11294,0.0,0.14286,0.17857,0.0,0.42857,9,0,0,9,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.12491,0.09277,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.14277,0.11294,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.15598,0.1093,0.14,0.14286,0.2857,0.0,0.4286,7,0,0,7,0,16,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.1383,0.0977,0.105,0.14286,0.14287,0.0,0.28571,8,0,0,8,0,17,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.10696,0.08741,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.16519,0.11355,0.14286,0.14286,0.17868,0.0,0.57143,5,0,0,5,0,19,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,58,0.8966,0.17393,0.09938,0.14286,0.14286,0.17868,0.0,0.42857,3,0,0,3,0,21,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.14732,0.12103,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,20,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[58,58,1.0,0.1517,0.07937,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,22,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"306660eeebbe61b5","q":"14. C8 (GER) ${ }^{\\mathrm{IMO} 3}$ Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that (i) each contestant solved at most six problems, and (ii) for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.","t":[{"b":2,"e":0.2857,"k":"falling","v":0.23215,"x":0.51785,"p":[[0,86,0.0,0.42409,0.40951,0.0,0.35714,0.85714,0.0,1.0,12,7,12,12,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,0,3,0,7],[4,86,0.0465,0.48658,0.29636,0.2857,0.28571,0.75,0.14286,1.0,0,5,0,0,0,4,0,0,13,0,0,3,0,0,3,0,0,1,0,0,3,0,5],[8,86,0.093,0.40178,0.22428,0.28571,0.28571,0.57143,0.14286,1.0,0,2,0,0,0,6,0,0,11,0,0,6,0,0,5,0,0,2,0,0,0,0,2],[12,86,0.1395,0.37943,0.28704,0.14286,0.28571,0.57111,0.14286,1.0,0,2,0,0,0,14,0,0,7,0,0,1,0,0,3,0,0,2,0,0,3,0,2],[16,86,0.186,0.44195,0.27283,0.24999,0.35714,0.60714,0.14286,1.0,0,1,0,0,0,8,0,0,8,0,0,5,0,0,3,0,0,1,0,0,6,0,1],[20,86,0.2326,0.36161,0.26722,0.14286,0.2857,0.60714,0.14286,1.0,0,1,0,0,0,14,0,0,8,0,0,1,0,0,1,0,0,5,0,0,2,0,1],[24,86,0.2791,0.3125,0.23265,0.14286,0.28571,0.32143,0.0,1.0,1,1,0,1,0,13,0,0,10,0,0,2,0,0,2,0,0,2,0,0,1,0,1],[28,86,0.3256,0.46418,0.32935,0.14286,0.28571,0.71429,0.0,1.0,1,6,0,1,0,9,0,0,7,0,0,2,0,0,3,0,0,3,0,0,1,0,6],[32,86,0.3721,0.43749,0.30916,0.14286,0.28571,0.71429,0.14286,1.0,0,4,0,0,0,10,0,0,9,0,0,2,0,0,2,0,0,2,0,0,3,0,4],[36,86,0.4186,0.50892,0.333,0.24999,0.42857,0.85714,0.0,1.0,1,6,0,1,0,7,0,0,7,0,0,2,0,0,4,0,0,1,0,0,4,0,6],[40,86,0.4651,0.47321,0.30813,0.2857,0.28571,0.71429,0.14286,1.0,0,6,0,0,0,6,0,0,12,0,0,3,0,0,0,0,0,5,0,0,0,0,6],[44,86,0.5116,0.3615,0.27201,0.14286,0.28571,0.4286,0.14,1.0,0,3,0,0,0,11,0,0,12,0,0,3,0,0,1,0,0,0,0,0,2,0,3],[48,86,0.5581,0.45079,0.3237,0.14286,0.28571,0.71429,0.14,1.0,0,6,0,0,0,10,0,0,9,0,0,1,0,0,3,0,0,2,0,0,1,0,6],[52,86,0.6047,0.49107,0.32525,0.14286,0.4286,0.71429,0.0,1.0,1,6,0,1,0,8,0,0,6,0,0,2,0,0,4,0,0,4,0,0,1,0,6],[56,86,0.6512,0.51785,0.31288,0.28571,0.42859,0.74996,0.14286,1.0,0,6,0,0,0,6,0,0,9,0,0,2,0,0,3,0,0,4,0,0,2,0,6],[60,86,0.6977,0.48202,0.323,0.14286,0.4998,0.71429,0.14,1.0,0,6,0,0,0,11,0,0,4,0,0,1,0,0,7,0,0,2,0,0,1,0,6],[64,86,0.7442,0.42409,0.27544,0.24999,0.28571,0.57143,0.0,1.0,1,3,0,1,0,7,0,0,9,0,0,4,0,0,4,0,0,3,0,0,1,0,3],[68,86,0.7907,0.33926,0.23889,0.14286,0.28571,0.4286,0.0,1.0,1,2,0,1,0,11,0,0,9,0,0,4,0,0,4,0,0,1,0,0,0,0,2],[72,86,0.8372,0.38838,0.27946,0.14286,0.28571,0.57143,0.14286,1.0,0,3,0,0,0,13,0,0,6,0,0,3,0,0,3,0,0,4,0,0,0,0,3],[76,86,0.8837,0.38829,0.26789,0.14286,0.28571,0.57143,0.0,1.0,1,3,0,1,0,11,0,0,5,0,0,3,0,0,9,0,0,0,0,0,0,0,3],[80,86,0.9302,0.31696,0.20434,0.14286,0.2857,0.42857,0.14286,1.0,0,1,0,0,0,14,0,0,6,0,0,7,0,0,3,0,0,1,0,0,0,0,1],[84,86,0.9767,0.26784,0.16266,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,12,0,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[86,86,1.0,0.23215,0.11152,0.14286,0.14286,0.28571,0.14286,0.4286,0,0,0,0,0,18,0,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"falling","v":0.26338,"x":0.52229,"p":[[0,90,0.0,0.49552,0.40086,0.10714,0.42857,1.0,0.0,1.0,8,10,8,8,0,2,0,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,10],[4,90,0.0444,0.51337,0.30692,0.28571,0.42857,0.74996,0.14286,1.0,0,6,0,0,0,6,0,0,8,0,0,3,0,0,5,0,0,2,0,0,2,0,6],[8,90,0.0889,0.40176,0.25612,0.14289,0.28571,0.57143,0.0,1.0,1,2,0,1,0,8,0,0,8,0,0,4,0,0,6,0,0,2,0,0,1,0,2],[12,90,0.1333,0.49107,0.29001,0.28571,0.42857,0.71429,0.14286,1.0,0,6,0,0,0,4,0,0,11,0,0,5,0,0,3,0,0,3,0,0,0,0,6],[16,90,0.1778,0.3615,0.19236,0.24999,0.28571,0.57111,0.14,0.71429,0,0,0,0,0,8,0,0,12,0,0,3,0,0,5,0,0,4,0,0,0,0,0],[20,90,0.2222,0.3839,0.22426,0.24999,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,8,0,0,12,0,0,0,0,0,9,0,0,1,0,0,1,0,1],[24,90,0.2667,0.36604,0.17471,0.2857,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,10,0,0,11,0,0,4,0,0,0,0,0,0,0,1],[28,90,0.3111,0.3482,0.24467,0.14286,0.2857,0.57143,0.14286,1.0,0,1,0,0,0,15,0,0,5,0,0,2,0,0,6,0,0,2,0,0,1,0,1],[32,90,0.3556,0.4107,0.2335,0.25,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,8,0,0,7,0,0,6,0,0,7,0,0,2,0,0,0,0,2],[36,90,0.4,0.46426,0.2988,0.2857,0.42857,0.57143,0.0,1.0,1,6,0,1,0,5,0,0,9,0,0,6,0,0,4,0,0,1,0,0,0,0,6],[40,90,0.4444,0.39722,0.25446,0.14289,0.42857,0.57111,0.0,1.0,1,2,0,1,0,9,0,0,5,0,0,8,0,0,4,0,0,2,0,0,1,0,2],[44,90,0.4889,0.45088,0.31157,0.14289,0.28571,0.60714,0.14286,1.0,0,6,0,0,0,9,0,0,9,0,0,2,0,0,4,0,0,2,0,0,0,0,6],[48,90,0.5333,0.42399,0.25132,0.14289,0.42857,0.57143,0.14,1.0,0,3,0,0,0,9,0,0,5,0,0,5,0,0,10,0,0,0,0,0,0,0,3],[52,90,0.5778,0.48658,0.29636,0.28571,0.35714,0.71429,0.14286,1.0,0,6,0,0,0,5,0,0,11,0,0,3,0,0,4,0,0,3,0,0,0,0,6],[56,90,0.6222,0.50444,0.26482,0.28571,0.4998,0.60714,0.14286,1.0,0,4,0,0,0,5,0,0,6,0,0,5,0,0,8,0,0,3,0,0,1,0,4],[60,90,0.6667,0.3705,0.23917,0.14286,0.28571,0.4642,0.14286,1.0,0,1,0,0,0,10,0,0,10,0,0,4,0,0,3,0,0,2,0,0,2,0,1],[64,90,0.7111,0.46874,0.33356,0.14286,0.42859,0.71429,0.0,1.0,1,6,0,1,0,11,0,0,3,0,0,3,0,0,4,0,0,3,0,0,1,0,6],[68,90,0.7556,0.52229,0.30222,0.2857,0.571,0.71429,0.0,1.0,1,6,0,1,0,6,0,0,3,0,0,5,0,0,8,0,0,2,0,0,1,0,6],[72,90,0.8,0.36161,0.23415,0.14286,0.28571,0.42858,0.14286,1.0,0,2,0,0,0,10,0,0,9,0,0,7,0,0,3,0,0,0,0,0,1,0,2],[76,90,0.8444,0.47767,0.29366,0.2857,0.35714,0.71429,0.14286,1.0,0,5,0,0,0,6,0,0,10,0,0,3,0,0,4,0,0,3,0,0,1,0,5],[80,90,0.8889,0.33473,0.19112,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,8,0,0,12,0,0,5,0,0,2,0,0,4,0,0,0,0,0],[84,90,0.9333,0.34808,0.20809,0.14289,0.28571,0.46418,0.0,0.857,2,0,0,2,0,7,0,0,10,0,0,5,0,0,5,0,0,2,0,0,1,0,0],[88,90,0.9778,0.39286,0.22868,0.14286,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,9,0,0,6,0,0,8,0,0,6,0,0,1,0,0,0,0,2],[90,90,1.0,0.26338,0.13412,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,12,0,0,17,0,0,0,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"c3d70aed81acfe5e","q":"Assume that in the  $\\triangle ABC$  there exists a point  $D$  on  $BC$  and a line  $l$  passing through  $A$  such that  $l$  is tangent to  $(ADC)$  and  $l$  bisects  $BD.$ Prove that  $a\\sqrt{2}\\geq b+c.$","t":[{"b":4,"e":1.0,"k":"rising","v":0.48213,"x":0.97768,"p":[[0,76,0.0,0.51785,0.26665,0.28571,0.57143,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,12,0,0,0,0,0,7,0,0,7,0,0,0,0,4],[4,76,0.0526,0.54909,0.29904,0.28571,0.57143,0.71429,0.0,1.0,1,7,0,1,0,1,0,0,12,0,0,0,0,0,6,0,0,5,0,0,0,0,7],[8,76,0.1053,0.67411,0.32583,0.28571,0.71429,1.0,0.0,1.0,1,14,0,1,0,1,0,0,7,0,0,1,0,0,5,0,0,3,0,0,0,0,14],[12,76,0.1579,0.48213,0.31083,0.28571,0.57143,0.71429,0.0,1.0,6,4,0,6,0,0,0,0,6,0,0,2,0,0,8,0,0,6,0,0,0,0,4],[16,76,0.2105,0.58927,0.28739,0.28571,0.57143,0.75,0.0,1.0,2,7,0,2,0,0,0,0,7,0,0,0,0,0,12,0,0,3,0,0,1,0,7],[20,76,0.2632,0.79464,0.25489,0.71429,0.9285,1.0,0.0,1.0,1,16,0,1,0,0,0,0,2,0,0,0,0,0,4,0,0,8,0,0,1,0,16],[24,76,0.3158,0.79017,0.21719,0.71429,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,10,0,0,1,0,14],[28,76,0.3684,0.76784,0.27142,0.57143,0.78571,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,1,0,15],[32,76,0.4211,0.86607,0.23128,0.71429,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,0,0,23],[36,76,0.4737,0.85268,0.19719,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,0,0,20],[40,76,0.5263,0.88839,0.21646,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,2,0,23],[44,76,0.5789,0.8883,0.24702,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,25],[48,76,0.6316,0.83481,0.25782,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,2,0,20],[52,76,0.6842,0.89731,0.22656,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,26],[56,76,0.7368,0.67854,0.36071,0.39286,0.78571,1.0,0.0,1.0,4,15,0,4,0,0,0,0,4,0,0,1,0,0,5,0,0,2,0,0,1,0,15],[60,76,0.7895,0.80798,0.24648,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,6,0,0,6,0,0,1,0,17],[64,76,0.8421,0.84372,0.25845,0.67857,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,0,0,22],[68,76,0.8947,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[72,76,0.9474,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[76,76,1.0,0.88393,0.22711,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,3,0,23]]},{"b":5,"e":1.0,"k":"rising","v":0.39286,"x":1.0,"p":[[0,120,0.0,0.39286,0.25505,0.2857,0.28571,0.57143,0.0,1.0,4,2,0,4,0,1,0,0,14,0,0,2,0,0,6,0,0,3,0,0,0,0,2],[4,120,0.0333,0.68299,0.35489,0.28571,0.85714,1.0,0.0,1.0,3,16,0,3,0,0,0,0,6,0,0,0,0,0,6,0,0,1,0,0,0,0,16],[8,120,0.0667,0.70088,0.32803,0.49968,0.71429,1.0,0.0,1.0,2,15,0,2,0,0,0,0,6,0,0,0,0,0,5,0,0,4,0,0,0,0,15],[12,120,0.1,0.70982,0.29984,0.57143,0.71429,1.0,0.0,1.0,1,14,0,1,0,0,0,0,6,0,0,0,0,0,6,0,0,5,0,0,0,0,14],[16,120,0.1333,0.57589,0.35081,0.28571,0.57143,1.0,0.0,1.0,3,11,0,3,0,0,0,0,11,0,0,0,0,0,5,0,0,2,0,0,0,0,11],[20,120,0.1667,0.78571,0.28572,0.57142,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,4,0,0,0,0,19],[24,120,0.2,0.67411,0.31183,0.5,0.64286,1.0,0.0,1.0,1,13,0,1,0,1,0,0,6,0,0,0,0,0,8,0,0,3,0,0,0,0,13],[28,120,0.2333,0.66963,0.3223,0.28571,0.64286,1.0,0.0,1.0,1,14,0,1,0,0,0,0,8,0,0,2,0,0,5,0,0,2,0,0,0,0,14],[32,120,0.2667,0.75446,0.29501,0.53571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,7,0,0,1,0,0,3,0,0,3,0,0,1,0,17],[36,120,0.3,0.75444,0.28176,0.571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,5,0,0,2,0,0,6,0,0,2,0,0,0,0,17],[40,120,0.3333,0.5982,0.27994,0.28571,0.57143,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,12,0,0,0,0,0,6,0,0,5,0,0,2,0,7],[44,120,0.3667,0.52679,0.33776,0.28571,0.42859,1.0,0.0,1.0,3,9,0,3,0,1,0,0,10,0,0,3,0,0,5,0,0,1,0,0,0,0,9],[48,120,0.4,0.61159,0.27254,0.28571,0.57143,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,10,0,0,1,0,0,7,0,0,6,0,0,0,0,8],[52,120,0.4333,0.61604,0.3102,0.28571,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,13,0,0,0,0,0,5,0,0,3,0,0,0,0,11],[56,120,0.4667,0.65178,0.33299,0.28593,0.57143,1.0,0.0,1.0,2,13,0,2,0,0,0,0,7,0,0,2,0,0,6,0,0,1,0,0,1,0,13],[60,120,0.5,0.64729,0.27661,0.39286,0.64286,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,7,0,0,1,0,0,7,0,0,6,0,0,1,0,9],[64,120,0.5333,0.86161,0.23003,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,5,0,0,0,0,22],[68,120,0.5667,0.87947,0.18249,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,1,0,21],[72,120,0.6,0.83929,0.22232,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,1,0,18],[76,120,0.6333,0.79911,0.2854,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,2,0,0,3,0,0,4,0,0,2,0,0,0,0,20],[80,120,0.6667,0.87499,0.17407,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,9,0,0,2,0,19],[84,120,0.7,0.8214,0.23693,0.67857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,4,0,0,2,0,18],[88,120,0.7333,0.73214,0.31083,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,4,0,0,1,0,0,6,0,0,2,0,0,1,0,16],[92,120,0.7667,0.81249,0.22143,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,5,0,0,1,0,17],[96,120,0.8,0.83036,0.21852,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,6,0,0,3,0,17],[100,120,0.8333,0.76339,0.3001,0.57143,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,2,0,0,1,0,0,5,0,0,5,0,0,0,0,17],[104,120,0.8667,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[108,120,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,120,0.9333,0.95088,0.18769,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[116,120,0.9667,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[120,120,1.0,0.96875,0.13236,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30]]}]},{"i":"272b42e61d1bcb6e","q":"Find all solutions  $ (x,y)\\in \\mathbb{R}\\times\\mathbb R$  of the following system:  $ \\begin{cases}x^3 \\plus{} 3xy^2 \\equal{} 49, \r\nx^2 \\plus{} 8xy \\plus{} y^2 \\equal{} 8y \\plus{} 17x.\\end{cases}$","t":[{"b":4,"e":0.14286,"k":"rising","v":0.19643,"x":0.91071,"p":[[0,169,0.0,0.26786,0.31693,0.14286,0.14286,0.14286,0.0,1.0,2,5,0,2,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[4,169,0.0237,0.27232,0.31412,0.14286,0.14286,0.14286,0.0,1.0,1,5,0,1,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,169,0.0473,0.30357,0.33455,0.14286,0.14286,0.14286,0.14286,1.0,0,6,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[12,169,0.071,0.19643,0.20748,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,169,0.0947,0.29464,0.31931,0.14286,0.14286,0.14286,0.14286,1.0,0,5,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[20,169,0.1183,0.32589,0.35756,0.14286,0.14286,0.14286,0.0,1.0,1,7,0,1,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[24,169,0.142,0.75446,0.38338,0.14286,1.0,1.0,0.14286,1.0,0,22,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,22],[28,169,0.1657,0.73214,0.3973,0.14286,1.0,1.0,0.14286,1.0,0,22,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22],[32,169,0.1893,0.60713,0.40406,0.14286,0.85714,1.0,0.0,1.0,1,13,0,1,0,12,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,13],[36,169,0.213,0.76786,0.36553,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,21],[40,169,0.2367,0.79911,0.3251,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,19],[44,169,0.2604,0.70089,0.38359,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,17],[48,169,0.284,0.66071,0.41304,0.14286,1.0,1.0,0.0,1.0,1,17,0,1,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,17],[52,169,0.3077,0.67411,0.39967,0.14286,0.92857,1.0,0.0,1.0,1,16,0,1,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,16],[56,169,0.3314,0.66964,0.40945,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,18],[60,169,0.355,0.73661,0.37646,0.14286,1.0,1.0,0.14286,1.0,0,19,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,19],[64,169,0.3787,0.76339,0.36177,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,19],[68,169,0.4024,0.64732,0.418,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,18],[72,169,0.426,0.66518,0.40816,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,18],[76,169,0.4497,0.61607,0.42021,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,17],[80,169,0.4734,0.79464,0.34799,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,21],[84,169,0.497,0.6875,0.40632,0.14286,1.0,1.0,0.0,1.0,1,18,0,1,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,18],[88,169,0.5207,0.70536,0.40711,0.14286,1.0,1.0,0.14286,1.0,0,21,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[92,169,0.5444,0.58482,0.41705,0.14286,0.85714,1.0,0.14286,1.0,0,14,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,14],[96,169,0.568,0.66964,0.40945,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,18],[100,169,0.5917,0.65625,0.41166,0.14286,1.0,1.0,0.0,1.0,1,17,0,1,0,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,17],[104,169,0.6154,0.60268,0.41914,0.14286,0.85714,1.0,0.0,1.0,1,15,0,1,0,13,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,15],[108,169,0.6391,0.70982,0.38711,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,18],[112,169,0.6627,0.61607,0.41869,0.14286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,16],[116,169,0.6864,0.6875,0.39679,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,17],[120,169,0.7101,0.63839,0.40245,0.14286,0.85714,1.0,0.0,1.0,1,15,0,1,0,11,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,15],[124,169,0.7337,0.65179,0.40867,0.14286,0.92857,1.0,0.0,1.0,1,16,0,1,0,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,16],[128,169,0.7574,0.66964,0.40945,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,18],[132,169,0.7811,0.57589,0.41876,0.14286,0.85714,1.0,0.0,1.0,1,13,0,1,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,13],[136,169,0.8047,0.54464,0.42773,0.14286,0.14286,1.0,0.14286,1.0,0,15,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[140,169,0.8284,0.47767,0.40658,0.14286,0.14286,1.0,0.14286,1.0,0,10,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,10],[144,169,0.8521,0.79911,0.34967,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,22],[148,169,0.8757,0.64732,0.40561,0.14286,0.85714,1.0,0.0,1.0,1,15,0,1,0,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,15],[152,169,0.8994,0.70982,0.39687,0.14286,1.0,1.0,0.0,1.0,1,19,0,1,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,19],[156,169,0.9231,0.68304,0.41301,0.14286,1.0,1.0,0.0,1.0,2,18,0,2,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,18],[160,169,0.9467,0.76786,0.36378,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,20],[164,169,0.9704,0.7366,0.37476,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,18],[168,169,0.9941,0.91071,0.21053,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,23],[169,169,1.0,0.91071,0.24936,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,27]]},{"b":6,"e":0.14286,"k":"falling","v":0.12946,"x":0.40625,"p":[[0,166,0.0,0.37946,0.38896,0.14286,0.14286,1.0,0.0,1.0,1,9,1,1,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[4,166,0.0241,0.32589,0.35756,0.14286,0.14286,0.14287,0.0,1.0,1,7,0,1,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[8,166,0.0482,0.29911,0.32608,0.14286,0.14286,0.14286,0.14286,1.0,0,5,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[12,166,0.0723,0.25,0.28347,0.14286,0.14286,0.14286,0.14286,1.0,0,4,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[16,166,0.0964,0.30357,0.33455,0.14286,0.14286,0.14286,0.14286,1.0,0,6,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[20,166,0.1205,0.29911,0.32608,0.14286,0.14286,0.14286,0.14286,1.0,0,5,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[24,166,0.1446,0.40625,0.3914,0.14286,0.14286,1.0,0.14286,1.0,0,9,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,9],[28,166,0.1687,0.32143,0.34993,0.14286,0.14286,0.14287,0.0,1.0,1,6,0,1,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[32,166,0.1928,0.26786,0.29179,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[36,166,0.2169,0.24107,0.28891,0.14286,0.14286,0.14286,0.0,1.0,2,4,0,2,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[40,166,0.241,0.24554,0.28624,0.14286,0.14286,0.14286,0.0,1.0,1,4,0,1,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[44,166,0.2651,0.34821,0.36759,0.14286,0.14286,0.32143,0.0,1.0,1,7,0,1,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7],[48,166,0.2892,0.27223,0.31416,0.14286,0.14286,0.14286,0.0,1.0,1,5,0,1,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[52,166,0.3133,0.24107,0.27534,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[56,166,0.3373,0.24107,0.2889,0.14286,0.14286,0.14286,0.0,1.0,2,4,0,2,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[60,166,0.3614,0.21429,0.25505,0.14286,0.14286,0.14286,0.0,1.0,2,3,0,2,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[64,166,0.3855,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,166,0.4096,0.21429,0.22588,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[72,166,0.4337,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[76,166,0.4578,0.24107,0.28891,0.14286,0.14286,0.14286,0.0,1.0,2,4,0,2,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[80,166,0.4819,0.24554,0.28624,0.14286,0.14286,0.14286,0.0,1.0,1,4,0,1,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[84,166,0.506,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,166,0.5301,0.19187,0.21013,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[92,166,0.5542,0.19188,0.21013,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[96,166,0.5783,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[100,166,0.6024,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,166,0.6265,0.16955,0.14915,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[108,166,0.6506,0.16071,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[112,166,0.6747,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[116,166,0.6988,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,166,0.7229,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,166,0.747,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[128,166,0.7711,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,166,0.7952,0.14286,1e-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],[136,166,0.8193,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,166,0.8434,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,166,0.8675,0.14286,1e-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],[148,166,0.8916,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[152,166,0.9157,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[156,166,0.9398,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[160,166,0.9639,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[164,166,0.988,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[166,166,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"aabd11a7a9917f08","q":"Find all nonnegative integer solutions to  $2^a + 3^b + 5^c = n!$ .\n\n*Proposed by Mark Sellke*","t":[{"b":4,"e":0.857,"k":"flat","v":0.82589,"x":0.98214,"p":[[0,140,0.0,0.82589,0.26663,0.82132,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,0,6,0,18],[4,140,0.0286,0.86606,0.18877,0.82132,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,6,0,18],[8,140,0.0571,0.87945,0.17538,0.82132,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,5,0,19],[12,140,0.0857,0.95536,0.11538,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[16,140,0.1143,0.90624,0.15818,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,22],[20,140,0.1429,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[24,140,0.1714,0.8616,0.19719,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,5,0,18],[28,140,0.2,0.91518,0.14222,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,21],[32,140,0.2286,0.92854,0.14732,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,24],[36,140,0.2571,0.96415,0.08029,1.0,1.0,1.0,0.71,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[40,140,0.2857,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[44,140,0.3143,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[48,140,0.3429,0.91071,0.12753,0.85711,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,20],[52,140,0.3714,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[56,140,0.4,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[60,140,0.4286,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[64,140,0.4571,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[68,140,0.4857,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[72,140,0.5143,0.95981,0.0892,1.0,1.0,1.0,0.714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[76,140,0.5429,0.94643,0.1171,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[80,140,0.5714,0.89286,0.16752,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,4,0,20],[84,140,0.6,0.93737,0.09439,0.85714,1.0,1.0,0.71,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[88,140,0.6286,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[92,140,0.6571,0.92857,0.15152,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[96,140,0.6857,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[100,140,0.7143,0.93749,0.15127,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[104,140,0.7429,0.92857,0.14726,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,23],[108,140,0.7714,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[112,140,0.8,0.90624,0.16602,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,7,0,21],[116,140,0.8286,0.90625,0.16213,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,10,0,19],[120,140,0.8571,0.83481,0.22335,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,0,6,0,16],[124,140,0.8857,0.87945,0.16016,0.857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,16],[128,140,0.9143,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[132,140,0.9429,0.92409,0.12873,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[136,140,0.9714,0.94643,0.15047,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[140,140,1.0,0.91964,0.16728,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,24]]},{"b":5,"e":0.28571,"k":"falling","v":0.3482,"x":0.9107,"p":[[0,154,0.0,0.77678,0.27418,0.57143,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,6,0,0,0,0,0,3,0,0,4,0,0,3,0,16],[4,154,0.026,0.70534,0.2765,0.57143,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,2,0,0,1,0,0,7,0,0,6,0,0,5,0,9],[8,154,0.0519,0.78118,0.22868,0.57132,0.85707,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,1,0,0,8,0,12],[12,154,0.0779,0.83926,0.24422,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,19],[16,154,0.1039,0.79463,0.24208,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,5,0,0,4,0,15],[20,154,0.1299,0.8125,0.25111,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,6,0,0,3,0,17],[24,154,0.1558,0.7098,0.32436,0.57132,0.85707,1.0,0.0,1.0,2,14,0,2,0,2,0,0,1,0,0,2,0,0,7,0,0,1,0,0,3,0,14],[28,154,0.1818,0.75444,0.31992,0.57142,0.85714,1.0,0.0,1.0,3,15,0,3,0,0,0,0,2,0,0,0,0,0,5,0,0,2,0,0,5,0,15],[32,154,0.2078,0.89732,0.1996,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,3,0,23],[36,154,0.2338,0.82143,0.24484,0.71429,1.0,1.0,0.1429,1.0,0,17,0,0,0,1,0,0,2,0,0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and Bob choose numbers from  $0,1,2,\\cdots,81$  in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all  $82$  numbers are chosen, let  $A$  be the sum of all the numbers Amy chooses, and let  $B$  be the sum of all the numbers Bob chooses. During the process, Amy tries to make  $\\gcd(A,B)$  as great as possible, and Bob tries to make  $\\gcd(A,B)$  as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine  $\\gcd(A,B)$  when all  $82$  numbers are chosen.","t":[{"b":1,"e":1.0,"k":"rising","v":0.16964,"x":0.64286,"p":[[0,80,0.0,0.19643,0.18814,0.14286,0.14286,0.1786,0.0,0.71429,6,0,6,6,0,18,0,0,4,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[4,80,0.05,0.63828,0.32352,0.46396,0.71429,1.0,0.14,1.0,0,9,0,0,0,8,0,0,0,0,0,0,0,0,6,0,0,6,0,0,3,0,9],[8,80,0.1,0.42409,0.2823,0.14286,0.35714,0.57143,0.14286,1.0,0,2,0,0,0,13,0,0,3,0,0,1,0,0,8,0,0,3,0,0,2,0,2],[12,80,0.15,0.45079,0.28605,0.14289,0.42857,0.57143,0.14,1.0,0,4,0,0,0,9,0,0,6,0,0,4,0,0,6,0,0,2,0,0,1,0,4],[16,80,0.2,0.38383,0.27074,0.14286,0.28571,0.57111,0.14,1.0,0,2,0,0,0,12,0,0,7,0,0,4,0,0,4,0,0,0,0,0,3,0,2],[20,80,0.25,0.53125,0.2993,0.2857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,7,0,0,6,0,0,1,0,0,4,0,0,7,0,0,3,0,4],[24,80,0.3,0.50892,0.31122,0.14286,0.49979,0.75,0.14286,1.0,0,4,0,0,0,10,0,0,2,0,0,4,0,0,4,0,0,4,0,0,4,0,4],[28,80,0.35,0.41518,0.29528,0.14286,0.28571,0.60714,0.14286,1.0,0,3,0,0,0,12,0,0,7,0,0,1,0,0,4,0,0,3,0,0,2,0,3],[32,80,0.4,0.48213,0.30252,0.14286,0.57121,0.71429,0.14286,1.0,0,3,0,0,0,12,0,0,1,0,0,2,0,0,5,0,0,7,0,0,2,0,3],[36,80,0.45,0.33929,0.28065,0.14286,0.14286,0.57143,0.0,1.0,1,2,0,1,0,18,0,0,2,0,0,0,0,0,5,0,0,4,0,0,0,0,2],[40,80,0.5,0.25892,0.22709,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,22,0,0,2,0,0,1,0,0,3,0,0,2,0,0,0,0,1],[44,80,0.55,0.24999,0.21427,0.14286,0.14286,0.21429,0.0,1.0,1,1,0,1,0,23,0,0,0,0,0,2,0,0,5,0,0,0,0,0,0,0,1],[48,80,0.6,0.24107,0.24074,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,27,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,2],[52,80,0.65,0.16964,0.09062,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,29,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,80,0.7,0.20527,0.14702,0.14286,0.14286,0.14287,0.14,0.71429,0,0,0,0,0,26,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[60,80,0.75,0.18265,0.12495,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,29,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[64,80,0.8,0.19196,0.12682,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,27,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[68,80,0.85,0.29456,0.26954,0.14286,0.14286,0.35713,0.14,1.0,0,2,0,0,0,23,0,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,2],[72,80,0.9,0.31249,0.23806,0.14286,0.14286,0.57111,0.14286,0.85714,0,0,0,0,0,20,0,0,1,0,0,2,0,0,4,0,0,4,0,0,1,0,0],[76,80,0.95,0.42409,0.26118,0.14286,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,13,0,0,1,0,0,2,0,0,9,0,0,5,0,0,1,0,1],[80,80,1.0,0.64286,0.21724,0.57143,0.64286,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,2,0,0,10,0,0,10,0,0,1,0,5]]},{"b":2,"e":1.0,"k":"rising","v":0.14732,"x":0.93749,"p":[[0,87,0.0,0.14732,0.14054,0.0,0.14286,0.2857,0.0,0.57143,11,0,11,11,0,12,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,87,0.046,0.73213,0.2442,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,3,0,0,6,0,0,7,0,0,4,0,10],[8,87,0.092,0.58473,0.35069,0.14286,0.64286,0.89286,0.14,1.0,0,8,0,0,0,10,0,0,2,0,0,0,0,0,4,0,0,3,0,0,5,0,8],[12,87,0.1379,0.52232,0.33619,0.14286,0.5,0.85714,0.14286,1.0,0,6,0,0,0,10,0,0,4,0,0,2,0,0,3,0,0,3,0,0,4,0,6],[16,87,0.1839,0.51339,0.32509,0.14286,0.57143,0.74996,0.14286,1.0,0,5,0,0,0,11,0,0,2,0,0,2,0,0,4,0,0,5,0,0,3,0,5],[20,87,0.2299,0.59374,0.29475,0.39286,0.57143,0.85714,0.0,1.0,1,5,0,1,0,5,0,0,2,0,0,1,0,0,8,0,0,6,0,0,4,0,5],[24,87,0.2759,0.60713,0.30094,0.28571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,6,0,0,3,0,0,1,0,0,5,0,0,7,0,0,4,0,6],[28,87,0.3218,0.8214,0.18561,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,0,6,0,13],[32,87,0.3678,0.75893,0.18013,0.71429,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,15,0,0,5,0,7],[36,87,0.4138,0.73213,0.20439,0.67857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,13,0,0,4,0,7],[40,87,0.4598,0.82141,0.15974,0.71429,0.85707,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,5,0,12],[44,87,0.5057,0.84821,0.15542,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,6,0,14],[48,87,0.5517,0.84375,0.20935,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,5,0,0,5,0,17],[52,87,0.5977,0.75892,0.22429,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,8,0,0,2,0,12],[56,87,0.6437,0.82143,0.18898,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,9,0,0,7,0,12],[60,87,0.6897,0.83036,0.15335,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,6,0,12],[64,87,0.7356,0.87946,0.14772,0.82132,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,8,0,16],[68,87,0.7816,0.80357,0.18472,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,0,7,0,11],[72,87,0.8276,0.78125,0.19556,0.67857,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,9,0,0,4,0,11],[76,87,0.8736,0.80578,0.23163,0.71429,0.85714,1.0,0.07143,1.0,0,13,0,0,1,1,0,0,0,0,0,0,0,0,3,0,0,8,0,0,6,0,13],[80,87,0.9195,0.91517,0.11769,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,19],[84,87,0.9655,0.91518,0.14223,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,22],[87,87,1.0,0.93749,0.10682,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22]]}]},{"i":"fad329ce12727427","q":"Given are two circles  $\\omega_1,\\omega_2$  which intersect at points  $X,Y$ . Let  $P$  be an arbitrary point on  $\\omega_1$ . Suppose that the lines  $PX,PY$  meet  $\\omega_2$  again at points  $A,B$  respectively. Prove that the circumcircles of all triangles  $PAB$  have the same radius.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.27679,"x":0.77678,"p":[[0,60,0.0,0.77678,0.30916,0.857,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,12,0,13],[4,60,0.0667,0.56695,0.38046,0.14286,0.57141,1.0,0.0,1.0,1,12,0,1,0,10,0,0,2,0,0,1,0,0,5,0,0,0,0,0,1,0,12],[8,60,0.1333,0.56248,0.333,0.2857,0.49979,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,5,0,0,5,0,0,4,0,0,0,0,0,4,0,8],[12,60,0.2,0.53122,0.39484,0.14286,0.57121,1.0,0.0,1.0,4,10,0,4,0,8,0,0,2,0,0,1,0,0,4,0,0,0,0,0,3,0,10],[16,60,0.2667,0.55357,0.39407,0.14286,0.4286,1.0,0.0,1.0,2,12,0,2,0,9,0,0,3,0,0,3,0,0,1,0,0,0,0,0,2,0,12],[20,60,0.3333,0.5357,0.38959,0.14286,0.49979,1.0,0.0,1.0,1,11,0,1,0,12,0,0,2,0,0,1,0,0,3,0,0,0,0,0,2,0,11],[24,60,0.4,0.38383,0.32037,0.14286,0.2857,0.4642,0.0,1.0,1,5,0,1,0,14,0,0,5,0,0,4,0,0,1,0,0,1,0,0,1,0,5],[28,60,0.4667,0.77677,0.3153,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,2,0,0,1,0,0,5,0,0,0,0,0,2,0,19],[32,60,0.5333,0.7098,0.33405,0.5713,0.85707,1.0,0.0,1.0,1,15,0,1,0,4,0,0,2,0,0,0,0,0,6,0,0,2,0,0,2,0,15],[36,60,0.6,0.64283,0.3407,0.42857,0.57143,1.0,0.0,1.0,3,13,0,3,0,1,0,0,2,0,0,6,0,0,5,0,0,2,0,0,0,0,13],[40,60,0.6667,0.5625,0.35163,0.14286,0.50001,1.0,0.14286,1.0,0,11,0,0,0,9,0,0,2,0,0,5,0,0,4,0,0,1,0,0,0,0,11],[44,60,0.7333,0.40625,0.36441,0.14286,0.14286,0.64286,0.0,1.0,3,7,0,3,0,14,0,0,2,0,0,2,0,0,3,0,0,0,0,0,1,0,7],[48,60,0.8,0.4866,0.35689,0.14286,0.42857,0.89286,0.0,1.0,2,8,0,2,0,9,0,0,3,0,0,6,0,0,2,0,0,0,0,0,2,0,8],[52,60,0.8667,0.33927,0.2389,0.14286,0.42857,0.42858,0.0,1.0,2,2,0,2,0,12,0,0,1,0,0,12,0,0,3,0,0,0,0,0,0,0,2],[56,60,0.9333,0.28571,0.21129,0.14286,0.14286,0.42857,0.0,1.0,3,1,0,3,0,14,0,0,1,0,0,11,0,0,2,0,0,0,0,0,0,0,1],[60,60,1.0,0.27679,0.15542,0.14286,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,14,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.37054,"x":0.74097,"p":[[0,71,0.0,0.60267,0.3775,0.14289,0.857,0.85714,0.0,1.0,5,7,3,5,0,4,0,0,1,0,0,1,0,0,3,0,0,1,0,0,10,0,7],[4,71,0.0563,0.37054,0.34784,0.14286,0.14293,0.57143,0.0,1.0,5,6,0,5,0,12,0,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,6],[8,71,0.1127,0.49999,0.35535,0.14286,0.4286,1.0,0.0,1.0,1,9,0,1,0,10,0,0,4,0,0,2,0,0,5,0,0,1,0,0,0,0,9],[12,71,0.169,0.63839,0.41493,0.14289,1.0,1.0,0.0,1.0,3,17,0,3,0,7,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,17],[16,71,0.2254,0.63382,0.36248,0.2857,0.64286,1.0,0.14,1.0,0,14,0,0,0,7,0,0,4,0,0,2,0,0,3,0,0,1,0,0,1,0,14],[20,71,0.2817,0.58027,0.36943,0.24999,0.50001,1.0,0.14,1.0,0,12,0,0,0,8,0,0,6,0,0,2,0,0,2,0,0,0,0,0,2,0,12],[24,71,0.338,0.58924,0.3421,0.2857,0.57121,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,7,0,0,0,0,0,5,0,0,3,0,0,0,0,11],[28,71,0.3944,0.62042,0.36714,0.24999,0.78564,1.0,0.0,1.0,1,11,0,1,0,7,0,0,4,0,0,0,0,0,3,0,0,1,0,0,5,0,11],[32,71,0.4507,0.6205,0.30222,0.39286,0.57143,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,5,0,0,5,0,0,5,0,0,2,0,0,3,0,9],[36,71,0.507,0.48649,0.36926,0.14286,0.28571,1.0,0.14,1.0,0,9,0,0,0,14,0,0,3,0,0,1,0,0,3,0,0,1,0,0,1,0,9],[40,71,0.5634,0.58928,0.36899,0.2857,0.5,1.0,0.0,1.0,1,12,0,1,0,6,0,0,6,0,0,3,0,0,1,0,0,1,0,0,2,0,12],[44,71,0.6197,0.63392,0.35703,0.28571,0.71429,1.0,0.0,1.0,2,11,0,2,0,4,0,0,5,0,0,0,0,0,3,0,0,3,0,0,4,0,11],[48,71,0.6761,0.74097,0.3264,0.42857,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,3,0,0,3,0,0,2,0,0,2,0,0,2,0,17],[52,71,0.7324,0.65177,0.32915,0.42857,0.57143,1.0,0.0,1.0,1,13,0,1,0,3,0,0,3,0,0,4,0,0,7,0,0,0,0,0,1,0,13],[56,71,0.7887,0.56247,0.31729,0.2857,0.571,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,9,0,0,2,0,0,6,0,0,0,0,0,3,0,8],[60,71,0.8451,0.40622,0.29256,0.14286,0.28571,0.571,0.14286,1.0,0,4,0,0,0,10,0,0,10,0,0,3,0,0,3,0,0,0,0,0,2,0,4],[64,71,0.9014,0.46872,0.32189,0.24999,0.28571,0.71429,0.14286,1.0,0,7,0,0,0,8,0,0,10,0,0,2,0,0,3,0,0,2,0,0,0,0,7],[68,71,0.9577,0.50445,0.32729,0.2857,0.42857,0.85704,0.0,1.0,1,7,0,1,0,6,0,0,7,0,0,5,0,0,3,0,0,1,0,0,2,0,7],[71,71,1.0,0.59821,0.2911,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,7,0,0,5,0,0,5,0,0,3,0,0,2,0,8]]}]},{"i":"f0819159baa7ada3","q":"In the acute-angled triangle  $ABC$ , the altitudes  $BP$  and  $CQ$  were drawn, and the point  $T$  is the intersection point  of the altitudes of  $\\Delta PAQ$ . It turned out that  $\\angle CTB = 90 {} ^ \\circ$ . Find the measure of  $\\angle BAC$ . \n\n(Mikhail Plotnikov)","t":[{"b":3,"e":0.14286,"k":"flat","v":0.13393,"x":0.18304,"p":[[0,134,0.0,0.16072,0.16656,0.14286,0.14286,0.14286,0.0,1.0,4,1,2,4,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,134,0.0299,0.17857,0.08748,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,27,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,134,0.0597,0.16509,0.0883,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,28,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,134,0.0896,0.16072,0.06916,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,134,0.1194,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,134,0.1493,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,134,0.1791,0.16955,0.08331,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,29,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,134,0.209,0.16965,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,134,0.2388,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,134,0.2687,0.17411,0.16262,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,134,0.2985,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,134,0.3284,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,134,0.3582,0.18304,0.1197,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,28,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[52,134,0.3881,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,134,0.4179,0.17849,0.1557,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[60,134,0.4478,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,134,0.4776,0.14286,1e-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],[68,134,0.5075,0.15608,0.10328,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[72,134,0.5373,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,134,0.5672,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,134,0.597,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,134,0.6269,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,134,0.6567,0.14277,0.06187,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,134,0.6866,0.14286,1e-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],[96,134,0.7164,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,134,0.7463,0.16964,0.10369,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,28,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[104,134,0.7761,0.16517,0.08066,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,29,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[108,134,0.806,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,134,0.8358,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[116,134,0.8657,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,134,0.8955,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,134,0.9254,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[128,134,0.9552,0.14286,1e-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],[132,134,0.9851,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.2857,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[134,134,1.0,0.14733,0.0563,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.13393,"x":0.17857,"p":[[0,95,0.0,0.13813,0.10402,0.14,0.14286,0.14286,0.0,0.42857,7,0,1,7,0,21,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,95,0.0421,0.16965,0.08328,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,95,0.0842,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,95,0.1263,0.16071,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,95,0.1684,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,95,0.2105,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,95,0.2526,0.15161,0.04975,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,95,0.2947,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,95,0.3368,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,95,0.3789,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,95,0.4211,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,95,0.4632,0.14733,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,95,0.5053,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,95,0.5474,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,95,0.5895,0.16045,0.0779,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,95,0.6316,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,95,0.6737,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,95,0.7158,0.17857,0.13363,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[72,95,0.7579,0.14724,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,95,0.8,0.15617,0.05488,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,95,0.8421,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,95,0.8842,0.16965,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[88,95,0.9263,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,95,0.9684,0.16072,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[95,95,1.0,0.14286,1e-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":"06aa61c343c41540","q":"Find out wich of the following polynomials are irreducible.\r\n\r\na)  $t^4+1$  over  $\\mathbb{R}$ ;\r\n\r\nb)  $t^4+1$  over  $\\mathbb{Q}$ ;\r\n\r\nc)  $t^3-7t^2+3t+3$  over  $\\mathbb{Q}$ ;\r\n\r\nd)  $t^4+7$  over  $\\mathbb{Z}_{17}$ ; \r\n\r\ne)  $t^3-5$  over  $\\mathbb{Z}_{11}$ ;\r\n\r\nf)  $t^6+7$  over  $\\mathbb{Q}(i)$ .","t":[{"b":6,"e":0.71429,"k":"falling","v":0.67411,"x":0.89732,"p":[[0,27,0.0,0.87946,0.12931,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,8,0,15],[4,27,0.1481,0.89732,0.1394,0.82143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,5,0,19],[8,27,0.2963,0.71875,0.09771,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,23,0,0,2,0,2],[12,27,0.4444,0.70089,0.09689,0.67857,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,20,0,0,3,0,1],[16,27,0.5926,0.72767,0.12555,0.67857,0.71429,0.74996,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,16,0,0,5,0,3],[20,27,0.7407,0.67411,0.07349,0.57143,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,21,0,0,1,0,0],[24,27,0.8889,0.68749,0.10971,0.57143,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,15,0,0,4,0,1],[27,27,1.0,0.68304,0.05906,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.87946,"x":1.0,"p":[[0,45,0.0,0.87946,0.17169,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,2,0,20],[4,45,0.0889,0.88839,0.13709,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,18],[8,45,0.1778,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,45,0.2667,0.97768,0.05187,1.0,1.0,1.0,0.8571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,45,0.3556,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,45,0.4444,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,45,0.5333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,45,0.6222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,45,0.7111,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,45,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,45,0.8889,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,45,0.9778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[45,45,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"a9f26db589be35ff","q":"Let  $n$  cards are placed in a circle. Each card has a white side and a black side. On each move, you pick one card with black side up, flip it over, and also flip over the two neighboring cards. Suppose initially, there are only one black-side-up card.\n(a)If  $n=2015$  , can you make all cards white-side-up through a finite number of moves?\n(b)If  $n=2016$   , can you make all cards white-side-up through a finite number of moves?","t":[{"b":0,"e":1.0,"k":"flat","v":0.88392,"x":1.0,"p":[[0,23,0.0,0.88392,0.21559,0.85711,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,23],[4,23,0.1739,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,23,0.3478,0.94642,0.19805,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[12,23,0.5217,0.96428,0.0945,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[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]]},{"b":2,"e":1.0,"k":"flat","v":0.94643,"x":0.99554,"p":[[0,28,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,28,0.1429,0.94643,0.19805,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[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],[12,28,0.4286,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[16,28,0.5714,0.96427,0.11299,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[20,28,0.7143,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[24,28,0.8571,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[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":"c3113a978ff02e0e","q":"Let  $ABCD$  be a convex quadrilateral. such that  $\\angle CAB = \\angle CDA$  and  $\\angle BCA = \\angle ACD$ . If  $M$  be the midpoint of  $AB$ , prove that  $\\angle BCM = \\angle DBA$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,70,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,70,0.0571,0.03125,0.10555,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,70,0.1143,0.04018,0.11425,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,70,0.1714,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,70,0.2286,0.03571,0.11294,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,70,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.05357,0.14174,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,70,0.4571,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,70,0.5143,0.08036,0.16728,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[40,70,0.5714,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,70,0.6286,0.02679,0.10374,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,70,0.6857,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,70,0.7429,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.03571,0.11294,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.06687,0.16356,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[64,70,0.9143,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.04464,0.12595,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[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":1,"e":0.0,"k":"flat","v":0.00893,"x":0.05357,"p":[[0,79,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,79,0.0506,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,79,0.1013,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,79,0.1519,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,79,0.2025,0.02679,0.10374,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,79,0.2532,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,79,0.3038,0.05357,0.14174,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,79,0.3544,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,79,0.4051,0.05357,0.14174,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[36,79,0.4557,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,79,0.5063,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,79,0.557,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,79,0.6076,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,79,0.6582,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,79,0.7089,0.04018,0.12492,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[60,79,0.7595,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,79,0.8101,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,79,0.8608,0.04018,0.12492,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[72,79,0.9114,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,79,0.962,0.05357,0.1915,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[79,79,1.0,0.03571,0.10101,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"457aad7be3a004f7","q":"Let  $a,b,c$  be constants and  $a,b,c$  are positive real numbers. Prove that the equations $2x+y+z=\\sqrt{c^2+z^2}+\\sqrt{c^2+y^2}$  $x+2y+z=\\sqrt{b^2+x^2}+\\sqrt{b^2+z^2}$  $x+y+2z=\\sqrt{a^2+x^2}+\\sqrt{a^2+y^2}$ have exactly one real solution  $(x,y,z)$  with  $x,y,z \\geq 0$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.73213,"x":0.95533,"p":[[0,45,0.0,0.85266,0.15357,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,10,0,13],[4,45,0.0889,0.76784,0.21944,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,3,0,0,8,0,10],[8,45,0.1778,0.82585,0.21053,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,3,0,17],[12,45,0.2667,0.75893,0.19704,0.67857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,9,0,0,7,0,8],[16,45,0.3556,0.73213,0.22518,0.57143,0.78571,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,4,0,0,8,0,8],[20,45,0.4444,0.79018,0.20511,0.57143,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,5,0,0,2,0,14],[24,45,0.5333,0.80804,0.21902,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,3,0,0,3,0,16],[28,45,0.6222,0.82588,0.17401,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,9,0,12],[32,45,0.7111,0.82143,0.20825,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,0,10,0,13],[36,45,0.8,0.82589,0.18466,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,3,0,15],[40,45,0.8889,0.92857,0.13363,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,22],[44,45,0.9778,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[45,45,1.0,0.95533,0.10981,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26]]},{"b":6,"e":0.71429,"k":"falling","v":0.58481,"x":0.86607,"p":[[0,57,0.0,0.82589,0.17762,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,6,0,13],[4,57,0.0702,0.82142,0.17498,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,8,0,12],[8,57,0.1404,0.80356,0.20125,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,7,0,0,3,0,14],[12,57,0.2105,0.86607,0.15542,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,8,0,15],[16,57,0.2807,0.84818,0.17109,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,4,0,16],[20,57,0.3509,0.81694,0.19639,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,7,0,13],[24,57,0.4211,0.80803,0.20079,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,4,0,14],[28,57,0.4912,0.79464,0.2141,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,5,0,0,5,0,13],[32,57,0.5614,0.74996,0.1856,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,6,0,0,6,0,8],[36,57,0.6316,0.68298,0.22515,0.571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,2,0,0,10,0,0,6,0,0,5,0,6],[40,57,0.7018,0.73213,0.21053,0.57143,0.78564,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,1,0,0,8,0,8],[44,57,0.7719,0.74552,0.24933,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,8,0,0,5,0,0,3,0,12],[48,57,0.8421,0.73216,0.23347,0.57132,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,7,0,0,2,0,11],[52,57,0.9123,0.58481,0.17627,0.42857,0.57121,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,14,0,0,2,0,0,13,0,0,0,0,2],[56,57,0.9825,0.62945,0.18851,0.5354,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,7,0,0,9,0,0,10,0,0,2,0,3],[57,57,1.0,0.63387,0.12342,0.571,0.64286,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,13,0,0,3,0,0]]}]},{"i":"6466e7b1242d1186","q":"Let  $ABCD$  be a tetrahedron.\n(a) Prove that the midpoints of the edges  $AB,AC,BD$ , and  $CD$  lie in a plane.\n(b) Find the point in that plane, whose sum of distances from the lines  $AD$  and  $BC$  is minimal.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.64284,"x":0.85714,"p":[[0,33,0.0,0.77677,0.24983,0.85714,0.85714,0.85714,0.0,0.85714,3,0,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0],[4,33,0.1212,0.7232,0.24725,0.7856,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,24,0,0],[8,33,0.2424,0.68747,0.29328,0.57132,0.85714,0.85714,0.0,0.85714,3,0,0,3,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,23,0,0],[12,33,0.3636,0.64284,0.33502,0.28571,0.85714,0.85714,0.0,0.85714,5,0,0,5,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,22,0,0],[16,33,0.4848,0.77678,0.20497,0.85714,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,27,0,0],[20,33,0.6061,0.74107,0.25614,0.85714,0.85714,0.85714,0.0,0.85714,2,0,0,2,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,26,0,0],[24,33,0.7273,0.73659,0.27918,0.857,0.85714,0.85714,0.0,1.0,3,1,0,3,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,25,0,1],[28,33,0.8485,0.8125,0.17655,0.85714,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,30,0,0],[32,33,0.9697,0.85714,2e-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],[33,33,1.0,0.79464,0.19865,0.85714,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,29,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.2857,"x":0.83035,"p":[[0,33,0.0,0.83035,0.14913,0.85714,0.85714,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0],[4,33,0.1212,0.50891,0.32524,0.2857,0.57121,0.85714,0.0,0.85714,5,0,0,5,0,0,0,0,10,0,0,0,0,0,4,0,0,0,0,0,13,0,0],[8,33,0.2424,0.45086,0.3136,0.2857,0.42836,0.85704,0.0,0.85714,6,0,0,6,0,1,0,0,9,0,0,0,0,0,7,0,0,0,0,0,9,0,0],[12,33,0.3636,0.32142,0.35534,0.0,0.28571,0.857,0.0,0.85714,14,0,0,14,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,9,0,0],[16,33,0.4848,0.33035,0.29545,0.0,0.2857,0.35714,0.0,0.85714,9,0,0,9,0,0,0,0,15,0,0,0,0,0,2,0,0,0,0,0,6,0,0],[20,33,0.6061,0.31248,0.35251,0.0,0.2857,0.64282,0.0,0.85714,15,0,0,15,0,0,0,0,7,0,0,0,0,0,2,0,0,0,0,0,8,0,0],[24,33,0.7273,0.4107,0.31892,0.2857,0.28571,0.857,0.0,0.85714,7,0,0,7,0,0,0,0,13,0,0,0,0,0,3,0,0,0,0,0,9,0,0],[28,33,0.8485,0.28571,0.36421,0.0,0.0,0.64286,0.0,0.85714,18,0,0,18,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,8,0,0],[32,33,0.9697,0.2857,0.29449,0.0,0.2857,0.57111,0.0,0.85714,13,0,0,13,0,0,0,0,10,0,0,0,0,0,5,0,0,0,0,0,4,0,0],[33,33,1.0,0.35713,0.33502,0.0,0.28571,0.57143,0.0,0.85714,12,0,0,12,0,0,0,0,7,0,0,0,0,0,6,0,0,0,0,0,7,0,0]]}]},{"i":"c57bc343d61a4ca2","q":"Let $f: \\mathbb{Z} \\rightarrow\\left\\{1,2, \\ldots, 10^{100}\\right\\}$ be a function satisfying  $$ \\operatorname{gcd}(f(x), f(y))=\\operatorname{gcd}(f(x), x-y) $$  for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x)=\\operatorname{gcd}(m+x, n)$ for all integers $x$.","t":[{"b":2,"e":1.0,"k":"rising","v":0.74106,"x":0.99554,"p":[[0,29,0.0,0.74106,0.32817,0.53571,0.85714,1.0,0.0,1.0,2,15,2,2,0,2,0,0,1,0,0,3,0,0,2,0,0,2,0,0,5,0,15],[4,29,0.1379,0.9375,0.1234,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[8,29,0.2759,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,29,0.4138,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,29,0.5517,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[20,29,0.6897,0.93304,0.13825,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[24,29,0.8276,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[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],[29,29,1.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]},{"b":3,"e":1.0,"k":"rising","v":0.74554,"x":1.0,"p":[[0,33,0.0,0.79911,0.29203,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,5,0,0,1,0,0,1,0,0,2,0,0,3,0,19],[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],[8,33,0.2424,0.74554,0.35307,0.57143,0.92857,1.0,0.0,1.0,4,16,0,4,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,0,5,0,16],[12,33,0.3636,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[16,33,0.4848,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[20,33,0.6061,0.87491,0.23919,0.85714,1.0,1.0,0.14,1.0,0,22,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,22],[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],[33,33,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"eaf8518f0810a4a4","q":"Let $\\mathcal{A}_{n}$ be the set of $n$-tuples $x=\\left(x_{1}, \\ldots, x_{n}\\right)$ with $x_{i} \\in\\{0,1,2\\}$. A triple $x, y, z$ of distinct elements of $\\mathcal{A}_{n}$ is called good if there is some $i$ such that $\\left\\{x_{i}, y_{i}, z_{i}\\right\\}=\\{0,1,2\\}$. A subset $A$ of $\\mathcal{A}_{n}$ is called good if every three distinct elements of $A$ form a good triple.\nProve that every good subset of $\\mathcal{A}_{n}$ has at most $2\\left(\\frac{3}{2}\\right)^{n}$ elements.\n\n## Proposed by Greece","t":[{"b":1,"e":0.42857,"k":"flat","v":0.22322,"x":0.55802,"p":[[0,59,0.0,0.41072,0.23351,0.28571,0.42857,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,7,0,0,9,0,0,4,0,0,5,0,0,0,0,1],[4,59,0.0678,0.32143,0.26486,0.14286,0.21429,0.42858,0.0,0.85714,6,0,0,6,0,10,0,0,1,0,0,8,0,0,0,0,0,6,0,0,1,0,0],[8,59,0.1356,0.3125,0.27067,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,4,0,0,5,0,0,7,0,0,3,0,0,1,0,0,3,0,0],[12,59,0.2034,0.43302,0.3184,0.14286,0.42857,0.71429,0.0,1.0,5,4,0,5,0,5,0,0,3,0,0,8,0,0,2,0,0,4,0,0,1,0,4],[16,59,0.2712,0.26338,0.25026,0.0,0.2857,0.42857,0.0,0.85714,11,0,0,11,0,4,0,0,6,0,0,5,0,0,3,0,0,2,0,0,1,0,0],[20,59,0.339,0.23661,0.2101,0.0,0.1429,0.42857,0.0,0.71429,9,0,0,9,0,9,0,0,3,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[24,59,0.4068,0.28124,0.18721,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,9,0,0,9,0,0,5,0,0,4,0,0,1,0,0,0,0,0],[28,59,0.4746,0.26786,0.26905,0.0,0.21428,0.42858,0.0,0.71429,14,0,0,14,0,2,0,0,1,0,0,8,0,0,3,0,0,4,0,0,0,0,0],[32,59,0.5424,0.27232,0.28203,0.0,0.21428,0.42857,0.0,1.0,11,1,0,11,0,5,0,0,6,0,0,3,0,0,4,0,0,0,0,0,2,0,1],[36,59,0.6102,0.33926,0.26424,0.14286,0.28571,0.42857,0.0,1.0,5,2,0,5,0,7,0,0,6,0,0,7,0,0,3,0,0,2,0,0,0,0,2],[40,59,0.678,0.22322,0.23941,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,8,0,0,6,0,0,2,0,0,2,0,0,2,0,0,1,0,0],[44,59,0.7458,0.3482,0.26228,0.14286,0.35714,0.57111,0.0,1.0,6,1,0,6,0,6,0,0,4,0,0,7,0,0,4,0,0,4,0,0,0,0,1],[48,59,0.8136,0.55802,0.24053,0.39286,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,4,0,0,10,0,0,5,0,0,1,0,4],[52,59,0.8814,0.54909,0.26027,0.28571,0.64286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,6,0,0,4,0,0,2,0,0,12,0,0,1,0,3],[56,59,0.9492,0.46875,0.25313,0.28571,0.5,0.71429,0.0,0.85714,3,0,0,3,0,3,0,0,5,0,0,5,0,0,5,0,0,9,0,0,2,0,0],[59,59,1.0,0.41963,0.20182,0.28571,0.42857,0.57111,0.0,1.0,1,1,0,1,0,4,0,0,6,0,0,12,0,0,5,0,0,3,0,0,0,0,1]]},{"b":5,"e":0.571,"k":"flat","v":0.21872,"x":0.42856,"p":[[0,70,0.0,0.35713,0.1675,0.28571,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,2,0,0,14,0,0,8,0,0,4,0,0,2,0,0,0,0,0],[4,70,0.0571,0.32143,0.24484,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,6,0,0,6,0,0,7,0,0,4,0,0,2,0,0,0,0,1],[8,70,0.1143,0.26786,0.26905,0.0,0.14286,0.42857,0.0,1.0,9,2,0,9,0,8,0,0,4,0,0,7,0,0,1,0,0,1,0,0,0,0,2],[12,70,0.1714,0.37499,0.36201,0.0,0.28571,0.60682,0.0,1.0,9,6,0,9,0,5,0,0,4,0,0,5,0,0,1,0,0,2,0,0,0,0,6],[16,70,0.2286,0.28572,0.27433,0.0,0.14286,0.42858,0.0,0.85714,9,0,0,9,0,8,0,0,3,0,0,5,0,0,3,0,0,1,0,0,3,0,0],[20,70,0.2857,0.28125,0.26841,0.0,0.2857,0.42857,0.0,1.0,10,1,0,10,0,4,0,0,8,0,0,4,0,0,1,0,0,4,0,0,0,0,1],[24,70,0.3429,0.42856,0.28347,0.24999,0.42857,0.60714,0.0,1.0,5,1,0,5,0,3,0,0,5,0,0,6,0,0,5,0,0,4,0,0,3,0,1],[28,70,0.4,0.33482,0.27107,0.10714,0.28571,0.42857,0.0,1.0,8,1,0,8,0,3,0,0,6,0,0,8,0,0,2,0,0,3,0,0,1,0,1],[32,70,0.4571,0.4241,0.25873,0.25,0.42857,0.57143,0.0,1.0,3,1,0,3,0,5,0,0,3,0,0,11,0,0,4,0,0,2,0,0,3,0,1],[36,70,0.5143,0.33482,0.32264,0.0,0.28571,0.71429,0.0,1.0,10,1,0,10,0,5,0,0,4,0,0,4,0,0,0,0,0,5,0,0,3,0,1],[40,70,0.5714,0.32143,0.3312,0.0,0.28571,0.57143,0.0,0.85714,14,0,0,14,0,1,0,0,3,0,0,3,0,0,4,0,0,2,0,0,5,0,0],[44,70,0.6286,0.31696,0.27371,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,7,0,0,5,0,0,6,0,0,3,0,0,1,0,0,2,0,1],[48,70,0.6857,0.36607,0.26229,0.14286,0.42857,0.57143,0.0,1.0,6,1,0,6,0,4,0,0,5,0,0,7,0,0,6,0,0,2,0,0,1,0,1],[52,70,0.7429,0.31686,0.24937,0.14214,0.35714,0.42857,0.0,1.0,7,1,0,7,0,6,0,0,3,0,0,10,0,0,3,0,0,2,0,0,0,0,1],[56,70,0.8,0.38384,0.29768,0.14286,0.28571,0.71429,0.0,0.85714,5,0,0,5,0,8,0,0,4,0,0,4,0,0,1,0,0,6,0,0,4,0,0],[60,70,0.8571,0.26339,0.23449,0.0,0.21431,0.42857,0.0,0.71429,9,0,0,9,0,7,0,0,5,0,0,5,0,0,3,0,0,3,0,0,0,0,0],[64,70,0.9143,0.27232,0.19351,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,7,0,0,8,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[68,70,0.9714,0.21872,0.26956,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,2,0,0,9,0,0,1,0,0,2,0,0,1,0,0,1,0,1],[70,70,1.0,0.26338,0.20548,0.10714,0.2857,0.42857,0.0,0.71429,8,0,0,8,0,6,0,0,6,0,0,8,0,0,3,0,0,1,0,0,0,0,0]]}]},{"i":"1f2d9e6dbedd95d4","q":"For positive integers $n$ and $k \\geqslant 2$ define $E_{k}(n)$ as the greatest exponent $r$ such that $k^{r}$ divides $n$ !. Prove that there are infinitely many $n$ such that $E_{10}(n)>E_{9}(n)$ and infinitely many $m$ such that $E_{10}(m)<E_{9}(m)$. (Brazil)","t":[{"b":3,"e":0.42857,"k":"flat","v":0.68304,"x":0.76339,"p":[[0,48,0.0,0.71429,0.09449,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,24,0,0,3,0,1],[4,48,0.0833,0.72768,0.07457,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,26,0,0,3,0,1],[8,48,0.1667,0.73659,0.12429,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,22,0,0,2,0,4],[12,48,0.25,0.74107,0.15746,0.71429,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,18,0,0,6,0,4],[16,48,0.3333,0.72767,0.12039,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,21,0,0,3,0,3],[20,48,0.4167,0.76339,0.20394,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,17,0,0,4,0,8],[24,48,0.5,0.70981,0.15766,0.71429,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,19,0,0,4,0,3],[28,48,0.5833,0.71871,0.12108,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,21,0,0,2,0,3],[32,48,0.6667,0.71875,0.13592,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,24,0,0,3,0,2],[36,48,0.75,0.68304,0.16649,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,26,0,0,0,0,2],[40,48,0.8333,0.72321,0.04971,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,3,0,0],[44,48,0.9167,0.69196,0.06298,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[48,48,1.0,0.68304,0.07771,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,27,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.70089,"x":0.77679,"p":[[0,60,0.0,0.70089,0.06546,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,0,1,0,0],[4,60,0.0667,0.76786,0.1171,0.71429,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,4,0,5],[8,60,0.1333,0.76339,0.13175,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,19,0,0,5,0,5],[12,60,0.2,0.77679,0.14698,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,16,0,0,5,0,7],[16,60,0.2667,0.75446,0.18638,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,15,0,0,5,0,7],[20,60,0.3333,0.74107,0.12595,0.71429,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,23,0,0,4,0,3],[24,60,0.4,0.77232,0.13296,0.71429,0.71429,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,23,0,0,1,0,7],[28,60,0.4667,0.76338,0.10481,0.71429,0.71429,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,0,7,0,3],[32,60,0.5333,0.74552,0.198,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,16,0,0,4,0,7],[36,60,0.6,0.75,0.17128,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,14,0,0,6,0,6],[40,60,0.6667,0.77232,0.13767,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,18,0,0,7,0,5],[44,60,0.7333,0.7589,0.16149,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,12,0,0,7,0,6],[48,60,0.8,0.73659,0.17897,0.71429,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,17,0,0,5,0,5],[52,60,0.8667,0.7454,0.13712,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,16,0,0,6,0,4],[56,60,0.9333,0.74551,0.2362,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,8,0,0,3,0,11],[60,60,1.0,0.74105,0.17656,0.71429,0.71429,0.85704,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,16,0,0,4,0,6]]}]},{"i":"3beba5f6be3e7e8e","q":"Call a rational number $r$ powerful if $r$ can be expressed in the form $\\frac{p^{k}}{q}$ for some relatively prime positive integers $p, q$ and some integer $k>1$. Let $a, b, c$ be positive\nrational numbers such that $a b c=1$. Suppose there exist positive integers $x, y, z$ such that $a^{x}+b^{y}+c^{z}$ is an integer. Prove that $a, b, c$ are all powerful.","t":[{"b":5,"e":0.57143,"k":"falling","v":0.58927,"x":0.95536,"p":[[0,53,0.0,0.78571,0.30723,0.53571,1.0,1.0,0.0,1.0,2,18,2,2,0,0,0,0,1,0,0,5,0,0,1,0,0,1,0,0,4,0,18],[4,53,0.0755,0.95536,0.14914,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[8,53,0.1509,0.89285,0.2369,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,4,0,24],[12,53,0.2264,0.84822,0.2257,0.67857,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,1,0,21],[16,53,0.3019,0.80803,0.28033,0.64286,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,6,0,0,0,0,0,2,0,0,3,0,19],[20,53,0.3774,0.875,0.24157,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,0,3,0,23],[24,53,0.4528,0.88391,0.18366,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,19],[28,53,0.5283,0.71428,0.25505,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,5,0,0,2,0,0,7,0,0,4,0,10],[32,53,0.6038,0.66964,0.35254,0.28571,0.85714,1.0,0.0,1.0,2,13,0,2,0,2,0,0,6,0,0,1,0,0,2,0,0,2,0,0,4,0,13],[36,53,0.6792,0.71428,0.29014,0.42857,0.78564,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,3,0,0,5,0,0,1,0,0,5,0,0,4,0,12],[40,53,0.7547,0.7857,0.30306,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,2,0,0,5,0,0,1,0,0,1,0,0,1,0,20],[44,53,0.8302,0.67856,0.30305,0.42857,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,7,0,0,7,0,0,1,0,0,2,0,0,2,0,13],[48,53,0.9057,0.71874,0.29122,0.42857,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,7,0,0,1,0,0,2,0,0,3,0,14],[52,53,0.9811,0.61605,0.27765,0.42857,0.49979,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,5,0,0,10,0,0,2,0,0,2,0,0,5,0,7],[53,53,1.0,0.58927,0.29613,0.42857,0.42857,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,4,0,0,12,0,0,2,0,0,1,0,0,3,0,8]]},{"b":6,"e":0.42857,"k":"falling","v":0.49552,"x":0.88392,"p":[[0,71,0.0,0.85266,0.2934,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,24],[4,71,0.0563,0.77678,0.31731,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,3,0,0,3,0,0,0,0,0,0,0,0,5,0,18],[8,71,0.1127,0.76338,0.24119,0.57143,0.857,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,6,0,0,5,0,12],[12,71,0.169,0.73661,0.30952,0.42857,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,2,0,0,6,0,0,2,0,0,2,0,0,2,0,16],[16,71,0.2254,0.80802,0.28929,0.57132,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,4,0,0,3,0,0,0,0,0,1,0,21],[20,71,0.2817,0.76784,0.27607,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,0,5,0,14],[24,71,0.338,0.73213,0.31085,0.57132,0.78571,1.0,0.0,1.0,1,15,0,1,0,2,0,0,3,0,0,1,0,0,3,0,0,6,0,0,1,0,15],[28,71,0.3944,0.81694,0.23213,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,2,0,0,4,0,17],[32,71,0.4507,0.78568,0.25507,0.571,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,3,0,0,5,0,15],[36,71,0.507,0.79017,0.23687,0.5354,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,4,0,0,4,0,15],[40,71,0.5634,0.88392,0.19706,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,20],[44,71,0.6197,0.72766,0.28874,0.42857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,8,0,0,3,0,0,1,0,0,2,0,15],[48,71,0.6761,0.79014,0.26725,0.571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,3,0,0,2,0,0,3,0,0,3,0,17],[52,71,0.7324,0.69643,0.31084,0.39286,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,7,0,0,4,0,0,1,0,0,2,0,0,4,0,13],[56,71,0.7887,0.75446,0.23753,0.57143,0.78571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,5,0,0,4,0,12],[60,71,0.8451,0.65624,0.30901,0.42857,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,3,0,0,8,0,0,2,0,0,3,0,0,4,0,10],[64,71,0.9014,0.69641,0.32685,0.39286,0.78571,1.0,0.0,1.0,1,15,0,1,0,1,0,0,6,0,0,2,0,0,4,0,0,2,0,0,1,0,15],[68,71,0.9577,0.50892,0.24468,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,7,0,0,11,0,0,1,0,0,6,0,0,3,0,2],[71,71,1.0,0.49552,0.22011,0.39286,0.4286,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,4,0,0,9,0,0,4,0,0,9,0,0,2,0,0]]}]},{"i":"9d10c7824df243e1","q":"Let $a, b, c$ be nonnegative real numbers such that $a^{2}+b^{2}+c^{2}+a b c=4$. Show that  $$ 0 \\leq a b+b c+c a-a b c \\leq 2 $$","t":[{"b":3,"e":0.14286,"k":"flat","v":0.27231,"x":0.83929,"p":[[0,103,0.0,0.27231,0.17983,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,10,0,0,13,0,0,0,0,0,5,0,0,1,0,0,0,0,0],[4,103,0.0388,0.76786,0.21053,0.71429,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,12,0,0,4,0,10],[8,103,0.0777,0.67411,0.25313,0.57143,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,2,0,0,0,0,0,4,0,0,14,0,0,4,0,5],[12,103,0.1165,0.76339,0.21902,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,15,0,0,6,0,8],[16,103,0.1553,0.72321,0.23941,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,0,0,0,6,0,0,11,0,0,1,0,10],[20,103,0.1942,0.72321,0.25985,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,8,0,0,4,0,10],[24,103,0.233,0.72768,0.22689,0.67857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,12,0,0,4,0,8],[28,103,0.2718,0.68302,0.27834,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,3,0,0,0,0,0,4,0,0,11,0,0,3,0,8],[32,103,0.3107,0.67411,0.26058,0.67857,0.71429,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,3,0,0,0,0,0,2,0,0,13,0,0,7,0,4],[36,103,0.3495,0.61607,0.23808,0.28571,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,0,0,0,2,0,0,16,0,0,2,0,3],[40,103,0.3883,0.70536,0.24728,0.71429,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,1,0,0,0,0,0,3,0,0,13,0,0,7,0,5],[44,103,0.4272,0.66518,0.27107,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,5,0,0,0,0,0,7,0,0,7,0,0,3,0,8],[48,103,0.466,0.66071,0.27837,0.57142,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,4,0,0,0,0,0,7,0,0,7,0,0,3,0,8],[52,103,0.5049,0.74107,0.19045,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,10,0,0,5,0,7],[56,103,0.5437,0.58036,0.30291,0.28571,0.64286,0.85714,0.0,1.0,3,4,0,3,0,1,0,0,6,0,0,0,0,0,6,0,0,7,0,0,5,0,4],[60,103,0.5825,0.64732,0.30301,0.39286,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,4,0,0,1,0,0,3,0,0,9,0,0,3,0,8],[64,103,0.6214,0.66062,0.21376,0.57143,0.71429,0.85714,0.14,1.0,0,3,0,0,0,1,0,0,3,0,0,2,0,0,7,0,0,10,0,0,6,0,3],[68,103,0.6602,0.72321,0.2111,0.71429,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,14,0,0,7,0,5],[72,103,0.699,0.74107,0.14913,0.71429,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,16,0,0,8,0,3],[76,103,0.7379,0.77679,0.2111,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,12,0,0,3,0,11],[80,103,0.7767,0.83929,0.24419,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,0,3,0,18],[84,103,0.8155,0.75893,0.22428,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,13,0,0,7,0,8],[88,103,0.8544,0.66071,0.32684,0.39286,0.71429,1.0,0.0,1.0,1,9,0,1,0,5,0,0,2,0,0,2,0,0,0,0,0,8,0,0,5,0,9],[92,103,0.8932,0.59375,0.28818,0.28571,0.71429,0.71429,0.0,1.0,2,4,0,2,0,3,0,0,4,0,0,0,0,0,2,0,0,16,0,0,1,0,4],[96,103,0.932,0.54018,0.31488,0.28571,0.71429,0.71429,0.0,1.0,3,3,0,3,0,3,0,0,7,0,0,0,0,0,1,0,0,11,0,0,4,0,3],[100,103,0.9709,0.65625,0.2854,0.39286,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,5,0,0,1,0,0,3,0,0,8,0,0,5,0,7],[103,103,1.0,0.2767,0.23408,0.14286,0.14286,0.28571,0.0,0.85714,2,0,0,2,0,17,0,0,7,0,0,0,0,0,0,0,0,5,0,0,1,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.29017,"x":0.7857,"p":[[0,79,0.0,0.29017,0.22441,0.14286,0.28571,0.28571,0.0,1.0,4,1,0,4,0,8,0,0,13,0,0,2,0,0,3,0,0,0,0,0,1,0,1],[4,79,0.0506,0.75,0.13832,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,15,0,0,7,0,4],[8,79,0.1013,0.67411,0.27487,0.53571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,2,0,0,6,0,0,6,0,0,3,0,9],[12,79,0.1519,0.69643,0.27141,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,3,0,0,2,0,0,3,0,0,9,0,0,5,0,8],[16,79,0.2025,0.70089,0.23787,0.67857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,4,0,0,2,0,0,1,0,0,11,0,0,8,0,5],[20,79,0.2532,0.7857,0.22017,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,7,0,0,7,0,11],[24,79,0.3038,0.74999,0.2342,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,10,0,0,6,0,9],[28,79,0.3544,0.75,0.23958,0.71429,0.78571,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,9,0,0,7,0,9],[32,79,0.4051,0.6875,0.24338,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,3,0,0,1,0,0,1,0,0,6,0,0,9,0,0,7,0,5],[36,79,0.4557,0.71429,0.26245,0.67857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,2,0,0,1,0,0,2,0,0,9,0,0,8,0,7],[40,79,0.5063,0.70982,0.26119,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,3,0,0,0,0,0,7,0,0,6,0,0,5,0,9],[44,79,0.557,0.70982,0.20973,0.71429,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,14,0,0,9,0,3],[48,79,0.6076,0.71875,0.22156,0.71429,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,16,0,0,6,0,5],[52,79,0.6582,0.62052,0.25407,0.57132,0.71429,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,0,0,0,6,0,0,12,0,0,4,0,3],[56,79,0.7089,0.74107,0.18363,0.71429,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,14,0,0,7,0,5],[60,79,0.7595,0.73661,0.22619,0.71429,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,11,0,0,9,0,6],[64,79,0.8101,0.66964,0.20958,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,2,0,0,3,0,0,12,0,0,8,0,2],[68,79,0.8608,0.67857,0.22868,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,0,0,0,4,0,0,14,0,0,5,0,4],[72,79,0.9114,0.625,0.32684,0.35714,0.71429,0.85714,0.0,1.0,2,6,0,2,0,6,0,0,0,0,0,1,0,0,1,0,0,11,0,0,5,0,6],[76,79,0.962,0.69195,0.17897,0.57143,0.71429,0.75,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,14,0,0,5,0,3],[79,79,1.0,0.59375,0.16793,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,7,0,0,9,0,0,12,0,0,1,0,1]]}]},{"i":"7d7f173f82ca1a55","q":"For positive integer  $n$ , let  $s(n)$  denote the sum of the digits of  $n$ . Find the smallest positive integer  $n$  satisfying  $s(n)=s(n+864)=20$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,68,0.0,0.98884,0.0362,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,29],[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],[12,68,0.1765,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,68,0.2353,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,68,0.2941,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,68,0.3529,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,68,0.4118,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,68,0.4706,0.93304,0.2172,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[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],[40,68,0.5882,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,68,0.6471,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[60,68,0.8824,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,68,0.9412,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,68,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,99,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,99,0.0404,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,99,0.0808,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,99,0.1212,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[20,99,0.202,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,99,0.2424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,99,0.2828,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,99,0.3232,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[36,99,0.3636,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[40,99,0.404,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[44,99,0.4444,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[48,99,0.4848,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[52,99,0.5253,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[56,99,0.5657,0.95089,0.19759,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[60,99,0.6061,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[64,99,0.6465,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[68,99,0.6869,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[72,99,0.7273,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[76,99,0.7677,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,99,0.8081,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,99,0.8485,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,99,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[92,99,0.9293,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,99,0.9697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"5f1e7e89426efc24","q":"Let $A B C$ be an acute triangle and $D$ a variable point on side $A C$. Point $E$ is on $B D$ such that $B E=\\frac{B C^{2}-C D \\cdot C A}{B D}$. As $D$ varies on side $A C$ prove that the circumcircle of $A D E$ passes through a fixed point other than 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 $f:[-\\pi/2,\\pi/2]\\to\\mathbb{R}$  be a twice differentiable function which satisfies \\[\\left(f''(x)-f(x)\\right)\\cdot\\tan(x)+2f'(x)\\geqslant 1,\\]for all  $x\\in(-\\pi/2,\\pi/2)$ . Prove that \\[\\int_{-\\pi/2}^{\\pi/2}f(x)\\cdot \\sin(x) \\ dx\\geqslant \\pi-2.\\]","t":[{"b":2,"e":0.14286,"k":"falling","v":0.16518,"x":0.75446,"p":[[0,219,0.0,0.60267,0.18466,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,6,0,0,11,0,0,7,0,0,3,0,2],[4,219,0.0183,0.69643,0.21354,0.57143,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,2,0,0,7,0,0,10,0,0,7,0,4],[8,219,0.0365,0.63839,0.21424,0.57143,0.64286,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,2,0,0,3,0,0,10,0,0,9,0,0,4,0,3],[12,219,0.0548,0.61161,0.21498,0.53571,0.57143,0.75,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,4,0,0,10,0,0,6,0,0,7,0,1],[16,219,0.0731,0.69642,0.16268,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,10,0,0,10,0,1],[20,219,0.0913,0.62933,0.23376,0.53571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,4,0,0,10,0,0,5,0,0,5,0,4],[24,219,0.1096,0.62946,0.23107,0.53571,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,4,0,0,3,0,0,7,0,0,8,0,0,7,0,2],[28,219,0.1279,0.66518,0.21902,0.4286,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,4,0,0,10,0,3],[32,219,0.1461,0.59374,0.23987,0.42857,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,4,0,0,6,0,0,9,0,0,5,0,0,3,0,4],[36,219,0.1644,0.66071,0.2165,0.57143,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,1,0,0,2,0,0,7,0,0,10,0,0,9,0,1],[40,219,0.1826,0.67857,0.18211,0.57143,0.71429,0.75,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,12,0,0,5,0,3],[44,219,0.2009,0.65179,0.27418,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,1,0,0,7,0,0,4,0,0,6,0,0,3,0,8],[48,219,0.2192,0.66964,0.19704,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,8,0,0,7,0,3],[52,219,0.2374,0.75446,0.22934,0.57143,0.78571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,7,0,0,6,0,10],[56,219,0.2557,0.64286,0.22304,0.53571,0.57143,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,6,0,0,9,0,0,5,0,0,7,0,3],[60,219,0.274,0.625,0.24679,0.42857,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,4,0,0,3,0,0,7,0,0,7,0,0,5,0,4],[64,219,0.2922,0.55804,0.26573,0.42857,0.57143,0.71429,0.0,1.0,3,2,0,3,0,0,0,0,4,0,0,4,0,0,9,0,0,5,0,0,5,0,2],[68,219,0.3105,0.53123,0.30562,0.28571,0.57121,0.71429,0.0,1.0,3,4,0,3,0,2,0,0,6,0,0,4,0,0,3,0,0,7,0,0,3,0,4],[72,219,0.3288,0.52679,0.2911,0.28571,0.57143,0.71429,0.0,1.0,4,2,0,4,0,2,0,0,3,0,0,2,0,0,9,0,0,6,0,0,4,0,2],[76,219,0.347,0.63393,0.25985,0.42857,0.71429,0.75,0.0,1.0,2,5,0,2,0,0,0,0,2,0,0,5,0,0,5,0,0,10,0,0,3,0,5],[80,219,0.3653,0.55357,0.25939,0.42857,0.57141,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,2,0,0,10,0,0,6,0,0,4,0,0,4,0,3],[84,219,0.3836,0.59821,0.25111,0.39286,0.64286,0.75,0.14286,1.0,0,3,0,0,0,2,0,0,6,0,0,3,0,0,5,0,0,8,0,0,5,0,3],[88,219,0.4018,0.65178,0.25985,0.53572,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,1,0,0,4,0,0,5,0,0,9,0,0,5,0,5],[92,219,0.4201,0.5625,0.30291,0.28571,0.64286,0.85704,0.0,1.0,2,3,0,2,0,4,0,0,4,0,0,2,0,0,4,0,0,7,0,0,6,0,3],[96,219,0.4384,0.57143,0.28571,0.28571,0.64286,0.85714,0.0,1.0,1,2,0,1,0,5,0,0,3,0,0,2,0,0,5,0,0,7,0,0,7,0,2],[100,219,0.4566,0.60713,0.30094,0.42857,0.57143,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,3,0,0,4,0,0,6,0,0,4,0,0,5,0,6],[104,219,0.4749,0.49553,0.2789,0.28571,0.42857,0.71429,0.0,1.0,3,4,0,3,0,0,0,0,8,0,0,7,0,0,5,0,0,4,0,0,1,0,4],[108,219,0.4932,0.46429,0.27664,0.28571,0.42859,0.71429,0.0,1.0,5,1,0,5,0,1,0,0,4,0,0,7,0,0,6,0,0,5,0,0,3,0,1],[112,219,0.5114,0.52677,0.31224,0.28571,0.57121,0.75,0.0,1.0,4,4,0,4,0,2,0,0,3,0,0,6,0,0,5,0,0,4,0,0,4,0,4],[116,219,0.5297,0.45982,0.23619,0.28571,0.42857,0.60714,0.0,0.85714,3,0,0,3,0,2,0,0,5,0,0,7,0,0,7,0,0,6,0,0,2,0,0],[120,219,0.5479,0.5625,0.27418,0.28571,0.57143,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,5,0,0,2,0,0,6,0,0,8,0,0,5,0,2],[124,219,0.5662,0.37054,0.29851,0.14286,0.28571,0.60714,0.0,1.0,6,1,0,6,0,7,0,0,5,0,0,2,0,0,4,0,0,5,0,0,2,0,1],[128,219,0.5845,0.33036,0.26107,0.0,0.35714,0.57143,0.0,0.85714,9,0,0,9,0,3,0,0,4,0,0,5,0,0,8,0,0,2,0,0,1,0,0],[132,219,0.6027,0.33482,0.26633,0.10714,0.28571,0.57143,0.0,0.85714,8,0,0,8,0,4,0,0,5,0,0,5,0,0,6,0,0,2,0,0,2,0,0],[136,219,0.621,0.52679,0.27534,0.28571,0.42857,0.75,0.14286,1.0,0,2,0,0,0,5,0,0,6,0,0,6,0,0,2,0,0,5,0,0,6,0,2],[140,219,0.6393,0.30804,0.28596,0.14286,0.14286,0.46429,0.0,1.0,5,1,0,5,0,14,0,0,2,0,0,3,0,0,2,0,0,3,0,0,2,0,1],[144,219,0.6575,0.37945,0.29796,0.0,0.42857,0.57143,0.0,1.0,9,1,0,9,0,3,0,0,1,0,0,5,0,0,8,0,0,4,0,0,1,0,1],[148,219,0.6758,0.34821,0.3008,0.10714,0.28571,0.57143,0.0,1.0,8,2,0,8,0,4,0,0,6,0,0,5,0,0,4,0,0,1,0,0,2,0,2],[152,219,0.6941,0.33482,0.26149,0.14286,0.28571,0.42858,0.0,1.0,3,2,0,3,0,12,0,0,3,0,0,7,0,0,3,0,0,2,0,0,0,0,2],[156,219,0.7123,0.41518,0.32015,0.14286,0.42857,0.71429,0.0,1.0,4,3,0,4,0,10,0,0,0,0,0,6,0,0,3,0,0,4,0,0,2,0,3],[160,219,0.7306,0.45982,0.32875,0.14286,0.42857,0.71429,0.0,1.0,4,3,0,4,0,6,0,0,5,0,0,3,0,0,2,0,0,5,0,0,4,0,3],[164,219,0.7489,0.49107,0.34058,0.14286,0.5,0.85714,0.0,1.0,3,5,0,3,0,7,0,0,4,0,0,2,0,0,5,0,0,2,0,0,4,0,5],[168,219,0.7671,0.40625,0.30537,0.14286,0.35714,0.71429,0.0,1.0,3,2,0,3,0,11,0,0,2,0,0,3,0,0,4,0,0,5,0,0,2,0,2],[172,219,0.7854,0.39723,0.28743,0.14286,0.28571,0.57143,0.0,0.85714,4,0,0,4,0,6,0,0,8,0,0,2,0,0,5,0,0,1,0,0,6,0,0],[176,219,0.8037,0.3392,0.25449,0.14286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1,0.51339,0.25719,0.28571,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,11,0,0,8,0,0,2,0,0,4,0,0,2,0,4],[132,135,0.9778,0.33036,0.0974,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,17,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[135,135,1.0,0.33036,0.09062,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,19,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"e4bc04b6fb9ecc5b","q":"Let  $n$  be a positive integer. Let  $A_n$  denote the set of primes  $p$  such that there exists positive integers  $a,b$  satisfying   $$ \\frac{a+b}{p} \\text{ and } \\frac{a^n + b^n}{p^2} $$  are both integers that are relatively prime to  $p$ . If  $A_n$  is finite, let  $f(n)$  denote  $|A_n|$ .\n\na) Prove that  $A_n$  is finite if and only if  $n \\not = 2$ .\n\nb) Let  $m,k$  be odd positive integers and let  $d$  be their gcd. Show that   $$ f(d) \\leq f(k) + f(m) - f(km) \\leq 2 f(d). $$","t":[{"b":3,"e":0.85714,"k":"flat","v":0.79909,"x":0.97768,"p":[[0,34,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,34,0.1176,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[8,34,0.2353,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,34,0.3529,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[16,34,0.4706,0.94641,0.09285,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[20,34,0.5882,0.79909,0.22265,0.67857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,4,0,0,7,0,13],[24,34,0.7059,0.81249,0.20026,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,11,0,11],[28,34,0.8235,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,34,0.9412,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[34,34,1.0,0.95088,0.07669,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22]]},{"b":6,"e":1.0,"k":"flat","v":0.85714,"x":0.9866,"p":[[0,50,0.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,50,0.08,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,50,0.16,0.92402,0.19915,1.0,1.0,1.0,0.14,1.0,0,26,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,26],[12,50,0.24,0.95981,0.10854,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[16,50,0.32,0.94643,0.17767,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[20,50,0.4,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[24,50,0.48,0.96874,0.12238,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[28,50,0.56,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,50,0.64,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[36,50,0.72,0.95311,0.09722,0.98214,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,1,24],[40,50,0.8,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[44,50,0.88,0.95535,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[48,50,0.96,0.85714,0.15152,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,7,0,14],[50,50,1.0,0.89731,0.14393,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,6,0,19]]}]},{"i":"222aa76607fc7f28","q":"Let ABCDEF be an equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are the extensions of AB, CD and EF has side lengths 200, 240 and 300 respectively. Find the side length of the hexagon.","t":[{"b":2,"e":0.1429,"k":"falling","v":0.02679,"x":0.92411,"p":[[0,209,0.0,0.92411,0.24218,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[4,209,0.0191,0.41509,0.35247,0.14286,0.2857,0.71429,0.0,1.0,3,6,0,3,0,12,0,0,3,0,0,4,0,0,0,0,0,3,0,0,1,0,6],[8,209,0.0383,0.32143,0.36596,0.14286,0.14286,0.5,0.0,1.0,7,6,0,7,0,15,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,6],[12,209,0.0574,0.30804,0.30952,0.10714,0.14286,0.42857,0.0,1.0,8,3,0,8,0,10,0,0,0,0,0,9,0,0,0,0,0,1,0,0,1,0,3],[16,209,0.0766,0.39732,0.41147,0.14286,0.14286,1.0,0.0,1.0,7,9,0,7,0,13,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,9],[20,209,0.0957,0.28125,0.31438,0.14286,0.14286,0.21432,0.0,1.0,4,4,0,4,0,20,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,4],[24,209,0.1148,0.51339,0.40384,0.10714,0.42859,1.0,0.0,1.0,8,10,0,8,0,3,0,0,1,0,0,5,0,0,1,0,0,3,0,0,1,0,10],[28,209,0.134,0.4375,0.3862,0.14286,0.1429,0.89286,0.0,1.0,3,8,0,3,0,14,0,0,1,0,0,3,0,0,0,0,0,1,0,0,2,0,8],[32,209,0.1531,0.28572,0.34069,0.0,0.14286,0.42858,0.0,1.0,11,4,0,11,0,9,0,0,3,0,0,2,0,0,0,0,0,3,0,0,0,0,4],[36,209,0.1722,0.51777,0.41619,0.14286,0.50001,1.0,0.0,1.0,6,11,0,6,0,8,0,0,0,0,0,2,0,0,1,0,0,3,0,0,1,0,11],[40,209,0.1914,0.33929,0.39407,0.0,0.14286,0.71429,0.0,1.0,11,7,0,11,0,9,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,7],[44,209,0.2105,0.29455,0.38292,0.0,0.14286,0.28571,0.0,1.0,11,7,0,11,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[48,209,0.2297,0.26784,0.34207,0.0,0.14286,0.42858,0.0,1.0,12,4,0,12,0,10,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,4],[52,209,0.2488,0.39732,0.42818,0.0,0.14286,1.0,0.0,1.0,9,10,0,9,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,10],[56,209,0.2679,0.32143,0.35535,0.0,0.14286,0.42857,0.0,1.0,9,5,0,9,0,10,0,0,2,0,0,4,0,0,0,0,0,1,0,0,1,0,5],[60,209,0.2871,0.3482,0.36234,0.0,0.14286,0.60682,0.0,1.0,9,5,0,9,0,9,0,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,5],[64,209,0.3062,0.32133,0.39289,0.0,0.14286,0.60682,0.0,1.0,11,7,0,11,0,11,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,7],[68,209,0.3254,0.31686,0.37755,0.0,0.14286,0.57111,0.0,1.0,12,6,0,12,0,8,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,6],[72,209,0.3445,0.35705,0.4418,0.0,0.14286,1.0,0.0,1.0,14,10,0,14,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,10],[76,209,0.3636,0.33482,0.36702,0.0,0.21428,0.50002,0.0,1.0,12,5,1,12,0,4,0,0,4,0,0,4,0,0,0,0,0,2,0,0,1,0,5],[80,209,0.3828,0.28572,0.35892,0.0,0.14286,0.5,0.0,1.0,14,4,0,14,0,7,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,4],[84,209,0.4019,0.25893,0.33775,0.0,0.14286,0.42857,0.0,1.0,13,4,0,13,0,8,0,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,4],[88,209,0.4211,0.31696,0.38752,0.0,0.14286,0.5,0.0,1.0,10,7,0,10,0,12,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,7],[92,209,0.4402,0.22768,0.3533,0.0,0.0,0.2857,0.0,1.0,17,5,0,17,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[96,209,0.4593,0.1875,0.2635,0.0,0.14286,0.21432,0.0,1.0,15,1,0,15,0,9,0,0,0,0,0,5,0,0,0,0,0,1,0,0,1,0,1],[100,209,0.4785,0.24554,0.33165,0.0,0.14286,0.42857,0.0,1.0,13,4,0,13,0,10,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,4],[104,209,0.4976,0.23213,0.3046,0.0,0.14286,0.1786,0.0,1.0,11,3,0,11,0,13,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,3],[108,209,0.5167,0.27232,0.39506,0.0,0.14286,0.21429,0.0,1.0,15,7,0,15,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[112,209,0.5359,0.24098,0.33014,0.0,0.14286,0.32143,0.0,1.0,13,4,0,13,0,10,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,4],[116,209,0.555,0.20534,0.305,0.0,0.0,0.32142,0.0,1.0,17,3,0,17,0,5,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,3],[120,209,0.5742,0.09821,0.19704,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[124,209,0.5933,0.21429,0.35175,0.0,0.0,0.14287,0.0,1.0,17,5,0,17,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[128,209,0.6124,0.1607,0.26182,0.0,0.0,0.1429,0.0,1.0,19,1,0,19,0,6,0,0,0,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[132,209,0.6316,0.17411,0.28956,0.0,0.07143,0.14286,0.0,1.0,16,2,0,16,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,2],[136,209,0.6507,0.14733,0.31438,0.0,0.0,0.14287,0.0,1.0,22,3,0,22,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[140,209,0.6699,0.20982,0.32338,0.0,0.07143,0.1786,0.0,1.0,16,3,0,16,0,8,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,3],[144,209,0.689,0.21875,0.34623,0.0,0.14286,0.14286,0.0,1.0,15,5,0,15,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[148,209,0.7081,0.12946,0.29528,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[152,209,0.7273,0.10714,0.21724,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[156,209,0.7464,0.08036,0.19865,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[160,209,0.7656,0.14732,0.28004,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,5,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,2],[164,209,0.7847,0.12499,0.2544,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,4,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0],[168,209,0.8038,0.19196,0.30222,0.0,0.0,0.21429,0.0,1.0,18,2,0,18,0,6,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,2],[172,209,0.823,0.11161,0.21939,0.0,0.0,0.14287,0.0,1.0,23,1,0,23,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[176,209,0.8421,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[180,209,0.8612,0.07589,0.22862,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[184,209,0.8804,0.03571,0.10102,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[188,209,0.8995,0.08027,0.20803,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[192,209,0.9187,0.21876,0.13825,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,23,0,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[196,209,0.9378,0.24107,0.19704,0.14286,0.14286,0.21432,0.14286,1.0,0,1,0,0,0,24,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[200,209,0.9569,0.21429,0.10714,0.14286,0.14286,0.28571,0.14286,0.4286,0,0,0,0,0,21,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[204,209,0.9761,0.18304,0.09606,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[208,209,0.9952,0.18741,0.13095,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,26,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[209,209,1.0,0.20974,0.14283,0.14286,0.14286,0.1786,0.0,0.71429,1,0,0,1,0,23,0,0,2,0,0,5,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.00446,"x":0.91964,"p":[[0,128,0.0,0.91964,0.24984,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[4,128,0.0312,0.50882,0.40404,0.14286,0.42857,1.0,0.0,1.0,6,11,0,6,0,6,0,0,2,0,0,3,0,0,2,0,0,2,0,0,0,0,11],[8,128,0.0625,0.51338,0.39907,0.14286,0.42857,1.0,0.0,1.0,6,10,0,6,0,6,0,0,0,0,0,6,0,0,1,0,0,1,0,0,2,0,10],[12,128,0.0938,0.34822,0.33108,0.14286,0.14286,0.5,0.0,1.0,6,4,0,6,0,12,0,0,0,0,0,6,0,0,0,0,0,4,0,0,0,0,4],[16,128,0.125,0.58036,0.42097,0.14286,0.71429,1.0,0.0,1.0,5,14,0,5,0,7,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,14],[20,128,0.1562,0.52232,0.35465,0.14286,0.42859,0.89286,0.0,1.0,2,8,0,2,0,9,0,0,0,0,0,7,0,0,0,0,0,5,0,0,1,0,8],[24,128,0.1875,0.40625,0.36615,0.14286,0.2857,0.75,0.0,1.0,5,6,0,5,0,10,0,0,4,0,0,3,0,0,0,0,0,2,0,0,2,0,6],[28,128,0.2188,0.38392,0.32426,0.14286,0.28574,0.4642,0.0,1.0,3,5,0,3,0,13,0,0,0,0,0,8,0,0,1,0,0,2,0,0,0,0,5],[32,128,0.25,0.3616,0.28789,0.14286,0.2143,0.60682,0.0,1.0,3,2,0,3,0,13,0,0,1,0,0,6,0,0,1,0,0,6,0,0,0,0,2],[36,128,0.2812,0.47768,0.28033,0.24999,0.42859,0.71429,0.0,1.0,1,3,1,1,0,7,0,0,2,0,0,10,0,0,2,0,0,5,0,0,2,0,3],[40,128,0.3125,0.44642,0.31288,0.14286,0.42857,0.71429,0.0,1.0,3,4,0,3,0,7,0,0,4,0,0,6,0,0,2,0,0,5,0,0,1,0,4],[44,128,0.3438,0.44195,0.31815,0.14286,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,14,0,0,1,0,0,5,0,0,2,0,0,5,0,0,0,0,5],[48,128,0.375,0.27233,0.22689,0.14286,0.14286,0.42858,0.0,0.71429,4,0,0,4,0,15,0,0,3,0,0,5,0,0,0,0,0,5,0,0,0,0,0],[52,128,0.4062,0.44195,0.3376,0.14286,0.42857,0.71429,0.0,1.0,2,5,0,2,0,12,0,0,1,0,0,5,0,0,2,0,0,3,0,0,2,0,5],[56,128,0.4375,0.52229,0.36528,0.14286,0.42857,1.0,0.0,1.0,1,9,0,1,0,11,0,0,1,0,0,4,0,0,3,0,0,1,0,0,2,0,9],[60,128,0.4688,0.35268,0.29877,0.14286,0.28574,0.4286,0.0,1.0,4,3,0,4,0,12,0,0,0,0,0,10,0,0,0,0,0,2,0,0,1,0,3],[64,128,0.5,0.53123,0.34299,0.14286,0.42859,0.85704,0.0,1.0,2,7,0,2,0,8,0,0,0,0,0,7,0,0,1,0,0,5,0,0,2,0,7],[68,128,0.5312,0.28125,0.28004,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,13,0,0,1,0,0,5,0,0,0,0,0,4,0,0,1,0,1],[72,128,0.5625,0.4375,0.34615,0.14286,0.42857,0.71429,0.0,1.0,4,5,0,4,0,9,0,0,2,0,0,6,0,0,0,0,0,4,0,0,2,0,5],[76,128,0.5938,0.44187,0.32023,0.14286,0.42857,0.71429,0.0,1.0,4,3,0,4,0,7,0,0,3,0,0,6,0,0,0,0,0,7,0,0,2,0,3],[80,128,0.625,0.3125,0.32623,0.14286,0.14286,0.60714,0.0,1.0,7,3,0,7,0,14,0,0,0,0,0,2,0,0,1,0,0,5,0,0,0,0,3],[84,128,0.6562,0.32143,0.28571,0.14286,0.14286,0.4286,0.0,1.0,5,2,0,5,0,13,0,0,0,0,0,7,0,0,1,0,0,4,0,0,0,0,2],[88,128,0.6875,0.42411,0.35443,0.14286,0.35714,0.71429,0.0,1.0,4,6,0,4,0,11,0,0,1,0,0,5,0,0,0,0,0,5,0,0,0,0,6],[92,128,0.7188,0.33928,0.32683,0.14286,0.14286,0.60714,0.0,1.0,6,3,0,6,0,13,0,0,0,0,0,4,0,0,1,0,0,4,0,0,1,0,3],[96,128,0.75,0.30355,0.30876,0.0,0.14286,0.571,0.0,1.0,10,2,0,10,0,8,0,0,1,0,0,4,0,0,3,0,0,4,0,0,0,0,2],[100,128,0.7812,0.42854,0.33692,0.14286,0.42857,0.71429,0.0,1.0,6,4,0,6,0,7,0,0,1,0,0,4,0,0,3,0,0,7,0,0,0,0,4],[104,128,0.8125,0.4509,0.37305,0.14286,0.42857,0.75,0.0,1.0,5,7,0,5,0,9,0,0,1,0,0,5,0,0,0,0,0,4,0,0,1,0,7],[108,128,0.8438,0.42857,0.37115,0.14286,0.42857,0.71429,0.0,1.0,7,6,0,7,0,8,0,0,0,0,0,4,0,0,1,0,0,6,0,0,0,0,6],[112,128,0.875,0.32142,0.36942,0.0,0.14286,0.46418,0.0,1.0,10,6,0,10,0,10,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,6],[116,128,0.9062,0.3125,0.36147,0.0,0.14286,0.50002,0.0,1.0,11,4,0,11,0,9,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,4],[120,128,0.9375,0.20982,0.29011,0.0,0.0,0.42857,0.0,1.0,17,2,0,17,0,4,0,0,1,0,0,6,0,0,1,0,0,1,0,0,0,0,2],[124,128,0.9688,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[128,128,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4cd5ebc72861c881","q":"Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that\n\n$$\n\\left(a+\\frac{1}{b}\\right)^{2}+\\left(b+\\frac{1}{c}\\right)^{2}+\\left(c+\\frac{1}{a}\\right)^{2} \\geq 3(a+b+c+1)\n$$\n\nWhen does equality hold?","t":[{"b":5,"e":0.14286,"k":"flat","v":0.13839,"x":0.24107,"p":[[0,180,0.0,0.22768,0.14223,0.14286,0.14286,0.2857,0.14286,0.71429,0,0,0,0,0,21,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[4,180,0.0222,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,180,0.0444,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,180,0.0667,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,180,0.0889,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,180,0.1111,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,180,0.1333,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,180,0.1556,0.16518,0.06298,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,180,0.1778,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,180,0.2,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,180,0.2222,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,180,0.2444,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,180,0.2667,0.16071,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,180,0.2889,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,180,0.3111,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,180,0.3333,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,180,0.3556,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,180,0.3778,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,180,0.4,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,180,0.4222,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,180,0.4444,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,180,0.4667,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,180,0.4889,0.16071,0.04725,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,180,0.5111,0.17411,0.1504,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[96,180,0.5333,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[100,180,0.5556,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,180,0.5778,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,180,0.6,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[112,180,0.6222,0.16518,0.08073,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,29,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[116,180,0.6444,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,180,0.6667,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,180,0.6889,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,180,0.7111,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,180,0.7333,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,180,0.7556,0.15624,0.08261,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,29,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[140,180,0.7778,0.16071,0.05923,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[144,180,0.8,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[148,180,0.8222,0.16518,0.06298,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[152,180,0.8444,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[156,180,0.8667,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[160,180,0.8889,0.15179,0.04972,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[164,180,0.9111,0.14286,1e-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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],[152,191,0.7958,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[156,191,0.8168,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[160,191,0.8377,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[164,191,0.8586,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[168,191,0.8796,0.16071,0.05923,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[172,191,0.9005,0.16071,0.05923,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[176,191,0.9215,0.14286,1e-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],[180,191,0.9424,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[184,191,0.9634,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[188,191,0.9843,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[191,191,1.0,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"26c1426cf6d794df","q":"Prove that for all real numbers  $x,y,z \\in [1,2]$  the following inequality always holds:\r\n\\[ (x+y+z)(\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z})\\geq 6(\\frac{x}{y+z}+\\frac{y}{z+x}+\\frac{z}{x+y}).  \\]\r\nWhen does the equality occur?","t":[{"b":2,"e":0.14286,"k":"flat","v":0.13839,"x":0.17411,"p":[[0,69,0.0,0.17411,0.13236,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,69,0.058,0.14733,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,69,0.1159,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,69,0.2319,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.16072,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.15161,0.04975,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.16955,0.10974,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[32,69,0.4638,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.14286,1e-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],[44,69,0.6377,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.16071,0.09942,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,69,0.7536,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,69,0.8696,0.17411,0.10555,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,29,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[64,69,0.9275,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.14286,"k":"flat","v":0.13839,"x":0.16964,"p":[[0,47,0.0,0.16071,0.06916,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.14286,1e-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],[8,47,0.1702,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,47,0.4255,0.15625,0.07457,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,47,0.5106,0.14286,1e-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],[28,47,0.5957,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.14286,1e-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],[36,47,0.766,0.16071,0.07784,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,47,0.8511,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"38514a43db951922","q":"Show that for any arbitrary triangle  $ABC$ , we have\n\\[\\sin\\left(\\frac{A}{2}\\right) \\cdot \\sin\\left(\\frac{B}{2}\\right) \\cdot \\sin\\left(\\frac{C}{2}\\right) \\leq \\frac{abc}{(a+b)(b+c)(c+a)}.\\]","t":[{"b":5,"e":1.0,"k":"rising","v":0.29904,"x":1.0,"p":[[0,56,0.0,0.29904,0.26325,0.0,0.14288,0.57141,0.0,0.71429,10,0,0,10,0,7,0,0,0,0,0,1,0,0,13,0,0,1,0,0,0,0,0],[4,56,0.0714,0.43746,0.34613,0.0,0.57141,0.57143,0.0,1.0,9,5,0,9,0,3,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,5],[8,56,0.1429,0.42856,0.28121,0.14286,0.57143,0.57143,0.0,1.0,7,1,0,7,0,3,0,0,1,0,0,0,0,0,16,0,0,4,0,0,0,0,1],[12,56,0.2143,0.43747,0.27417,0.14286,0.57143,0.57143,0.0,0.71429,7,0,0,7,0,3,0,0,0,0,0,0,0,0,15,0,0,7,0,0,0,0,0],[16,56,0.2857,0.47764,0.28706,0.49968,0.57143,0.57143,0.0,1.0,7,3,0,7,0,0,0,0,1,0,0,0,0,0,21,0,0,0,0,0,0,0,3],[20,56,0.3571,0.41518,0.31412,0.14286,0.57143,0.57143,0.0,1.0,6,3,0,6,0,7,0,0,1,0,0,0,0,0,12,0,0,3,0,0,0,0,3],[24,56,0.4286,0.47319,0.35969,0.10714,0.57143,0.60714,0.0,1.0,8,6,0,8,0,3,0,0,1,0,0,0,0,0,12,0,0,1,0,0,1,0,6],[28,56,0.5,0.49552,0.29662,0.1429,0.57143,0.57143,0.0,1.0,4,4,0,4,0,5,0,0,0,0,0,0,0,0,17,0,0,2,0,0,0,0,4],[32,56,0.5714,0.49551,0.31131,0.14286,0.57143,0.71429,0.0,1.0,5,4,0,5,0,4,0,0,1,0,0,0,0,0,13,0,0,5,0,0,0,0,4],[36,56,0.6429,0.87945,0.25534,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,25],[40,56,0.7143,0.84374,0.31817,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,25],[44,56,0.7857,0.82143,0.36422,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,25],[48,56,0.8571,0.72319,0.3606,0.57132,1.0,1.0,0.0,1.0,4,18,0,4,0,1,0,0,1,0,0,0,0,0,7,0,0,1,0,0,0,0,18],[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],[56,56,1.0,0.97321,0.10375,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29]]},{"b":7,"e":1.0,"k":"rising","v":0.29908,"x":1.0,"p":[[0,66,0.0,0.35714,0.25754,0.14286,0.50001,0.57143,0.0,0.71429,6,0,0,6,0,8,0,0,1,0,0,1,0,0,13,0,0,3,0,0,0,0,0],[4,66,0.0606,0.29908,0.2657,0.0,0.14286,0.57143,0.0,0.71429,9,0,0,9,0,9,0,0,0,0,0,0,0,0,12,0,0,2,0,0,0,0,0],[8,66,0.1212,0.48209,0.33263,0.14286,0.57143,0.57143,0.0,1.0,7,5,0,7,0,3,0,0,0,0,0,0,0,0,15,0,0,2,0,0,0,0,5],[12,66,0.1818,0.38392,0.2976,0.14286,0.57143,0.57143,0.0,1.0,7,2,0,7,0,7,0,0,0,0,0,0,0,0,15,0,0,1,0,0,0,0,2],[16,66,0.2424,0.4062,0.32751,0.0,0.571,0.57143,0.0,1.0,10,3,0,10,0,2,0,0,1,0,0,0,0,0,14,0,0,2,0,0,0,0,3],[20,66,0.303,0.38837,0.29929,0.0,0.57143,0.57143,0.0,1.0,9,1,0,9,0,4,0,0,0,0,0,0,0,0,14,0,0,4,0,0,0,0,1],[24,66,0.3636,0.38392,0.35792,0.10714,0.14286,0.57143,0.0,1.0,8,5,0,8,0,9,0,0,0,0,0,0,0,0,8,0,0,2,0,0,0,0,5],[28,66,0.4242,0.33926,0.31286,0.0,0.21429,0.57143,0.0,1.0,10,2,0,10,0,6,0,0,1,0,0,0,0,0,11,0,0,2,0,0,0,0,2],[32,66,0.4848,0.33926,0.28957,0.0,0.571,0.57143,0.0,1.0,11,1,0,11,0,3,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,1],[36,66,0.5455,0.42853,0.31337,0.14286,0.57121,0.57143,0.0,1.0,7,2,0,7,0,5,0,0,0,0,0,1,0,0,13,0,0,2,0,0,2,0,2],[40,66,0.6061,0.47762,0.25655,0.14286,0.57143,0.57143,0.0,0.85714,3,0,0,3,0,6,0,0,0,0,0,0,0,0,17,0,0,3,0,0,3,0,0],[44,66,0.6667,0.45084,0.30325,0.14286,0.57143,0.57143,0.0,1.0,7,3,0,7,0,3,0,0,0,0,0,0,0,0,18,0,0,1,0,0,0,0,3],[48,66,0.7273,0.46426,0.29013,0.32143,0.57143,0.57143,0.0,1.0,8,2,0,8,0,0,0,0,0,0,0,1,0,0,18,0,0,3,0,0,0,0,2],[52,66,0.7879,0.40177,0.2911,0.10714,0.57143,0.57143,0.0,1.0,8,1,0,8,0,4,0,0,0,0,0,0,0,0,17,0,0,1,0,0,1,0,1],[56,66,0.8485,0.32587,0.31789,0.0,0.21428,0.57143,0.0,1.0,13,1,0,13,0,3,0,0,1,0,0,0,0,0,9,0,0,5,0,0,0,0,1],[60,66,0.9091,0.89286,0.25505,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,26],[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":"da199e63c3d081ff","q":"Given a triangle $A B C$, let $H$ and $O$ be its orthocentre and circumcentre, respectively. Let $K$ be the midpoint of the line segment $A H$. Let further $\\ell$ be a line through $O$, and let $P$ and $Q$ be the orthogonal projections of $B$ and $C$ onto $\\ell$, respectively. Prove that $K P+K Q \\geq B C$.\n\n# Russia, Vasily Mokin","t":[{"b":0,"e":0.14286,"k":"flat","v":0.06697,"x":0.17411,"p":[[0,39,0.0,0.08036,0.19541,0.0,0.0,0.03571,0.0,0.85714,24,0,1,24,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[4,39,0.1026,0.08036,0.1234,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.12945,0.19675,0.0,0.0,0.2857,0.0,0.5714,21,0,0,21,0,2,0,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[12,39,0.3077,0.10268,0.15663,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,39,0.4103,0.12498,0.18116,0.0,0.0,0.1786,0.0,0.571,19,0,0,19,0,5,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[20,39,0.5128,0.16962,0.22706,0.0,0.14286,0.14287,0.0,0.71429,15,0,0,15,0,10,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,0],[24,39,0.6154,0.17411,0.19145,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,5,0,0,3,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[28,39,0.7179,0.08482,0.1551,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,1,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.06697,0.15561,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,39,0.9231,0.09375,0.18766,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[39,39,1.0,0.08036,0.15126,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.03572,"x":0.27213,"p":[[0,110,0.0,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,1,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,110,0.0364,0.23213,0.31287,0.0,0.14286,0.32143,0.0,1.0,15,3,0,15,0,6,0,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,3],[8,110,0.0727,0.19197,0.249,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,8,0,0,3,0,0,4,0,0,0,0,0,2,0,0,0,0,1],[12,110,0.1091,0.23215,0.28065,0.0,0.14286,0.28571,0.0,1.0,12,2,0,12,0,8,0,0,5,0,0,1,0,0,3,0,0,1,0,0,0,0,2],[16,110,0.1455,0.25893,0.3163,0.0,0.14286,0.42857,0.0,1.0,13,3,0,13,0,6,0,0,4,0,0,3,0,0,1,0,0,2,0,0,0,0,3],[20,110,0.1818,0.2142,0.26003,0.0,0.14286,0.32143,0.0,1.0,13,1,0,13,0,8,0,0,3,0,0,3,0,0,2,0,0,2,0,0,0,0,1],[24,110,0.2182,0.15177,0.20803,0.0,0.14286,0.17857,0.0,1.0,14,1,0,14,0,10,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[28,110,0.2545,0.20981,0.33877,0.0,0.0,0.32143,0.0,1.0,20,4,0,20,0,2,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,4],[32,110,0.2909,0.2009,0.23652,0.0,0.14286,0.42857,0.0,1.0,14,1,0,14,0,6,0,0,2,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[36,110,0.3273,0.18304,0.29502,0.0,0.0,0.32143,0.0,1.0,20,2,0,20,0,3,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,2],[40,110,0.3636,0.16964,0.2299,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,7,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[44,110,0.4,0.21875,0.31539,0.0,0.0,0.32143,0.0,1.0,17,3,0,17,0,4,0,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,3],[48,110,0.4364,0.21429,0.26726,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,5,0,0,3,0,0,4,0,0,2,0,0,2,0,0,0,0,1],[52,110,0.4727,0.15624,0.23784,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,3,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[56,110,0.5091,0.27213,0.29318,0.105,0.14286,0.42857,0.0,1.0,8,3,0,8,0,12,0,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,3],[60,110,0.5455,0.23213,0.28514,0.0,0.07143,0.42858,0.0,1.0,16,1,0,16,0,4,0,0,0,0,0,5,0,0,4,0,0,2,0,0,0,0,1],[64,110,0.5818,0.23214,0.29179,0.0,0.14286,0.42857,0.0,1.0,12,3,0,12,0,9,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,3],[68,110,0.6182,0.19187,0.21904,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,7,0,0,2,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[72,110,0.6545,0.1339,0.19535,0.0,0.0,0.17857,0.0,0.71429,18,0,0,18,0,6,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[76,110,0.6909,0.13839,0.21866,0.0,0.0,0.17857,0.0,1.0,18,1,0,18,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[80,110,0.7273,0.12053,0.14772,0.0,0.07143,0.17857,0.0,0.57143,16,0,0,16,0,8,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[84,110,0.7636,0.16964,0.25111,0.0,0.0,0.28571,0.0,1.0,19,1,0,19,0,2,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,1],[88,110,0.8,0.06696,0.13356,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[92,110,0.8364,0.11607,0.15746,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,4,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[96,110,0.8727,0.17411,0.21349,0.0,0.0,0.42857,0.0,0.57143,17,0,0,17,0,4,0,0,1,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[100,110,0.9091,0.12054,0.16793,0.0,0.0,0.1429,0.0,0.57143,18,0,0,18,0,7,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[104,110,0.9455,0.08929,0.17035,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,3,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[108,110,0.9818,0.04911,0.12169,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[110,110,1.0,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.143,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"728e82875e4bf5b5","q":"Solve in the real numbers the equation  $ (n+1)^x+(n+3)^x+\\left( n^2+2n\\right)^x=n^x+(n+2)^x+\\left( n^2+4n+3\\right)^x, $  wher  $ n\\ge 2 $  is a fixed natural number.","t":[{"b":2,"e":1.0,"k":"rising","v":0.56239,"x":0.97768,"p":[[0,89,0.0,0.56239,0.33691,0.14286,0.64286,0.85704,0.14,1.0,0,7,0,0,0,10,0,0,1,0,0,3,0,0,2,0,0,6,0,0,3,0,7],[4,89,0.0449,0.94643,0.13243,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,27],[8,89,0.0899,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[12,89,0.1348,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[16,89,0.1798,0.95982,0.12993,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,27],[20,89,0.2247,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[24,89,0.2697,0.92856,0.16752,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[28,89,0.3146,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[32,89,0.3596,0.91516,0.13771,0.85711,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,22],[36,89,0.4045,0.94643,0.17035,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[40,89,0.4494,0.93302,0.13829,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,25],[44,89,0.4944,0.8884,0.19143,0.85713,1.0,1.0,0.143,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0,20],[48,89,0.5393,0.84375,0.25843,0.82132,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,19],[52,89,0.5843,0.8125,0.21852,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,7,0,0,4,0,15],[56,89,0.6292,0.84374,0.26813,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,0,2,0,22],[60,89,0.6742,0.90175,0.15751,0.82132,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,22],[64,89,0.7191,0.8616,0.22724,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,0,2,0,21],[68,89,0.764,0.7857,0.2342,0.67856,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,5,0,0,6,0,13],[72,89,0.809,0.89284,0.20205,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,22],[76,89,0.8539,0.74552,0.26663,0.57143,0.85707,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0,6,0,0,6,0,11],[80,89,0.8989,0.84818,0.1854,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,7,0,15],[84,89,0.9438,0.75443,0.26545,0.5713,0.78564,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,3,0,0,6,0,0,4,0,0,1,0,15],[88,89,0.9888,0.89285,0.18211,0.85711,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,4,0,21],[89,89,1.0,0.84374,0.26574,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,21]]},{"b":5,"e":0.71429,"k":"flat","v":0.63392,"x":0.98214,"p":[[0,93,0.0,0.69643,0.36202,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,7,0,0,2,0,0,2,0,0,0,0,0,4,0,0,0,0,17],[4,93,0.043,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,93,0.086,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[12,93,0.129,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,93,0.172,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[20,93,0.2151,0.89285,0.21724,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,23],[24,93,0.2581,0.89731,0.20277,0.85711,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,4,0,22],[28,93,0.3011,0.91963,0.147,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,23],[32,93,0.3441,0.90177,0.15749,0.85711,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,21],[36,93,0.3871,0.95089,0.13651,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[40,93,0.4301,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[44,93,0.4731,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27],[48,93,0.5161,0.87946,0.2055,0.82143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,6,0,0,4,0,20],[52,93,0.5591,0.80802,0.28708,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,19],[56,93,0.6022,0.89732,0.20897,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,23],[60,93,0.6452,0.90625,0.19103,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,24],[64,93,0.6882,0.91517,0.18851,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,24],[68,93,0.7312,0.90179,0.21852,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,25],[72,93,0.7742,0.89286,0.23419,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,24],[76,93,0.8172,0.86607,0.24728,0.82132,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,4,0,0,2,0,22],[80,93,0.8602,0.69191,0.27227,0.571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,1,0,0,3,0,0,6,0,0,7,0,0,2,0,10],[84,93,0.9032,0.68746,0.25864,0.5354,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,5,0,0,5,0,0,9,0,0,0,0,10],[88,93,0.9462,0.74994,0.27202,0.571,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,1,0,0,4,0,0,4,0,0,3,0,0,5,0,13],[92,93,0.9892,0.69639,0.29181,0.42857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,4,0,0,3,0,0,4,0,0,5,0,0,2,0,12],[93,93,1.0,0.63392,0.27879,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,5,0,0,6,0,0,0,0,0,9,0,0,3,0,7]]}]},{"i":"7e6b45e1933d5a63","q":"Let  $a_1, a_2, \\cdots, a_{2022}$  be non-negative integers such that their sum is  $2022$ . Let  $x$  denote the number of index  $i$  which satisfies  $a_i+a_{i+1}\\geqslant 3$ . Let  $y$  denote the number of index  $j$  which satisfies  $a_j\\ne a_{j+1}$ . Here, indices are taken modulo  $2022$ .\n\nDetermine the maximum possible value of  $x+y$ .\n\n*Duanyang ZHANG, High School Affiliated to Renmin University of 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0,0,1,0,4],[140,291,0.4811,0.58035,0.3008,0.28571,0.64286,0.85714,0.0,1.0,1,4,0,1,0,4,0,0,4,0,0,5,0,0,2,0,0,5,0,0,7,0,4],[144,291,0.4948,0.4375,0.29867,0.14286,0.42857,0.71429,0.0,1.0,1,4,0,1,0,9,0,0,5,0,0,7,0,0,1,0,0,4,0,0,1,0,4],[148,291,0.5086,0.36159,0.28117,0.14286,0.2857,0.57111,0.0,1.0,3,2,0,3,0,11,0,0,4,0,0,5,0,0,3,0,0,3,0,0,1,0,2],[152,291,0.5223,0.38393,0.29329,0.14286,0.35714,0.57141,0.0,1.0,4,2,0,4,0,8,0,0,4,0,0,7,0,0,3,0,0,1,0,0,3,0,2],[156,291,0.5361,0.30348,0.24425,0.14286,0.2857,0.42858,0.0,1.0,5,1,0,5,0,9,0,0,7,0,0,5,0,0,2,0,0,3,0,0,0,0,1],[160,291,0.5498,0.49106,0.31122,0.24999,0.42857,0.71429,0.0,1.0,1,5,0,1,0,7,0,0,6,0,0,4,0,0,2,0,0,6,0,0,1,0,5],[164,291,0.5636,0.33025,0.2955,0.14214,0.28571,0.57111,0.0,1.0,7,1,0,7,0,8,0,0,5,0,0,3,0,0,2,0,0,4,0,0,2,0,1],[168,291,0.5773,0.45081,0.30754,0.14289,0.42857,0.71429,0.0,1.0,2,3,0,2,0,9,0,0,3,0,0,5,0,0,2,0,0,6,0,0,2,0,3],[172,291,0.5911,0.35258,0.24225,0.14286,0.28571,0.46418,0.14,1.0,0,1,0,0,0,14,0,0,5,0,0,5,0,0,3,0,0,3,0,0,1,0,1],[176,291,0.6048,0.39284,0.29666,0.14286,0.35714,0.57143,0.0,1.0,5,2,0,5,0,6,0,0,5,0,0,5,0,0,4,0,0,3,0,0,2,0,2],[180,291,0.6186,0.26785,0.22797,0.14286,0.14286,0.28571,0.0,0.857,3,0,0,3,0,16,0,0,6,0,0,2,0,0,0,0,0,4,0,0,1,0,0],[184,291,0.6323,0.39286,0.26964,0.14286,0.28571,0.57143,0.0,1.0,3,1,0,3,0,6,0,0,8,0,0,6,0,0,2,0,0,3,0,0,3,0,1],[188,291,0.646,0.33929,0.28065,0.14286,0.21428,0.4286,0.0,1.0,2,3,0,2,0,14,0,0,4,0,0,5,0,0,2,0,0,2,0,0,0,0,3],[192,291,0.6598,0.27661,0.23951,0.14214,0.21428,0.42857,0.0,1.0,6,1,0,6,0,10,0,0,5,0,0,7,0,0,2,0,0,0,0,0,1,0,1],[196,291,0.6735,0.3616,0.25749,0.14286,0.28571,0.57111,0.0,0.85714,2,0,0,2,0,12,0,0,3,0,0,6,0,0,3,0,0,3,0,0,3,0,0],[200,291,0.6873,0.25893,0.24856,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,5,0,0,5,0,0,1,0,0,1,0,0,1,0,1],[204,291,0.701,0.29018,0.26362,0.14286,0.14288,0.42857,0.0,1.0,5,1,0,5,0,14,0,0,2,0,0,4,0,0,3,0,0,2,0,0,1,0,1],[208,291,0.7148,0.30803,0.29689,0.10714,0.14286,0.42857,0.0,1.0,8,1,0,8,0,9,0,0,3,0,0,5,0,0,0,0,0,4,0,0,2,0,1],[212,291,0.7285,0.33473,0.2849,0.14286,0.28571,0.46429,0.0,1.0,6,1,0,6,0,9,0,0,3,0,0,6,0,0,2,0,0,3,0,0,2,0,1],[216,291,0.7423,0.22765,0.23103,0.10714,0.14286,0.32143,0.0,0.85714,8,0,0,8,0,13,0,0,3,0,0,4,0,0,2,0,0,0,0,0,2,0,0],[220,291,0.756,0.25891,0.2185,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,10,0,0,5,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[224,291,0.7698,0.28125,0.289,0.14286,0.14286,0.32165,0.0,1.0,7,1,0,7,0,12,0,0,5,0,0,1,0,0,2,0,0,1,0,0,3,0,1],[228,291,0.7835,0.30348,0.27612,0.14286,0.14286,0.42857,0.0,1.0,4,2,0,4,0,15,0,0,1,0,0,6,0,0,2,0,0,1,0,0,1,0,2],[232,291,0.7973,0.16518,0.20858,0.0,0.14286,0.17857,0.0,0.85714,13,0,0,13,0,11,0,0,3,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[236,291,0.811,0.27232,0.24578,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,10,0,0,3,0,0,8,0,0,2,0,0,0,0,0,1,0,1],[240,291,0.8247,0.19643,0.21651,0.14286,0.14286,0.14286,0.0,0.85714,7,0,0,7,0,18,0,0,3,0,0,0,0,0,2,0,0,0,0,0,2,0,0],[244,291,0.8385,0.13839,0.19061,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,11,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[248,291,0.8522,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,12,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[252,291,0.866,0.04464,0.1357,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[256,291,0.8797,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[260,291,0.8935,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[264,291,0.9072,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[268,291,0.921,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[272,291,0.9347,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[276,291,0.9485,0.02223,0.06281,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[280,291,0.9622,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[284,291,0.9759,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[288,291,0.9897,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[291,291,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ecc39478317c41ce","q":"Show that the equation\n\n$$\nx^{2}+y^{2}+z^{2}=(x-y)(y-z)(z-x)\n$$\n\nhas infinitely many solutions in integers $x, y, z$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.19194,"x":0.95982,"p":[[0,87,0.0,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[4,87,0.046,0.88839,0.29393,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[8,87,0.092,0.84375,0.34323,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[12,87,0.1379,0.85267,0.33214,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,25],[16,87,0.1839,0.79016,0.37963,0.82132,1.0,1.0,0.0,1.0,5,23,0,5,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,23],[20,87,0.2299,0.60263,0.43848,0.0,0.85707,1.0,0.0,1.0,10,14,0,10,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,14],[24,87,0.2759,0.8125,0.38038,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[28,87,0.3218,0.80804,0.34922,0.71429,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,23],[32,87,0.3678,0.77692,0.39582,0.82143,1.0,1.0,0.0,1.0,6,23,0,6,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,23],[36,87,0.4138,0.73659,0.40581,0.57132,1.0,1.0,0.0,1.0,7,20,0,7,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,20],[40,87,0.4598,0.80803,0.35465,0.85711,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,23],[44,87,0.5057,0.68301,0.40207,0.2857,1.0,1.0,0.0,1.0,6,17,0,6,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,0,1,0,17],[48,87,0.5517,0.69196,0.44192,0.10714,1.0,1.0,0.0,1.0,8,20,0,8,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,20],[52,87,0.5977,0.74106,0.36324,0.57143,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,2,0,18],[56,87,0.6437,0.69642,0.42671,0.25,1.0,1.0,0.0,1.0,7,20,0,7,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,20],[60,87,0.6897,0.77232,0.38773,0.82143,1.0,1.0,0.0,1.0,6,21,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,21],[64,87,0.7356,0.66965,0.40158,0.2857,0.92857,1.0,0.0,1.0,5,16,0,5,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,3,0,16],[68,87,0.7816,0.67855,0.41803,0.21429,0.85714,1.0,0.0,1.0,8,15,0,8,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,15],[72,87,0.8276,0.77678,0.38785,0.85711,1.0,1.0,0.0,1.0,6,21,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,21],[76,87,0.8736,0.52678,0.46351,0.0,0.71429,1.0,0.0,1.0,13,13,0,13,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,13],[80,87,0.9195,0.35715,0.4432,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,9],[84,87,0.9655,0.19195,0.33427,0.0,0.0,0.28571,0.0,1.0,22,3,0,22,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,3],[87,87,1.0,0.19194,0.28925,0.0,0.0,0.42857,0.0,0.85714,21,0,0,21,0,0,0,0,2,0,0,3,0,0,2,0,0,2,0,0,2,0,0]]},{"b":1,"e":1.0,"k":"flat","v":0.51338,"x":1.0,"p":[[0,57,0.0,0.92411,0.23415,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,28],[4,57,0.0702,0.89284,0.26002,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,26],[8,57,0.1404,0.89732,0.25812,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,25],[12,57,0.2105,0.76786,0.40994,0.82143,1.0,1.0,0.0,1.0,7,23,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,23],[16,57,0.2807,0.80356,0.34396,0.82143,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,21],[20,57,0.3509,0.73661,0.41971,0.57143,1.0,1.0,0.0,1.0,7,21,0,7,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,21],[24,57,0.4211,0.56249,0.44022,0.0,0.64286,1.0,0.0,1.0,9,14,0,9,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,14],[28,57,0.4912,0.58482,0.43354,0.0,0.71429,1.0,0.0,1.0,10,14,0,10,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,1,0,14],[32,57,0.5614,0.51338,0.46271,0.0,0.64279,1.0,0.0,1.0,13,12,0,13,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,4,0,12],[36,57,0.6316,0.62053,0.45261,0.0,0.85714,1.0,0.0,1.0,11,13,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,13],[40,57,0.7018,0.70088,0.42911,0.25,1.0,1.0,0.0,1.0,7,21,0,7,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,21],[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],[48,57,0.8421,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,57,0.9123,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,57,0.9825,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[57,57,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"c806b9aa7b7c5009","q":"Let  $\\mathcal P$  be a finite set of squares on an infinite chessboard. Kelvin the Frog notes that  $\\mathcal P$  may be tiled with only  $1 \\times 2$  dominoes, while Alex the Kat notes that  $\\mathcal P$  may be tiled with only  $2 \\times 1$  dominoes. The dominoes cannot be rotated in each tiling. Prove that the area of  $\\mathcal P$  is a multiple of 4.","t":[{"b":1,"e":0.42857,"k":"volatile","v":0.23214,"x":0.84821,"p":[[0,15,0.0,0.66071,0.41458,0.2857,1.0,1.0,0.0,1.0,5,18,0,5,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,18],[4,15,0.2667,0.84821,0.31529,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,25],[8,15,0.5333,0.83036,0.32427,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,24],[12,15,0.8,0.25,0.18898,0.0,0.35714,0.42857,0.0,0.4286,9,0,0,9,0,6,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.23214,0.21053,0.0,0.28571,0.42857,0.0,0.57143,13,0,0,13,0,3,0,0,0,0,0,15,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.77232,"x":1.0,"p":[[0,57,0.0,0.89732,0.27947,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[4,57,0.0702,0.77232,0.36045,0.42857,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,22],[8,57,0.1404,0.77679,0.37617,0.64286,1.0,1.0,0.0,1.0,3,23,0,3,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,23],[12,57,0.2105,0.92411,0.2172,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,27],[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],[20,57,0.3509,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,57,0.5614,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[36,57,0.6316,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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":"8de0941639dba9b6","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}$ be positive real numbers, and let $S_{k}$ be the sum of products of $a_{1}, a_{2}, \\ldots, a_{n}$ taken $k$ at a time.\nShow that\n\n$$\nS_{k} S_{n-k} \\geq\\binom{ n}{k}^{2} a_{1} a_{2} \\ldots a_{n}, \\quad \\text { for } \\quad k=1,2, \\ldots, n-1\n$$","t":[{"b":0,"e":0.28571,"k":"falling","v":0.15178,"x":0.40625,"p":[[0,6,0.0,0.40625,0.46717,0.0,0.07143,1.0,0.0,1.0,16,12,0,16,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[4,6,0.6667,0.29464,0.37617,0.0,0.14285,0.35714,0.0,1.0,16,6,0,16,0,0,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,6],[6,6,1.0,0.15178,0.13803,0.0,0.21428,0.28571,0.0,0.28571,14,0,0,14,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.34375,"x":1.0,"p":[[0,38,0.0,0.34375,0.42387,0.0,0.14286,1.0,0.0,1.0,15,9,0,15,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[4,38,0.1053,0.92411,0.23954,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[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],[16,38,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"7d724b4e7bd62959","q":"Let  $a,b,c$  be real numbers. Consider the quadratic equation in  $\\cos{x}$  \\[ a \\cos^2{x}+b \\cos{x}+c=0.  \\] Using the numbers  $a,b,c$  form a quadratic equation in  $\\cos{2x}$  whose roots are the same as those of the original equation. Compare the equation in  $\\cos{x}$  and  $\\cos{2x}$  for  $a=4$ ,  $b=2$ ,  $c=-1$ .","t":[{"b":3,"e":1.0,"k":"volatile","v":0.53125,"x":1.0,"p":[[0,4,0.0,0.53125,0.38004,0.28571,0.35714,1.0,0.0,1.0,4,12,0,4,0,1,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,12],[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]]},{"b":6,"e":0.2857,"k":"falling","v":0.28571,"x":0.53571,"p":[[0,7,0.0,0.50892,0.36586,0.2857,0.35714,1.0,0.0,1.0,4,10,0,4,0,1,0,0,11,0,0,5,0,0,0,0,0,0,0,0,1,0,10],[4,7,0.5714,0.53571,0.37287,0.2857,0.28571,1.0,0.0,1.0,3,12,0,3,0,1,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,12],[7,7,1.0,0.28571,0.12877,0.2857,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,1,0,0,18,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"eea89cb955da6ce5","q":"Let  $\\triangle ABC$  which  $\\angle ABC$  are right angle, Let  $D$  be point on  $AB$  (  $D \\neq A , B$  ), Let  $E$  be point on line  $AB$  which  $B$  is the midpoint of  $DE$ , Let  $I$  be incenter of  $\\triangle ACE$  and  $J$  be  $A$ -excenter of  $\\triangle ACD$ . Prove that perpendicular bisector of  $BC$  bisects  $IJ$","t":[{"b":2,"e":0.14286,"k":"flat","v":0.00893,"x":0.03125,"p":[[0,20,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.0533,"p":[[0,48,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.0533,0.08536,0.0,0.0,0.14,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f46db3eeaf4b4038","q":"Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\\ldots+a_{0}$ be a polynomial with complex coefficients such that $a_{i} \\neq 0$ for all $i$. Prove that $|r| \\leq 2 \\max _{i=0}^{n-1}\\left|\\frac{a_{i-1}}{a_{i}}\\right|$ for all roots $r$ of all such polynomials $p$. Here we let $|z|$ denote the absolute value of the complex number $z$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.84821,"x":0.94643,"p":[[0,5,0.0,0.94643,0.14174,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,27],[4,5,0.8,0.84821,0.17835,0.67857,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,6,0,16],[5,5,1.0,0.85266,0.16937,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,5,0,16]]},{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":0.97768,"p":[[0,10,0.0,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[4,10,0.4,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[8,10,0.8,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[10,10,1.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29]]}]},{"i":"4fb8dc2d8a8b79b0","q":"Let  $P$  be a point inside a square  $ABCD$  such that  $PA:PB:PC$  is  $1:2:3$ . Determine the angle  $\\angle BPA$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,21,0.0,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[4,21,0.1905,0.95534,0.1871,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[8,21,0.381,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,21,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,21,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":1.0,"k":"flat","v":0.76784,"x":0.8683,"p":[[0,19,0.0,0.78571,0.34626,0.57143,1.0,1.0,0.0,1.0,3,21,0,3,0,2,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,21],[4,19,0.2105,0.79018,0.27603,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,0,0,0,7,1,0,2,0,0,1,1,17],[8,19,0.4211,0.76784,0.26667,0.57143,0.85714,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,0,0,0,11,0,0,3,0,0,0,0,16],[12,19,0.6316,0.8683,0.22312,0.82143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,3,1,20],[16,19,0.8421,0.8393,0.21048,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,2,0,19],[19,19,1.0,0.80133,0.17564,0.57143,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,9,1,0,4,0,0,7,0,11]]}]},{"i":"87930de844799c32","q":"$(x_{n})_{-\\infty<n<\\infty}$  is a sequence of real numbers which satisfies  $x_{n+1}=\\frac{x_{n}^2+10}{7}$  for every  $n \\in \\mathbb{Z}$ . If there exist a real upperbound for this sequence, find all the values  $x_{0}$  can take.","t":[{"b":0,"e":1.0,"k":"flat","v":0.98661,"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],[4,14,0.2857,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,14,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,14,0.8571,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[14,14,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.97321,"x":0.99554,"p":[[0,27,0.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,27,0.1481,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,27,0.2963,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,27,0.4444,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,27,0.5926,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,27,0.7407,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[24,27,0.8889,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[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":"7730e605bcd51d28","q":"The vertices of a triangle  $DEF$  lie on different sides of a triangle  $ABC$ . The lengths of the tangents from the incenter of  $DEF$  to the excircles of  $ABC$  are equal. Prove that  $4S_{DEF} \\ge S_{ABC}$ .\n\n*Note: By  $S_{XYZ}$  we denote the area of triangle  $XYZ$ .*","t":[{"b":4,"e":0.42857,"k":"falling","v":0.23214,"x":0.58473,"p":[[0,75,0.0,0.49554,0.22299,0.28571,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,5,0,0,3,0,0,3,0,0,9,0,0,11,0,0,0,0,0],[4,75,0.0533,0.5625,0.31731,0.39286,0.50001,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,2,0,0,8,0,0,2,0,0,4,0,0,4,0,6],[8,75,0.1067,0.26786,0.21354,0.14286,0.2143,0.32143,0.0,0.85714,4,0,0,4,0,12,0,0,8,0,0,4,0,0,2,0,0,0,0,0,2,0,0],[12,75,0.16,0.58473,0.29107,0.42857,0.64286,0.85714,0.0,1.0,2,3,0,2,0,3,0,0,2,0,0,5,0,0,4,0,0,6,0,0,7,0,3],[16,75,0.2133,0.39731,0.25934,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,8,0,0,2,0,0,8,0,0,5,0,0,4,0,0,1,0,1],[20,75,0.2667,0.44639,0.23889,0.2857,0.42857,0.57143,0.0,1.0,2,1,0,2,0,3,0,0,7,0,0,7,0,0,7,0,0,3,0,0,2,0,1],[24,75,0.32,0.3549,0.25472,0.14286,0.2857,0.4286,0.0,1.0,1,2,0,1,0,11,0,0,7,0,0,7,0,0,2,0,0,1,0,0,0,1,2],[28,75,0.3733,0.46651,0.25378,0.25,0.42859,0.66082,0.14286,1.0,0,1,0,0,0,8,0,0,4,0,0,5,0,0,6,1,0,4,0,0,3,0,1],[32,75,0.4267,0.39732,0.27833,0.14286,0.28571,0.60714,0.0,1.0,3,2,0,3,0,6,0,0,9,0,0,4,0,0,2,0,0,5,0,0,1,0,2],[36,75,0.48,0.48212,0.31082,0.14289,0.42857,0.74996,0.0,1.0,2,2,0,2,0,7,0,0,5,0,0,3,0,0,3,0,0,4,0,0,6,0,2],[40,75,0.5333,0.42857,0.25254,0.25,0.42857,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,7,0,0,5,0,0,3,0,0,7,0,0,1,0,1],[44,75,0.5867,0.23214,0.16656,0.14286,0.21428,0.32143,0.0,0.57143,6,0,0,6,0,10,0,0,8,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[48,75,0.64,0.26785,0.14617,0.14286,0.2857,0.42857,0.0,0.71429,1,0,0,1,0,13,0,0,9,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[52,75,0.6933,0.25,0.15152,0.14286,0.14286,0.42857,0.0,0.57143,2,0,0,2,0,16,0,0,3,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[56,75,0.7467,0.28129,0.15359,0.14286,0.2857,0.42857,0.0,0.71429,1,0,0,1,0,12,0,0,9,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[60,75,0.8,0.30802,0.19267,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,9,0,0,7,0,0,8,0,0,3,0,0,2,0,0,0,0,0],[64,75,0.8533,0.29018,0.14054,0.14289,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,8,0,0,10,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[68,75,0.9067,0.23214,0.16656,0.14286,0.14286,0.28571,0.0,0.71429,4,0,0,4,0,14,0,0,7,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[72,75,0.96,0.36159,0.14277,0.2857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,8,0,0,14,0,0,3,0,0,1,0,0,0,0,0],[75,75,1.0,0.23214,0.16269,0.14286,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,9,0,0,10,0,0,5,0,0,2,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.23215,"x":0.55802,"p":[[0,47,0.0,0.5067,0.24443,0.2857,0.64286,0.71429,0.0,0.71429,2,0,1,2,0,3,0,1,4,0,0,3,0,0,3,0,0,16,0,0,0,0,0],[4,47,0.0851,0.42409,0.2912,0.14286,0.42857,0.60714,0.0,1.0,5,1,0,5,0,5,0,0,3,0,0,5,0,0,6,0,0,4,0,0,3,0,1],[8,47,0.1702,0.33026,0.2845,0.14286,0.14286,0.60714,0.0,0.85714,3,0,0,3,0,16,0,0,2,0,0,1,0,0,2,0,0,5,0,0,3,0,0],[12,47,0.2553,0.37045,0.25478,0.14286,0.28571,0.46429,0.0,1.0,1,1,0,1,0,10,0,0,8,0,0,5,0,0,3,0,0,1,0,0,3,0,1],[16,47,0.3404,0.36606,0.25237,0.14286,0.28571,0.57143,0.0,0.85714,3,0,0,3,0,9,0,0,5,0,0,5,0,0,4,0,0,4,0,0,2,0,0],[20,47,0.4255,0.36606,0.25237,0.14286,0.28571,0.57111,0.0,0.85714,1,0,0,1,0,13,0,0,3,0,0,6,0,0,3,0,0,3,0,0,3,0,0],[24,47,0.5106,0.31241,0.27073,0.14214,0.2857,0.42858,0.0,0.85714,7,0,0,7,0,8,0,0,4,0,0,6,0,0,2,0,0,2,0,0,3,0,0],[28,47,0.5957,0.23215,0.21651,0.14286,0.14286,0.28571,0.0,0.85714,6,0,0,6,0,14,0,0,5,0,0,4,0,0,1,0,0,0,0,0,2,0,0],[32,47,0.6809,0.41516,0.29093,0.14286,0.42857,0.71429,0.0,1.0,3,1,0,3,0,9,0,0,3,0,0,4,0,0,4,0,0,5,0,0,3,0,1],[36,47,0.766,0.55802,0.3038,0.39286,0.57141,0.85714,0.0,1.0,3,4,0,3,0,2,0,0,3,0,0,6,0,0,4,0,0,5,0,0,5,0,4],[40,47,0.8511,0.50221,0.26213,0.28571,0.571,0.71429,0.0,1.0,3,1,0,3,0,2,0,0,4,0,0,5,0,1,6,0,0,7,0,0,3,0,1],[44,47,0.9362,0.41517,0.26572,0.14286,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,11,0,0,3,0,0,4,0,0,4,0,0,6,0,0,3,0,0],[47,47,1.0,0.3615,0.24226,0.14286,0.2857,0.57143,0.0,0.85714,3,0,0,3,0,9,0,0,5,0,0,4,0,0,6,0,0,4,0,0,1,0,0]]}]},{"i":"b7c3ce7c9b7815e6","q":"The sum of several (not necessarily different) real numbers from  $[0,1]$  doesn't exceed  $S$ .\nFind the maximum value of  $S$  such that it is always possible to partition these numbers into two groups with sums not greater than  $9$ .\n\n*(I. Gorodnin)*","t":[{"b":1,"e":0.57143,"k":"rising","v":0.22768,"x":0.68747,"p":[[0,47,0.0,0.22768,0.1551,0.14286,0.28571,0.28571,0.0,0.71429,7,0,3,7,0,4,0,0,18,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,47,0.0851,0.68747,0.2354,0.571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,3,0,0,5,0,0,7,0,0,8,0,5],[8,47,0.1702,0.62944,0.2283,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,5,0,0,5,0,0,7,0,0,8,0,2],[12,47,0.2553,0.61604,0.21261,0.42857,0.57143,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,5,0,0,7,0,0,7,0,0,6,0,2],[16,47,0.3404,0.64731,0.20198,0.4286,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,9,0,0,7,0,2],[20,47,0.4255,0.66963,0.23266,0.53539,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,3,0,0,5,0,0,8,0,0,6,0,5],[24,47,0.5106,0.64729,0.24481,0.42857,0.64286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,8,0,0,4,0,0,5,0,0,5,0,6],[28,47,0.5957,0.56695,0.21572,0.42857,0.57121,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,6,0,0,9,0,0,5,0,0,6,0,0,4,0,2],[32,47,0.6809,0.57143,0.21724,0.28571,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,9,0,0,3,0,0,6,0,0,7,0,0,7,0,0],[36,47,0.766,0.63393,0.27418,0.28571,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,10,0,0,4,0,6],[40,47,0.8511,0.61159,0.2506,0.39286,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,8,0,0,4,0,0,5,0,0,5,0,0,6,0,4],[44,47,0.9362,0.61161,0.2402,0.42857,0.64286,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,6,0,0,3,0,0,6,0,0,9,0,0,3,0,4],[47,47,1.0,0.53125,0.24804,0.28571,0.5,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,11,0,0,4,0,0,4,0,0,7,0,0,2,0,3]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.63837,"p":[[0,57,0.0,0.23214,0.14174,0.14286,0.28571,0.28571,0.0,0.57143,6,0,1,6,0,5,0,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,57,0.0702,0.63837,0.25997,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,6,0,0,3,0,0,5,0,0,7,0,0,4,0,6],[8,57,0.1404,0.60267,0.31081,0.39286,0.64286,0.85714,0.0,1.0,3,5,0,3,0,1,0,0,4,0,0,3,0,0,5,0,0,4,0,0,7,0,5],[12,57,0.2105,0.59822,0.27994,0.42857,0.64286,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,4,0,0,4,0,0,5,0,0,7,0,0,5,0,4],[16,57,0.2807,0.58482,0.27976,0.39286,0.57143,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,6,0,0,5,0,0,4,0,0,6,0,0,5,0,4],[20,57,0.3509,0.55802,0.26572,0.42857,0.57121,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,8,0,0,6,0,0,2,0,0,6,0,3],[24,57,0.4211,0.49106,0.29,0.28571,0.57143,0.71429,0.0,1.0,5,1,0,5,0,2,0,0,3,0,0,3,0,0,7,0,0,8,0,0,3,0,1],[28,57,0.4912,0.43749,0.26711,0.28571,0.35714,0.57143,0.0,1.0,3,2,0,3,0,0,0,0,13,0,0,6,0,0,4,0,0,0,0,0,4,0,2],[32,57,0.5614,0.44642,0.3067,0.28571,0.42857,0.71429,0.0,1.0,6,3,0,6,0,0,0,0,8,0,0,6,0,0,1,0,0,7,0,0,1,0,3],[36,57,0.6316,0.42407,0.25871,0.28571,0.35714,0.60714,0.0,0.85714,5,0,0,5,0,0,0,0,11,0,0,1,0,0,7,0,0,6,0,0,2,0,0],[40,57,0.7018,0.33926,0.27604,0.0,0.42857,0.571,0.0,1.0,10,1,0,10,0,1,0,0,3,0,0,9,0,0,6,0,0,1,0,0,1,0,1],[44,57,0.7719,0.38392,0.36497,0.0,0.28571,0.75,0.0,1.0,10,3,0,10,0,4,0,0,4,0,0,3,0,0,1,0,0,2,0,0,5,0,3],[48,57,0.8421,0.09822,0.19045,0.0,0.0,0.1429,0.0,0.85714,23,0,0,23,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[52,57,0.9123,0.04464,0.12078,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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],[57,57,1.0,0.02677,0.10367,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"0476f4e70f97359d","q":"Given an integer  $n\\geq 2$  and a closed unit disc, evaluate the maximum of the product of the lengths of all  $\\frac{n(n-1)}{2}$  segments determined by  $n$  points in that disc.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.14268,"x":0.31696,"p":[[0,26,0.0,0.28125,0.26362,0.14286,0.14286,0.32142,0.14286,1.0,0,3,0,0,0,23,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,3],[4,26,0.1538,0.31696,0.33261,0.14286,0.14286,0.17857,0.14286,1.0,0,6,0,0,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[8,26,0.3077,0.16955,0.14915,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,26,0.4615,0.19195,0.13171,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[16,26,0.6154,0.14723,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.16071,0.07784,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.18741,"x":0.29009,"p":[[0,13,0.0,0.26768,0.28523,0.14286,0.14286,0.14286,0.14,1.0,0,4,0,0,0,26,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[4,13,0.3077,0.29009,0.31033,0.14286,0.14286,0.14286,0.14,1.0,0,5,0,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[8,13,0.6154,0.28125,0.31029,0.14286,0.14286,0.14286,0.14286,1.0,0,5,0,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[12,13,0.9231,0.20535,0.18186,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,28,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[13,13,1.0,0.18741,0.14035,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0]]}]},{"i":"632f642af299fbf6","q":"Find pairs of positive integers  $(n, k)$  satisfying \\[(n + 1)^k - 1 = n!\\]","t":[{"b":0,"e":0.28571,"k":"flat","v":0.26339,"x":0.59375,"p":[[0,41,0.0,0.41071,0.23351,0.2857,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,1,0,0,18,0,0,9,0,0,0,0,0,0,0,0,0,0,4],[4,41,0.0976,0.4866,0.30064,0.2857,0.28571,0.57145,0.2857,1.0,0,8,0,0,0,0,0,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,8],[8,41,0.1951,0.45981,0.27602,0.28571,0.28571,0.46418,0.14286,1.0,0,5,0,0,0,1,0,0,18,0,0,5,0,0,1,0,0,0,0,0,2,0,5],[12,41,0.2927,0.57143,0.30723,0.28571,0.4286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,13,0,0,6,0,0,1,0,0,1,0,0,2,0,9],[16,41,0.3902,0.59375,0.30328,0.28571,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,11,0,0,7,0,0,1,0,0,1,0,0,3,0,9],[20,41,0.4878,0.59375,0.33902,0.28571,0.42859,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,9,0,0,6,0,0,1,0,0,0,0,0,1,0,12],[24,41,0.5854,0.44196,0.24578,0.28571,0.28571,0.42858,0.2857,1.0,0,3,0,0,0,0,0,0,19,0,0,6,0,0,1,0,0,0,0,0,3,0,3],[28,41,0.6829,0.37946,0.22759,0.28571,0.28571,0.32143,0.0,1.0,1,3,1,1,0,0,0,0,23,0,0,3,0,0,1,0,0,1,0,0,0,0,3],[32,41,0.7805,0.40625,0.25781,0.28571,0.28571,0.28571,0.2857,1.0,0,5,0,0,0,0,0,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[36,41,0.878,0.26339,0.06298,0.2857,0.28571,0.28571,0.0,0.28571,1,0,1,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.28571,0.06186,0.2857,0.28571,0.28571,0.0,0.4286,1,0,1,1,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.2991,0.08268,0.2857,0.28571,0.28571,0.0,0.57143,1,0,1,1,0,0,0,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.33035,"x":0.54911,"p":[[0,79,0.0,0.46875,0.27718,0.28571,0.28571,0.42857,0.2857,1.0,0,6,0,0,0,0,0,0,18,0,0,7,0,0,0,0,0,0,0,0,1,0,6],[4,79,0.0506,0.42857,0.24223,0.2857,0.28571,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,19,0,0,8,0,0,0,0,0,0,0,0,1,0,4],[8,79,0.1013,0.41071,0.23891,0.2857,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,1,0,0,20,0,0,5,0,0,2,0,0,0,0,0,0,0,4],[12,79,0.1519,0.54911,0.31362,0.28571,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,15,0,0,6,0,0,0,0,0,0,0,0,2,0,9],[16,79,0.2025,0.39732,0.22513,0.2857,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,3,0,0,17,0,0,6,0,0,2,0,0,0,0,0,2,0,2],[20,79,0.2532,0.41964,0.25985,0.28571,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,2,0,0,20,0,0,3,0,0,1,0,0,1,0,0,1,0,4],[24,79,0.3038,0.44196,0.27284,0.2857,0.28571,0.46429,0.14286,1.0,0,4,0,0,0,4,0,0,14,0,0,6,0,0,1,0,0,1,0,0,2,0,4],[28,79,0.3544,0.41072,0.23622,0.28571,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,1,0,0,19,0,0,7,0,0,1,0,0,0,0,0,0,0,4],[32,79,0.4051,0.38839,0.19638,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,3,0,0,14,0,0,11,0,0,1,0,0,1,0,0,0,0,2],[36,79,0.4557,0.41964,0.21706,0.28571,0.28571,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,17,0,0,11,0,0,0,0,0,0,0,0,1,0,3],[40,79,0.5063,0.41516,0.23515,0.28571,0.28571,0.46418,0.14286,1.0,0,2,0,0,0,2,0,0,19,0,0,3,0,0,2,0,0,2,0,0,2,0,2],[44,79,0.557,0.44643,0.28959,0.28571,0.28571,0.46431,0.14286,1.0,0,5,0,0,0,5,0,0,13,0,0,6,0,0,1,0,0,0,0,0,2,0,5],[48,79,0.6076,0.38393,0.23266,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,3,0,0,19,0,0,6,0,0,0,0,0,0,0,0,1,0,3],[52,79,0.6582,0.38839,0.20277,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,17,0,0,9,0,0,1,0,0,0,0,0,1,0,2],[56,79,0.7089,0.41518,0.24053,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,3,0,0,16,0,0,7,0,0,0,0,0,1,0,0,3,0,2],[60,79,0.7595,0.41964,0.26471,0.2857,0.28571,0.4286,0.14286,1.0,0,4,0,0,0,4,0,0,16,0,0,5,0,0,1,0,0,1,0,0,1,0,4],[64,79,0.8101,0.45089,0.26271,0.2857,0.28571,0.46431,0.14286,1.0,0,4,0,0,0,1,0,0,18,0,0,5,0,0,1,0,0,1,0,0,2,0,4],[68,79,0.8608,0.41517,0.24836,0.2857,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,22,0,0,2,0,0,1,0,0,1,0,0,2,0,3],[72,79,0.9114,0.35268,0.17122,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,18,0,0,9,0,0,0,0,0,0,0,0,1,0,1],[76,79,0.962,0.34374,0.20471,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,6,0,0,18,0,0,4,0,0,1,0,0,0,0,0,2,0,1],[79,79,1.0,0.33035,0.17655,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,5,0,0,19,0,0,6,0,0,0,0,0,0,0,0,1,0,1]]}]},{"i":"2a25f356495dbcec","q":"How many 6-tuples  $ (a_1,a_2,a_3,a_4,a_5,a_6)$  are there such that each of  $ a_1,a_2,a_3,a_4,a_5,a_6$  is from the set  $ \\{1,2,3,4\\}$  and the six expressions\r\n\\[ a_j^2 \\minus{} a_ja_{j \\plus{} 1} \\plus{} a_{j \\plus{} 1}^2\\]\r\nfor  $ j \\equal{} 1,2,3,4,5,6$  (where  $ a_7$  is to be taken as  $ a_1$ ) are all equal to one another?","t":[{"b":0,"e":1.0,"k":"flat","v":0.87947,"x":1.0,"p":[[0,89,0.0,0.87947,0.30327,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[4,89,0.0449,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,89,0.0899,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,89,0.1348,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,89,0.1798,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,89,0.2247,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,89,0.2697,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,89,0.3146,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,89,0.3596,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,89,0.4045,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,89,0.4494,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,89,0.4944,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,89,0.5393,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,89,0.5843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,89,0.6292,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,89,0.6742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,89,0.7191,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,89,0.764,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,89,0.809,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,89,0.8539,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,89,0.8989,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,89,0.9438,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[89,89,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":0.71429,"k":"flat","v":0.91071,"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],[4,51,0.0784,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,51,0.1569,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,51,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,51,0.3137,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,51,0.3922,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,51,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,51,0.549,0.99553,0.02486,1.0,1.0,1.0,0.8571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,51,0.6275,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,51,0.7059,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,51,0.7843,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[44,51,0.8627,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[48,51,0.9412,0.92857,0.13832,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,25],[51,51,1.0,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22]]}]},{"i":"952555991800a3a5","q":"Find all  $a\\in\\mathbb{R}$  for which there exists a non-constant function  $f: (0,1]\\rightarrow\\mathbb{R}$  such that \\[a+f(x+y-xy)+f(x)f(y)\\leq f(x)+f(y)\\] for all  $x,y\\in(0,1].$","t":[{"b":1,"e":0.0,"k":"falling","v":0.04911,"x":0.88393,"p":[[0,112,0.0,0.41518,0.29366,0.14286,0.42857,0.58932,0.0,1.0,4,2,0,4,0,7,0,0,3,0,0,5,0,1,4,1,0,3,0,0,2,0,2],[4,112,0.0357,0.88393,0.20341,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,18],[8,112,0.0714,0.75892,0.29975,0.71429,0.85714,1.0,0.0,1.0,3,11,0,3,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,8,0,11],[12,112,0.1071,0.66518,0.35285,0.57143,0.78571,0.89286,0.0,1.0,5,8,0,5,0,2,0,0,0,0,0,0,0,0,2,0,0,7,0,0,8,0,8],[16,112,0.1429,0.79911,0.25719,0.71429,0.85714,1.0,0.0,1.0,2,11,1,2,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,12,0,11],[20,112,0.1786,0.88393,0.15335,0.85714,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,8,0,17],[24,112,0.2143,0.72768,0.26332,0.71429,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,0,0,0,1,0,0,3,0,0,10,0,0,8,0,7],[28,112,0.25,0.77232,0.29851,0.71429,0.85714,1.0,0.0,1.0,3,14,0,3,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,5,0,14],[32,112,0.2857,0.71651,0.31768,0.55325,0.78571,1.0,0.0,1.0,3,12,0,3,0,1,0,0,0,0,0,3,0,1,1,0,0,7,0,0,4,0,12],[36,112,0.3214,0.75446,0.27718,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,2,0,0,1,0,0,2,0,0,0,0,0,7,0,0,9,0,10],[40,112,0.3571,0.70982,0.35262,0.57143,0.85714,1.0,0.0,1.0,4,13,0,4,0,2,0,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,13],[44,112,0.3929,0.70982,0.34531,0.71429,0.85714,1.0,0.0,1.0,4,11,0,4,0,2,0,0,0,0,0,1,0,0,0,0,0,7,0,0,7,0,11],[48,112,0.4286,0.66071,0.35129,0.35714,0.78571,1.0,0.0,1.0,2,10,0,2,0,6,0,0,0,0,0,1,0,0,2,0,0,5,0,0,6,0,10],[52,112,0.4643,0.59375,0.36441,0.25,0.71429,0.85714,0.0,1.0,6,6,0,6,0,2,0,0,1,0,0,2,0,0,1,0,0,7,0,0,7,0,6],[56,112,0.5,0.58271,0.39701,0.14286,0.82357,0.85714,0.0,1.0,7,6,0,7,0,4,0,0,0,0,0,0,0,0,1,0,0,3,1,0,10,0,6],[60,112,0.5357,0.54017,0.39405,0.14286,0.71429,0.85714,0.0,1.0,7,6,0,7,0,5,0,0,0,0,0,1,0,0,1,0,0,5,0,0,7,0,6],[64,112,0.5714,0.58482,0.41857,0.0,0.78571,1.0,0.0,1.0,9,10,0,9,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,0,6,0,10],[68,112,0.6071,0.54463,0.39357,0.14286,0.71429,0.89286,0.0,1.0,6,8,0,6,0,6,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,8],[72,112,0.6429,0.60268,0.40679,0.10714,0.71429,1.0,0.0,1.0,8,11,0,8,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,11],[76,112,0.6786,0.58034,0.3387,0.35714,0.64286,0.85714,0.0,1.0,3,6,0,3,0,5,0,0,0,0,0,5,0,0,3,0,0,4,0,0,6,0,6],[80,112,0.7143,0.62054,0.35823,0.42857,0.71429,1.0,0.0,1.0,6,9,0,6,0,1,0,0,0,0,0,2,0,0,3,0,0,9,0,0,2,0,9],[84,112,0.75,0.49554,0.4134,0.0,0.57143,0.85714,0.0,1.0,10,7,0,10,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,5,0,7],[88,112,0.7857,0.52678,0.39999,0.0,0.64286,0.85714,0.0,1.0,9,5,0,9,0,2,0,0,1,0,0,2,0,0,2,0,0,1,0,0,10,0,5],[92,112,0.8214,0.32589,0.36287,0.0,0.14286,0.60714,0.0,1.0,11,4,0,11,0,9,0,0,0,0,0,1,0,0,3,0,0,3,0,0,1,0,4],[96,112,0.8571,0.39732,0.38088,0.0,0.35714,0.71429,0.0,1.0,11,5,0,11,0,4,0,0,1,0,0,3,0,0,3,0,0,3,0,0,2,0,5],[100,112,0.8929,0.23214,0.33645,0.0,0.0,0.35714,0.0,1.0,17,2,0,17,0,6,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,2],[104,112,0.9286,0.27455,0.36514,0.0,0.14286,0.46429,0.0,1.0,13,4,0,13,1,9,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,4],[108,112,0.9643,0.12054,0.26513,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[112,112,1.0,0.04911,0.13651,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.16518,"x":0.85714,"p":[[0,134,0.0,0.39951,0.26233,0.19643,0.42857,0.57143,0.0,1.0,4,1,0,4,0,4,0,1,6,0,0,5,0,0,7,0,0,2,0,0,2,0,1],[4,134,0.0299,0.79017,0.25749,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,2,0,0,0,0,0,0,0,0,1,0,0,9,0,0,7,0,12],[8,134,0.0597,0.85713,0.22018,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,9,0,16],[12,134,0.0896,0.73214,0.30878,0.57143,0.85714,1.0,0.0,1.0,2,11,0,2,0,2,0,0,0,0,0,3,0,0,3,0,0,2,0,0,9,0,11],[16,134,0.1194,0.84375,0.14445,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,11,0,11],[20,134,0.1493,0.79018,0.25996,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,5,0,0,5,0,15],[24,134,0.1791,0.74107,0.25111,0.67857,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,4,0,0,1,0,0,5,0,0,12,0,7],[28,134,0.209,0.76112,0.24979,0.57107,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,3,0,0,7,1,11],[32,134,0.2388,0.85714,0.21429,0.85714,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,9,0,16],[36,134,0.2687,0.75445,0.2059,0.57143,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,3,0,0,13,0,6],[40,134,0.2985,0.69196,0.32362,0.67857,0.78571,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,0,0,0,2,0,0,1,0,0,8,0,0,8,0,8],[44,134,0.3284,0.74553,0.3108,0.71429,0.85714,1.0,0.0,1.0,3,11,0,3,0,1,0,0,0,0,0,2,0,0,1,0,0,5,0,0,9,0,11],[48,134,0.3582,0.69197,0.30537,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,5,0,0,0,0,0,1,0,0,0,0,0,11,0,0,6,0,8],[52,134,0.3881,0.74107,0.30606,0.71429,0.85714,1.0,0.0,1.0,3,9,0,3,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,0,12,0,9],[56,134,0.4179,0.71651,0.298,0.57143,0.78571,1.0,0.0,1.0,2,10,0,2,0,2,0,0,0,0,0,2,0,0,3,1,0,6,0,0,6,0,10],[60,134,0.4478,0.68973,0.33149,0.42857,0.71429,1.0,0.0,1.0,3,11,0,3,0,2,0,0,0,0,0,4,0,0,1,0,0,7,0,0,3,1,11],[64,134,0.4776,0.66518,0.35823,0.42857,0.71429,1.0,0.0,1.0,4,12,0,4,0,2,0,0,1,0,0,3,0,0,1,0,0,6,0,0,3,0,12],[68,134,0.5075,0.57812,0.33804,0.39286,0.71429,0.85714,0.0,1.0,4,6,0,4,0,3,0,0,1,0,0,5,0,0,2,0,0,7,0,0,3,1,6],[72,134,0.5373,0.71418,0.29898,0.57143,0.85714,1.0,0.0,1.0,2,9,0,2,0,2,0,0,1,0,0,0,0,0,5,0,0,5,0,0,8,0,9],[76,134,0.5672,0.72766,0.31615,0.57132,0.85714,1.0,0.0,1.0,3,11,0,3,0,1,0,0,0,0,0,3,0,0,2,0,0,4,0,0,8,0,11],[80,134,0.597,0.57588,0.32827,0.42857,0.57143,0.85714,0.0,1.0,3,6,0,3,0,4,0,0,0,0,0,7,0,0,3,0,0,4,0,0,5,0,6],[84,134,0.6269,0.52679,0.38867,0.14286,0.64286,0.85714,0.0,1.0,6,7,0,6,0,6,0,0,0,0,0,3,0,0,1,0,0,4,0,0,5,0,7],[88,134,0.6567,0.58929,0.34209,0.28571,0.71429,0.85714,0.0,1.0,4,5,0,4,0,3,0,0,2,0,0,3,0,0,2,0,0,5,0,0,8,0,5],[92,134,0.6866,0.63392,0.35524,0.42857,0.71429,1.0,0.0,1.0,4,9,0,4,0,3,0,0,0,0,0,4,0,0,2,0,0,4,0,0,6,0,9],[96,134,0.7164,0.4866,0.38607,0.14286,0.5,0.85714,0.0,1.0,7,6,0,7,0,6,0,0,0,0,0,3,0,0,3,0,0,2,0,0,5,0,6],[100,134,0.7463,0.45089,0.37476,0.0,0.42857,0.75,0.0,1.0,10,5,0,10,0,2,0,0,0,0,0,5,0,0,4,0,0,3,0,0,3,0,5],[104,134,0.7761,0.61607,0.34151,0.42857,0.71429,0.89286,0.0,1.0,4,8,0,4,0,3,0,0,0,0,0,3,0,0,3,0,0,8,0,0,3,0,8],[108,134,0.806,0.45536,0.37701,0.0,0.57143,0.85714,0.0,1.0,9,3,0,9,0,5,0,0,0,0,0,0,0,0,5,0,0,4,0,0,6,0,3],[112,134,0.8358,0.38838,0.36462,0.0,0.21428,0.71429,0.0,1.0,9,4,0,9,0,7,0,0,1,0,0,2,0,0,3,0,0,4,0,0,2,0,4],[116,134,0.8657,0.35268,0.38463,0.10714,0.14286,0.85714,0.0,1.0,8,5,0,8,0,13,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,5],[120,134,0.8955,0.27679,0.3387,0.0,0.14286,0.5,0.0,1.0,12,3,0,12,0,10,0,0,0,0,0,2,0,0,0,0,0,5,0,0,0,0,3],[124,134,0.9254,0.29911,0.32608,0.0,0.14286,0.57143,0.0,0.85714,13,0,0,13,0,6,0,0,0,0,0,2,0,0,4,0,0,3,0,0,4,0,0],[128,134,0.9552,0.16518,0.27689,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,8,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,1],[132,134,0.9851,0.26786,0.32684,0.0,0.14286,0.57143,0.0,1.0,15,1,0,15,0,5,0,0,1,0,0,2,0,0,2,0,0,4,0,0,2,0,1],[134,134,1.0,0.17857,0.34256,0.0,0.0,0.14286,0.0,1.0,23,3,0,23,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,3]]}]},{"i":"599499766c98a2da","q":"An acute triangle $A B C$ is given and let $H$ be its orthocenter. Let $\\omega$ be the circle through $B, C$ and $H$, and let $\\Gamma$ be the circle with diameter $A H$. Let $X \\neq H$ be the other intersection point of $\\omega$ and $\\Gamma$, and let $\\gamma$ be the reflection of $\\Gamma$ over $A X$.\n\nSuppose $\\gamma$ and $\\omega$ intersect again at $Y \\neq X$, and line $A H$ and $\\omega$ intersect again at $Z \\neq H$. Show that the circle through $A, Y, Z$ passes through the midpoint of segment $B C$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.30802,"p":[[0,77,0.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.23661,0.27107,0.0,0.14286,0.32143,0.0,0.85714,10,0,0,10,0,12,0,0,2,0,0,2,0,0,2,0,0,1,0,0,3,0,0],[8,77,0.1039,0.24999,0.29449,0.0,0.14286,0.5711,0.0,0.85714,11,0,0,11,0,12,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0],[12,77,0.1558,0.30802,0.26988,0.14286,0.14286,0.42857,0.0,0.85714,5,0,0,5,0,12,0,0,3,0,0,6,0,0,1,0,0,1,0,0,4,0,0],[16,77,0.2078,0.19643,0.27606,0.0,0.14286,0.1786,0.0,0.85714,14,0,0,14,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0,4,0,0],[20,77,0.2597,0.21429,0.27433,0.0,0.14286,0.32142,0.0,0.85714,14,0,0,14,0,8,0,0,2,0,0,2,0,0,3,0,0,0,0,0,3,0,0],[24,77,0.3117,0.25892,0.31223,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,10,0,0,4,0,0,0,0,0,1,0,0,0,0,0,6,0,0],[28,77,0.3636,0.2009,0.30485,0.0,0.07143,0.1429,0.0,0.85714,16,0,0,16,0,9,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,0],[32,77,0.4156,0.22321,0.29218,0.0,0.14286,0.2857,0.0,0.85714,13,0,0,13,0,10,0,0,2,0,0,1,0,0,1,0,0,1,0,0,4,0,0],[36,77,0.4675,0.17848,0.28123,0.0,0.0,0.14286,0.0,0.85714,17,0,0,17,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0,4,0,0],[40,77,0.5195,0.14284,0.19882,0.0,0.07143,0.17857,0.0,0.85714,16,0,0,16,0,8,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[44,77,0.5714,0.13393,0.20806,0.0,0.0,0.14287,0.0,0.85714,18,0,0,18,0,7,0,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[48,77,0.6234,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,77,0.6753,0.08036,0.12339,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,77,0.7273,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,77,0.7792,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,77,0.8312,0.07589,0.11836,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,77,0.8831,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,77,0.9351,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[77,77,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"flat","v":0.09379,"x":0.39284,"p":[[0,80,0.0,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,80,0.05,0.39284,0.30513,0.14286,0.42857,0.60714,0.0,0.85714,5,0,0,5,0,9,0,0,1,0,0,5,0,0,4,0,0,2,0,0,6,0,0],[8,80,0.1,0.3482,0.305,0.14286,0.14288,0.57143,0.0,0.85714,6,0,0,6,0,11,0,0,1,0,0,3,0,0,4,0,0,2,0,0,5,0,0],[12,80,0.15,0.26783,0.28513,0.0,0.14286,0.4642,0.0,1.0,12,1,0,12,0,6,0,0,2,0,0,4,0,0,5,0,0,1,0,0,1,0,1],[16,80,0.2,0.20536,0.23941,0.0,0.14286,0.32143,0.0,0.85714,14,0,0,14,0,6,0,0,4,0,0,2,0,0,5,0,0,0,0,0,1,0,0],[20,80,0.25,0.25893,0.27765,0.0,0.14286,0.42857,0.0,0.85714,10,0,0,10,0,9,0,0,4,0,0,3,0,0,1,0,0,2,0,0,3,0,0],[24,80,0.3,0.17857,0.24484,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,7,0,0,6,0,0,0,0,0,1,0,0,1,0,0,2,0,0],[28,80,0.35,0.1875,0.2635,0.0,0.07143,0.28571,0.0,0.85714,16,0,0,16,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,0,3,0,0],[32,80,0.4,0.16518,0.22047,0.0,0.14286,0.2857,0.0,0.85714,15,0,0,15,0,8,0,0,4,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[36,80,0.45,0.17409,0.21644,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,6,0,0,5,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[40,80,0.5,0.09822,0.14914,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,10,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[44,80,0.55,0.27231,0.32995,0.0,0.14286,0.4642,0.0,0.85714,13,0,0,13,0,8,0,0,1,0,0,2,0,0,1,0,0,1,0,0,6,0,0],[48,80,0.6,0.11606,0.20649,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[52,80,0.65,0.12499,0.2362,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,6,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[56,80,0.7,0.09379,0.13662,0.0,0.0,0.14286,0.0,0.43,19,0,0,19,0,8,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[60,80,0.75,0.13393,0.23402,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,8,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[64,80,0.8,0.21427,0.29231,0.0,0.0,0.42857,0.0,0.85714,18,0,0,18,0,2,0,0,3,0,0,3,0,0,2,0,0,1,0,0,3,0,0],[68,80,0.85,0.26783,0.32486,0.0,0.14286,0.571,0.0,0.85714,12,0,0,12,0,10,0,0,1,0,0,0,0,0,3,0,0,0,0,0,6,0,0],[72,80,0.9,0.20979,0.29008,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,4,0,0,4,0,0,0,0,0,3,0,0,1,0,0,3,0,0],[76,80,0.95,0.17856,0.23143,0.0,0.14286,0.2857,0.0,0.85714,14,0,0,14,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[80,80,1.0,0.11607,0.1729,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,6,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"12854dbb62db3dd9","q":"Choose positive integers  $b_1, b_2, \\dotsc$  satisfying\n\\[1=\\frac{b_1}{1^2} > \\frac{b_2}{2^2} > \\frac{b_3}{3^2} > \\frac{b_4}{4^2} > \\dotsb\\]\nand let  $r$  denote the largest real number satisfying  $\\tfrac{b_n}{n^2} \\geq r$  for all positive integers  $n$ . What are the possible values of  $r$  across all possible choices of the sequence  $(b_n)$ ?\n\n*Carl Schildkraut and Milan Haiman*","t":[{"b":3,"e":1.0,"k":"rising","v":0.54464,"x":0.94643,"p":[[0,42,0.0,0.54464,0.1729,0.42859,0.57143,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,0,0,0,7,0,0,18,0,0,3,0,0,1,0,1],[4,42,0.0952,0.86607,0.24206,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,22],[8,42,0.1905,0.82588,0.28736,0.78571,1.0,1.0,0.0,1.0,1,20,0,1,0,2,0,0,0,0,0,1,0,0,4,0,0,0,0,0,4,0,20],[12,42,0.2857,0.84821,0.29653,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,22],[16,42,0.381,0.875,0.22517,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,1,0,22],[20,42,0.4762,0.74107,0.33396,0.57143,0.85714,1.0,0.0,1.0,2,14,0,2,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,14],[24,42,0.5714,0.79017,0.28344,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,2,0,0,9,0,14],[28,42,0.6667,0.94195,0.11219,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[32,42,0.7619,0.90625,0.12682,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,19],[36,42,0.8571,0.91964,0.14698,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,21],[40,42,0.9524,0.92411,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,22],[42,42,1.0,0.94643,0.13716,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27]]},{"b":7,"e":0.0,"k":"falling","v":0.33929,"x":0.95089,"p":[[0,50,0.0,0.51339,0.25719,0.53569,0.57143,0.57143,0.0,1.0,4,2,3,4,0,2,0,0,0,0,0,2,0,0,18,0,0,3,0,0,1,0,2],[4,50,0.08,0.88839,0.25688,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,24],[8,50,0.16,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[12,50,0.24,0.92856,0.18213,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,26],[16,50,0.32,0.91515,0.20163,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,24],[20,50,0.4,0.90625,0.18073,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,22],[24,50,0.48,0.80356,0.33646,0.67857,1.0,1.0,0.0,1.0,2,22,0,2,0,3,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,22],[28,50,0.56,0.73661,0.33714,0.57142,0.92857,1.0,0.0,1.0,2,16,0,2,0,3,0,0,1,0,0,1,0,0,2,0,0,5,0,0,2,0,16],[32,50,0.64,0.71429,0.35535,0.53571,0.85714,1.0,0.0,1.0,2,14,0,2,0,5,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,14],[36,50,0.72,0.71204,0.3898,0.46396,0.92857,1.0,0.0,1.0,4,16,0,4,1,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,16],[40,50,0.8,0.75445,0.35398,0.67846,0.92857,1.0,0.0,1.0,3,16,0,3,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,16],[44,50,0.88,0.69643,0.37072,0.57143,0.85714,1.0,0.0,1.0,5,12,0,5,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,12],[48,50,0.96,0.60713,0.4165,0.14286,0.78571,1.0,0.0,1.0,7,13,0,7,0,3,0,0,1,0,0,0,0,0,3,0,0,2,0,0,3,0,13],[50,50,1.0,0.33929,0.42371,0.0,0.14286,0.89286,0.0,1.0,15,8,0,15,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,8]]}]},{"i":"d1a59928f9795225","q":"For any natural number  $ n\\ge 2, $  define  $ m(n) $  to be the minimum number of elements of a set  $ S $  that simultaneously satisfy: $ \\text{(i)}\\quad \\{ 1,n\\} \\subset S\\subset \\{ 1,2,\\ldots ,n\\} $  $ \\text{(ii)}\\quad $  any element of  $ S, $  distinct from  $ 1, $  is equal to the sum of two (not necessarily distinct) elements from  $ S. $ **a)** Prove that  $ m(n)\\ge 1+\\left\\lfloor \\log_2 n \\right\\rfloor ,\\quad\\forall n\\in\\mathbb{N}_{\\ge 2} . $ **b)** Prove that there are infinitely many natural numbers  $ n\\ge 2 $  such that  $ m(n)=m(n+1). $  $ \\lfloor\\rfloor $  denotes the usual integer part.","t":[{"b":3,"e":0.71429,"k":"falling","v":0.5802,"x":0.94643,"p":[[0,44,0.0,0.94643,0.14174,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,27],[4,44,0.0909,0.857,0.20836,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,4,0,0,2,0,20],[8,44,0.1818,0.94196,0.14664,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,26],[12,44,0.2727,0.89732,0.20589,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,24],[16,44,0.3636,0.91964,0.2141,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,27],[20,44,0.4545,0.92411,0.17122,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,25],[24,44,0.5455,0.73657,0.196,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,10,0,0,0,0,10],[28,44,0.6364,0.70979,0.20668,0.57143,0.64286,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,4,0,0,4,0,8],[32,44,0.7273,0.66513,0.21608,0.53575,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,8,0,0,12,0,0,3,0,0,1,0,8],[36,44,0.8182,0.62942,0.24187,0.42857,0.49979,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,16,0,0,5,0,0,1,0,0,2,0,8],[40,44,0.9091,0.5802,0.14245,0.4286,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,11,0,0,8,0,0,1,0,1],[44,44,1.0,0.58475,0.13997,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,12,0,0,8,0,0,1,0,1]]},{"b":4,"e":0.71429,"k":"falling","v":0.71423,"x":0.98661,"p":[[0,36,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,36,0.1111,0.84374,0.21537,0.57143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,1,0,20],[8,36,0.2222,0.85265,0.21576,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,3,0,20],[12,36,0.3333,0.87052,0.21239,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,0,2,0,22],[16,36,0.4444,0.88839,0.19144,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,3,0,22],[20,36,0.5556,0.86159,0.21868,0.67857,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,1,0,22],[24,36,0.6667,0.85714,0.19885,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,3,0,19],[28,36,0.7778,0.90625,0.18423,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,24],[32,36,0.8889,0.80802,0.16215,0.71429,0.71429,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,4,0,11],[36,36,1.0,0.71423,0.15572,0.57143,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,9,0,0,10,0,2]]}]},{"i":"534aee680455dede","q":"In a planar rectangular coordinate system, a sequence of points  ${A_n}$  on the positive half of the y-axis and a sequence of points  ${B_n}$  on the curve  $y=\\sqrt{2x}$   $(x\\ge0)$  satisfy the condition  $|OA_n|=|OB_n|=\\frac{1}{n}$ . The x-intercept of line  $A_nB_n$  is  $a_n$ , and the x-coordinate of point  $B_n$  is  $b_n$ ,  $n\\in\\mathbb{N}$ . Prove that\n(1)  $a_n>a_{n+1}>4$ ,  $n\\in\\mathbb{N}$ ;\n(2) There is  $n_0\\in\\mathbb{N}$ , such that for any  $n>n_0$ ,  $\\frac{b_2}{b_1}+\\frac{b_3}{b_2}+\\ldots +\\frac{b_n}{b_{n-1}}+\\frac{b_{n+1}}{b_n}<n-2004$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.85713,"x":1.0,"p":[[0,92,0.0,0.85713,0.1713,0.82143,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,9,0,15],[4,92,0.0435,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,92,0.087,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[12,92,0.1304,0.97321,0.10374,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[16,92,0.1739,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,92,0.2174,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,92,0.2609,0.94196,0.14223,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[28,92,0.3043,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,92,0.3478,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,92,0.3913,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[40,92,0.4348,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,92,0.4783,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[48,92,0.5217,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[52,92,0.5652,0.92856,0.12376,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[56,92,0.6087,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[60,92,0.6522,0.95088,0.09857,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[64,92,0.6957,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[68,92,0.7391,0.92411,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[72,92,0.7826,0.95535,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[76,92,0.8261,0.87498,0.15467,0.85711,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,11,0,15],[80,92,0.8696,0.94647,0.1169,0.96429,1.0,1.0,0.43,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[84,92,0.913,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,27],[88,92,0.9565,0.92857,0.08748,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,18],[92,92,1.0,0.87052,0.15306,0.85714,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,13,0,13]]},{"b":3,"e":0.4286,"k":"flat","v":0.84813,"x":0.99554,"p":[[0,54,0.0,0.87053,0.21535,0.82132,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,4,0,0,4,0,20],[4,54,0.0741,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[8,54,0.1481,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,54,0.2222,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[20,54,0.3704,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[24,54,0.4444,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[28,54,0.5185,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,54,0.5926,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[36,54,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,54,0.7407,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[44,54,0.8148,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[48,54,0.8889,0.95089,0.12168,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[52,54,0.963,0.84813,0.22598,0.71429,1.0,1.0,0.14,1.0,0,18,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,0,5,0,18],[54,54,1.0,0.87053,0.16114,0.85711,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,10,0,15]]}]},{"i":"48a6ecf6c41a0a4c","q":"In a family there are four children of di\ufb00erent ages, each age being a positive integer not less than  $2$  and not greater than  $16$ . A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?","t":[{"b":1,"e":1.0,"k":"rising","v":0.76785,"x":0.97768,"p":[[0,182,0.0,0.79911,0.30693,0.67857,1.0,1.0,0.0,1.0,2,19,1,2,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,0,3,0,19],[4,182,0.022,0.8125,0.18707,0.82143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,0,17,0,7],[8,182,0.044,0.82143,0.14286,0.71429,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,16,0,6],[12,182,0.0659,0.78124,0.22299,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,5,0,0,13,0,8],[16,182,0.0879,0.77232,0.25218,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,7,0,0,11,0,9],[20,182,0.1099,0.80802,0.19759,0.82132,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,0,17,0,7],[24,182,0.1319,0.86606,0.16728,0.85711,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,12],[28,182,0.1538,0.80357,0.23623,0.82143,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,15,0,9],[32,182,0.1758,0.80804,0.26633,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,0,12,0,13],[36,182,0.1978,0.76785,0.26666,0.67857,0.85714,1.0,0.0,1.0,1,10,1,1,0,2,0,0,0,0,0,1,0,0,4,0,0,3,0,0,11,0,10],[40,182,0.2198,0.85714,0.20825,0.85714,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,11,0,15],[44,182,0.2418,0.91518,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[48,182,0.2637,0.86606,0.15126,0.857,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,14,0,12],[52,182,0.2857,0.86607,0.20806,0.85714,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,15],[56,182,0.3077,0.83929,0.20438,0.85714,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,16,0,10],[60,182,0.3297,0.83929,0.20438,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,5,0,0,9,0,14],[64,182,0.3516,0.87499,0.14617,0.85714,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,16,0,12],[68,182,0.3736,0.83913,0.13731,0.82132,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,16,0,8],[72,182,0.3956,0.86161,0.12103,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,15,0,10],[76,182,0.4176,0.87053,0.12556,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,18,0,10],[80,182,0.4396,0.89285,0.13363,0.85714,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,17,0,13],[84,182,0.4615,0.89285,0.10102,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,12],[88,182,0.4835,0.90624,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[92,182,0.5055,0.85268,0.15355,0.85714,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,18,0,9],[96,182,0.5275,0.8482,0.15947,0.85714,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,18,0,9],[100,182,0.5495,0.84374,0.14445,0.85713,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,23,0,5],[104,182,0.5714,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[108,182,0.5934,0.9241,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[112,182,0.6154,0.87946,0.11356,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,14,0,12],[116,182,0.6374,0.875,0.12242,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,16,0,11],[120,182,0.6593,0.91964,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,18],[124,182,0.6813,0.91963,0.10063,0.85714,1.0,1.0,0.5714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[128,182,0.7033,0.90625,0.10479,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,15],[132,182,0.7253,0.94642,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[136,182,0.7473,0.91517,0.10013,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,16],[140,182,0.7692,0.89285,0.09449,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,12],[144,182,0.7912,0.90625,0.12169,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,18],[148,182,0.8132,0.91518,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,17],[152,182,0.8352,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[156,182,0.8571,0.90625,0.13175,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[160,182,0.8791,0.90179,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,14],[164,182,0.9011,0.91964,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,19],[168,182,0.9231,0.88839,0.10555,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,16,0,12],[172,182,0.9451,0.89732,0.1394,0.85711,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,7,0,18],[176,182,0.967,0.90177,0.09741,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[180,182,0.989,0.9241,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18],[182,182,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]},{"b":5,"e":1.0,"k":"flat","v":0.77678,"x":0.96429,"p":[[0,173,0.0,0.875,0.25939,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,24],[4,173,0.0231,0.77678,0.3008,0.85711,0.85714,1.0,0.0,1.0,2,11,1,2,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,14,0,11],[8,173,0.0462,0.85267,0.19061,0.85711,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,12],[12,173,0.0694,0.85268,0.16554,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,15,0,11],[16,173,0.0925,0.9241,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[20,173,0.1156,0.88392,0.0974,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,16,0,11],[24,173,0.1387,0.8125,0.19045,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,13,0,8],[28,173,0.1618,0.82588,0.23072,0.85711,0.85714,1.0,0.0,1.0,2,9,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,9],[32,173,0.185,0.87946,0.10779,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,19,0,10],[36,173,0.2081,0.85268,0.145,0.82143,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,13,0,11],[40,173,0.2312,0.85267,0.2004,0.85714,0.85714,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,15,0,12],[44,173,0.2543,0.89285,0.10714,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,14],[48,173,0.2775,0.89286,0.10714,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,15,0,13],[52,173,0.3006,0.86594,0.12356,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,14,0,11],[56,173,0.3237,0.85267,0.12103,0.857,0.85714,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,17,0,8],[60,173,0.3468,0.87054,0.12037,0.85714,0.85714,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,20,0,9],[64,173,0.3699,0.88392,0.11538,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,13],[68,173,0.3931,0.875,0.15465,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,12,0,14],[72,173,0.4162,0.90624,0.09182,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,14],[76,173,0.4393,0.89285,0.10714,0.85713,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,14],[80,173,0.4624,0.8215,0.18561,0.85714,0.85714,0.85786,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,0,18,0,7],[84,173,0.4855,0.84821,0.16727,0.857,0.85714,1.0,0.1429,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,15,0,10],[88,173,0.5087,0.875,0.11152,0.85714,0.85714,1.0,0.5714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,15,0,11],[92,173,0.5318,0.88392,0.11538,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,16,0,12],[96,173,0.5549,0.86607,0.10062,0.85711,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,16,0,9],[100,173,0.578,0.89285,0.12372,0.85714,0.85714,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,17,0,13],[104,173,0.6012,0.85714,0.14725,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,17,0,10],[108,173,0.6243,0.82129,0.18219,0.71429,0.85714,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,16,0,7],[112,173,0.6474,0.83927,0.13716,0.857,0.85714,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,19,0,7],[116,173,0.6705,0.91071,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15],[120,173,0.6936,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[124,173,0.7168,0.89731,0.11971,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,13,0,15],[128,173,0.7399,0.92857,0.08749,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,18],[132,173,0.763,0.875,0.12242,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,16,0,11],[136,173,0.7861,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[140,173,0.8092,0.90177,0.09741,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[144,173,0.8324,0.91517,0.07874,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[148,173,0.8555,0.87946,0.18249,0.85714,0.85714,1.0,0.0,1.0,1,14,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,14],[152,173,0.8786,0.93303,0.15562,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,22],[156,173,0.9017,0.92856,0.10715,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[160,173,0.9249,0.87054,0.13054,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,14,0,12],[164,173,0.948,0.89285,0.18558,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,19],[168,173,0.9711,0.85267,0.22724,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,9,0,17],[172,173,0.9942,0.95536,0.12078,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[173,173,1.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27]]}]},{"i":"8e92cb950ce51cac","q":"Let  $n$  be a positive integer.  Suppose we are given  $2^n+1$  distinct sets, each containing finitely many objects.  Place each set into one of two categories, the red sets and the blue sets, so that there is at least one set in each category.  We define the *symmetric difference* of two sets as the set of objects belonging to exactly one of the two sets.  Prove that there are at least  $2^n$  different sets which can be obtained as the symmetric difference of a red set and a blue set.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.14286,"x":0.48662,"p":[[0,38,0.0,0.20089,0.21387,0.14286,0.14286,0.14286,0.0,1.0,1,2,1,1,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,38,0.1053,0.48662,0.38107,0.14286,0.21428,1.0,0.14286,1.0,0,9,0,0,0,16,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,9],[8,38,0.2105,0.33482,0.32065,0.14286,0.14286,0.5,0.14286,1.0,0,3,0,0,0,23,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,3],[12,38,0.3158,0.19643,0.15047,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,26,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[16,38,0.4211,0.20982,0.22583,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[20,38,0.5263,0.2008,0.18164,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[24,38,0.6316,0.30804,0.32559,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,3],[28,38,0.7368,0.17402,0.13238,0.14286,0.14286,0.14286,0.14,0.85714,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[32,38,0.8421,0.16071,0.1275,0.14286,0.14286,0.14286,0.0,0.857,1,0,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[36,38,0.9474,0.16509,0.0883,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"flat","v":0.16509,"x":0.50893,"p":[[0,53,0.0,0.16509,0.152,0.14286,0.14286,0.14286,0.0,1.0,1,1,1,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,53,0.0755,0.36607,0.35524,0.14286,0.14286,0.5,0.14286,1.0,0,7,0,0,0,22,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,7],[8,53,0.1509,0.50893,0.36585,0.14286,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,14,0,0,1,0,0,2,0,0,1,0,0,4,0,0,2,0,8],[12,53,0.2264,0.37927,0.33057,0.14286,0.14286,0.60682,0.14,1.0,0,5,0,0,0,19,0,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,5],[16,53,0.3019,0.36143,0.33321,0.14286,0.14286,0.71429,0.14,1.0,0,5,0,0,0,21,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,5],[20,53,0.3774,0.32579,0.30147,0.14286,0.14286,0.60682,0.14,1.0,0,2,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,2],[24,53,0.4528,0.35267,0.29662,0.14286,0.14286,0.60682,0.14286,1.0,0,3,0,0,0,19,0,0,2,0,0,2,0,0,1,0,0,5,0,0,0,0,3],[28,53,0.5283,0.50444,0.34066,0.14286,0.4998,0.71429,0.14286,1.0,0,7,0,0,0,13,0,0,0,0,0,3,0,0,3,0,0,6,0,0,0,0,7],[32,53,0.6038,0.34372,0.28538,0.14286,0.14286,0.571,0.14286,1.0,0,2,0,0,0,19,0,0,2,0,0,2,0,0,3,0,0,2,0,0,2,0,2],[36,53,0.6792,0.30795,0.28151,0.14286,0.14286,0.42858,0.14,1.0,0,3,0,0,0,22,0,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,3],[40,53,0.7547,0.32589,0.27947,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,19,0,0,3,0,0,4,0,0,0,0,0,3,0,0,0,0,3],[44,53,0.8302,0.33474,0.25664,0.14286,0.14286,0.46431,0.14,1.0,0,1,0,0,0,19,0,0,0,0,0,5,0,0,1,0,0,6,0,0,0,0,1],[48,53,0.9057,0.36596,0.25244,0.14286,0.21429,0.57143,0.14,1.0,0,1,0,0,0,16,0,0,1,0,0,4,0,0,5,0,0,5,0,0,0,0,1],[52,53,0.9811,0.2857,0.19231,0.14286,0.14286,0.32143,0.14286,0.71429,0,0,0,0,0,17,0,0,7,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[53,53,1.0,0.23213,0.16653,0.14286,0.14286,0.2857,0.14286,0.71429,0,0,0,0,0,23,0,0,3,0,0,3,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"df62c5f4e7ad2d75","q":"Let  $n$  be a positive integer. Suppose that we have disks of radii  $1, 2, . . . , n.$  Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the\ndisks using a vertical stick. We want to make stacks of disks that satisfy the following conditions:\n  $i)$  Of each size exactly one disk lies in the stack.\n  $ii)$  If we look at the stack from directly above, we can see the edges of all of the  $n$  disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.)\nDetermine the number of distinct stacks of disks satisfying these conditions.\n(Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)","t":[{"b":4,"e":0.42857,"k":"falling","v":0.32143,"x":0.55355,"p":[[0,28,0.0,0.55355,0.26666,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,10,0,0,1,0,0,11,0,0,2,0,0,2,0,5],[4,28,0.1429,0.32143,0.09449,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,25,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[8,28,0.2857,0.40179,0.20652,0.28571,0.28571,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,19,0,0,2,0,0,8,0,0,0,0,0,0,0,2],[12,28,0.4286,0.32589,0.11425,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,21,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[16,28,0.5714,0.40179,0.09062,0.28571,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,10,0,0,18,0,0,4,0,0,0,0,0,0,0,0],[20,28,0.7143,0.37946,0.1411,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,17,0,0,12,0,0,2,0,0,0,0,0,0,0,1],[24,28,0.8571,0.36161,0.08737,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,14,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[28,28,1.0,0.38839,0.1394,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,12,0,0,17,0,0,1,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.57143,"k":"falling","v":0.33482,"x":0.5714,"p":[[0,28,0.0,0.5714,0.23145,0.571,0.57143,0.57143,0.0,1.0,1,4,1,1,0,0,0,0,6,0,0,0,0,0,19,0,0,0,0,0,2,0,4],[4,28,0.1429,0.3482,0.15945,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,25,0,0,0,0,0,5,0,0,0,0,0,0,0,1],[8,28,0.2857,0.36604,0.13329,0.2857,0.28571,0.571,0.14286,0.57143,0,0,0,0,0,1,0,0,21,0,0,1,0,0,9,0,0,0,0,0,0,0,0],[12,28,0.4286,0.41515,0.19017,0.28571,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,15,0,0,2,0,0,11,0,0,0,0,0,2,0,0],[16,28,0.5714,0.43303,0.16554,0.28571,0.28571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,17,0,0,0,0,0,13,0,0,1,0,0,1,0,0],[20,28,0.7143,0.45089,0.17896,0.28571,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0,0,1,0,0],[24,28,0.8571,0.33928,0.10565,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,25,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[28,28,1.0,0.33482,0.12682,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,27,0,0,1,0,0,3,0,0,0,0,0,1,0,0]]}]},{"i":"d5a7eeba1ee3c83c","q":"Let  $ ABC$  be a triangle with incenter  $ I$  and let  $ \\Gamma$  be a circle centered at  $ I$ , whose radius is greater than the inradius and does not pass through any vertex. Let  $ X_{1}$  be the intersection point of  $ \\Gamma$  and line  $ AB$ , closer to  $ B$ ;  $ X_{2}$ ,  $ X_{3}$  the points of intersection of  $ \\Gamma$  and line  $ BC$ , with  $ X_{2}$  closer to  $ B$ ; and let  $ X_{4}$  be the point of intersection of  $ \\Gamma$  with line  $ CA$  closer to  $ C$ . Let  $ K$  be the intersection point of lines  $ X_{1}X_{2}$  and  $ X_{3}X_{4}$ . Prove that  $ AK$  bisects segment  $ X_{2}X_{3}$ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.30356,"x":0.47313,"p":[[0,146,0.0,0.31463,0.23686,0.14286,0.14288,0.4642,0.0,1.0,1,1,1,1,0,16,0,1,2,0,0,4,0,0,6,0,0,0,0,0,1,0,1],[4,146,0.0274,0.4597,0.17412,0.39293,0.571,0.57143,0.14,0.85714,0,0,0,0,0,5,0,0,3,0,0,6,0,0,17,0,0,0,0,0,1,0,0],[8,146,0.0548,0.4508,0.24263,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,4,0,0,7,0,0,7,0,0,9,0,0,1,0,0,0,0,3],[12,146,0.0822,0.40177,0.16144,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,11,0,0,7,0,0,10,0,0,1,0,0,0,0,0],[16,146,0.1096,0.4598,0.13233,0.42857,0.4998,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,5,0,0,9,0,0,16,0,0,0,0,0,0,0,0],[20,146,0.137,0.47313,0.18703,0.39286,0.571,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,3,0,0,4,0,0,19,0,0,0,0,0,0,0,1],[24,146,0.1644,0.42631,0.15817,0.28571,0.42857,0.57143,0.14286,0.7857,0,0,0,0,0,4,0,0,6,0,0,10,0,0,11,0,0,0,1,0,0,0,0],[28,146,0.1918,0.38834,0.13933,0.2857,0.42857,0.4642,0.14286,0.57143,0,0,0,0,0,4,0,0,9,0,0,11,0,0,8,0,0,0,0,0,0,0,0],[32,146,0.2192,0.3616,0.11835,0.2857,0.35714,0.4286,0.14286,0.57143,0,0,0,0,0,3,0,0,13,0,0,12,0,0,4,0,0,0,0,0,0,0,0],[36,146,0.2466,0.33933,0.1505,0.2857,0.28571,0.4286,0.0,0.57143,2,0,0,2,0,3,0,0,13,0,0,9,0,0,5,0,0,0,0,0,0,0,0],[40,146,0.274,0.42408,0.13112,0.28571,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,8,0,0,11,0,0,11,0,0,0,0,0,0,0,0],[44,146,0.3014,0.43301,0.12616,0.39286,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,6,0,0,13,0,0,11,0,0,0,0,0,0,0,0],[48,146,0.3288,0.3481,0.14267,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,15,0,0,7,0,0,6,0,0,0,0,0,0,0,0],[52,146,0.3562,0.39289,0.13832,0.28571,0.42857,0.42893,0.0,0.57143,1,0,0,1,0,2,0,0,8,0,0,14,0,0,7,0,0,0,0,0,0,0,0],[56,146,0.3836,0.41066,0.15356,0.2857,0.42857,0.571,0.14286,0.57143,0,0,0,0,0,5,0,1,4,0,0,10,0,1,11,0,0,0,0,0,0,0,0],[60,146,0.411,0.36606,0.13331,0.28571,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,4,0,0,12,0,0,10,0,0,6,0,0,0,0,0,0,0,0],[64,146,0.4384,0.33036,0.15335,0.2857,0.28571,0.42858,0.0,0.57143,2,0,0,2,0,5,0,0,10,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[68,146,0.4658,0.37054,0.14223,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,10,0,0,12,0,0,6,0,0,0,0,0,0,0,0],[72,146,0.4932,0.37051,0.1509,0.2857,0.42857,0.4642,0.14286,0.57143,0,0,0,0,0,6,0,0,9,0,0,9,0,0,8,0,0,0,0,0,0,0,0],[76,146,0.5205,0.36142,0.15584,0.2857,0.28571,0.4642,0.0,0.57143,1,0,0,1,0,4,0,0,12,0,0,7,0,0,8,0,0,0,0,0,0,0,0],[80,146,0.5479,0.34375,0.1063,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,15,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[84,146,0.5753,0.35703,0.15576,0.2857,0.35714,0.4286,0.0,0.57143,1,0,0,1,0,5,0,0,10,0,0,9,0,0,7,0,0,0,0,0,0,0,0],[88,146,0.6027,0.42408,0.1576,0.28571,0.4286,0.57143,0.0,0.57143,1,0,0,1,0,2,0,0,8,0,0,7,0,0,14,0,0,0,0,0,0,0,0],[92,146,0.6301,0.30356,0.13713,0.1429,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,9,0,0,14,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[96,146,0.6575,0.37946,0.12166,0.28571,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,3,0,0,10,0,0,14,0,0,5,0,0,0,0,0,0,0,0],[100,146,0.6849,0.40624,0.15197,0.2857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,4,0,0,9,0,0,7,0,0,12,0,0,0,0,0,0,0,0],[104,146,0.7123,0.43751,0.12335,0.28571,0.42859,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,8,0,0,11,0,0,12,0,0,0,0,0,0,0,0],[108,146,0.7397,0.3973,0.15037,0.2857,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,5,0,0,7,0,0,10,0,0,10,0,0,0,0,0,0,0,0],[112,146,0.7671,0.42407,0.1311,0.28571,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,2,0,0,8,0,0,11,0,0,11,0,0,0,0,0,0,0,0],[116,146,0.7945,0.38383,0.13584,0.2857,0.42857,0.4642,0.14,0.57143,0,0,0,0,0,3,0,0,12,0,0,9,0,0,8,0,0,0,0,0,0,0,0],[120,146,0.8219,0.37051,0.16695,0.28571,0.42857,0.571,0.0,0.57143,1,0,0,1,0,6,0,0,7,0,0,9,0,0,9,0,0,0,0,0,0,0,0],[124,146,0.8493,0.39285,0.13361,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,2,0,0,7,0,0,16,0,0,6,0,0,0,0,0,0,0,0],[128,146,0.8767,0.4196,0.13799,0.28571,0.42857,0.57141,0.14286,0.57143,0,0,0,0,0,2,0,0,10,0,0,8,0,0,12,0,0,0,0,0,0,0,0],[132,146,0.9041,0.38837,0.15248,0.28571,0.42857,0.4642,0.0,0.57143,1,0,0,1,0,4,0,0,6,0,0,13,0,0,8,0,0,0,0,0,0,0,0],[136,146,0.9315,0.37052,0.19185,0.2857,0.42857,0.4286,0.0,1.0,1,1,0,1,0,6,0,0,8,0,0,10,0,0,6,0,0,0,0,0,0,0,1],[140,146,0.9589,0.36606,0.1554,0.2857,0.42857,0.4286,0.0,0.57143,2,0,0,2,0,3,0,0,8,0,0,13,0,0,6,0,0,0,0,0,0,0,0],[144,146,0.9863,0.38392,0.13569,0.28571,0.35714,0.4642,0.0,0.57143,1,0,0,1,0,0,0,0,15,0,0,8,0,0,8,0,0,0,0,0,0,0,0],[146,146,1.0,0.40623,0.14331,0.28571,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,4,0,0,7,0,0,11,0,0,10,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14,"k":"falling","v":0.18741,"x":0.48225,"p":[[0,51,0.0,0.43516,0.2379,0.25002,0.49979,0.57143,0.0,1.0,1,1,1,1,0,7,0,0,5,0,0,3,0,0,11,2,0,0,1,0,1,0,1],[4,51,0.0784,0.43291,0.14947,0.28571,0.42859,0.57143,0.14,0.57143,0,0,0,0,0,3,0,0,8,0,0,6,0,0,15,0,0,0,0,0,0,0,0],[8,51,0.1569,0.48225,0.17336,0.39286,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,9,0,0,12,0,0,0,2,0,0,0,1],[12,51,0.2353,0.4553,0.19042,0.28571,0.571,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,8,0,0,3,0,0,17,0,0,0,0,0,0,0,1],[16,51,0.3137,0.45981,0.16646,0.28571,0.4293,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,7,0,0,8,0,0,14,0,0,0,0,0,0,0,1],[20,51,0.3922,0.48002,0.17721,0.42857,0.571,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,8,0,0,15,0,0,0,1,0,0,0,1],[24,51,0.4706,0.44865,0.12781,0.28571,0.4286,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,8,0,0,8,0,1,14,0,0,0,0,0,0,0,0],[28,51,0.549,0.45313,0.1928,0.28571,0.5005,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,7,0,0,5,0,0,14,0,0,0,1,0,0,0,1],[32,51,0.6275,0.44863,0.19,0.28571,0.4998,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,5,0,1,6,0,0,15,0,0,0,0,0,0,0,1],[36,51,0.7059,0.35937,0.16314,0.1429,0.42857,0.4286,0.07143,0.57143,0,0,0,0,1,8,0,0,4,0,0,12,0,0,7,0,0,0,0,0,0,0,0],[40,51,0.7843,0.37051,0.17442,0.25,0.42857,0.5711,0.0,0.57143,1,0,0,1,0,7,0,0,6,0,0,8,0,0,10,0,0,0,0,0,0,0,0],[44,51,0.8627,0.3525,0.17512,0.14286,0.42857,0.57143,0.14,0.57143,0,0,0,0,0,11,0,0,4,0,0,8,0,0,9,0,0,0,0,0,0,0,0],[48,51,0.9412,0.35703,0.18305,0.14286,0.42857,0.571,0.0,0.57143,1,0,0,1,0,10,0,0,2,0,1,8,0,1,9,0,0,0,0,0,0,0,0],[51,51,1.0,0.18741,0.13095,0.14286,0.14286,0.1429,0.0,0.57143,2,0,0,2,0,24,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"54c8ea8eb01a2420","q":"Let  $n\\geq 3$  and  $A=\\{1,2,\\dots ,n\\}$ . For any function  $f:A\\rightarrow A$  we define   $$ A_f=\\{|f(1)-f(2)|,|f(2)-f(3)|,\\dots ,|f(n-1)-f(n)|,|f(n)-f(1)|\\}. $$   Determine the smallest and greatest value of the cardinal of  $A_f$  as  $f$  goes through all bijective functions from  $A$  to  $A$ .\n\n*Silviu Cristea*","t":[{"b":1,"e":1.0,"k":"flat","v":0.90178,"x":0.95535,"p":[[0,33,0.0,0.95534,0.10977,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[4,33,0.1212,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,20],[8,33,0.2424,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,22],[12,33,0.3636,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[16,33,0.4848,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[20,33,0.6061,0.94642,0.10569,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[24,33,0.7273,0.9375,0.09407,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[28,33,0.8485,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[32,33,0.9697,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[33,33,1.0,0.90178,0.10972,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,13]]},{"b":2,"e":0.85714,"k":"flat","v":0.92411,"x":0.97321,"p":[[0,38,0.0,0.94196,0.11214,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,22],[4,38,0.1053,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[8,38,0.2105,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[12,38,0.3158,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[16,38,0.4211,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[20,38,0.5263,0.95981,0.08923,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[24,38,0.6316,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[28,38,0.7368,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[32,38,0.8421,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[36,38,0.9474,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[38,38,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]}]},{"i":"2fee74b11412acdc","q":"Let  $a, b, c$  be positive reals such that  $abc=1$ . Show that \\[\\frac{1}{a^5(b+2c)^2} + \\frac{1}{b^5(c+2a)^2} + \\frac{1}{c^5(a+2b)^2} \\ge \\frac{1}{3}.\\]","t":[{"b":3,"e":0.42857,"k":"falling","v":0.11607,"x":0.47321,"p":[[0,110,0.0,0.47321,0.16536,0.42857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,2,0,0,15,0,0,11,0,0,1,0,0,0,0,1],[4,110,0.0364,0.30357,0.26184,0.14286,0.14286,0.42857,0.0,1.0,4,1,0,4,0,14,0,0,1,0,0,8,0,0,1,0,0,1,0,0,2,0,1],[8,110,0.0727,0.43304,0.35801,0.14286,0.42857,0.75,0.0,1.0,6,5,0,6,0,7,0,0,2,0,0,5,0,0,2,0,0,2,0,0,3,0,5],[12,110,0.1091,0.34375,0.28316,0.14286,0.28571,0.46429,0.0,1.0,5,1,0,5,0,11,0,0,0,0,0,8,0,0,2,0,0,3,0,0,2,0,1],[16,110,0.1455,0.33927,0.29612,0.14286,0.14286,0.57111,0.0,1.0,5,2,0,5,0,12,0,0,1,0,0,5,0,0,3,0,0,3,0,0,1,0,2],[20,110,0.1818,0.29018,0.23279,0.14286,0.14286,0.42857,0.0,1.0,4,1,0,4,0,13,0,0,2,0,0,9,0,0,1,0,0,2,0,0,0,0,1],[24,110,0.2182,0.24553,0.27019,0.0,0.14286,0.42857,0.0,0.85714,10,0,0,10,0,12,0,0,0,0,0,3,0,0,3,0,0,2,0,0,2,0,0],[28,110,0.2545,0.38839,0.31791,0.14286,0.42857,0.57143,0.0,1.0,6,3,0,6,0,8,0,0,0,0,0,7,0,0,5,0,0,1,0,0,2,0,3],[32,110,0.2909,0.27679,0.26471,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,3,0,0,6,0,0,2,0,0,0,0,0,2,0,1],[36,110,0.3273,0.26339,0.20858,0.14286,0.14286,0.42857,0.0,0.85714,6,0,0,6,0,11,0,0,2,0,0,10,0,0,2,0,0,0,0,0,1,0,0],[40,110,0.3636,0.42857,0.30723,0.14286,0.42857,0.57143,0.0,1.0,5,3,0,5,0,6,0,0,0,0,0,9,0,0,5,0,0,2,0,0,2,0,3],[44,110,0.4,0.3125,0.23808,0.14286,0.21428,0.42857,0.0,1.0,3,1,0,3,0,13,0,0,3,0,0,6,0,0,4,0,0,2,0,0,0,0,1],[48,110,0.4364,0.2857,0.27431,0.14286,0.14286,0.42857,0.0,1.0,6,2,0,6,0,14,0,0,0,0,0,5,0,0,4,0,0,1,0,0,0,0,2],[52,110,0.4727,0.32588,0.30562,0.0,0.14286,0.60714,0.0,0.85714,9,0,0,9,0,8,0,0,1,0,0,4,0,0,2,0,0,5,0,0,3,0,0],[56,110,0.5091,0.38839,0.24545,0.14289,0.42857,0.57143,0.0,1.0,5,1,0,5,0,4,0,0,2,0,0,11,0,0,6,0,0,3,0,0,0,0,1],[60,110,0.5455,0.29464,0.25739,0.14286,0.14286,0.42857,0.0,1.0,4,1,0,4,0,14,0,0,3,0,0,6,0,0,0,0,0,3,0,0,1,0,1],[64,110,0.5818,0.27231,0.2633,0.0,0.14286,0.42858,0.0,1.0,9,1,0,9,0,9,0,0,2,0,0,5,0,0,4,0,0,2,0,0,0,0,1],[68,110,0.6182,0.41964,0.2765,0.14286,0.42857,0.57143,0.0,1.0,2,1,0,2,0,10,0,0,1,0,0,6,0,0,6,0,0,3,0,0,3,0,1],[72,110,0.6545,0.33036,0.32031,0.0,0.21429,0.46429,0.0,1.0,9,2,0,9,0,7,0,0,2,0,0,6,0,0,2,0,0,1,0,0,3,0,2],[76,110,0.6909,0.29911,0.25595,0.10714,0.28571,0.46431,0.0,1.0,8,1,0,8,0,7,0,0,3,0,0,6,0,0,6,0,0,1,0,0,0,0,1],[80,110,0.7273,0.39732,0.31488,0.14286,0.42857,0.57143,0.0,1.0,4,4,0,4,0,10,0,0,1,0,0,6,0,0,4,0,0,3,0,0,0,0,4],[84,110,0.7636,0.34822,0.34244,0.0,0.14288,0.57143,0.0,1.0,9,3,0,9,0,8,0,0,1,0,0,3,0,0,5,0,0,0,0,0,3,0,3],[88,110,0.8,0.37052,0.32704,0.0,0.42857,0.71429,0.0,1.0,10,1,0,10,0,4,0,0,1,0,0,5,0,0,3,0,0,5,0,0,3,0,1],[92,110,0.8364,0.33929,0.26904,0.14286,0.42857,0.57143,0.0,1.0,7,1,0,7,0,7,0,0,1,0,0,7,0,0,6,0,0,3,0,0,0,0,1],[96,110,0.8727,0.27233,0.27976,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,9,0,0,1,0,0,5,0,0,2,0,0,4,0,0,0,0,1],[100,110,0.9091,0.34821,0.35344,0.0,0.14286,0.60714,0.0,1.0,11,3,0,11,0,6,0,0,1,0,0,2,0,0,4,0,0,3,0,0,2,0,3],[104,110,0.9455,0.32143,0.25254,0.14286,0.35714,0.46429,0.0,0.85714,7,0,0,7,0,7,0,0,2,0,0,8,0,0,4,0,0,3,0,0,1,0,0],[108,110,0.9818,0.28569,0.22584,0.14286,0.14286,0.42857,0.0,0.85714,5,0,0,5,0,12,0,0,2,0,0,7,0,0,4,0,0,1,0,0,1,0,0],[110,110,1.0,0.11607,0.0974,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.18304,"x":0.45982,"p":[[0,123,0.0,0.45982,0.1461,0.42857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,0,0,0,17,0,0,10,0,0,2,0,0,0,0,0],[4,123,0.0325,0.29909,0.26087,0.14286,0.14286,0.42857,0.0,0.85714,5,0,0,5,0,13,0,0,2,0,0,5,0,0,3,0,0,1,0,0,3,0,0],[8,123,0.065,0.4107,0.3004,0.14286,0.42857,0.60714,0.0,1.0,5,3,0,5,0,6,0,0,2,0,0,8,0,0,3,0,0,5,0,0,0,0,3],[12,123,0.0976,0.33927,0.25691,0.14286,0.42857,0.42857,0.0,1.0,5,1,0,5,0,9,0,0,1,0,0,10,0,0,2,0,0,4,0,0,0,0,1],[16,123,0.1301,0.39284,0.32142,0.14286,0.35714,0.57111,0.0,1.0,4,3,0,4,0,10,0,0,2,0,0,7,0,0,2,0,0,0,0,0,4,0,3],[20,123,0.1626,0.32589,0.29285,0.14286,0.14286,0.46431,0.0,1.0,6,2,0,6,0,11,0,0,1,0,0,6,0,0,3,0,0,2,0,0,1,0,2],[24,123,0.1951,0.35268,0.27429,0.14286,0.35714,0.57143,0.0,0.85714,5,0,0,5,0,10,0,0,1,0,0,5,0,0,6,0,0,2,0,0,3,0,0],[28,123,0.2276,0.39286,0.29667,0.14286,0.42857,0.46431,0.0,1.0,3,3,0,3,0,10,0,0,1,0,0,10,0,0,2,0,0,1,0,0,2,0,3],[32,123,0.2602,0.34374,0.30063,0.14286,0.14286,0.57143,0.0,1.0,6,1,0,6,0,12,0,0,0,0,0,1,0,0,6,0,0,5,0,0,1,0,1],[36,123,0.2927,0.28572,0.27894,0.10714,0.14286,0.42857,0.0,1.0,8,2,0,8,0,10,0,0,1,0,0,8,0,0,1,0,0,2,0,0,0,0,2],[40,123,0.3252,0.35267,0.28117,0.14286,0.35714,0.57111,0.0,1.0,5,2,0,5,0,9,0,0,2,0,0,7,0,0,5,0,0,1,0,0,1,0,2],[44,123,0.3577,0.33036,0.25111,0.14286,0.42857,0.42857,0.0,0.85714,7,0,0,7,0,5,0,0,3,0,0,11,0,0,2,0,0,2,0,0,2,0,0],[48,123,0.3902,0.37499,0.2829,0.14286,0.42857,0.57143,0.0,1.0,4,2,0,4,0,10,0,0,0,0,0,8,0,0,5,0,0,2,0,0,1,0,2],[52,123,0.4228,0.35714,0.27894,0.14286,0.35714,0.57143,0.0,1.0,4,2,0,4,0,11,0,0,1,0,0,6,0,0,5,0,0,3,0,0,0,0,2],[56,123,0.4553,0.25445,0.2307,0.10714,0.14286,0.42857,0.0,0.85714,8,0,0,8,0,11,0,0,0,0,0,9,0,0,2,0,0,1,0,0,1,0,0],[60,123,0.4878,0.36157,0.27887,0.14286,0.42857,0.4642,0.0,1.0,4,2,0,4,0,10,0,0,1,0,0,9,0,0,3,0,0,2,0,0,1,0,2],[64,123,0.5203,0.33929,0.28516,0.14286,0.14286,0.46431,0.0,1.0,4,2,0,4,0,13,0,0,0,0,0,7,0,0,3,0,0,2,0,0,1,0,2],[68,123,0.5528,0.27232,0.26812,0.14286,0.14286,0.42857,0.0,0.85714,5,0,0,5,0,17,0,0,0,0,0,4,0,0,2,0,0,0,0,0,4,0,0],[72,123,0.5854,0.29909,0.24576,0.14286,0.21429,0.42857,0.0,1.0,5,1,0,5,0,11,0,0,2,0,0,10,0,0,1,0,0,1,0,0,1,0,1],[76,123,0.6179,0.31697,0.28288,0.0,0.21429,0.57143,0.0,1.0,9,1,0,9,0,7,0,0,1,0,0,4,0,0,7,0,0,3,0,0,0,0,1],[80,123,0.6504,0.25447,0.30667,0.0,0.14286,0.42857,0.0,1.0,12,1,0,12,0,10,0,0,0,0,0,3,0,0,1,0,0,3,0,0,2,0,1],[84,123,0.6829,0.28561,0.34444,0.0,0.14143,0.46418,0.0,1.0,15,1,0,15,0,4,0,0,1,0,0,4,0,0,1,0,0,1,0,0,5,0,1],[88,123,0.7154,0.26786,0.30252,0.0,0.14286,0.46429,0.0,1.0,11,2,0,11,0,9,0,0,2,0,0,2,0,0,3,0,0,3,0,0,0,0,2],[92,123,0.748,0.33482,0.33429,0.0,0.21429,0.46429,0.0,1.0,10,4,0,10,0,6,0,0,1,0,0,7,0,0,2,0,0,2,0,0,0,0,4],[96,123,0.7805,0.32589,0.27019,0.14286,0.35714,0.42857,0.0,1.0,7,2,0,7,0,6,0,0,3,0,0,11,0,0,1,0,0,2,0,0,0,0,2],[100,123,0.813,0.3482,0.27878,0.14286,0.35714,0.42857,0.0,1.0,5,1,0,5,0,9,0,0,2,0,0,9,0,0,2,0,0,1,0,0,3,0,1],[104,123,0.8455,0.28124,0.23819,0.14286,0.14286,0.42857,0.0,0.71429,7,0,0,7,0,11,0,0,0,0,0,7,0,0,4,0,0,3,0,0,0,0,0],[108,123,0.878,0.26777,0.25195,0.105,0.14286,0.42857,0.0,1.0,8,1,0,8,0,10,0,0,2,0,0,7,0,0,2,0,0,2,0,0,0,0,1],[112,123,0.9106,0.18304,0.27487,0.0,0.14286,0.14286,0.0,1.0,15,2,0,15,0,10,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[116,123,0.9431,0.23214,0.27837,0.0,0.14286,0.42857,0.0,0.85714,13,0,0,13,0,8,0,0,1,0,0,5,0,0,1,0,0,1,0,0,3,0,0],[120,123,0.9756,0.25893,0.22142,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,6,0,0,5,0,0,8,0,0,3,0,0,0,0,0,1,0,0],[123,123,1.0,0.29911,0.22968,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,7,0,0,5,0,0,10,0,0,2,0,0,1,0,0,0,0,1]]}]},{"i":"7ab7912d43f589e4","q":"A circle that passes through the vertices  $ B,C $  of a triangle  $ ABC, $  cuts the segments  $ AB,AC $  (endpoints excluded) in  $ N, $  respectively,  $ M. $  Consider the point  $ P $  on the segment  $ MN $  and  $ Q $  on the segment  $ BC $  (endpoints excluded on both segments) such that the angles  $ \\angle BAC,\\angle PAQ $  have the same bisector. Show that:**a)**  $ \\frac{PM}{PN} =\\frac{QB}{QC} . $ **b)** The midpoints of the segments  $ BM,CN,PQ $  are collinear.","t":[{"b":6,"e":0.85714,"k":"flat","v":0.81695,"x":0.90178,"p":[[0,35,0.0,0.875,0.05923,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,5],[4,35,0.1143,0.85267,0.16554,0.85714,0.85714,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,6],[8,35,0.2286,0.86169,0.11564,0.85714,0.85714,0.895,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,20,0,8],[12,35,0.3429,0.81695,0.15253,0.85714,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,26,0,2],[16,35,0.4571,0.87053,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[20,35,0.5714,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[24,35,0.6857,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[28,35,0.8,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[32,35,0.9143,0.84821,0.15947,0.85714,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,4],[35,35,1.0,0.87053,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,5]]},{"b":7,"e":0.85714,"k":"flat","v":0.85714,"x":0.90179,"p":[[0,19,0.0,0.90179,0.07523,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,11],[4,19,0.2105,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[8,19,0.4211,0.87499,0.04725,0.85714,0.85714,0.85714,0.857,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[12,19,0.6316,0.87946,0.05187,0.85714,0.85714,0.85714,0.857,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[16,19,0.8421,0.85714,2e-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],[19,19,1.0,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6]]}]},{"i":"314fa017206bc850","q":"Circles $\\alpha$ and $\\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\\alpha$ and $\\beta$. A third circle of the same radius passes through $P$ and intersects $\\alpha$ and $\\beta$ in the points $X$ and $Y$, respectively.\nShow that the line $X Y$ is parallel to the line $A B$.","t":[{"b":6,"e":0.28571,"k":"flat","v":0.25446,"x":0.35268,"p":[[0,26,0.0,0.35268,0.2082,0.28571,0.28571,0.28571,0.2857,1.0,0,3,0,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,26,0.1538,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.30357,0.09943,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,26,0.4615,0.27232,0.05486,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.25446,0.08552,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.27679,0.04971,0.28571,0.28571,0.28571,0.0,0.286,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.28126,0.02485,0.28571,0.28571,0.28571,0.1429,0.286,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"volatile","v":0.27678,"x":1.0,"p":[[0,11,0.0,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.33036,0.1729,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,11,0.7273,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[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":"66bb6d81197f1421","q":"An acute-angled triangle  $ABC$  with an altitude  $AD$  and orthocenter  $H$  are given.  $AD$  intersects the circumcircle of  $ABC$   $\\omega$  at  $P$ .  $K$  is a point on segment  $BC$  such that  $KC=BD$ . The circumcircle of  $KPH$  intersects  $\\omega$  at  $Q$  and  $BC$  at  $N$ . A line perpendicular to  $PQ$  and passing through  $N$  intersects  $AD$  at  $T$ . Prove that the center of  $\\omega$  lies on line  $TK$ .\n*U. Maksimenkau*","t":[{"b":1,"e":0.571,"k":"flat","v":0.34374,"x":0.45088,"p":[[0,38,0.0,0.35714,0.19233,0.14286,0.28571,0.46428,0.0,0.71429,1,0,0,1,0,8,0,0,8,0,0,7,0,0,5,0,0,3,0,0,0,0,0],[4,38,0.1053,0.40177,0.18362,0.28571,0.42857,0.4642,0.0,0.71429,2,0,0,2,0,2,0,0,8,0,0,12,0,0,4,0,0,4,0,0,0,0,0],[8,38,0.2105,0.45088,0.16408,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,12,0,0,7,0,0,3,0,0,1,0,0],[12,38,0.3158,0.45084,0.13878,0.42857,0.42859,0.57111,0.14286,0.71429,0,0,0,0,0,2,0,0,5,0,0,13,0,0,10,0,0,2,0,0,0,0,0],[16,38,0.4211,0.38381,0.13582,0.28571,0.42857,0.42858,0.14,0.57143,0,0,0,0,0,4,0,0,9,0,0,12,0,0,7,0,0,0,0,0,0,0,0],[20,38,0.5263,0.41962,0.14695,0.28571,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,2,0,0,10,0,0,10,0,0,8,0,0,2,0,0,0,0,0],[24,38,0.6316,0.34374,0.14662,0.28571,0.35714,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,10,0,0,13,0,0,2,0,0,1,0,0,0,0,0],[28,38,0.7368,0.40623,0.16013,0.28571,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,4,0,0,8,0,0,12,0,0,5,0,0,3,0,0,0,0,0],[32,38,0.8421,0.41965,0.11812,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,16,0,0,6,0,0,1,0,0,0,0,0],[36,38,0.9474,0.442,0.12035,0.39286,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,15,0,0,7,0,0,2,0,0,0,0,0],[38,38,1.0,0.38838,0.16839,0.28571,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,3,0,0,10,0,0,11,0,0,4,0,0,3,0,0,0,0,0]]},{"b":2,"e":0.57143,"k":"flat","v":0.31696,"x":0.41068,"p":[[0,35,0.0,0.40183,0.17291,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,12,0,0,7,0,0,7,0,0,3,0,0,0,0,0],[4,35,0.1143,0.3304,0.16148,0.24999,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,11,0,0,8,0,0,4,0,0,1,0,0,0,0,0],[8,35,0.2286,0.36164,0.15966,0.2857,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,6,0,0,11,0,0,9,0,0,4,0,0,2,0,0,0,0,0],[12,35,0.3429,0.41068,0.13711,0.28571,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,3,0,0,7,0,0,14,0,0,7,0,0,1,0,0,0,0,0],[16,35,0.4571,0.33929,0.14174,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,4,0,0,13,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[20,35,0.5714,0.35715,0.19561,0.1429,0.42857,0.46431,0.0,0.71429,2,0,0,2,0,8,0,0,4,0,0,10,0,0,6,0,0,2,0,0,0,0,0],[24,35,0.6857,0.34375,0.13296,0.2857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,8,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[28,35,0.8,0.36161,0.12364,0.2857,0.28571,0.4286,0.14286,0.57143,0,0,0,0,0,3,0,0,14,0,0,10,0,0,5,0,0,0,0,0,0,0,0],[32,35,0.9143,0.31696,0.1461,0.2857,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,5,0,0,12,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[35,35,1.0,0.36607,0.11812,0.28571,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,3,0,0,12,0,0,13,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"3e44a1636ff6e95b","q":"Let $m, n$ be positive integers with $m>1$. Anastasia partitions the integers $1,2, \\ldots, 2 m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to $n$.\n(Netherlands)","t":[{"b":0,"e":0.0,"k":"falling","v":0.07366,"x":0.68747,"p":[[0,72,0.0,0.60268,0.24675,0.39286,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,3,0,0,1,0,0,16,0,0,0,0,4],[4,72,0.0556,0.68747,0.27535,0.571,0.71429,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,3,0,0,2,0,0,2,0,0,11,0,0,5,0,7],[8,72,0.1111,0.56247,0.25985,0.42857,0.57143,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,4,0,0,4,0,0,8,0,0,9,0,0,0,0,4],[12,72,0.1667,0.58926,0.25191,0.42857,0.57143,0.71429,0.0,1.0,2,3,0,2,0,0,0,0,2,0,0,9,0,0,4,0,0,8,0,0,4,0,3],[16,72,0.2222,0.51786,0.22798,0.42857,0.4286,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,4,0,0,11,0,0,4,0,0,9,0,0,0,0,2],[20,72,0.2778,0.43304,0.23003,0.28571,0.42857,0.60714,0.0,0.71429,4,0,0,4,0,1,0,0,6,0,0,8,0,0,5,0,0,8,0,0,0,0,0],[24,72,0.3333,0.48212,0.24678,0.28571,0.42857,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,9,0,0,7,0,0,4,0,0,5,0,0,2,0,2],[28,72,0.3889,0.34815,0.27644,0.10714,0.28571,0.571,0.0,1.0,8,2,0,8,0,2,0,0,7,0,0,5,0,0,7,0,0,1,0,0,0,0,2],[32,72,0.4444,0.51785,0.2829,0.39286,0.50001,0.71429,0.0,1.0,4,2,0,4,0,0,0,0,4,0,0,8,0,0,5,0,0,4,0,0,5,0,2],[36,72,0.5,0.47312,0.22986,0.28571,0.4998,0.57143,0.0,1.0,3,1,0,3,0,0,0,0,6,0,0,7,0,0,11,0,0,2,0,0,2,0,1],[40,72,0.5556,0.46424,0.26242,0.28571,0.42859,0.71429,0.0,0.85714,4,0,0,4,0,1,0,0,6,0,0,7,0,0,4,0,0,6,0,0,4,0,0],[44,72,0.6111,0.41963,0.29867,0.14286,0.42857,0.71429,0.0,1.0,6,2,1,6,0,3,0,0,4,0,0,8,0,0,1,0,0,7,0,0,1,0,2],[48,72,0.6667,0.43301,0.24867,0.2857,0.4286,0.71429,0.0,0.85714,4,0,0,4,0,2,0,0,7,0,0,5,0,0,5,0,0,8,0,0,1,0,0],[52,72,0.7222,0.36602,0.25485,0.14286,0.42857,0.571,0.0,0.85714,6,0,0,6,0,4,0,0,4,0,0,9,0,0,4,0,0,3,0,0,2,0,0],[56,72,0.7778,0.34819,0.24466,0.1429,0.28571,0.571,0.0,0.85714,6,0,0,6,0,3,0,0,9,0,0,5,0,0,4,0,0,4,0,0,1,0,0],[60,72,0.8333,0.38382,0.26837,0.25,0.42857,0.4642,0.0,1.0,6,1,0,6,0,2,0,0,6,0,0,10,0,0,2,0,0,3,0,0,2,0,1],[64,72,0.8889,0.29462,0.25486,0.0,0.28571,0.4286,0.0,0.71429,9,0,0,9,0,6,0,0,3,0,0,7,0,0,2,0,0,5,0,0,0,0,0],[68,72,0.9444,0.18749,0.2214,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,7,0,0,4,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[72,72,1.0,0.07366,0.13301,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,4,0,0,2,0,1,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"falling","v":0.25,"x":0.65179,"p":[[0,76,0.0,0.65179,0.20806,0.67857,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,2,0,0,1,0,0,20,0,0,1,0,3],[4,76,0.0526,0.55356,0.22798,0.42857,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,2,0,0,6,0,0,8,0,0,10,0,0,2,0,1],[8,76,0.1053,0.56692,0.23003,0.42857,0.571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,6,0,0,7,0,0,5,0,0,8,0,0,2,0,3],[12,76,0.1579,0.42853,0.26484,0.24999,0.571,0.60714,0.0,0.71429,7,0,0,7,0,1,0,0,2,0,0,5,0,0,9,0,0,8,0,0,0,0,0],[16,76,0.2105,0.48209,0.24934,0.28571,0.571,0.60714,0.0,1.0,2,2,0,2,0,3,0,0,6,0,0,3,0,0,10,0,0,6,0,0,0,0,2],[20,76,0.2632,0.5,0.2369,0.42857,0.4286,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,4,0,0,11,0,0,5,0,0,6,0,0,1,0,2],[24,76,0.3158,0.4107,0.24935,0.25,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,3,0,0,3,0,0,9,0,0,5,0,0,6,0,0,1,0,0],[28,76,0.3684,0.44641,0.23351,0.28571,0.42857,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,9,0,0,7,0,0,3,0,0,8,0,0,0,0,1],[32,76,0.4211,0.49992,0.23143,0.28571,0.571,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,5,0,0,6,0,0,8,0,0,6,0,0,2,0,1],[36,76,0.4737,0.47319,0.22141,0.39286,0.42857,0.60714,0.0,1.0,2,1,0,2,0,2,0,0,4,0,0,10,0,0,6,0,0,7,0,0,0,0,1],[40,76,0.5263,0.47764,0.23583,0.28571,0.42859,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,6,0,0,7,0,0,6,0,0,7,0,0,1,0,1],[44,76,0.5789,0.44643,0.24679,0.2857,0.42857,0.71429,0.0,1.0,1,1,0,1,0,5,0,0,7,0,0,8,0,0,2,0,0,6,0,0,2,0,1],[48,76,0.6316,0.33472,0.18431,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,4,0,0,11,0,0,10,0,0,1,0,0,3,0,0,0,0,0],[52,76,0.6842,0.41505,0.21544,0.2857,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,8,0,0,5,0,0,7,0,0,6,0,0,0,0,0],[56,76,0.7368,0.32142,0.23144,0.14286,0.28571,0.42858,0.0,0.85714,4,0,0,4,0,9,0,0,6,0,0,6,0,0,3,0,0,3,0,0,1,0,0],[60,76,0.7895,0.35715,0.25,0.14289,0.28571,0.46431,0.0,0.85714,4,0,0,4,0,7,0,0,6,0,0,7,0,0,2,0,0,4,0,0,2,0,0],[64,76,0.8421,0.39282,0.223,0.25,0.42857,0.571,0.0,1.0,2,1,0,2,0,6,0,0,5,0,0,10,0,0,5,0,0,3,0,0,0,0,1],[68,76,0.8947,0.39727,0.18804,0.28571,0.42857,0.571,0.0,0.71429,1,0,0,1,0,4,0,0,10,0,0,7,0,0,6,0,0,4,0,0,0,0,0],[72,76,0.9474,0.34822,0.18536,0.14297,0.28571,0.4286,0.0,0.71429,1,0,0,1,0,8,0,0,8,0,0,9,0,0,3,0,0,3,0,0,0,0,0],[76,76,1.0,0.25,0.15152,0.14286,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,10,0,0,14,0,0,3,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"581069b0a9b567f1","q":"Let $n$ be a natural number such that $n=a^{2}+b^{2}+c^{2}$, for some natural numbers $a, b, c$. Prove that\n\n$$\n9 n=\\left(p_{1} a+q_{1} b+r_{1} c\\right)^{2}+\\left(p_{2} a+q_{2} b+r_{2} c\\right)^{2}+\\left(p_{3} a+q_{3} b+r_{3} c\\right)^{2}\n$$\n\nwhere $p_{j}$ 's, $q_{j}$ 's, $r_{j}$ 's are all nonzero integers. Further, if 3 does not divide at least one of $a, b, c$, prove that $9 n$ can be expressed in the form $x^{2}+y^{2}+z^{2}$, where $x, y, z$ are natural numbers none of which is divisible by 3 .","t":[{"b":3,"e":1.0,"k":"rising","v":0.70089,"x":1.0,"p":[[0,36,0.0,0.70089,0.29529,0.39286,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,8,0,0,3,0,0,2,0,0,1,0,0,7,0,11],[4,36,0.1111,0.88839,0.11143,0.85714,0.85714,1.0,0.5714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,13],[8,36,0.2222,0.88838,0.15043,0.85714,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,13,0,15],[12,36,0.3333,0.95089,0.1411,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[16,36,0.4444,0.95088,0.10484,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[20,36,0.5556,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[24,36,0.6667,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,36,0.7778,0.95535,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[32,36,0.8889,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,36,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":0.85714,"k":"rising","v":0.65178,"x":0.86158,"p":[[0,29,0.0,0.65178,0.29001,0.39286,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,7,0,0,3,0,0,3,0,0,4,0,0,7,0,7],[4,29,0.1379,0.82143,0.21129,0.85714,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,19,0,8],[8,29,0.2759,0.80802,0.16984,0.85714,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,22,0,4],[12,29,0.4138,0.78124,0.19557,0.71429,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,18,0,4],[16,29,0.5517,0.86158,0.09775,0.85714,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,23,0,6],[20,29,0.6897,0.82139,0.13366,0.82132,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,19,0,5],[24,29,0.8276,0.85714,0.08748,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,23,0,5],[28,29,0.9655,0.81683,0.1395,0.85714,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,23,0,3],[29,29,1.0,0.85266,0.09099,0.85714,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,22,0,5]]}]},{"i":"5ebca36b77a90f52","q":"Let  $\\{a_n \\}_{n=0}^{\\infty}$  be a sequence given recrusively such that  $a_0=1$  and   $$ a_{n+1}=\\frac{7a_n+\\sqrt{45a_n^2-36}}{2} $$   for  $n\\geq 0$  \n\nShow that :\na)  $a_n$  is a positive integer.\nb)  $a_n a_{n+1}-1$  is a square of an integer.\n\n*Proposed by  Stefan Gyurki (Matej Bel University, Banska Bystrica).*","t":[{"b":0,"e":0.57143,"k":"falling","v":0.48211,"x":0.96429,"p":[[0,148,0.0,0.91518,0.15093,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,6,0,21],[4,148,0.027,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[8,148,0.0541,0.85267,0.30406,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,23],[12,148,0.0811,0.90625,0.18073,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,1,0,24],[16,148,0.1081,0.88839,0.16262,0.71429,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,3,0,20],[20,148,0.1351,0.90178,0.1448,0.85711,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,20],[24,148,0.1622,0.90178,0.20652,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,23],[28,148,0.1892,0.8125,0.27765,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,4,0,0,1,0,0,4,0,0,2,0,19],[32,148,0.2162,0.85714,0.19885,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,5,0,18],[36,148,0.2432,0.78125,0.27429,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,5,0,14],[40,148,0.2703,0.90625,0.16214,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,21],[44,148,0.2973,0.84375,0.24836,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,0,0,21],[48,148,0.3243,0.76326,0.2566,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,2,0,0,4,0,14],[52,148,0.3514,0.85714,0.20516,0.82143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,6,0,18],[56,148,0.3784,0.8125,0.27534,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,4,0,0,3,0,0,2,0,0,1,0,20],[60,148,0.4054,0.73659,0.28819,0.5354,0.78571,1.0,0.0,1.0,1,14,0,1,0,1,0,0,1,0,0,5,0,0,3,0,0,5,0,0,2,0,14],[64,148,0.4324,0.69641,0.31289,0.53539,0.71429,1.0,0.0,1.0,2,13,0,2,0,0,0,0,4,0,0,2,0,0,6,0,0,3,0,0,2,0,13],[68,148,0.4595,0.8125,0.25862,0.57143,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,3,0,18],[72,148,0.4865,0.78124,0.25251,0.57143,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,7,0,13],[76,148,0.5135,0.79911,0.27166,0.67857,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,3,0,0,3,0,0,4,0,0,3,0,17],[80,148,0.5405,0.69195,0.28147,0.42857,0.64286,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,5,0,0,7,0,0,1,0,0,4,0,11],[84,148,0.5676,0.84375,0.2212,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,4,0,17],[88,148,0.5946,0.77677,0.21707,0.71429,0.78571,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,6,0,10],[92,148,0.6216,0.76784,0.27142,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,4,0,0,5,0,0,4,0,0,1,0,16],[96,148,0.6486,0.79017,0.23416,0.57132,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,3,0,0,4,0,15],[100,148,0.6757,0.69195,0.23176,0.42859,0.64286,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,4,0,0,3,0,9],[104,148,0.7027,0.76784,0.26905,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,3,0,0,7,0,0,3,0,0,1,0,16],[108,148,0.7297,0.84822,0.21409,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,4,0,0,3,0,19],[112,148,0.7568,0.77229,0.23923,0.5713,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,5,0,0,1,0,15],[116,148,0.7838,0.76783,0.23625,0.57132,0.78564,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,5,0,0,2,0,14],[120,148,0.8108,0.72316,0.28336,0.571,0.71429,1.0,0.0,1.0,2,13,0,2,0,0,0,0,0,0,0,4,0,0,7,0,0,5,0,0,1,0,13],[124,148,0.8378,0.76784,0.24158,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,4,0,0,4,0,0,5,0,0,7,0,11],[128,148,0.8649,0.70536,0.29437,0.42859,0.71429,1.0,0.0,1.0,2,13,0,2,0,0,0,0,0,0,0,7,0,0,5,0,0,4,0,0,1,0,13],[132,148,0.8919,0.74552,0.22795,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,5,0,0,4,0,11],[136,148,0.9189,0.76336,0.26873,0.571,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,6,0,0,5,0,0,2,0,0,3,0,15],[140,148,0.9459,0.65624,0.2809,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,2,0,0,7,0,0,4,0,0,6,0,0,2,0,9],[144,148,0.973,0.58034,0.28558,0.42857,0.57143,0.75,0.0,1.0,2,6,0,2,0,2,0,0,2,0,0,5,0,0,10,0,0,3,0,0,2,0,6],[148,148,1.0,0.48211,0.24155,0.39286,0.42857,0.57143,0.0,1.0,2,2,0,2,0,1,0,0,5,0,0,12,0,0,6,0,0,1,0,0,3,0,2]]},{"b":4,"e":0.42857,"k":"falling","v":0.3125,"x":0.93304,"p":[[0,121,0.0,0.93304,0.18893,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[4,121,0.0331,0.89732,0.17215,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,5,0,21],[8,121,0.0661,0.8482,0.24208,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,2,0,20],[12,121,0.0992,0.79464,0.2285,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,6,0,0,8,0,12],[16,121,0.1322,0.8125,0.27067,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,3,0,0,1,0,0,2,0,0,6,0,17],[20,121,0.1653,0.77219,0.24967,0.71321,0.78571,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,3,0,0,2,0,0,9,0,0,3,0,13],[24,121,0.1983,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22],[28,121,0.2314,0.875,0.18123,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,2,0,20],[32,121,0.2645,0.76786,0.28959,0.57143,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,1,0,0,2,0,0,5,0,0,3,0,0,4,0,15],[36,121,0.2975,0.80802,0.25658,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,0,6,0,16],[40,121,0.3306,0.82143,0.22587,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,3,0,0,4,0,17],[44,121,0.3636,0.82142,0.2287,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,5,0,0,3,0,17],[48,121,0.3967,0.82143,0.21724,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,3,0,0,7,0,15],[52,121,0.4298,0.71429,0.23145,0.4286,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,2,0,0,6,0,9],[56,121,0.4628,0.77231,0.2572,0.57132,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,3,0,0,5,0,14],[60,121,0.4959,0.78123,0.22012,0.57143,0.85707,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,6,0,0,4,0,13],[64,121,0.5289,0.75893,0.27534,0.53572,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,6,0,0,2,0,0,5,0,0,2,0,15],[68,121,0.562,0.80795,0.27129,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,3,0,0,2,0,0,5,0,0,2,0,18],[72,121,0.595,0.75892,0.24075,0.4286,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,4,0,0,4,0,13],[76,121,0.6281,0.70536,0.27185,0.53571,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,1,0,0,5,0,0,3,0,0,8,0,0,3,0,10],[80,121,0.6612,0.74104,0.2867,0.5354,0.9285,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,6,0,0,7,0,0,0,0,0,1,0,16],[84,121,0.6942,0.73213,0.21944,0.57132,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,6,0,0,7,0,8],[88,121,0.7273,0.74106,0.25863,0.57132,0.78571,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,6,0,0,5,0,0,4,0,0,4,0,12],[92,121,0.7603,0.74107,0.24598,0.4286,0.78571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,4,0,0,4,0,12],[96,121,0.7934,0.69643,0.25692,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,9,0,0,3,0,0,6,0,0,4,0,9],[100,121,0.8264,0.74997,0.27895,0.57132,0.78571,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,4,0,0,6,0,0,4,0,0,1,0,15],[104,121,0.8595,0.74999,0.26487,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,7,0,0,6,0,0,1,0,0,2,0,15],[108,121,0.8926,0.80357,0.26426,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,0,2,0,17],[112,121,0.9256,0.79018,0.22583,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,5,0,0,4,0,14],[116,121,0.9587,0.72319,0.26231,0.57132,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,3,0,0,5,0,0,9,0,0,0,0,12],[120,121,0.9917,0.38393,0.30813,0.14286,0.28571,0.42858,0.0,1.0,4,5,0,4,0,7,0,0,6,0,0,8,0,0,2,0,0,0,0,0,0,0,5],[121,121,1.0,0.3125,0.27067,0.14286,0.2857,0.42858,0.0,1.0,7,2,0,7,0,7,0,0,4,0,0,9,0,0,1,0,0,2,0,0,0,0,2]]}]},{"i":"1ccabff6359badc0","q":"Let $m, n$ be distinct positive integers. Prove that\n\n$$\n\\operatorname{gcd}(m, n)+\\operatorname{gcd}(m+1, n+1)+\\operatorname{gcd}(m+2, n+2) \\leq 2|m-n|+1\n$$\n\nFurther, determine when equality holds.","t":[{"b":0,"e":1.0,"k":"flat","v":0.72767,"x":0.98661,"p":[[0,79,0.0,0.95088,0.13179,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[4,79,0.0506,0.75446,0.16841,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,12,0,0,3,0,8],[8,79,0.1013,0.73213,0.14618,0.57143,0.71429,0.71429,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,16,0,0,1,0,6],[12,79,0.1519,0.72767,0.16116,0.71429,0.71429,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,17,0,0,3,0,5],[16,79,0.2025,0.89732,0.1394,0.82132,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,5,0,19],[20,79,0.2532,0.92411,0.15146,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,25],[24,79,0.3038,0.91964,0.1234,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,21],[28,79,0.3544,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[32,79,0.4051,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[36,79,0.4557,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[40,79,0.5063,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[44,79,0.557,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[48,79,0.6076,0.97766,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[52,79,0.6582,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[56,79,0.7089,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[60,79,0.7595,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[64,79,0.8101,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[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],[72,79,0.9114,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[76,79,0.962,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[79,79,1.0,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28]]},{"b":3,"e":1.0,"k":"flat","v":0.70089,"x":1.0,"p":[[0,36,0.0,0.92411,0.15561,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,25],[4,36,0.1111,0.77231,0.14225,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,0,1,0,8],[8,36,0.2222,0.71427,0.09451,0.71429,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,24,0,0,1,0,2],[12,36,0.3333,0.70089,0.06546,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,25,0,0,2,0,0],[16,36,0.4444,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[20,36,0.5556,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,36,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,36,0.7778,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,36,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,36,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"44ddaa173b843f6c","q":"Pentagon  $ABCDE$  has circle  $S$  inscribed into it. Side  $BC$  is tangent to  $S$  at point  $K$ . If  $AB=BC=CD$ , prove that angle  $EKB$  is a right angle.","t":[{"b":0,"e":0.57143,"k":"falling","v":0.26777,"x":0.69641,"p":[[0,107,0.0,0.68747,0.17292,0.57142,0.64286,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,2,0,0,13,0,1],[4,107,0.0374,0.69641,0.15047,0.57143,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,5,0,0,13,0,0],[8,107,0.0748,0.61159,0.19638,0.57132,0.57143,0.85704,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,3,0,0,13,0,0,3,0,0,9,0,0],[12,107,0.1121,0.66069,0.16268,0.57143,0.57143,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,3,0,0,11,0,0],[16,107,0.1495,0.6339,0.16729,0.57143,0.57143,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,3,0,0,15,0,0,3,0,0,9,0,0],[20,107,0.1869,0.58479,0.17261,0.57132,0.57143,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,1,0,0,18,0,0,2,0,0,6,0,0],[24,107,0.2243,0.6071,0.13364,0.57142,0.57143,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,3,0,0,20,0,0,3,0,0,5,0,0],[28,107,0.2617,0.63836,0.21424,0.57143,0.57143,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,1,0,0,1,0,0,15,0,0,1,0,0,12,0,0],[32,107,0.2991,0.58477,0.12556,0.5713,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,22,0,0,3,0,0,3,0,0],[36,107,0.3364,0.53122,0.17581,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,3,0,0,7,0,0,17,0,0,0,0,0,4,0,0],[40,107,0.3738,0.5446,0.14479,0.53539,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,3,0,0,19,0,0,3,0,0,2,0,0],[44,107,0.4112,0.57588,0.15355,0.42859,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,8,0,0,18,0,0,0,0,0,4,0,1],[48,107,0.4486,0.53125,0.16458,0.42857,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,5,0,0,17,0,0,0,0,0,4,0,0],[52,107,0.486,0.52677,0.19045,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,4,0,0,16,0,0,3,0,0,3,0,0],[56,107,0.5234,0.59818,0.19377,0.53539,0.57143,0.74996,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,6,0,0,14,0,0,2,0,0,8,0,0],[60,107,0.5607,0.59815,0.20341,0.57132,0.57143,0.74996,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,1,0,0,15,0,0,3,0,0,8,0,0],[64,107,0.5981,0.60263,0.20119,0.571,0.57143,0.74996,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,0,0,0,14,0,0,4,0,0,8,0,0],[68,107,0.6355,0.54017,0.19475,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,5,0,0,14,0,0,2,0,0,5,0,0],[72,107,0.6729,0.52232,0.15407,0.39286,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,1,0,0,18,0,0,4,0,0,1,0,0],[76,107,0.7103,0.49106,0.2257,0.39286,0.57121,0.57143,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,7,0,0,12,0,0,1,0,0,3,0,1],[80,107,0.7477,0.4464,0.17766,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,8,0,0,7,0,0,12,0,0,0,0,0,2,0,0],[84,107,0.785,0.47766,0.166,0.28571,0.4998,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,7,0,0,13,0,0,1,0,0,2,0,0],[88,107,0.8224,0.53569,0.19884,0.4286,0.57143,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,2,0,0,4,0,0,16,0,0,5,0,0,1,0,1],[92,107,0.8598,0.45972,0.20447,0.28571,0.49979,0.57143,0.14,0.85714,0,0,0,0,0,4,0,0,8,0,0,4,0,0,12,0,0,1,0,0,3,0,0],[96,107,0.8972,0.45088,0.20237,0.28571,0.5712,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,11,0,0,2,0,0,14,0,0,1,0,0,2,0,0],[100,107,0.9346,0.36155,0.15554,0.2857,0.28571,0.571,0.14286,0.57143,0,0,0,0,0,5,0,0,15,0,0,2,0,0,10,0,0,0,0,0,0,0,0],[104,107,0.972,0.29908,0.1444,0.2857,0.28571,0.32143,0.0,0.57143,2,0,0,2,0,5,0,0,17,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[107,107,1.0,0.26777,0.08577,0.25,0.2857,0.28571,0.14,0.4286,0,0,0,0,0,8,0,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"falling","v":0.23661,"x":0.66961,"p":[[0,68,0.0,0.66961,0.15336,0.57143,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,7,0,0,10,0,0],[4,68,0.0588,0.53571,0.20203,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,9,0,0,10,0,0,0,0,0,7,0,0],[8,68,0.1176,0.58479,0.2183,0.42857,0.57143,0.75,0.0,0.85714,1,0,0,1,0,0,0,0,5,0,0,3,0,0,11,0,0,4,0,0,8,0,0],[12,68,0.1765,0.59358,0.22353,0.53465,0.57143,0.85704,0.14,1.0,0,1,0,0,0,2,0,0,4,0,0,2,0,0,13,0,0,2,0,0,8,0,1],[16,68,0.2353,0.56693,0.23278,0.42857,0.57143,0.85704,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,3,0,0,14,0,0,0,0,0,9,0,0],[20,68,0.2941,0.57589,0.22724,0.39286,0.57143,0.75,0.0,0.85714,1,0,0,1,0,0,0,0,7,0,0,1,0,0,11,0,0,4,0,0,8,0,0],[24,68,0.3529,0.60712,0.18211,0.5354,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,4,0,0,11,0,0,6,0,0,7,0,0],[28,68,0.4118,0.65176,0.20807,0.5354,0.64286,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,4,0,0,8,0,0,2,0,0,14,0,0],[32,68,0.4706,0.62053,0.20394,0.57143,0.57143,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,1,0,0,11,0,0,4,0,0,10,0,0],[36,68,0.5294,0.5357,0.21129,0.42857,0.57143,0.60714,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,4,0,0,13,0,0,3,0,0,5,0,0],[40,68,0.5882,0.54909,0.18935,0.42857,0.57143,0.60714,0.1429,0.85714,0,0,0,0,0,1,0,0,5,0,0,5,0,0,13,0,0,3,0,0,5,0,0],[44,68,0.6471,0.52231,0.21902,0.39286,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,4,0,0,11,0,0,5,0,0,4,0,0],[48,68,0.7059,0.47317,0.2065,0.28571,0.57121,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,8,0,0,2,0,0,16,0,0,1,0,0,1,0,1],[52,68,0.7647,0.43749,0.23127,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,12,0,0,5,0,0,6,0,0,2,0,0,4,0,0],[56,68,0.8235,0.41516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$\\Gamma_{1}$ and $\\Gamma_{2}$ intersect at $P$ and $Q$. Let $A$ be a point on $\\Gamma_{1}$ not equal to $P$ or $Q$. The lines $A P$ and $A Q$ intersect $\\Gamma_{2}$ again at $B$ and $C$ respectively.\nProve that the altitude from $A$ in triangle $A B C$ passes through a point that is independent of the choice of $A$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.52677,"p":[[0,54,0.0,0.27676,0.33678,0.0,0.07143,0.4286,0.0,1.0,16,3,6,16,0,1,0,0,3,0,0,5,0,0,2,0,0,1,0,0,1,0,3],[4,54,0.0741,0.48213,0.36377,0.21427,0.42857,0.85704,0.0,1.0,8,7,0,8,0,0,0,0,3,0,0,8,0,0,3,0,0,1,0,0,2,0,7],[8,54,0.1481,0.52677,0.3719,0.21427,0.57121,0.85714,0.0,1.0,8,7,0,8,0,0,0,0,2,0,0,5,0,0,3,0,0,4,0,0,3,0,7],[12,54,0.2222,0.50446,0.38462,0.14286,0.42857,1.0,0.0,1.0,7,9,0,7,0,3,0,0,1,0,0,7,0,0,3,0,0,0,0,0,2,0,9],[16,54,0.2963,0.45088,0.41049,0.0,0.42857,0.89286,0.0,1.0,12,8,0,12,0,0,0,0,2,0,0,5,0,0,1,0,0,2,0,0,2,0,8],[20,54,0.3704,0.29017,0.33212,0.0,0.2857,0.42857,0.0,1.0,15,4,0,15,0,0,0,0,3,0,0,9,0,0,1,0,0,0,0,0,0,0,4],[24,54,0.4444,0.34819,0.30078,0.0,0.28571,0.42857,0.0,1.0,9,3,0,9,0,1,0,0,7,0,0,8,0,0,3,0,0,0,0,0,1,0,3],[28,54,0.5185,0.44196,0.33189,0.24999,0.42857,0.60714,0.0,1.0,7,5,0,7,0,1,0,0,3,0,0,12,0,0,1,0,0,1,0,0,2,0,5],[32,54,0.5926,0.24107,0.24598,0.0,0.2857,0.42857,0.0,0.85714,15,0,0,15,0,0,0,0,3,0,0,10,0,0,3,0,0,0,0,0,1,0,0],[36,54,0.6667,0.40177,0.34706,0.0,0.42857,0.57143,0.0,1.0,10,5,0,10,0,0,0,0,4,0,0,8,0,0,3,0,0,1,0,0,1,0,5],[40,54,0.7407,0.21879,0.33691,0.0,0.0,0.42895,0.0,1.0,21,2,0,21,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,0,2,0,2],[44,54,0.8148,0.04017,0.12987,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[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],[52,54,0.963,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.61159,"p":[[0,42,0.0,0.31696,0.34944,0.0,0.2857,0.42857,0.0,1.0,13,5,5,13,0,1,0,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,5],[4,42,0.0952,0.61159,0.34669,0.42857,0.57141,1.0,0.0,1.0,5,9,0,5,0,0,0,0,0,0,0,9,0,0,3,0,0,1,0,0,5,0,9],[8,42,0.1905,0.47766,0.37047,0.21427,0.42857,0.89275,0.0,1.0,8,8,0,8,0,0,0,0,4,0,0,8,0,0,2,0,0,1,0,0,1,0,8],[12,42,0.2857,0.44194,0.35239,0.0,0.42857,0.60714,0.0,1.0,9,5,0,9,0,1,0,0,2,0,0,6,0,0,6,0,0,1,0,0,2,0,5],[16,42,0.381,0.38391,0.33962,0.0,0.42857,0.57143,0.0,1.0,11,3,0,11,0,0,0,0,3,0,0,8,0,0,3,0,0,1,0,0,3,0,3],[20,42,0.4762,0.40177,0.38038,0.0,0.42857,0.64286,0.0,1.0,12,6,0,12,0,0,0,0,3,0,0,6,0,0,3,0,0,0,0,0,2,0,6],[24,42,0.5714,0.10714,0.24223,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,1],[28,42,0.6667,0.09374,0.20391,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[32,42,0.7619,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.06696,0.14718,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,42,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6669c1aee6c3c33f","q":"The lengths of the sides of a triangle are integers, whereas the radius of its circumscribed circle is a prime number. Prove that the triangle is right-angled.","t":[{"b":3,"e":0.14286,"k":"falling","v":0.1875,"x":0.80357,"p":[[0,41,0.0,0.80357,0.30671,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,0,3,0,20],[4,41,0.0976,0.74543,0.33469,0.53539,1.0,1.0,0.14,1.0,0,18,0,0,0,5,0,0,2,0,0,1,0,0,2,0,0,3,0,0,1,0,18],[8,41,0.1951,0.71875,0.32436,0.53571,0.85714,1.0,0.0,1.0,1,14,0,1,0,3,0,0,3,0,0,1,0,0,2,0,0,5,0,0,3,0,14],[12,41,0.2927,0.63391,0.33491,0.28571,0.57143,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,4,0,0,3,0,0,6,0,0,1,0,0,0,0,13],[16,41,0.3902,0.54911,0.36789,0.14286,0.50001,0.89286,0.0,1.0,4,8,0,4,0,5,0,0,2,0,0,5,0,0,1,0,0,3,0,0,4,0,8],[20,41,0.4878,0.69188,0.28613,0.53569,0.71429,1.0,0.14,1.0,0,10,0,0,0,3,0,0,2,0,0,3,0,0,6,0,0,3,0,0,5,0,10],[24,41,0.5854,0.71874,0.27546,0.57132,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,3,0,0,2,0,0,4,0,0,6,0,0,4,0,11],[28,41,0.6829,0.60264,0.30037,0.42857,0.71429,0.85704,0.0,1.0,2,5,0,2,0,4,0,0,1,0,0,2,0,0,6,0,0,8,0,0,4,0,5],[32,41,0.7805,0.51784,0.3004,0.2857,0.57143,0.71429,0.0,1.0,2,5,0,2,0,5,0,0,4,0,0,1,0,0,11,0,0,3,0,0,1,0,5],[36,41,0.878,0.49998,0.39122,0.14286,0.42857,0.89286,0.0,1.0,6,8,0,6,0,6,0,0,2,0,0,3,0,0,1,0,0,3,0,0,3,0,8],[40,41,0.9756,0.1875,0.14914,0.14286,0.14286,0.1786,0.0,0.85714,3,0,0,3,0,21,0,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[41,41,1.0,0.20089,0.15916,0.14286,0.14286,0.2857,0.0,0.85714,4,0,0,4,0,17,0,0,8,0,0,2,0,0,0,0,0,0,0,0,1,0,0]]},{"b":5,"e":0.42857,"k":"falling","v":0.6116,"x":0.84375,"p":[[0,34,0.0,0.84375,0.24836,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,4,0,19],[4,34,0.1176,0.71875,0.32827,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,6,0,0,1,0,0,0,0,0,5,0,0,0,0,0,7,0,13],[8,34,0.2353,0.74999,0.29015,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,0,0,0,2,0,0,4,0,0,3,0,0,6,0,13],[12,34,0.3529,0.7633,0.3066,0.67857,0.85714,1.0,0.0,1.0,2,14,0,2,0,2,0,0,0,0,0,1,0,0,3,0,0,4,0,0,6,0,14],[16,34,0.4706,0.79451,0.29441,0.67536,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,1,0,0,1,0,0,3,0,0,2,0,0,5,0,17],[20,34,0.5882,0.82589,0.27833,0.82143,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,6,0,18],[24,34,0.7059,0.82588,0.19476,0.67857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,5,0,15],[28,34,0.8235,0.79911,0.20159,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,6,0,12],[32,34,0.9412,0.65164,0.18873,0.57143,0.57143,0.75,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,7,0,0,5,0,3],[34,34,1.0,0.6116,0.19959,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,7,0,0,9,0,0,7,0,0,3,0,3]]}]},{"i":"fb9808aa6008b962","q":"2. N2 (ROM) ${ }^{\\mathrm{IMO} 4}$ Let $n \\geq 2$ be a positive integer, with divisors $1=d_{1}<$ $d_{2}<\\cdots<d_{k}=n$. Prove that $d_{1} d_{2}+d_{2} d_{3}+\\cdots+d_{k-1} d_{k}$ is always less than $n^{2}$, and determine when it is a divisor of $n^{2}$.","t":[{"b":0,"e":0.28571,"k":"falling","v":0.1384,"x":0.58482,"p":[[0,73,0.0,0.32143,0.37287,0.0,0.14286,0.60714,0.0,1.0,14,4,0,14,0,4,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,4],[4,73,0.0548,0.58482,0.27747,0.28571,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,8,0,0,3,0,0,6,0,0,3,0,0,5,0,5],[8,73,0.1096,0.49554,0.2923,0.28571,0.42859,0.71429,0.0,1.0,3,4,0,3,0,1,0,0,9,0,0,4,0,0,4,0,0,6,0,0,1,0,4],[12,73,0.1644,0.49554,0.27196,0.28571,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,5,0,0,8,0,0,4,0,0,6,0,0,4,0,0,1,0,4],[16,73,0.2192,0.4375,0.35524,0.14286,0.28571,0.85714,0.0,1.0,3,5,0,3,0,10,0,0,6,0,0,1,0,0,1,0,0,2,0,0,4,0,5],[20,73,0.274,0.375,0.32291,0.14286,0.28571,0.57143,0.0,1.0,6,3,0,6,0,6,0,0,8,0,0,2,0,0,3,0,0,1,0,0,3,0,3],[24,73,0.3288,0.26777,0.22522,0.14286,0.2857,0.32143,0.0,0.85714,6,0,0,6,0,9,0,0,9,0,0,3,0,0,3,0,0,0,0,0,2,0,0],[28,73,0.3836,0.26339,0.25532,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,11,0,0,5,0,0,5,0,0,0,0,0,2,0,0,1,0,1],[32,73,0.4384,0.25891,0.24072,0.14286,0.14288,0.28571,0.0,1.0,6,1,0,6,0,11,0,0,8,0,0,3,0,0,1,0,0,1,0,0,1,0,1],[36,73,0.4932,0.29016,0.23819,0.10714,0.28571,0.42857,0.0,0.85714,8,0,0,8,0,6,0,0,5,0,0,6,0,0,5,0,0,1,0,0,1,0,0],[40,73,0.5479,0.33929,0.28959,0.14286,0.28571,0.46431,0.0,1.0,7,2,0,7,0,4,0,0,10,0,0,3,0,0,2,0,0,3,0,0,1,0,2],[44,73,0.6027,0.37499,0.32092,0.14286,0.28571,0.46418,0.0,1.0,4,4,0,4,0,9,0,0,7,0,0,4,0,0,1,0,0,1,0,0,2,0,4],[48,73,0.6575,0.38393,0.33012,0.14286,0.2857,0.57143,0.0,1.0,4,5,0,4,0,9,0,0,7,0,0,3,0,0,2,0,0,1,0,0,1,0,5],[52,73,0.7123,0.33027,0.25118,0.14286,0.28571,0.46431,0.0,1.0,2,2,0,2,0,12,0,0,7,0,0,3,0,0,5,0,0,1,0,0,0,0,2],[56,73,0.7671,0.28125,0.27313,0.10714,0.21428,0.42857,0.0,1.0,8,2,0,8,0,8,0,0,6,0,0,5,0,0,1,0,0,2,0,0,0,0,2],[60,73,0.8219,0.27232,0.27976,0.10714,0.14286,0.28571,0.0,1.0,8,2,0,8,0,9,0,0,8,0,0,1,0,0,1,0,0,3,0,0,0,0,2],[64,73,0.8767,0.20089,0.21086,0.10714,0.14286,0.28571,0.0,1.0,8,1,0,8,0,13,0,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[68,73,0.9315,0.17857,0.12877,0.14286,0.14286,0.28571,0.0,0.57143,6,0,0,6,0,15,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[72,73,0.9863,0.17857,0.11845,0.14286,0.14286,0.2857,0.0,0.42857,6,0,0,6,0,14,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.1384,0.09094,0.14286,0.14286,0.1429,0.0,0.28571,7,0,0,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.09375,"x":0.375,"p":[[0,54,0.0,0.33481,0.35464,0.0,0.21428,0.57143,0.0,1.0,13,3,0,13,0,3,0,0,3,0,0,1,0,0,5,0,0,2,0,0,2,0,3],[4,54,0.0741,0.35714,0.31134,0.14286,0.28571,0.57143,0.0,1.0,4,3,0,4,0,10,0,0,8,0,0,1,0,0,2,0,0,2,0,0,2,0,3],[8,54,0.1481,0.35266,0.30718,0.10714,0.28571,0.57143,0.0,1.0,8,2,0,8,0,6,0,0,3,0,0,4,0,0,5,0,0,3,0,0,1,0,2],[12,54,0.2222,0.29909,0.28872,0.14286,0.14286,0.42857,0.0,1.0,6,3,0,6,0,12,0,0,3,0,0,4,0,0,4,0,0,0,0,0,0,0,3],[16,54,0.2963,0.30795,0.30122,0.14286,0.14286,0.42858,0.0,1.0,6,3,0,6,0,11,0,0,6,0,0,2,0,0,1,0,0,3,0,0,0,0,3],[20,54,0.3704,0.26775,0.27839,0.0,0.14286,0.32143,0.0,1.0,9,1,0,9,0,9,0,0,6,0,0,1,0,0,2,0,0,3,0,0,1,0,1],[24,54,0.4444,0.23661,0.23854,0.0,0.21428,0.32143,0.0,0.71429,12,0,0,12,0,4,0,0,8,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[28,54,0.5185,0.375,0.31491,0.14286,0.2857,0.57143,0.0,1.0,3,4,0,3,0,11,0,0,6,0,0,3,0,0,2,0,0,2,0,0,1,0,4],[32,54,0.5926,0.18303,0.20277,0.0,0.14286,0.2857,0.0,0.85714,10,0,0,10,0,13,0,0,4,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[36,54,0.6667,0.17857,0.21129,0.0,0.14286,0.1429,0.0,1.0,9,1,0,9,0,16,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[40,54,0.7407,0.1607,0.19802,0.0,0.14286,0.28571,0.0,0.85714,14,0,0,14,0,8,0,0,6,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[44,54,0.8148,0.26339,0.27225,0.14286,0.14286,0.32143,0.0,0.85714,6,0,0,6,0,16,0,0,2,0,0,2,0,0,0,0,0,3,0,0,3,0,0],[48,54,0.8889,0.10715,0.12372,0.0,0.07143,0.14292,0.0,0.42857,16,0,0,16,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.13393,0.09407,0.10714,0.14286,0.1429,0.0,0.28571,8,0,0,8,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.09375,0.08459,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fdf0e8fe1a201b8a","q":"Let $x$ and $y$ be positive real numbers such that $x+y^{2016} \\geq 1$. Prove that $x^{2016}+y>$ 1 - 1/100.","t":[{"b":1,"e":0.0,"k":"flat","v":0.00893,"x":0.4241,"p":[[0,44,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,6,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.40625,0.32754,0.10714,0.42857,0.60714,0.0,1.0,8,4,1,8,0,2,0,0,3,0,0,10,0,0,1,0,0,3,0,0,1,0,4],[8,44,0.1818,0.32143,0.31135,0.0,0.28571,0.42858,0.0,1.0,11,3,0,11,0,2,0,0,5,0,0,7,0,0,2,0,0,2,0,0,0,0,3],[12,44,0.2727,0.38393,0.34523,0.0,0.28571,0.71429,0.0,1.0,10,3,0,10,0,2,0,0,5,0,0,4,0,0,1,0,0,5,0,0,2,0,3],[16,44,0.3636,0.375,0.33834,0.0,0.35714,0.57143,0.0,1.0,9,5,0,9,0,3,0,0,4,0,0,7,0,0,3,0,0,1,0,0,0,0,5],[20,44,0.4545,0.4241,0.34345,0.0,0.42857,0.71429,0.0,1.0,9,3,0,9,0,2,0,0,3,0,0,4,0,0,3,0,0,6,0,0,2,0,3],[24,44,0.5455,0.31249,0.30185,0.0,0.21431,0.46418,0.0,1.0,10,1,0,10,0,6,0,0,2,0,0,6,0,0,2,0,0,3,0,0,2,0,1],[28,44,0.6364,0.32142,0.34809,0.0,0.28571,0.4642,0.0,1.0,14,3,0,14,0,1,0,0,3,0,0,6,0,0,1,0,0,2,0,0,2,0,3],[32,44,0.7273,0.30803,0.26752,0.0,0.35714,0.42857,0.0,1.0,11,1,0,11,0,1,0,0,4,0,0,9,0,0,4,0,0,2,0,0,0,0,1],[36,44,0.8182,0.19643,0.17405,0.0,0.14286,0.28571,0.0,0.57143,11,0,0,11,0,6,0,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[40,44,0.9091,0.34821,0.26949,0.14286,0.28571,0.57143,0.0,0.85714,6,0,0,6,0,6,0,0,6,0,0,4,0,0,5,0,0,2,0,0,3,0,0],[44,44,1.0,0.08036,0.13803,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"rising","v":0.02232,"x":0.38839,"p":[[0,56,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,5,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,56,0.0714,0.28125,0.26119,0.0,0.28571,0.42857,0.0,1.0,11,1,0,11,0,2,0,0,6,0,0,8,0,0,3,0,0,0,0,0,1,0,1],[8,56,0.1429,0.29017,0.30195,0.0,0.21429,0.57143,0.0,1.0,13,1,0,13,0,3,0,0,3,0,0,4,0,0,4,0,0,3,0,0,1,0,1],[12,56,0.2143,0.33927,0.32092,0.0,0.28571,0.57143,0.0,1.0,10,3,0,10,0,4,0,0,4,0,0,3,0,0,7,0,0,0,0,0,1,0,3],[16,56,0.2857,0.38839,0.31183,0.14286,0.28571,0.60714,0.0,1.0,6,3,0,6,0,3,0,0,10,0,0,4,0,0,1,0,0,3,0,0,2,0,3],[20,56,0.3571,0.25892,0.2586,0.0,0.2857,0.42857,0.0,0.85714,12,0,0,12,0,2,0,0,9,0,0,4,0,0,0,0,0,4,0,0,1,0,0],[24,56,0.4286,0.31696,0.3209,0.0,0.28571,0.46429,0.0,1.0,11,3,0,11,0,3,0,0,6,0,0,4,0,0,3,0,0,1,0,0,1,0,3],[28,56,0.5,0.33482,0.28928,0.0,0.28571,0.57143,0.0,1.0,9,1,0,9,0,4,0,0,5,0,0,4,0,0,3,0,0,6,0,0,0,0,1],[32,56,0.5714,0.29018,0.28456,0.0,0.28571,0.42857,0.0,1.0,11,2,0,11,0,1,0,0,11,0,0,2,0,0,3,0,0,2,0,0,0,0,2],[36,56,0.6429,0.29017,0.24609,0.0,0.28571,0.42858,0.0,1.0,9,1,0,9,0,3,0,0,8,0,0,6,0,0,4,0,0,1,0,0,0,0,1],[40,56,0.7143,0.35714,0.31744,0.0,0.28571,0.60714,0.0,1.0,9,3,0,9,0,2,0,0,8,0,0,4,0,0,1,0,0,5,0,0,0,0,3],[44,56,0.7857,0.2857,0.25998,0.0,0.28571,0.42857,0.0,0.857,11,0,0,11,0,2,0,0,7,0,0,5,0,0,3,0,0,3,0,0,1,0,0],[48,56,0.8571,0.31249,0.33585,0.0,0.28571,0.57111,0.0,1.0,13,3,0,13,0,2,0,0,5,0,0,3,0,0,3,0,0,2,0,0,1,0,3],[52,56,0.9286,0.37051,0.30692,0.14286,0.28571,0.60714,0.0,1.0,7,2,0,7,0,5,0,0,6,0,0,3,0,0,3,0,0,5,0,0,1,0,2],[56,56,1.0,0.30357,0.25939,0.10714,0.28571,0.46429,0.0,1.0,8,1,0,8,0,4,0,0,10,0,0,2,0,0,4,0,0,3,0,0,0,0,1]]}]},{"i":"901923da26f7cb54","q":"Find all natural numbers  $ n\\ge 4 $  that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers  $ a,b,c $  that verify the equation  $$  (a-b)^n+(b-c)^n+(c-a)^n=0,  $$  represent the vertices of an equilateral triangle in the complex plane.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.2232,"x":0.45535,"p":[[0,71,0.0,0.2232,0.25983,0.0,0.2857,0.28571,0.0,1.0,13,2,1,13,0,2,0,0,13,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[4,71,0.0563,0.40622,0.23447,0.28571,0.35714,0.60714,0.0,0.71429,4,0,0,4,0,1,0,0,11,0,0,4,0,0,4,0,0,8,0,0,0,0,0],[8,71,0.1127,0.43749,0.20496,0.28571,0.28571,0.60714,0.0,0.85714,1,0,0,1,0,0,0,0,16,0,0,3,0,0,4,0,0,7,0,0,1,0,0],[12,71,0.169,0.45535,0.2126,0.28571,0.42857,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,13,0,0,7,0,0,2,0,0,8,0,0,0,0,1],[16,71,0.2254,0.45088,0.22047,0.2857,0.28571,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,17,0,0,1,0,0,2,0,0,10,0,0,1,0,0],[20,71,0.2817,0.33924,0.12745,0.2857,0.28571,0.32143,0.14286,0.71429,0,0,0,0,0,2,0,0,22,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[24,71,0.338,0.37052,0.16697,0.2857,0.28571,0.4642,0.14286,0.71429,0,0,0,0,0,2,0,0,21,0,0,1,0,0,4,0,0,4,0,0,0,0,0],[28,71,0.3944,0.34821,0.16342,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,2,0,0,20,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[32,71,0.4507,0.37053,0.1885,0.2857,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,22,0,0,4,0,0,0,0,0,3,0,0,2,0,0],[36,71,0.507,0.33927,0.14615,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,0,0,0,24,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[40,71,0.5634,0.30356,0.16266,0.2857,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,2,0,0,20,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[44,71,0.6197,0.32589,0.11426,0.2857,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,0,0,0,23,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[48,71,0.6761,0.31696,0.13709,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,2,0,0,23,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[52,71,0.7324,0.29017,0.09771,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,0,0,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[56,71,0.7887,0.29018,0.14054,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,1,0,0,24,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[60,71,0.8451,0.30802,0.10167,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,0,0,0,27,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[64,71,0.9014,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,71,0.9577,0.31696,0.08548,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[71,71,1.0,0.29464,0.03459,0.2857,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.1607,"x":0.375,"p":[[0,88,0.0,0.1607,0.17765,0.0,0.07143,0.28571,0.0,0.571,16,0,1,16,0,2,0,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[4,88,0.0455,0.375,0.23891,0.2857,0.28571,0.57143,0.0,0.85714,5,0,0,5,0,0,0,0,14,0,0,4,0,0,2,0,0,6,0,0,1,0,0],[8,88,0.0909,0.35702,0.23695,0.25,0.28571,0.571,0.0,0.857,4,0,0,4,0,4,0,0,12,0,0,3,0,0,3,0,0,5,0,0,1,0,0],[12,88,0.1364,0.32141,0.18897,0.2857,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,3,0,0,14,0,0,5,0,0,4,0,0,2,0,0,0,0,0],[16,88,0.1818,0.35712,0.17495,0.2857,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,1,0,0,18,0,0,4,0,0,4,0,0,3,0,0,0,0,0],[20,88,0.2273,0.29018,0.22724,0.0,0.28571,0.42858,0.0,0.71429,9,0,0,9,0,0,0,0,14,0,0,2,0,0,4,0,0,3,0,0,0,0,0],[24,88,0.2727,0.24997,0.21125,0.0,0.28571,0.32143,0.0,0.71429,10,0,0,10,0,2,0,0,12,0,0,4,0,0,2,0,0,2,0,0,0,0,0],[28,88,0.3182,0.26774,0.22519,0.14214,0.2857,0.32143,0.0,0.85714,7,0,0,7,0,7,0,0,10,0,0,3,0,0,2,0,0,2,0,0,1,0,0],[32,88,0.3636,0.28122,0.23,0.14286,0.28571,0.28571,0.0,1.0,7,1,0,7,0,4,0,0,14,0,0,1,0,0,4,0,0,1,0,0,0,0,1],[36,88,0.4091,0.29915,0.27518,0.0,0.2857,0.42857,0.0,1.0,9,1,0,9,0,2,0,0,12,0,0,4,0,0,1,0,0,0,0,0,3,0,1],[40,88,0.4545,0.20973,0.15969,0.105,0.2857,0.28571,0.0,0.71429,8,0,0,8,0,6,0,0,15,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[44,88,0.5,0.26777,0.15472,0.25,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,5,0,0,21,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[48,88,0.5455,0.31248,0.19702,0.2857,0.28571,0.28571,0.0,0.85714,5,0,0,5,0,0,0,0,20,0,0,1,0,0,4,0,0,1,0,0,1,0,0],[52,88,0.5909,0.29018,0.14934,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,3,0,0,20,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[56,88,0.6364,0.29464,0.1234,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,2,0,0,23,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[60,88,0.6818,0.26786,0.12242,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,3,0,0,23,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[64,88,0.7273,0.27222,0.17266,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,4,0,0,17,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[68,88,0.7727,0.29462,0.138,0.2857,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,1,0,0,23,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[72,88,0.8182,0.27677,0.12843,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,4,0,0,23,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[76,88,0.8636,0.28126,0.12103,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,5,0,0,23,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[80,88,0.9091,0.25446,0.07771,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,88,0.9545,0.27233,0.06546,0.2857,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,2,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,88,1.0,0.25891,0.08323,0.25,0.2857,0.28571,0.14286,0.571,0,0,0,0,0,8,0,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"b5010c2ea9dbb765","q":"Find the greatest common divisor of the numbers \\[ 2^{561}\\minus{}2, 3^{561}\\minus{}3, \\ldots, 561^{561}\\minus{}561.\\]","t":[{"b":4,"e":0.71429,"k":"flat","v":0.73661,"x":0.94643,"p":[[0,30,0.0,0.73661,0.26271,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,7,0,0,3,0,12],[4,30,0.1333,0.94196,0.13767,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,26],[8,30,0.2667,0.94643,0.12753,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,26],[12,30,0.4,0.93749,0.14702,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[16,30,0.5333,0.90179,0.15335,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,1,0,22],[20,30,0.6667,0.87054,0.18681,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,0,0,0,21],[24,30,0.8,0.87052,0.18683,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,1,0,20],[28,30,0.9333,0.87052,0.1689,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,2,0,19],[30,30,1.0,0.84374,0.17629,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,3,0,16]]},{"b":5,"e":0.71429,"k":"flat","v":0.79464,"x":0.89732,"p":[[0,46,0.0,0.79464,0.21998,0.71429,0.78571,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,9,0,0,2,0,14],[4,46,0.087,0.89732,0.16457,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,1,0,22],[8,46,0.1739,0.88839,0.15865,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,0,0,21],[12,46,0.2609,0.87499,0.18125,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,0,0,21],[16,46,0.3478,0.875,0.15872,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,1,0,19],[20,46,0.4348,0.88839,0.15865,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,0,0,21],[24,46,0.5217,0.86607,0.16342,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,2,0,18],[28,46,0.6087,0.89732,0.1439,0.71429,1.0,1.0,0.5714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,21],[32,46,0.6957,0.89732,0.15251,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,21],[36,46,0.7826,0.87945,0.16796,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,2,0,20],[40,46,0.8696,0.87052,0.17987,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,2,0,19],[44,46,0.9565,0.84372,0.1651,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,1,0,16],[46,46,1.0,0.82589,0.19144,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,12,0,0,3,0,14]]}]},{"i":"a7d412c4a42b1b69","q":"Suppose that positive integers  $m,n,k$  satisfy the equations   $$ m^2+1=2n^2, 2m^2+1=11k^2. $$   Find the residue when  $n$  is divided by  $17$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.10268,"x":0.26339,"p":[[0,160,0.0,0.22321,0.14258,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,0,18,0,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[4,160,0.025,0.26339,0.18594,0.14286,0.14286,0.42857,0.0,0.857,1,0,0,1,0,19,0,0,1,0,0,8,0,0,2,0,0,0,0,0,1,0,0],[8,160,0.05,0.20527,0.15546,0.14286,0.14286,0.17857,0.0,0.71429,3,0,0,3,0,21,0,0,1,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[12,160,0.075,0.20982,0.1636,0.14286,0.14286,0.32142,0.0,0.57143,5,0,0,5,0,17,0,0,2,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[16,160,0.1,0.18295,0.13943,0.14286,0.14286,0.2857,0.0,0.57143,6,0,0,6,0,16,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,160,0.125,0.22312,0.15131,0.14286,0.14286,0.32143,0.0,0.57143,4,0,0,4,0,15,0,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[24,160,0.15,0.18304,0.16458,0.14286,0.14286,0.14287,0.0,0.57143,7,0,0,7,0,18,0,0,0,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[28,160,0.175,0.18304,0.14826,0.14286,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,16,0,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[32,160,0.2,0.17411,0.14167,0.14286,0.14286,0.17857,0.0,0.42857,7,0,0,7,0,17,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[36,160,0.225,0.1875,0.14032,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,18,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[40,160,0.25,0.13839,0.13592,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,15,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[44,160,0.275,0.16063,0.14618,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,16,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[48,160,0.3,0.15179,0.1234,0.10714,0.14286,0.14289,0.0,0.4286,8,0,0,8,0,17,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,160,0.325,0.14277,0.14725,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,16,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[56,160,0.35,0.1517,0.13804,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,17,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[60,160,0.375,0.16499,0.14333,0.105,0.14286,0.17857,0.0,0.571,8,0,0,8,0,16,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[64,160,0.4,0.19643,0.12242,0.14286,0.14286,0.14287,0.0,0.57143,1,0,0,1,0,24,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[68,160,0.425,0.19196,0.17353,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,14,0,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[72,160,0.45,0.20088,0.15913,0.14286,0.14286,0.17857,0.0,0.57143,4,0,0,4,0,20,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[76,160,0.475,0.16062,0.13244,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,21,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[80,160,0.5,0.1874,0.14913,0.14286,0.14286,0.17857,0.0,0.5714,5,0,0,5,0,19,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[84,160,0.525,0.16964,0.18011,0.0,0.14286,0.14286,0.0,0.857,9,0,0,9,0,16,0,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[88,160,0.55,0.14732,0.13115,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,17,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[92,160,0.575,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,160,0.6,0.15624,0.10921,0.14286,0.14286,0.14286,0.0,0.571,4,0,0,4,0,24,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[100,160,0.625,0.16518,0.13415,0.14286,0.14286,0.14286,0.0,0.71429,5,0,0,5,0,21,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[104,160,0.65,0.15179,0.14258,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,17,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[108,160,0.675,0.14723,0.12103,0.0,0.14286,0.17857,0.0,0.42857,9,0,0,9,0,15,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[112,160,0.7,0.15607,0.12558,0.14,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,19,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[116,160,0.725,0.11598,0.11536,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,18,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[120,160,0.75,0.12482,0.11707,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,18,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[124,160,0.775,0.125,0.11152,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,18,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[128,160,0.8,0.11161,0.11143,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,19,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[132,160,0.825,0.13357,0.08699,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,160,0.85,0.10268,0.09606,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[140,160,0.875,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,160,0.9,0.11608,0.09062,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[148,160,0.925,0.12491,0.08562,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[152,160,0.95,0.15177,0.13329,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,23,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[156,160,0.975,0.12938,0.12036,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,18,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[160,160,1.0,0.13375,0.07934,0.14214,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.08036,"x":0.24554,"p":[[0,61,0.0,0.2008,0.08651,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,18,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,61,0.0656,0.24554,0.19638,0.14286,0.14286,0.32143,0.0,1.0,3,1,0,3,0,16,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[8,61,0.1311,0.14728,0.13602,0.0,0.14286,0.14286,0.0,0.43,10,0,0,10,0,15,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,61,0.1967,0.20982,0.16746,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,16,0,0,5,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[16,61,0.2623,0.18749,0.15742,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,22,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[20,61,0.3279,0.13393,0.10677,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,20,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,61,0.3934,0.19634,0.12757,0.14286,0.14286,0.28571,0.0,0.4286,3,0,0,3,0,20,0,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[28,61,0.459,0.14286,0.14725,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,13,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[32,61,0.5246,0.16071,0.09942,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,22,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,61,0.5902,0.18304,0.12492,0.14286,0.14286,0.17857,0.0,0.4286,4,0,0,4,0,20,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[40,61,0.6557,0.11607,0.10972,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,61,0.7213,0.15152,0.11814,0.14,0.14286,0.14286,0.0,0.4286,7,0,0,7,0,19,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,61,0.7869,0.11161,0.10555,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,61,0.8525,0.14259,0.09449,0.14,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,21,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,61,0.918,0.13393,0.08703,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,61,0.9836,0.1025,0.0891,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[61,61,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a797623d70b76377","q":"Let  $A$  be a set of  $n$  points in the space. From the family of all segments with endpoints in  $A$ ,  $q$  segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of  $m$  yellow segments, where  $m \\geq \\frac{2q}{n}$ , arranged in order of increasing length.","t":[{"b":0,"e":0.14286,"k":"rising","v":0.08036,"x":0.47767,"p":[[0,50,0.0,0.10268,0.16457,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,6,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,50,0.08,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.36598,0.24736,0.14286,0.28571,0.57143,0.0,1.0,2,1,0,2,0,10,0,0,5,0,0,5,0,0,6,0,0,2,0,0,1,0,1],[16,50,0.32,0.47767,0.29147,0.28571,0.42857,0.71429,0.0,1.0,1,4,0,1,0,5,0,0,7,0,0,8,0,0,2,0,0,2,0,0,3,0,4],[20,50,0.4,0.35714,0.28571,0.14286,0.28571,0.57143,0.0,1.0,6,2,0,6,0,5,0,0,7,0,0,5,0,0,5,0,0,0,0,0,2,0,2],[24,50,0.48,0.36161,0.29011,0.14286,0.28571,0.42858,0.0,1.0,5,2,0,5,0,5,0,0,10,0,0,5,0,0,1,0,0,1,0,0,3,0,2],[28,50,0.56,0.39732,0.25688,0.2857,0.28571,0.42857,0.0,1.0,1,2,0,1,0,5,0,0,14,0,0,5,0,0,0,0,0,3,0,0,2,0,2],[32,50,0.64,0.11161,0.22794,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[36,50,0.72,0.15178,0.23402,0.0,0.07143,0.17857,0.0,1.0,16,1,0,16,0,8,0,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[40,50,0.8,0.17848,0.23959,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,8,0,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[44,50,0.88,0.12946,0.11495,0.0,0.14286,0.2857,0.0,0.28571,12,0,0,12,0,11,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.24553,0.07349,0.24999,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.28116,0.10415,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,4,0,0,24,0,0,1,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04464,"x":0.1875,"p":[[0,25,0.0,0.08037,0.15128,0.0,0.0,0.03571,0.0,0.4286,24,0,2,24,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.16518,0.20238,0.0,0.14286,0.14286,0.0,0.85714,12,0,0,12,0,13,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[8,25,0.32,0.1875,0.23266,0.0,0.14286,0.1786,0.0,0.85714,11,0,0,11,0,13,0,0,3,0,0,2,0,0,0,0,0,1,0,0,2,0,0],[12,25,0.48,0.10268,0.17582,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,25,0.64,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.143,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2711a9a88876d900","q":"Let $A B C D$ be a cyclic quadrilateral with $A B=7$ and $C D=8$. Points $P$ and $Q$ are selected on line segment $A B$ so that $A P=B Q=3$. Points $R$ and $S$ are selected on line segment $C D$ so that $C R=D S=2$. Prove that $P Q R S$ is a cyclic quadrilateral.","t":[{"b":4,"e":1.0,"k":"flat","v":0.77677,"x":0.95536,"p":[[0,15,0.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[4,15,0.2667,0.92848,0.17536,1.0,1.0,1.0,0.14,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,26],[8,15,0.5333,0.77677,0.24207,0.67857,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,9,0,0,0,0,15],[12,15,0.8,0.875,0.17035,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,9,0,0,2,0,19],[15,15,1.0,0.86161,0.14934,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,17]]},{"b":5,"e":0.71429,"k":"flat","v":0.84365,"x":0.91964,"p":[[0,13,0.0,0.89732,0.20589,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,23],[4,13,0.3077,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[8,13,0.6154,0.91071,0.13716,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,22],[12,13,0.9231,0.86161,0.16554,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,0,0,18],[13,13,1.0,0.84365,0.22997,0.71429,1.0,1.0,0.14,1.0,0,20,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,7,0,0,0,0,20]]}]},{"i":"57bd4d20712c9fa4","q":"$\\triangle ABC$  is a triangle with sides  $a,b,c$ . Each side of  $\\triangle ABC$  is divided in  $n$  equal segments. Let  $S$  be the sum of the squares of the distances from each vertex to each of the points of division on its opposite side. Show that  $\\frac{S}{a^2+b^2+c^2}$  is a rational number.","t":[{"b":6,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,28,0.0,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[4,28,0.1429,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24],[8,28,0.2857,0.89286,0.20203,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,22],[12,28,0.4286,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[16,28,0.5714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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],[24,28,0.8571,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[28,28,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.83928,"x":1.0,"p":[[0,13,0.0,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,23],[4,13,0.3077,0.91071,0.16269,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,19],[8,13,0.6154,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[12,13,0.9231,0.83928,0.11152,0.82132,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,18,0,6],[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":"7e3b1c71c53ef9cf","q":"For a positive integer $k$, call an integer a pure $k$-th power if it can be represented as $m^{k}$ for some integer $m$. Show that for every positive integer $n$ there exist $n$ distinct positive integers such that their sum is a pure 2009-th power, and their product is a pure 2010-th power.","t":[{"b":2,"e":0.57143,"k":"falling","v":0.27232,"x":0.84375,"p":[[0,60,0.0,0.84375,0.24836,0.71429,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,1,0,0,2,0,0,0,0,0,7,0,0,1,0,20],[4,60,0.0667,0.66071,0.28291,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,8,0,0,1,0,0,1,0,0,11,0,0,1,0,9],[8,60,0.1333,0.57588,0.29984,0.28571,0.5712,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,8,0,0,5,0,0,2,0,0,6,0,0,3,0,6],[12,60,0.2,0.54685,0.32074,0.28571,0.57121,0.71429,0.0,1.0,3,6,0,3,1,1,0,0,6,0,0,3,0,0,3,0,0,8,0,0,1,0,6],[16,60,0.2667,0.54018,0.36724,0.28571,0.50001,1.0,0.0,1.0,5,9,0,5,0,1,0,0,8,0,0,2,0,0,1,0,0,5,0,0,1,0,9],[20,60,0.3333,0.44643,0.3531,0.14286,0.28571,0.71429,0.0,1.0,6,4,0,6,0,5,0,0,6,0,0,1,0,0,1,0,0,6,0,0,3,0,4],[24,60,0.4,0.40177,0.3203,0.14286,0.28571,0.60714,0.0,1.0,6,4,0,6,0,3,0,0,9,0,0,4,0,0,2,0,0,3,0,0,1,0,4],[28,60,0.4667,0.4732,0.30605,0.28571,0.28571,0.71429,0.0,1.0,3,4,0,3,0,2,0,0,12,0,0,1,0,0,2,0,0,7,0,0,1,0,4],[32,60,0.5333,0.45981,0.33261,0.28571,0.42857,0.71429,0.0,1.0,6,4,0,6,0,1,0,0,8,0,0,4,0,0,2,0,0,4,0,0,3,0,4],[36,60,0.6,0.55804,0.32015,0.28571,0.57143,0.75,0.0,1.0,3,7,0,3,0,2,0,0,5,0,0,3,0,0,6,0,0,5,0,0,1,0,7],[40,60,0.6667,0.46875,0.35035,0.2857,0.28571,0.75,0.0,1.0,5,6,0,5,0,2,0,0,11,0,0,1,0,0,1,0,0,4,0,0,2,0,6],[44,60,0.7333,0.37054,0.32115,0.14286,0.28571,0.71429,0.0,1.0,7,3,0,7,0,5,0,0,8,0,0,2,0,0,0,0,0,7,0,0,0,0,3],[48,60,0.8,0.38838,0.30353,0.28571,0.28571,0.4642,0.0,1.0,6,4,0,6,0,1,0,0,12,0,0,5,0,0,1,0,0,3,0,0,0,0,4],[52,60,0.8667,0.39731,0.25186,0.2857,0.28571,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,14,0,0,2,0,0,3,0,0,4,0,0,2,0,1],[56,60,0.9333,0.39732,0.23887,0.2857,0.28571,0.46431,0.0,1.0,2,2,0,2,0,2,0,0,15,0,0,5,0,0,2,0,0,4,0,0,0,0,2],[60,60,1.0,0.27232,0.2,0.14286,0.28571,0.28571,0.0,0.85714,6,0,0,6,0,3,0,0,18,0,0,2,0,0,0,0,0,2,0,0,1,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.64732,"x":0.92857,"p":[[0,45,0.0,0.75446,0.26301,0.67857,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,2,0,0,3,0,0,2,0,0,9,0,0,2,0,13],[4,45,0.0889,0.72321,0.28107,0.71429,0.71429,1.0,0.0,1.0,2,9,0,2,0,2,0,0,0,0,0,0,0,0,2,0,0,13,0,0,4,0,9],[8,45,0.1778,0.77679,0.21706,0.71429,0.78571,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,0,7,0,9],[12,45,0.2667,0.70087,0.23519,0.53539,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,5,0,0,2,0,0,9,0,0,7,0,6],[16,45,0.3556,0.76784,0.21055,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,15,0,0,4,0,9],[20,45,0.4444,0.75446,0.27487,0.71429,0.78571,1.0,0.0,1.0,1,12,0,1,0,1,0,0,3,0,0,0,0,0,1,0,0,10,0,0,4,0,12],[24,45,0.5333,0.83482,0.21162,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,7,0,14],[28,45,0.6222,0.64732,0.28568,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,2,0,0,5,0,0,4,0,0,9,0,0,0,0,9],[32,45,0.7111,0.76783,0.27608,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,4,0,0,6,0,0,3,0,0,0,0,17],[36,45,0.8,0.88839,0.18466,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,21],[40,45,0.8889,0.875,0.18123,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,8,0,0,3,0,19],[44,45,0.9778,0.92857,0.11845,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[45,45,1.0,0.87053,0.19678,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,3,0,20]]}]},{"i":"007956eddc315931","q":"In decimal representation   $$ \\text {34!=295232799039a041408476186096435b0000000}. $$   Find the numbers  $a$  and  $b$ .","t":[{"b":1,"e":0.0,"k":"falling","v":0.125,"x":0.86607,"p":[[0,136,0.0,0.7366,0.28819,0.42857,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,2,0,0,7,0,0,1,0,0,4,0,0,3,0,14],[4,136,0.0294,0.69643,0.3567,0.28571,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,9,0,0,2,0,0,0,0,0,0,0,0,2,0,17],[8,136,0.0588,0.76786,0.31083,0.42857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,20],[12,136,0.0882,0.86607,0.2788,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[16,136,0.1176,0.76786,0.31894,0.4286,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,20],[20,136,0.1471,0.70536,0.42097,0.2857,1.0,1.0,0.0,1.0,6,21,0,6,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,21],[24,136,0.1765,0.70982,0.38875,0.28571,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,20],[28,136,0.2059,0.71874,0.38381,0.28571,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,20],[32,136,0.2353,0.58036,0.43877,0.10714,0.78571,1.0,0.0,1.0,8,16,0,8,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,16],[36,136,0.2647,0.62946,0.41933,0.2857,1.0,1.0,0.0,1.0,7,17,0,7,0,0,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,17],[40,136,0.2941,0.63393,0.41945,0.28571,1.0,1.0,0.0,1.0,6,17,0,6,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,17],[44,136,0.3235,0.54018,0.44282,0.0,0.57143,1.0,0.0,1.0,10,14,0,10,0,0,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,14],[48,136,0.3529,0.60714,0.43448,0.10717,0.92857,1.0,0.0,1.0,8,16,0,8,0,1,0,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,16],[52,136,0.3824,0.55357,0.41764,0.25,0.35714,1.0,0.0,1.0,6,14,0,6,0,2,0,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,14],[56,136,0.4118,0.4866,0.40542,0.24999,0.28571,1.0,0.0,1.0,7,11,0,7,0,1,0,0,11,0,0,1,0,0,0,0,0,0,0,0,1,0,11],[60,136,0.4412,0.73212,0.37586,0.39286,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,20],[64,136,0.4706,0.51339,0.42236,0.10714,0.35714,1.0,0.0,1.0,8,13,0,8,0,1,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,13],[68,136,0.5,0.70981,0.41724,0.2857,1.0,1.0,0.0,1.0,6,21,0,6,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,21],[72,136,0.5294,0.59375,0.42873,0.28571,0.85714,1.0,0.0,1.0,7,16,0,7,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,16],[76,136,0.5588,0.66518,0.40344,0.28571,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,18],[80,136,0.5882,0.66518,0.40186,0.28571,1.0,1.0,0.0,1.0,5,17,0,5,0,0,0,0,6,0,0,2,0,0,0,0,0,0,0,0,2,0,17],[84,136,0.6176,0.67411,0.43188,0.28571,1.0,1.0,0.0,1.0,7,20,0,7,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,20],[88,136,0.6471,0.54464,0.44383,0.0,0.42859,1.0,0.0,1.0,9,15,0,9,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,15],[92,136,0.6765,0.51339,0.45717,0.0,0.42857,1.0,0.0,1.0,12,14,0,12,0,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,14],[96,136,0.7059,0.49107,0.44883,0.0,0.5,1.0,0.0,1.0,13,11,0,13,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,11],[100,136,0.7353,0.59821,0.42624,0.28571,0.85714,1.0,0.0,1.0,7,16,0,7,0,0,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,16],[104,136,0.7647,0.50892,0.43292,0.0,0.28571,1.0,0.0,1.0,9,12,0,9,0,2,0,0,6,0,0,0,0,0,0,0,0,2,0,0,1,0,12],[108,136,0.7941,0.51339,0.47227,0.0,0.42857,1.0,0.0,1.0,13,15,0,13,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,15],[112,136,0.8235,0.4375,0.43586,0.0,0.28571,1.0,0.0,1.0,13,11,0,13,0,0,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,11],[116,136,0.8529,0.29463,0.43144,0.0,0.0,0.67857,0.0,1.0,21,8,0,21,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,8],[120,136,0.8824,0.44643,0.46941,0.0,0.28571,1.0,0.0,1.0,15,13,1,15,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,13],[124,136,0.9118,0.35714,0.43595,0.0,0.0,1.0,0.0,1.0,17,9,1,17,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,9],[128,136,0.9412,0.41964,0.4642,0.0,0.14286,1.0,0.0,1.0,16,12,0,16,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,12],[132,136,0.9706,0.375,0.4598,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[136,136,1.0,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4]]},{"b":4,"e":0.0,"k":"falling","v":0.26786,"x":0.9375,"p":[[0,27,0.0,0.76339,0.31866,0.53572,1.0,1.0,0.0,1.0,2,18,1,2,0,0,0,0,3,0,0,3,0,0,1,0,0,4,0,0,1,0,18],[4,27,0.1481,0.74107,0.32623,0.28571,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,10,0,0,1,0,0,0,0,0,1,0,0,2,0,18],[8,27,0.2963,0.78125,0.32534,0.28571,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,22],[12,27,0.4444,0.84375,0.28652,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,24],[16,27,0.5926,0.82143,0.29451,0.64286,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,23],[20,27,0.7407,0.9375,0.19541,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[24,27,0.8889,0.80357,0.35129,0.85714,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[27,27,1.0,0.26786,0.43412,0.0,0.0,0.67857,0.0,1.0,23,8,0,23,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8]]}]},{"i":"576136bbb798cb41","q":"Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_{1}, b_{2}, \\ldots, b_{K}$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_{1}, b_{2}, \\ldots, b_{K}$.","t":[{"b":1,"e":0.71429,"k":"rising","v":0.25446,"x":0.44639,"p":[[0,37,0.0,0.25446,0.18117,0.2857,0.28571,0.28571,0.0,0.85714,7,0,3,7,0,0,0,0,23,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[4,37,0.1081,0.33481,0.2438,0.2857,0.28571,0.28571,0.0,1.0,4,2,0,4,0,1,0,0,21,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[8,37,0.2162,0.36605,0.21408,0.28571,0.28571,0.32144,0.0,1.0,2,1,0,2,0,0,0,0,22,0,0,2,0,0,1,0,0,3,0,0,1,0,1],[12,37,0.3243,0.29909,0.19677,0.2857,0.28571,0.28571,0.0,1.0,4,1,0,4,0,2,0,0,21,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[16,37,0.4324,0.36607,0.2111,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,2,0,0,24,0,0,0,0,0,3,0,0,0,0,0,1,0,2],[20,37,0.5405,0.35713,0.23144,0.28571,0.28571,0.32143,0.0,1.0,2,2,0,2,0,2,0,0,20,0,0,3,0,0,1,0,0,1,0,0,1,0,2],[24,37,0.6486,0.34821,0.23673,0.2857,0.28571,0.28571,0.0,1.0,3,2,0,3,0,1,0,0,21,0,0,2,0,0,1,0,0,1,0,0,1,0,2],[28,37,0.7568,0.37932,0.30435,0.24999,0.28571,0.571,0.0,1.0,6,4,0,6,0,2,0,0,13,0,0,0,0,0,6,0,0,1,0,0,0,0,4],[32,37,0.8649,0.44639,0.30038,0.2857,0.28571,0.71429,0.0,1.0,3,3,0,3,0,4,0,0,10,0,0,2,0,0,3,0,0,5,0,0,2,0,3],[36,37,0.973,0.37052,0.38772,0.0,0.21428,0.85714,0.0,1.0,11,5,0,11,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,0,4,0,5],[37,37,1.0,0.44195,0.37518,0.0,0.28571,0.75,0.0,1.0,9,5,0,9,0,2,0,0,6,0,0,0,0,0,3,0,0,4,0,0,3,0,5]]},{"b":2,"e":0.57143,"k":"rising","v":0.27232,"x":0.59371,"p":[[0,46,0.0,0.27232,0.22406,0.2857,0.28571,0.28571,0.0,1.0,7,2,5,7,0,0,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,46,0.087,0.30803,0.22047,0.2857,0.28571,0.28571,0.0,1.0,4,2,0,4,0,1,0,0,23,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[8,46,0.1739,0.35266,0.2314,0.2857,0.28571,0.28571,0.0,1.0,3,2,0,3,0,0,0,0,22,0,0,1,0,0,3,0,0,0,0,0,1,0,2],[12,46,0.2609,0.50891,0.32915,0.28571,0.28571,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,13,0,0,2,0,0,1,0,0,2,0,0,4,0,6],[16,46,0.3478,0.34818,0.30077,0.10717,0.28571,0.571,0.0,1.0,8,3,0,8,0,1,0,0,13,0,0,1,0,0,3,0,0,3,0,0,0,0,3],[20,46,0.4348,0.50891,0.33108,0.28571,0.28571,0.85714,0.0,1.0,2,6,0,2,0,3,0,0,12,0,0,1,0,0,1,0,0,4,0,0,3,0,6],[24,46,0.5217,0.45977,0.31079,0.2857,0.4998,0.71429,0.0,1.0,5,3,0,5,0,2,0,0,8,0,0,1,0,0,6,0,0,5,0,0,2,0,3],[28,46,0.6087,0.4464,0.3046,0.25002,0.42857,0.71429,0.0,1.0,4,3,0,4,0,4,0,0,7,0,0,3,0,0,5,0,0,4,0,0,2,0,3],[32,46,0.6957,0.41953,0.27657,0.25,0.35714,0.60714,0.0,1.0,4,1,0,4,0,4,0,0,8,0,0,2,0,0,6,0,0,5,0,0,2,0,1],[36,46,0.7826,0.45087,0.37133,0.10714,0.28571,0.75,0.0,1.0,8,6,0,8,0,2,0,0,7,0,0,0,0,0,4,0,0,3,0,0,2,0,6],[40,46,0.8696,0.4598,0.36549,0.14286,0.42859,0.71429,0.0,1.0,7,6,0,7,0,4,0,0,4,0,0,2,0,0,3,0,0,5,0,0,1,0,6],[44,46,0.9565,0.59371,0.34276,0.2857,0.64286,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,4,0,0,4,0,0,3,0,0,4,0,0,4,0,8],[46,46,1.0,0.45089,0.34646,0.1429,0.28571,0.75,0.0,1.0,5,5,0,5,0,4,0,0,8,0,0,3,0,0,1,0,0,3,0,0,3,0,5]]}]},{"i":"887c233ef0c54b45","q":"Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that:\n\n$$\n\\frac{a^{2}+a+1}{b^{2}+b+1}=n^{2}+n+1\n$$","t":[{"b":0,"e":0.71429,"k":"flat","v":0.72768,"x":0.89286,"p":[[0,17,0.0,0.83036,0.22711,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,17],[4,17,0.2353,0.72768,0.27747,0.71429,0.71429,1.0,0.0,1.0,2,11,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,16,0,0,0,0,11],[8,17,0.4706,0.79464,0.25985,0.71429,0.85714,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,0,0,0,0,0,0,12,0,0,0,0,16],[12,17,0.7059,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[16,17,0.9412,0.86607,0.14258,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,17],[17,17,1.0,0.89286,0.13832,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,20]]},{"b":5,"e":0.71429,"k":"flat","v":0.64732,"x":0.78125,"p":[[0,42,0.0,0.73659,0.33524,0.67846,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,4,0,0,0,0,0,1,0,0,7,0,0,1,0,16],[4,42,0.0952,0.72321,0.27418,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,16,0,0,1,0,10],[8,42,0.1905,0.64732,0.35533,0.28571,0.71429,1.0,0.0,1.0,5,11,0,5,0,0,0,0,4,0,0,0,0,0,0,0,0,12,0,0,0,0,11],[12,42,0.2857,0.78125,0.15561,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,0,0,0,0,9],[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],[20,42,0.4762,0.75,0.09449,0.71429,0.71429,0.71429,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[24,42,0.5714,0.74552,0.09935,0.71429,0.71429,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,0,0,4],[28,42,0.6667,0.76786,0.11152,0.71429,0.71429,0.71429,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,6],[32,42,0.7619,0.74999,0.09449,0.71429,0.71429,0.71429,0.714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,4],[36,42,0.8571,0.74107,0.16146,0.71429,0.71429,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,24,0,0,0,0,6],[40,42,0.9524,0.73661,0.0724,0.71429,0.71429,0.71429,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,1,0,2],[42,42,1.0,0.73661,0.12428,0.71429,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,27,0,0,0,0,4]]}]},{"i":"28069210b40868b2","q":"Does there exist an angle $\\alpha \\in(0, \\pi / 2)$ such that $\\sin \\alpha, \\cos \\alpha, \\tan \\alpha$ and $\\cot \\alpha$, taken in some order, are consecutive terms of an arithmetic progression?","t":[{"b":5,"e":0.85714,"k":"flat","v":0.83926,"x":0.98661,"p":[[0,90,0.0,0.83926,0.21654,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,8,0,15],[4,90,0.0444,0.87052,0.25093,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,23],[8,90,0.0889,0.9241,0.19557,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,25],[12,90,0.1333,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[16,90,0.1778,0.89286,0.24485,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,25],[20,90,0.2222,0.95535,0.12079,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[24,90,0.2667,0.9375,0.14698,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,4,0,25],[28,90,0.3111,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[32,90,0.3556,0.95089,0.18766,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[36,90,0.4,0.92411,0.16745,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,4,0,24],[40,90,0.4444,0.95088,0.09857,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[44,90,0.4889,0.94197,0.18161,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[48,90,0.5333,0.95534,0.12599,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[52,90,0.5778,0.96428,0.10714,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,27],[56,90,0.6222,0.91962,0.14702,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,22],[60,90,0.6667,0.90179,0.25862,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,27],[64,90,0.7111,0.91071,0.1948,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,24],[68,90,0.7556,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[72,90,0.8,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[76,90,0.8444,0.90177,0.1871,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,4,0,23],[80,90,0.8889,0.95982,0.12492,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,28],[84,90,0.9333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[88,90,0.9778,0.9241,0.19228,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,26],[90,90,1.0,0.94196,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22]]},{"b":7,"e":0.0,"k":"flat","v":0.66967,"x":0.86607,"p":[[0,55,0.0,0.80355,0.27608,0.82132,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,0,11,0,13],[4,55,0.0727,0.86605,0.2574,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,0,6,0,21],[8,55,0.1455,0.72767,0.356,0.5354,0.85714,1.0,0.0,1.0,3,15,0,3,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,0,6,0,15],[12,55,0.2182,0.84374,0.24579,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,0,0,0,6,0,19],[16,55,0.2909,0.80802,0.29583,0.82143,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,0,0,0,2,0,0,1,0,0,1,0,0,6,0,18],[20,55,0.3636,0.86607,0.23128,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,17],[24,55,0.4364,0.80803,0.27574,0.85714,0.85714,1.0,0.0,1.0,1,12,0,1,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,15,0,12],[28,55,0.5091,0.77679,0.29867,0.82143,0.85714,1.0,0.0,1.0,3,11,0,3,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,13,0,11],[32,55,0.5818,0.74105,0.32818,0.85711,0.85714,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,17,0,8],[36,55,0.6545,0.80357,0.27606,0.85714,0.85714,1.0,0.0,1.0,2,12,0,2,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,13,0,12],[40,55,0.7273,0.82142,0.24223,0.85714,0.85714,1.0,0.0,1.0,1,11,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,15,0,11],[44,55,0.8,0.86159,0.17309,0.85711,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,10,0,15],[48,55,0.8727,0.6964,0.33835,0.42857,0.85714,1.0,0.0,1.0,4,9,0,4,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,0,11,0,9],[52,55,0.9455,0.82141,0.16753,0.85711,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,18,0,7],[55,55,1.0,0.66967,0.33583,0.42964,0.85714,0.85714,0.0,1.0,5,4,0,5,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,17,0,4]]}]},{"i":"31dba312d9210df8","q":"Triangle  $\\triangle ABC$  has  $\\angle{A}=90^\\circ$  with  $BC=12$ . Square  $BCDE$  is drawn such that  $A$  is in its interior. The line through  $A$  tangent to the circumcircle of  $\\triangle ABC$  intersects  $CD$  and  $BE$  at  $P$  and  $Q$ , respectively. If  $PA=4\\cdot QA$ , and the area of  $\\triangle ABC$  can be expressed as  $\\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , then compute  $m+n$ .\n\n*Proposed by Andy Xu*","t":[{"b":6,"e":0.85714,"k":"flat","v":0.58036,"x":0.71429,"p":[[0,38,0.0,0.58482,0.19678,0.42857,0.42857,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,2,0,0,10,0,0],[4,38,0.1053,0.66964,0.20958,0.42857,0.78571,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,2,0,0,15,0,1],[8,38,0.2105,0.58036,0.18536,0.42857,0.42859,0.75,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,18,0,0,2,0,0,4,0,0,8,0,0],[12,38,0.3158,0.61607,0.23266,0.42857,0.5,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,15,0,0,1,0,0,1,0,0,14,0,0],[16,38,0.4211,0.64286,0.21724,0.42857,0.57143,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,1,0,0,12,0,2],[20,38,0.5263,0.70982,0.19393,0.42857,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,3,0,0,19,0,0],[24,38,0.6316,0.68304,0.20119,0.42857,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,3,0,0,17,0,0],[28,38,0.7368,0.67857,0.19561,0.42857,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,6,0,0,13,0,1],[32,38,0.8421,0.62054,0.19759,0.42857,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,0,7,0,0,10,0,0],[36,38,0.9474,0.71429,0.19885,0.42857,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,3,0,0,18,0,1],[38,38,1.0,0.58482,0.2212,0.42857,0.42857,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,17,0,0,0,0,0,4,0,0,10,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.55804,"x":0.75,"p":[[0,35,0.0,0.55804,0.21535,0.42857,0.42857,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,19,0,0,0,0,0,5,0,0,6,0,1],[4,35,0.1143,0.70536,0.23673,0.42857,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,11,0,0,0,0,0,3,0,0,11,0,6],[8,35,0.2286,0.68304,0.22227,0.42857,0.78571,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,1,0,0,12,0,4],[12,35,0.3429,0.65179,0.20806,0.42857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,6,0,0,10,0,2],[16,35,0.4571,0.75,0.18898,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,7,0,0,14,0,4],[20,35,0.5714,0.73214,0.20124,0.42857,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,4,0,0,16,0,3],[24,35,0.6857,0.69196,0.19597,0.42857,0.78571,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,4,0,0,15,0,1],[28,35,0.8,0.69643,0.21053,0.42857,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,18,0,1],[32,35,0.9143,0.67857,0.20516,0.42857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,6,0,0,12,0,2],[35,35,1.0,0.73214,0.1915,0.64286,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,7,0,0,14,0,3]]}]},{"i":"8d1e04da7d2887ba","q":"Different positive  $a, b, c$  are such that  $a^{239} = ac- 1$  and  $b^{239} = bc- 1$ .Prove that  $238^2 (ab)^{239} <1$ .","t":[{"b":1,"e":0.71429,"k":"rising","v":0.52679,"x":0.70982,"p":[[0,22,0.0,0.52679,0.30397,0.24999,0.71429,0.71429,0.0,1.0,4,2,0,4,0,4,0,0,1,0,0,4,0,0,1,0,0,14,0,0,2,0,2],[4,22,0.1818,0.64286,0.21724,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,0,0,0,2,0,0,1,0,0,22,0,0,1,0,2],[8,22,0.3636,0.69643,0.25939,0.71429,0.71429,0.75,0.0,1.0,1,7,0,1,0,3,0,0,0,0,0,1,0,0,0,0,0,19,0,0,1,0,7],[12,22,0.5455,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[16,22,0.7273,0.70982,0.05629,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,0,1,0,0],[20,22,0.9091,0.67409,0.09607,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,27,0,0,0,0,0],[22,22,1.0,0.69643,0.06916,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0]]},{"b":2,"e":0.71429,"k":"flat","v":0.49552,"x":0.70982,"p":[[0,27,0.0,0.54464,0.29329,0.42857,0.71429,0.71429,0.0,1.0,6,1,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,17,0,0,2,0,1],[4,27,0.1481,0.49552,0.25249,0.35714,0.49979,0.71429,0.0,1.0,1,1,0,1,0,7,0,0,0,0,0,8,0,0,2,0,0,13,0,0,0,0,1],[8,27,0.2963,0.67415,0.12483,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,23,0,0,3,0,0],[12,27,0.4444,0.68304,0.13709,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,22,0,0,3,0,1],[16,27,0.5926,0.66518,0.11633,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,25,0,0,1,0,0],[20,27,0.7407,0.66518,0.11071,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,24,0,0,1,0,0],[24,27,0.8889,0.70982,0.08364,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,27,0,0,3,0,0],[27,27,1.0,0.66071,0.10565,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,25,0,0,0,0,0]]}]},{"i":"e97a93a47581e6c3","q":"Function  $f(n), n \\in \\mathbb N$ , is defined as follows:\nLet  $\\frac{(2n)!}{n!(n+1000)!} = \\frac{A(n)}{B(n)}$  , where  $A(n), B(n)$  are coprime positive integers; if  $B(n) = 1$ , then  $f(n) = 1$ ; if  $B(n) \\neq 1$ , then  $f(n)$  is the largest prime factor of  $B(n)$ . Prove that the values of  $f(n)$  are finite, and find the maximum value of  $f(n).$","t":[{"b":0,"e":0.57143,"k":"flat","v":0.54013,"x":0.68752,"p":[[0,30,0.0,0.63388,0.18189,0.5354,0.57143,0.74996,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,6,0,0,9,0,0,7,0,0,7,0,1],[4,30,0.1333,0.64279,0.19564,0.5354,0.57143,0.75,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,6,0,0,9,0,0,7,0,0,5,0,3],[8,30,0.2667,0.68752,0.1614,0.57143,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,8,0,0,6,0,3],[12,30,0.4,0.68747,0.19046,0.571,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,7,0,0,7,0,0,7,0,4],[16,30,0.5333,0.59374,0.16409,0.57142,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,4,0,0,13,0,0,11,0,0,2,0,0],[20,30,0.6667,0.64729,0.17852,0.5354,0.64286,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,8,0,0,8,0,0,11,0,0,1,0,4],[24,30,0.8,0.62054,0.17353,0.42859,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,8,0,0,8,0,0,12,0,0,0,0,3],[28,30,0.9333,0.65604,0.17806,0.571,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,11,0,0,5,0,2],[30,30,1.0,0.54013,0.16649,0.42857,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,7,0,0,11,0,0,10,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.25884,"x":0.7098,"p":[[0,68,0.0,0.62498,0.17767,0.57132,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,13,0,0,5,0,0],[4,68,0.0588,0.69192,0.16018,0.57143,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,9,0,0,8,0,2],[8,68,0.1176,0.66515,0.20395,0.571,0.64286,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,5,0,0,9,0,0,6,0,0,6,0,4],[12,68,0.1765,0.62497,0.23075,0.57132,0.71429,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,0,0,0,4,0,0,5,0,0,14,0,0,5,0,1],[16,68,0.2353,0.70532,0.15543,0.57143,0.71429,0.74996,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,12,0,0,4,0,4],[20,68,0.2941,0.70976,0.16168,0.57142,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,11,0,0,9,0,2],[24,68,0.3529,0.6428,0.18213,0.5713,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,3,0,0,15,0,0,8,0,0,1,0,4],[28,68,0.4118,0.70536,0.13333,0.57143,0.71429,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,15,0,0,7,0,1],[32,68,0.4706,0.70532,0.17475,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,10,0,0,10,0,2],[36,68,0.5294,0.66516,0.21609,0.57143,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,2,0,0,1,0,0,4,0,0,13,0,0,10,0,0],[40,68,0.5882,0.61606,0.18708,0.42857,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,11,0,0,6,0,0],[44,68,0.6471,0.64732,0.21423,0.53572,0.71429,0.74996,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,5,0,0,4,0,0,12,0,0,6,0,2],[48,68,0.7059,0.69195,0.13884,0.57143,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,12,0,0,9,0,0],[52,68,0.7647,0.67856,0.15153,0.57143,0.71429,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,12,0,0,7,0,1],[56,68,0.8235,0.67409,0.18293,0.57132,0.71429,0.75,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,5,0,0,3,0,0,14,0,0,6,0,2],[60,68,0.8824,0.7098,0.18379,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,8,0,0,10,0,3],[64,68,0.9412,0.49106,0.30501,0.14289,0.57121,0.71429,0.0,0.85714,4,0,0,4,0,5,0,0,2,0,0,4,0,0,3,0,0,7,0,0,7,0,0],[68,68,1.0,0.25884,0.26111,0.14286,0.14286,0.42857,0.0,0.85714,6,0,0,6,0,16,0,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,0]]}]},{"i":"8d580deb9be32731","q":"Points $\\mathrm{A}_{1}, \\mathrm{~A}_{2}, \\ldots, \\mathrm{A}_{\\mathrm{n}}$ are equally spaced on the side $\\mathrm{BC}$ of the triangle $\\mathrm{ABC}$ (so that $\\mathrm{BA}_{1}=$ $\\left.A_{1} A_{2}=\\ldots=A_{n-1} A_{n}=A_{n} C\\right)$. Similarly, points $B_{1}, B_{2}, \\ldots, B_{n}$ are equally spaced on the side CA, and points $\\mathrm{C}_{1}, \\mathrm{C}_{2}, \\ldots, \\mathrm{C}_{\\mathrm{n}}$ are equally spaced on the side $\\mathrm{AB}$. Show that $\\left(\\mathrm{AA}_{1}{ }^{2}+\\mathrm{AA}_{2}{ }^{2}+\\ldots+\\right.$ $\\left.\\mathrm{AA}_{\\mathrm{n}}{ }^{2}+\\mathrm{BB}_{1}{ }^{2}+\\mathrm{BB}_{2}{ }^{2}+\\ldots+\\mathrm{BB}_{\\mathrm{n}}{ }^{2}+\\mathrm{C}_{1}{ }^{2}+\\ldots+\\mathrm{CC}_{\\mathrm{n}}{ }^{2}\\right)$ is a rational multiple of $\\left(\\mathrm{AB}^{2}+\\mathrm{BC}^{2}+\\right.$ $\\left.\\mathrm{CA}^{2}\\right)$.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.95536,"x":0.99107,"p":[[0,7,0.0,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[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],[7,7,1.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]},{"b":6,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,12,0.0,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[4,12,0.3333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"f540c56fd7df911e","q":"Consider digits  $\\underline{A}, \\underline{B}, \\underline{C}, \\underline{D}$ , with  $\\underline{A} \\neq 0,$  such that  $\\underline{A} \\underline{B} \\underline{C} \\underline{D} = (\\underline{C} \\underline{D} ) ^2 - (\\underline{A} \\underline{B})^2.$  Compute the sum of all distinct possible values of  $\\underline{A} + \\underline{B} + \\underline{C} + \\underline{D}$ .\n\n*Proposed by Kyle Lee*","t":[{"b":2,"e":0.42857,"k":"flat","v":0.53572,"x":0.61161,"p":[[0,66,0.0,0.58036,0.17835,0.42857,0.50001,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,15,0,0,1,0,0,13,0,0,0,0,2],[4,66,0.0606,0.57589,0.14934,0.42857,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,0,17,0,0,0,0,0],[8,66,0.1212,0.53572,0.15152,0.42857,0.42857,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,19,0,0,0,0,0,11,0,0,1,0,0],[12,66,0.1818,0.58036,0.17474,0.42857,0.42857,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,13,0,0,0,0,2],[16,66,0.2424,0.57589,0.145,0.42857,0.64286,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,13,0,0,2,0,0,16,0,0,0,0,0],[20,66,0.303,0.55804,0.1488,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,15,0,0,0,0,0],[24,66,0.3636,0.58482,0.1488,0.42857,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,13,0,0,0,0,0,18,0,0,0,0,0],[28,66,0.4242,0.57143,0.15972,0.42857,0.42857,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,14,0,0,0,0,1],[32,66,0.4848,0.60268,0.13709,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,19,0,0,0,0,0],[36,66,0.5455,0.54911,0.15198,0.42857,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,15,0,0,1,0,0,14,0,0,0,0,0],[40,66,0.6061,0.58036,0.14698,0.42857,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,13,0,0,1,0,0,17,0,0,0,0,0],[44,66,0.6667,0.59822,0.15745,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,17,0,0,0,0,1],[48,66,0.7273,0.55804,0.1488,0.42857,0.42859,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,15,0,0,0,0,0],[52,66,0.7879,0.54911,0.14773,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,17,0,0,0,0,0,14,0,0,0,0,0],[56,66,0.8485,0.54911,0.15198,0.42857,0.4286,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,15,0,0,1,0,0,14,0,0,0,0,0],[60,66,0.9091,0.61159,0.15251,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,18,0,0,0,0,1],[64,66,0.9697,0.61161,0.1439,0.42857,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,0,21,0,0,0,0,0],[66,66,1.0,0.54911,0.16409,0.42857,0.42857,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,18,0,0,0,0,0,12,0,0,0,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.54018,"x":0.66963,"p":[[0,50,0.0,0.66963,0.20025,0.42857,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,2,0,0,14,0,0,0,0,6],[4,50,0.08,0.58482,0.15714,0.42857,0.64286,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,15,0,0,1,0,0,15,0,0,0,0,1],[8,50,0.16,0.61607,0.15335,0.42857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,19,0,0,0,0,1],[12,50,0.24,0.60714,0.14286,0.42857,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,1,0,0,20,0,0,0,0,0],[16,50,0.32,0.59821,0.14032,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,19,0,0,0,0,0],[20,50,0.4,0.55357,0.14174,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,14,0,0,0,0,0],[24,50,0.48,0.58929,0.17034,0.42857,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,13,0,0,0,0,2],[28,50,0.56,0.58036,0.15542,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,14,0,0,0,0,1],[32,50,0.64,0.54018,0.13709,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,12,0,0,0,0,0],[36,50,0.72,0.57143,0.14286,0.42857,0.57144,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0],[40,50,0.8,0.57589,0.145,0.42857,0.64286,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,13,0,0,2,0,0,16,0,0,0,0,0],[44,50,0.88,0.58482,0.13997,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,17,0,0,0,0,0],[48,50,0.96,0.55357,0.14174,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,14,0,0,0,0,0],[50,50,1.0,0.60714,0.13832,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,20,0,0,0,0,0]]}]},{"i":"90d0a61f3129566c","q":"$x$ ,  $y$  and  $z$  are positive reals such that  $x+y+z=xyz$ . Find the minimum value of:\r\n\\[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \\]","t":[{"b":6,"e":0.85714,"k":"rising","v":0.64286,"x":0.87946,"p":[[0,22,0.0,0.65625,0.33093,0.28571,0.64286,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,11,0,0,5,0,0,0,0,0,0,0,0,2,0,14],[4,22,0.1818,0.87054,0.25595,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,25],[8,22,0.3636,0.70982,0.32436,0.28571,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,10,0,0,2,0,0,1,0,0,1,0,0,2,0,16],[12,22,0.5455,0.64286,0.33312,0.28571,0.5,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,12,0,0,4,0,0,1,0,0,0,0,0,1,0,14],[16,22,0.7273,0.75,0.31743,0.28571,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,9,0,0,1,0,0,1,0,0,1,0,0,2,0,18],[20,22,0.9091,0.86607,0.07936,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,25,0,5],[22,22,1.0,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5]]},{"b":7,"e":0.2857,"k":"falling","v":0.31696,"x":0.72768,"p":[[0,36,0.0,0.72768,0.3038,0.42857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,7,0,0,4,0,0,2,0,0,0,0,0,4,0,15],[4,36,0.1111,0.63839,0.3369,0.28571,0.5,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,13,0,0,3,0,0,1,0,0,0,0,0,1,0,14],[8,36,0.2222,0.57143,0.31542,0.28571,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,14,0,0,5,0,0,1,0,0,1,0,0,1,0,10],[12,36,0.3333,0.52232,0.29148,0.28571,0.42857,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,15,0,0,6,0,0,2,0,0,0,0,0,2,0,7],[16,36,0.4444,0.44195,0.25091,0.28571,0.28571,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,18,0,0,8,0,0,1,0,0,0,0,0,0,0,5],[20,36,0.5556,0.32589,0.07349,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.37946,0.21011,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,21,0,0,7,0,0,0,0,0,0,0,0,0,0,3],[28,36,0.7778,0.37946,0.14987,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,11,0,0,1,0,0,1,0,0,0,0,1],[32,36,0.8889,0.34375,0.18509,0.2857,0.28571,0.28571,0.0,1.0,1,2,0,1,0,0,0,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[36,36,1.0,0.31696,0.05906,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"94c178419b8689fe","q":"2. (BUL 3) Let $a_{0}, a_{1}, \\ldots, a_{n}, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:  $$ \\begin{aligned} a_{0} & =a_{n+1}=0 \\\\ \\left|a_{k-1}-2 a_{k}+a_{k+1}\\right| & \\leq 1 \\quad(k=1,2, \\ldots, n) \\end{aligned} $$  Prove that $\\left|a_{k}\\right| \\leq \\frac{k(n+1-k)}{2}(k=0,1, \\ldots, n+1)$.","t":[{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.63393,"p":[[0,27,0.0,0.63393,0.46694,0.0,1.0,1.0,0.0,1.0,11,19,0,11,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,19],[4,27,0.1481,0.40179,0.47974,0.0,0.0,1.0,0.0,1.0,18,12,0,18,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12],[8,27,0.2963,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,27,0.4444,0.04911,0.18423,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,27,0.5926,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":1.0,"k":"rising","v":0.45534,"x":1.0,"p":[[0,13,0.0,0.45534,0.47169,0.0,0.21429,1.0,0.0,1.0,16,12,0,16,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,12],[4,13,0.3077,0.76786,0.37244,0.71429,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,21],[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":"4db181c849f9fc07","q":"The positive integers \\( a \\) and \\( b \\) are coprime and such that there exist positive integers \\( m_2 \\) and \\( m_5 \\) for which \\( am_2 + b \\) is a perfect square of a positive integer, and \\( am_5 + b \\) is a perfect fifth power of a positive integer. Does there always exist a positive integer \\( n \\) for which \\( an + b \\) is a perfect \\( k \\)-th power of a positive integer, if:\na) \\( k = 7 \\);\nb) \\( k = 10 \\)?","t":[{"b":3,"e":1.0,"k":"flat","v":0.94197,"x":0.9866,"p":[[0,27,0.0,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[4,27,0.1481,0.94197,0.12299,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[8,27,0.2963,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[12,27,0.4444,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[16,27,0.5926,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,27,0.7407,0.9866,0.04167,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,27,0.8889,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[27,27,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":7,"e":1.0,"k":"flat","v":0.95982,"x":0.9866,"p":[[0,39,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,39,0.1026,0.97321,0.10375,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[8,39,0.2051,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,39,0.3077,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,39,0.4103,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[20,39,0.5128,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[24,39,0.6154,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,39,0.7179,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[32,39,0.8205,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,39,0.9231,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[39,39,1.0,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"3096db588d57a29e","q":"Which statement is not true for at least one prime  $p$ ? $ \\textbf{(A)}\\ \\text{If } x^2+x+3 \\equiv 0 \\pmod p \\text{ has a solution, then }  \\qquad x^2+x+25 \\equiv 0 \\pmod p \\text{ has a solution.}\n  \\textbf{(B)}\\ \\text{If } x^2+x+3 \\equiv 0 \\pmod p \\text{ does not have a solution, then}  \\qquad x^2+x+25 \\equiv 0 \\pmod p \\text{ has no solution}\n  \\qquad\\textbf{(C)}\\ \\text{If } x^2+x+25 \\equiv 0 \\pmod p \\text{ has a solution, then}  \\qquad x^2+x+3 \\equiv 0 \\pmod p \\text{ has a solution}.\n  \\qquad\\textbf{(D)}\\ \\text{If } x^2+x+25 \\equiv 0 \\pmod p \\text{ does not have a solution, then}  \\qquad x^2+x+3 \\equiv 0 \\pmod p \\text{ has no solution. }\n  \\qquad\\textbf{(E)}\\ \\text{None}\n$","t":[{"b":2,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,112,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,112,0.0357,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,112,0.0714,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,112,0.1071,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,112,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,112,0.1786,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,112,0.2143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,112,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[36,112,0.3214,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,112,0.3571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,112,0.3929,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,112,0.4286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,112,0.4643,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,112,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,112,0.5357,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,112,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,112,0.6071,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,112,0.6429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,112,0.6786,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,112,0.7143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,112,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,112,0.7857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,112,0.8214,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,112,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,112,0.8929,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,112,0.9286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[108,112,0.9643,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[112,112,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,97,0.0,0.99106,0.03461,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,97,0.0412,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,97,0.0825,0.97767,0.05189,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,97,0.1237,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,97,0.1649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,97,0.2062,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,97,0.2474,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,97,0.2887,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,97,0.3299,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,97,0.3711,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,97,0.4124,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,97,0.4536,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,97,0.4948,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,97,0.5361,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,97,0.5773,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[60,97,0.6186,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,97,0.6598,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,97,0.701,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,97,0.7423,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,97,0.7835,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,97,0.8247,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,97,0.866,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,97,0.9072,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,97,0.9485,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,97,0.9897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[97,97,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"6c8db9dfc2ee177c","q":"Let  $p\\geqslant 5$  be a prime number. Prove that the set  $\\{1,2,\\ldots,p - 1\\}$  can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo  $p{}$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.13393,"p":[[0,26,0.0,0.13393,0.18189,0.0,0.0,0.32143,0.0,0.4286,19,0,0,19,0,4,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.10714,"x":0.22313,"p":[[0,55,0.0,0.10714,0.17857,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,7,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[4,55,0.0727,0.16964,0.18363,0.0,0.14286,0.42857,0.0,0.57143,13,0,0,13,0,10,0,0,0,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[8,55,0.1455,0.14286,0.17857,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,12,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[12,55,0.2182,0.14286,0.20516,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,13,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[16,55,0.2909,0.20089,0.17445,0.14286,0.14286,0.32142,0.0,0.71429,7,0,0,7,0,15,0,0,2,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[20,55,0.3636,0.15625,0.13054,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,20,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.1607,0.19802,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,19,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[28,55,0.5091,0.12946,0.10326,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,21,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.22313,0.18193,0.14286,0.14286,0.32143,0.0,0.71429,4,0,0,4,0,18,0,0,2,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[36,55,0.6545,0.1875,0.10972,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,27,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[40,55,0.7273,0.13393,0.07087,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.16964,0.12595,0.14286,0.14286,0.14286,0.0,0.71429,2,0,0,2,0,27,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[48,55,0.8727,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"010942e4efec5c8f","q":"Let  $a$ ,  $b$ ,  $c$  be positive real numbers.  Prove that\r\n\\[ \\biggl(1+\\frac{a}{b}\\biggr) \\biggl(1+\\frac{b}{c}\\biggr) \\biggl(1+\\frac{c}{a}\\biggr) \\ge 2 \\biggl(1+\\frac{a+b+c}{\\sqrt[3]{abc}}\\biggr).  \\]","t":[{"b":3,"e":0.0,"k":"flat","v":0.10268,"x":0.3125,"p":[[0,54,0.0,0.16518,0.11355,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,23,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.12054,0.13882,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,10,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.18304,0.18638,0.0,0.14286,0.42857,0.0,0.4286,15,0,0,15,0,2,0,0,6,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.26786,0.16656,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,9,0,0,3,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.26338,0.17894,0.14286,0.28571,0.42857,0.0,0.571,6,0,0,6,0,9,0,0,2,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[20,54,0.3704,0.28125,0.15765,0.14286,0.28571,0.42857,0.0,0.42857,4,0,0,4,0,8,0,0,5,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.22768,0.20782,0.10714,0.14286,0.32143,0.0,1.0,8,1,0,8,0,9,0,0,7,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[28,54,0.5185,0.3125,0.16917,0.14286,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,7,0,0,1,0,0,19,0,0,1,0,0,0,0,0,0,0,0],[32,54,0.5926,0.21429,0.16752,0.14286,0.14286,0.42857,0.0,0.57143,7,0,0,7,0,12,0,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[36,54,0.6667,0.19643,0.16269,0.10714,0.14286,0.42857,0.0,0.42857,8,0,0,8,0,13,0,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.25446,0.15865,0.14286,0.21428,0.42857,0.0,0.42857,4,0,0,4,0,12,0,0,3,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.14286,0.16751,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,6,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.17857,0.16751,0.0,0.14286,0.28571,0.0,0.42857,13,0,0,13,0,4,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.11607,0.16917,0.0,0.0,0.28571,0.0,0.42857,21,0,0,21,0,1,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.10268,0.15251,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,2,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.05357,"x":0.15616,"p":[[0,28,0.0,0.15616,0.12557,0.14214,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,19,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.125,0.20748,0.0,0.0,0.2857,0.0,1.0,20,1,0,20,0,2,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,28,0.2857,0.10268,0.1394,0.0,0.0,0.28571,0.0,0.42857,20,0,0,20,0,2,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.09375,0.14555,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,4,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.12946,0.15303,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,4,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.07589,0.13825,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.10714,0.15567,0.0,0.0,0.28571,0.0,0.42857,21,0,0,21,0,1,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7a57ee57151edfe8","q":"Let  $p > 3$  be a prime number, and let  $F_p$  denote the (fi\fnite) set of residue classes modulo  $p$ .\nLet  $S_d$  denote the set of  $2$ -variable polynomials  $P(x, y)$  with coefficients in  $F_p$ , total degree  $\\le d$ , and satisfying  $P(x, y) = P(y,- x -y)$ . Show that   $$ |S_d| = p^{\\lceil (d+1)(d+2)/6 \\rceil} $$  .\n*The total degree of a  $2$ -variable polynomial  $P(x, y)$  is the largest value of  $i + j$  among monomials  $x^iy^j$* appearing in  $P$ .","t":[{"b":2,"e":0.42857,"k":"falling","v":0.42408,"x":0.81696,"p":[[0,37,0.0,0.74107,0.26592,0.42857,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,2,0,0,1,0,15],[4,37,0.1081,0.81696,0.2321,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,7,0,0,2,0,17],[8,37,0.2162,0.75446,0.26543,0.53571,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,5,0,0,1,0,15],[12,37,0.3243,0.75447,0.28174,0.53572,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,2,0,0,0,0,17],[16,37,0.4324,0.67857,0.31542,0.39286,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,6,0,0,4,0,0,3,0,0,0,0,0,5,0,12],[20,37,0.5405,0.59821,0.25111,0.42857,0.57143,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,10,0,0,6,0,0,3,0,0,1,0,7],[24,37,0.6486,0.63838,0.28344,0.42857,0.49979,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,5,0,0,11,0,0,2,0,0,3,0,0,0,0,11],[28,37,0.7568,0.54017,0.24152,0.42857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,4,0,0,12,0,0,5,0,0,3,0,0,2,0,4],[32,37,0.8649,0.49552,0.17849,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,14,0,0,9,0,0,2,0,0,0,0,2],[36,37,0.973,0.42857,0.16366,0.42857,0.42857,0.4286,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,19,0,0,4,0,0,1,0,0,0,0,1],[37,37,1.0,0.42408,0.121,0.39286,0.42857,0.571,0.14286,0.57143,0,0,0,0,0,2,0,0,6,0,0,15,0,0,9,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"falling","v":0.48213,"x":0.87499,"p":[[0,47,0.0,0.76786,0.25692,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,5,0,0,2,0,15],[4,47,0.0851,0.75893,0.29329,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,1,0,0,4,0,0,5,0,0,1,0,0,3,0,16],[8,47,0.1702,0.79464,0.23402,0.67857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,7,0,0,1,0,16],[12,47,0.2553,0.77679,0.28557,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,3,0,0,2,0,0,4,0,0,3,0,0,2,0,17],[16,47,0.3404,0.83483,0.22045,0.71429,1.0,1.0,0.286,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,3,0,18],[20,47,0.4255,0.87499,0.21356,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,0,2,0,22],[24,47,0.5106,0.83929,0.23891,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,2,0,0,4,0,19],[28,47,0.5957,0.74107,0.28221,0.42857,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,4,0,0,5,0,0,5,0,0,1,0,0,1,0,16],[32,47,0.6809,0.72321,0.26229,0.42857,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,3,0,0,2,0,13],[36,47,0.766,0.70536,0.24206,0.42857,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,7,0,0,2,0,10],[40,47,0.8511,0.73214,0.27837,0.42857,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,9,0,0,4,0,0,1,0,0,0,0,16],[44,47,0.9362,0.78569,0.22018,0.57143,0.78571,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,7,0,0,2,0,14],[47,47,1.0,0.48213,0.17404,0.42857,0.42857,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,7,0,0,14,0,0,7,0,0,1,0,0,2,0,1]]}]},{"i":"1058cd48654cebff","q":"Let  ${c\\equiv c\\left(O, R\\right)}$  be a circle with center  ${O}$  and radius  ${R}$  and  ${A, B}$  be two points on it, not belonging to the same diameter. The bisector of angle ${\\angle{ABO}}$  intersects the circle  ${c}$  at point  ${C}$ , the circumcircle of the triangle  $AOB$  , say  ${c_1}$  at point  ${K}$  and the circumcircle of the triangle  $AOC$  , say  ${{c}_{2}}$  at point  ${L}$ . Prove that point  ${K}$  is the circumcircle of the triangle  $AOC$  and that point  ${L}$  is the incenter of the triangle  $AOB$ .\n\nEvangelos Psychas (Greece)","t":[{"b":3,"e":0.2857,"k":"falling","v":0.20078,"x":0.58031,"p":[[0,55,0.0,0.49996,0.31338,0.2857,0.4998,0.74996,0.0,1.0,2,5,0,2,0,5,0,0,6,0,0,3,0,0,7,0,0,1,0,0,3,0,5],[4,55,0.0727,0.58031,0.32914,0.28571,0.57143,0.85704,0.0,1.0,4,7,0,4,0,1,0,0,4,0,0,2,0,0,7,0,0,4,0,0,3,0,7],[8,55,0.1455,0.48659,0.27165,0.28571,0.571,0.71429,0.0,1.0,4,2,0,4,0,0,0,0,7,0,0,4,0,0,8,0,0,5,0,0,2,0,2],[12,55,0.2182,0.33479,0.28705,0.10714,0.28571,0.46418,0.0,1.0,8,2,0,8,0,2,0,0,11,0,0,3,0,0,3,0,0,2,0,0,1,0,2],[16,55,0.2909,0.42854,0.31132,0.1429,0.4286,0.60714,0.0,1.0,6,3,0,6,0,4,0,0,3,0,0,5,0,0,6,0,0,4,0,0,1,0,3],[20,55,0.3636,0.33027,0.2754,0.14214,0.2857,0.57143,0.0,1.0,7,1,0,7,0,5,0,0,8,0,0,3,0,0,5,0,0,1,0,0,2,0,1],[24,55,0.4364,0.3839,0.29327,0.0,0.42857,0.57143,0.0,1.0,9,1,0,9,0,1,0,0,4,0,0,4,0,0,8,0,0,4,0,0,1,0,1],[28,55,0.5091,0.27232,0.25843,0.0,0.2857,0.42858,0.0,0.85714,11,0,0,11,0,3,0,0,8,0,0,3,0,0,3,0,0,3,0,0,1,0,0],[32,55,0.5818,0.30357,0.20748,0.14289,0.28571,0.42858,0.0,0.85714,4,0,0,4,0,7,0,0,11,0,0,4,0,0,4,0,0,1,0,0,1,0,0],[36,55,0.6545,0.28549,0.26969,0.0,0.2857,0.571,0.0,1.0,9,1,0,9,0,6,0,0,8,0,0,0,0,0,5,0,0,3,0,0,0,0,1],[40,55,0.7273,0.22756,0.21386,0.0,0.21428,0.2857,0.0,0.85714,10,0,0,10,0,6,0,0,9,0,0,3,0,0,3,0,0,0,0,0,1,0,0],[44,55,0.8,0.20078,0.18159,0.105,0.14286,0.2857,0.0,0.714,8,0,0,8,0,12,0,0,7,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[48,55,0.8727,0.2811,0.22414,0.14286,0.2857,0.28571,0.0,1.0,6,1,0,6,0,5,0,0,14,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[52,55,0.9455,0.33017,0.23279,0.14286,0.2857,0.4286,0.0,0.85714,5,0,0,5,0,6,0,0,7,0,0,7,0,0,3,0,0,3,0,0,1,0,0],[55,55,1.0,0.23642,0.1944,0.14214,0.1429,0.28571,0.0,0.71429,6,0,0,6,0,11,0,0,9,0,0,2,0,0,2,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.28115,"x":0.61163,"p":[[0,48,0.0,0.3973,0.31283,0.14286,0.35714,0.60714,0.0,1.0,7,2,1,7,0,4,0,0,5,0,0,3,0,0,5,0,0,4,0,0,2,0,2],[4,48,0.0833,0.61163,0.27944,0.42857,0.57143,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,5,0,0,3,0,0,7,0,0,5,0,0,5,0,5],[8,48,0.1667,0.49554,0.37625,0.24999,0.42859,1.0,0.0,1.0,6,9,0,6,0,2,0,0,7,0,0,4,0,0,0,0,0,4,0,0,0,0,9],[12,48,0.25,0.34367,0.26217,0.24999,0.28571,0.57143,0.0,1.0,6,2,0,6,0,2,0,0,14,0,0,0,0,0,7,0,0,1,0,0,0,0,2],[16,48,0.3333,0.37052,0.25965,0.25,0.28571,0.57111,0.0,1.0,4,2,0,4,0,4,0,0,11,0,0,4,0,0,4,0,0,3,0,0,0,0,2],[20,48,0.4167,0.40624,0.26752,0.2857,0.28571,0.57143,0.0,1.0,3,3,0,3,0,3,0,0,12,0,0,3,0,0,6,0,0,2,0,0,0,0,3],[24,48,0.5,0.40614,0.25289,0.25,0.28571,0.71429,0.0,0.857,3,0,0,3,0,5,0,0,9,0,0,2,0,0,4,0,0,8,0,0,1,0,0],[28,48,0.5833,0.37497,0.26664,0.1429,0.28571,0.57143,0.0,0.85714,5,0,0,5,0,4,0,0,9,0,0,3,0,0,6,0,0,1,0,0,4,0,0],[32,48,0.6667,0.37943,0.26147,0.2857,0.28571,0.57141,0.0,1.0,6,1,0,6,0,1,0,0,10,0,0,4,0,0,6,0,0,3,0,0,1,0,1],[36,48,0.75,0.39282,0.27196,0.2857,0.28571,0.57111,0.0,1.0,5,2,0,5,0,2,0,0,10,0,0,4,0,0,6,0,0,2,0,0,1,0,2],[40,48,0.8333,0.39276,0.31141,0.14286,0.28571,0.57111,0.0,1.0,5,3,0,5,0,5,0,0,8,0,0,5,0,0,2,0,0,1,0,0,3,0,3],[44,48,0.9167,0.32132,0.20209,0.2857,0.28571,0.28571,0.0,1.0,3,1,0,3,0,2,0,0,20,0,0,3,0,0,2,0,0,0,0,0,1,0,1],[48,48,1.0,0.28115,0.1577,0.14296,0.2857,0.42857,0.0,0.57143,4,0,0,4,0,5,0,0,14,0,0,6,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"f69c632f8266bd40","q":"Let  $M \\geq 1$  be a real number. Determine all natural numbers  $n$  for which there exist distinct natural numbers  $a$ ,  $b$ ,  $c > M$ , such that $n = (a,b) \\cdot (b,c) + (b,c) \\cdot (c,a) + (c,a) \\cdot (a,b)$ (where  $(x,y)$  denotes the greatest common divisor of natural numbers  $x$  and  $y$ ).","t":[{"b":1,"e":0.35714,"k":"flat","v":0.37945,"x":0.57589,"p":[[0,40,0.0,0.37945,0.20704,0.2857,0.35714,0.57143,0.0,0.71429,3,0,3,3,0,3,0,0,10,0,0,6,0,0,6,0,0,4,0,0,0,0,0],[4,40,0.1,0.40401,0.15423,0.42857,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,2,0,0,2,0,0,21,0,1,2,0,0,2,0,0,0,0,0],[8,40,0.2,0.4732,0.14913,0.42857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,17,0,0,6,0,0,5,0,0,0,0,0],[12,40,0.3,0.47767,0.21901,0.28571,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,4,0,0,6,0,0,9,0,0,5,0,0,4,0,0,4,0,0],[16,40,0.4,0.53572,0.17128,0.42857,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,16,0,0,4,0,0,7,0,0,3,0,0],[20,40,0.5,0.51784,0.21053,0.42857,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,3,0,0,4,0,0,10,0,0,3,0,0,9,0,0,3,0,0],[24,40,0.6,0.49551,0.20819,0.42857,0.42857,0.60714,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,12,0,0,5,0,0,4,0,0,4,0,0],[28,40,0.7,0.48659,0.18161,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,8,0,0,8,0,0,3,0,0,3,0,0],[32,40,0.8,0.57589,0.22442,0.42857,0.64286,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,6,0,0,5,0,0,4,0,0,10,0,0,6,0,0],[36,40,0.9,0.52232,0.21011,0.28571,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,8,0,0,8,0,0,3,0,0,8,0,0,4,0,0],[40,40,1.0,0.39062,0.13121,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,8,0,1,4,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.01786,"x":0.42857,"p":[[0,101,0.0,0.34372,0.21384,0.2857,0.28571,0.4642,0.0,0.85714,5,0,3,5,0,2,0,0,11,0,0,6,0,0,6,0,0,1,0,0,1,0,0],[4,101,0.0396,0.4107,0.20747,0.39286,0.42857,0.4642,0.0,0.85714,4,0,1,4,0,1,0,0,3,0,0,16,0,0,4,0,0,3,0,0,1,0,0],[8,101,0.0792,0.42857,0.18898,0.28571,0.42857,0.57143,0.0,0.71429,2,0,1,2,0,2,0,0,5,0,0,13,0,0,5,0,0,5,0,0,0,0,0],[12,101,0.1188,0.41071,0.19149,0.28571,0.42857,0.42857,0.0,0.857,3,0,0,3,0,0,0,0,7,0,0,15,0,0,3,0,0,3,0,0,1,0,0],[16,101,0.1584,0.36161,0.25501,0.14286,0.42857,0.57143,0.0,0.85714,7,0,0,7,0,2,0,0,6,0,0,8,0,0,3,0,0,5,0,0,1,0,0],[20,101,0.198,0.27901,0.2175,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,2,0,0,7,0,1,11,0,0,0,0,0,1,0,0,1,0,0],[24,101,0.2376,0.33928,0.21354,0.24999,0.42857,0.42857,0.0,0.71429,6,0,0,6,0,2,0,0,6,0,0,14,0,0,0,0,0,4,0,0,0,0,0],[28,101,0.2772,0.2767,0.21416,0.105,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,4,0,0,9,0,0,6,0,0,3,0,0,2,0,0,0,0,0],[32,101,0.3168,0.26784,0.25189,0.0,0.28571,0.42857,0.0,0.85714,12,0,0,12,0,2,0,0,6,0,0,5,0,0,5,0,0,1,0,0,1,0,0],[36,101,0.3564,0.20535,0.25488,0.0,0.07143,0.42857,0.0,0.85714,16,0,0,16,0,4,0,0,1,0,0,8,0,0,1,0,0,0,0,0,2,0,0],[40,101,0.396,0.25893,0.21558,0.0,0.2857,0.42857,0.0,0.71429,10,0,0,10,0,3,0,0,7,0,0,9,0,0,1,0,0,2,0,0,0,0,0],[44,101,0.4356,0.1875,0.19704,0.0,0.14286,0.28571,0.0,0.71429,14,0,0,14,0,3,0,0,9,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[48,101,0.4752,0.18302,0.19635,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,5,0,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[52,101,0.5149,0.24107,0.25614,0.0,0.21428,0.42857,0.0,0.85714,15,0,1,15,0,1,0,0,3,0,0,8,0,0,3,0,0,1,0,0,1,0,0],[56,101,0.5545,0.16964,0.15335,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,6,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[60,101,0.5941,0.23215,0.20748,0.0,0.1429,0.42857,0.0,0.57143,10,0,0,10,0,8,0,0,2,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[64,101,0.6337,0.19643,0.19804,0.0,0.14286,0.28571,0.0,0.71429,13,0,0,13,0,4,0,0,8,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[68,101,0.6733,0.19643,0.21651,0.0,0.14286,0.32143,0.0,0.71429,14,0,0,14,0,5,0,0,5,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[72,101,0.7129,0.09821,0.16146,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,6,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[76,101,0.7525,0.06696,0.12364,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[80,101,0.7921,0.06473,0.12289,0.0,0.0,0.08929,0.0,0.42857,23,0,0,23,1,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[84,101,0.8317,0.0625,0.14698,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[88,101,0.8713,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,101,0.9109,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,101,0.9505,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,101,0.9901,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[101,101,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce99804dd75069a1","q":"Let $\\sqrt{3}=1 . b_{1} b_{2} b_{3} \\cdots(2)$ be the binary representation of $\\sqrt{3}$. Prove that for any positive integer $n$, at least one of the digits $b_{n}, b_{n+1}, \\ldots, b_{2 n}$ equals 1 .","t":[{"b":3,"e":0.571,"k":"falling","v":0.43523,"x":0.89286,"p":[[0,38,0.0,0.87933,0.19281,0.82132,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,3,0,21],[4,38,0.1053,0.85714,0.22016,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,5,0,0,4,0,19],[8,38,0.2105,0.78124,0.27661,0.67846,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,2,0,0,4,0,0,1,0,0,5,0,0,3,0,16],[12,38,0.3158,0.89286,0.20203,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,2,0,23],[16,38,0.4211,0.76782,0.26428,0.71429,0.85707,1.0,0.0,1.0,1,12,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,0,6,0,12],[20,38,0.5263,0.74554,0.24152,0.57143,0.78571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,6,0,0,6,0,10],[24,38,0.6316,0.78792,0.25223,0.71421,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,1,0,0,0,0,0,0,4,0,0,7,0,0,4,0,14],[28,38,0.7368,0.77902,0.22114,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,7,0,0,5,1,11],[32,38,0.8421,0.683,0.23073,0.5713,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,2,0,0,3,0,0,9,0,0,5,0,0,6,0,6],[36,38,0.9474,0.5357,0.26244,0.39286,0.4998,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,5,0,0,8,0,0,6,0,0,4,0,0,2,0,4],[38,38,1.0,0.43523,0.27451,0.25,0.42859,0.57143,0.0,1.0,3,2,0,3,1,4,0,0,4,0,0,7,0,0,7,0,0,2,0,0,2,0,2]]},{"b":4,"e":0.42857,"k":"flat","v":0.83481,"x":0.88616,"p":[[0,16,0.0,0.86161,0.16164,0.71429,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,5,0,16],[4,16,0.25,0.88616,0.15958,0.76786,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,1,0,4,0,19],[8,16,0.5,0.87945,0.18937,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,4,0,20],[12,16,0.75,0.86607,0.2111,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,0,5,0,19],[16,16,1.0,0.83481,0.20858,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,8,0,15]]}]},{"i":"9096d374901bb40a","q":"Prove that for every polynomial $P(x)$ with real coefficients there exist a positive integer $m$ and polynomials $P_{1}(x), P_{2}(x), \\ldots, P_{m}(x)$ with real coefficients such that\n\n$$\nP(x)=\\left(P_{1}(x)\\right)^{3}+\\left(P_{2}(x)\\right)^{3}+\\cdots+\\left(P_{m}(x)\\right)^{3} .\n$$","t":[{"b":1,"e":0.42857,"k":"flat","v":0.58929,"x":0.8616,"p":[[0,54,0.0,0.71874,0.30824,0.57132,0.85714,1.0,0.0,1.0,2,11,0,2,0,0,0,0,5,0,0,0,0,0,3,0,0,4,0,0,7,0,11],[4,54,0.0741,0.65179,0.41793,0.28571,0.85714,1.0,0.0,1.0,7,15,0,7,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,5,0,15],[8,54,0.1481,0.58929,0.44571,0.0,0.85714,1.0,0.0,1.0,10,14,0,10,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,4,0,14],[12,54,0.2222,0.62945,0.40542,0.28571,0.85714,1.0,0.0,1.0,6,13,0,6,0,1,0,0,5,0,0,0,0,0,1,0,0,1,0,0,5,0,13],[16,54,0.2963,0.75446,0.35035,0.64286,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,0,5,0,17],[20,54,0.3704,0.62946,0.40699,0.28571,0.85714,1.0,0.0,1.0,5,15,0,5,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,0,3,0,15],[24,54,0.4444,0.73214,0.34947,0.42857,0.92857,1.0,0.0,1.0,3,16,0,3,0,0,0,0,4,0,0,3,0,0,0,0,0,1,0,0,5,0,16],[28,54,0.5185,0.76338,0.36529,0.57132,1.0,1.0,0.0,1.0,4,20,0,4,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,20],[32,54,0.5926,0.83036,0.27765,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,0,3,0,20],[36,54,0.6667,0.79911,0.30485,0.82143,0.85714,1.0,0.0,1.0,3,15,0,3,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,9,0,15],[40,54,0.7407,0.8616,0.26362,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,6,0,21],[44,54,0.8148,0.64286,0.35892,0.39286,0.64286,1.0,0.0,1.0,3,14,0,3,0,1,0,0,4,0,0,6,0,0,2,0,0,1,0,0,1,0,14],[48,54,0.8889,0.69642,0.32879,0.42857,0.78571,1.0,0.0,1.0,2,12,0,2,0,2,0,0,3,0,0,2,0,0,1,0,0,6,0,0,4,0,12],[52,54,0.963,0.77232,0.23381,0.71429,0.78571,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,9,0,0,4,0,12],[54,54,1.0,0.6473,0.26723,0.4286,0.71429,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,1,0,0,6,0,0,6,0,0,8,0,0,2,0,7]]},{"b":5,"e":1.0,"k":"rising","v":0.61161,"x":0.99107,"p":[[0,62,0.0,0.75,0.30093,0.42857,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,4,0,0,4,0,0,2,0,0,1,0,0,5,0,15],[4,62,0.0645,0.61161,0.41686,0.24999,0.78571,1.0,0.0,1.0,7,14,0,7,0,1,0,0,4,0,0,1,0,0,0,0,0,3,0,0,2,0,14],[8,62,0.129,0.72768,0.34692,0.42857,0.85714,1.0,0.0,1.0,3,14,0,3,0,1,0,0,3,0,0,2,0,0,0,0,0,2,0,0,7,0,14],[12,62,0.1935,0.75893,0.38867,0.82143,1.0,1.0,0.0,1.0,6,19,0,6,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,19],[16,62,0.2581,0.77679,0.36235,0.82143,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,18],[20,62,0.3226,0.93304,0.18893,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[24,62,0.3871,0.94196,0.19186,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[28,62,0.4516,0.91964,0.19541,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,24],[32,62,0.5161,0.94197,0.20158,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[36,62,0.5806,0.95089,0.15815,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[40,62,0.6452,0.94196,0.1851,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[44,62,0.7097,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[48,62,0.7742,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,62,0.8387,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[56,62,0.9032,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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],[62,62,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]}]},{"i":"2bf34f888478b1a4","q":"Prove that:\n\na) the sequence  $a_n=\\frac{1}{n+1}+\\frac{1}{n+2}+\\ldots+\\frac{1}{n+n},\\ n\\ge 1$  is monotonic.\n\nb) there is a sequence  $(a_n)_{n\\ge 1}\\in \\{0,1\\}$  such that:\n\n\\[\\lim_{n\\to \\infty} \\left(\\frac{a_1}{n+1}+\\frac{a_2}{n+2}+\\ldots +\\frac{a_n}{n+n}\\right)=\\frac{1}{2}\\]\n\n*Radu Gologan*","t":[{"b":3,"e":0.14286,"k":"flat","v":0.14277,"x":0.26786,"p":[[0,47,0.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.24554,0.25564,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,26,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[8,47,0.1702,0.23652,0.25157,0.14286,0.14286,0.14287,0.14,1.0,0,2,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[12,47,0.2553,0.23661,0.24382,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,27,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[16,47,0.3404,0.21848,0.22021,0.14286,0.14286,0.14286,0.14,1.0,0,2,0,0,0,28,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[20,47,0.4255,0.20089,0.23107,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[24,47,0.5106,0.20295,0.20915,0.14286,0.14286,0.14286,0.07143,1.0,0,2,0,0,1,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,47,0.5957,0.21857,0.23692,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,24,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,1],[32,47,0.6809,0.19616,0.2106,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[36,47,0.766,0.26786,0.28738,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,25,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,3],[40,47,0.8511,0.21875,0.21124,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[44,47,0.9362,0.16072,0.15465,0.14286,0.14286,0.14286,0.0,0.85714,4,0,0,4,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[47,47,1.0,0.18733,0.1766,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1]]},{"b":7,"e":0.14286,"k":"flat","v":0.14277,"x":0.28116,"p":[[0,38,0.0,0.14277,0.06187,0.14286,0.14286,0.14286,0.0,0.4286,2,0,1,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.2633,0.26275,0.14286,0.14286,0.14287,0.14,1.0,0,2,0,0,0,25,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,2],[8,38,0.2105,0.28116,0.28009,0.14286,0.14286,0.1786,0.14,1.0,0,3,0,0,0,24,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,3],[12,38,0.3158,0.2767,0.28112,0.14286,0.14286,0.14286,0.14,1.0,0,2,0,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,2],[16,38,0.4211,0.2009,0.19516,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[20,38,0.5263,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.19179,0.16219,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[28,38,0.7368,0.19187,0.16985,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,29,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[32,38,0.8421,0.16965,0.10972,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,38,0.9474,0.15161,0.04975,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.16964,0.14914,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"53e69e6fd15603ce","q":"Let  $n$  be a non-negative integer. Find all non-negative integers  $a$ ,  $b$ ,  $c$ ,  $d$  such that \\[a^{2}+b^{2}+c^{2}+d^{2}= 7 \\cdot 4^{n}.\\]","t":[{"b":1,"e":0.57143,"k":"falling","v":0.47321,"x":0.99107,"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],[4,34,0.1176,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,34,0.2353,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,34,0.3529,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[16,34,0.4706,0.9375,0.16342,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,26],[20,34,0.5882,0.92857,0.17128,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,26],[24,34,0.7059,0.88393,0.2299,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,24],[28,34,0.8235,0.91072,0.15872,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,23],[32,34,0.9412,0.74991,0.30322,0.4286,1.0,1.0,0.14,1.0,0,17,0,0,0,2,0,0,3,0,0,4,0,0,2,0,0,3,0,0,1,0,17],[34,34,1.0,0.47321,0.30397,0.28571,0.42857,0.71429,0.0,1.0,2,4,0,2,0,5,0,0,8,0,0,3,0,0,4,0,0,4,0,0,2,0,4]]},{"b":4,"e":1.0,"k":"flat","v":0.95089,"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],[8,28,0.2857,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,28,0.4286,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,28,0.5714,0.97308,0.091,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[20,28,0.7143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,28,0.8571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,28,1.0,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23]]}]},{"i":"73037ccf123fc36d","q":"The quadrilateral  $ABCD$  is inscribed in the circle \u03c9. The diagonals  $AC$  and  $BD$  intersect at the point  $O$ . On the segments  $AO$  and  $DO$ , the points  $E$  and  $F$  are chosen, respectively. The straight line  $EF$  intersects \u03c9 at the points  $E_1$  and  $F_1$ . The circumscribed circles of the triangles  $ADE$  and  $BCF$  intersect the segment  $EF$  at the points  $E_2$  and  $F_2$  respectively (assume that all the points  $E, F, E_1, F_1, E_2$  and  $F_2$  are different). Prove that  $E_1E_2 = F_1F_2$ .  $(N. Sedrakyan)$","t":[{"b":0,"e":0.2857,"k":"flat","v":0.22767,"x":0.48209,"p":[[0,40,0.0,0.22767,0.2047,0.10714,0.14286,0.28571,0.0,0.71429,8,0,0,8,0,10,0,0,7,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[4,40,0.1,0.45978,0.23885,0.28571,0.42857,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,9,0,0,5,0,0,4,0,0,9,0,0,0,0,1],[8,40,0.2,0.40166,0.19386,0.2857,0.42857,0.571,0.0,0.71429,2,0,0,2,0,3,0,0,8,0,0,9,0,0,6,0,0,4,0,0,0,0,0],[12,40,0.3,0.47764,0.26148,0.28571,0.571,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,5,0,0,4,0,0,6,0,0,9,0,0,1,0,1],[16,40,0.4,0.41075,0.23351,0.2857,0.35714,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,10,0,0,7,0,0,2,0,0,5,0,0,1,0,1],[20,40,0.5,0.41067,0.25689,0.24999,0.42857,0.57143,0.0,1.0,4,1,0,4,0,4,0,0,6,0,0,5,0,0,6,0,0,6,0,0,0,0,1],[24,40,0.6,0.47098,0.24015,0.2857,0.42859,0.71429,0.0,0.71429,2,0,0,2,0,3,0,0,7,0,0,5,0,1,0,0,0,14,0,0,0,0,0],[28,40,0.7,0.48209,0.19147,0.42857,0.4998,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,0,0,0,12,0,0,11,0,0,4,0,0,1,0,0],[32,40,0.8,0.35711,0.19881,0.14286,0.28571,0.42858,0.14286,0.857,0,0,0,0,0,10,0,0,7,0,0,9,0,0,2,0,0,3,0,0,1,0,0],[36,40,0.9,0.35714,0.1821,0.14286,0.42857,0.42857,0.14286,0.857,0,0,0,0,0,9,0,0,6,0,0,13,0,0,1,0,0,2,0,0,1,0,0],[40,40,1.0,0.34375,0.15093,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,13,0,0,9,0,0,2,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"flat","v":0.20089,"x":0.45978,"p":[[0,45,0.0,0.20089,0.20472,0.10714,0.14288,0.2857,0.0,1.0,8,1,0,8,0,11,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[4,45,0.0889,0.39721,0.20128,0.28571,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,3,0,0,7,0,0,14,0,0,2,0,0,2,0,0,2,0,0],[8,45,0.1778,0.35714,0.23958,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,5,0,0,8,0,0,8,0,0,2,0,0,4,0,0,0,0,1],[12,45,0.2667,0.4553,0.22426,0.2857,0.571,0.60714,0.0,0.71429,2,0,0,2,0,4,0,0,5,0,0,4,0,0,9,0,0,8,0,0,0,0,0],[16,45,0.3556,0.45978,0.24673,0.28571,0.42859,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,5,0,0,8,0,0,6,0,0,4,0,0,1,0,2],[20,45,0.4444,0.37053,0.23106,0.14289,0.28571,0.57143,0.0,0.85714,1,0,0,1,0,8,0,0,11,0,0,3,0,0,3,0,0,4,0,0,2,0,0],[24,45,0.5333,0.38834,0.21195,0.24999,0.42857,0.571,0.0,1.0,1,1,0,1,0,7,0,0,7,0,0,6,0,0,9,0,0,1,0,0,0,0,1],[28,45,0.6222,0.35704,0.20211,0.24999,0.28571,0.4286,0.0,0.85714,2,0,0,2,0,6,0,0,9,0,0,8,0,0,4,0,0,2,0,0,1,0,0],[32,45,0.7111,0.32143,0.20516,0.14286,0.2857,0.42857,0.0,0.8571,4,0,0,4,0,6,0,0,7,0,0,12,0,0,0,0,0,2,0,0,1,0,0],[36,45,0.8,0.26777,0.20129,0.14286,0.2857,0.42857,0.0,0.71429,6,0,0,6,0,7,0,0,10,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[40,45,0.8889,0.27679,0.15126,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,13,0,0,12,0,0,5,0,0,0,0,0,2,0,0,0,0,0],[44,45,0.9778,0.25893,0.09062,0.14286,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,10,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.28125,0.12103,0.14286,0.2857,0.42857,0.14286,0.57143,0,0,0,0,0,11,0,0,12,0,0,8,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"3f918d0d03fcc1f5","q":"Let  $a, b, c, d$  be odd positive integers and pairwise coprime. For a positive integer  $n$ , let   $$ f(n) = \\left[\\frac{n}{a}\n\\right]+\\left[\\frac{n}{b}\\right]+\\left[\\frac{n}{c}\\right]+\\left[\\frac{n}{d}\\right] $$   Prove that   $$ \\sum_{n=1}^{abcd}(-1)^{f(n)}=1 $$","t":[{"b":4,"e":1.0,"k":"flat","v":0.29463,"x":0.77232,"p":[[0,43,0.0,0.4107,0.36899,0.10714,0.35714,0.71429,0.0,1.0,8,6,0,8,0,6,0,0,2,0,0,4,0,0,3,0,0,2,0,0,1,0,6],[4,43,0.093,0.67411,0.32972,0.42857,0.71429,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,4,0,0,7,0,0,1,0,0,2,0,0,0,0,15],[8,43,0.186,0.77232,0.3251,0.42857,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,3,0,0,4,0,0,0,0,0,2,0,0,2,0,19],[12,43,0.2791,0.71878,0.35259,0.39286,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,4,0,0,3,0,0,1,0,0,1,0,0,1,0,18],[16,43,0.3721,0.77232,0.31916,0.42859,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,6,0,0,2,0,0,0,0,0,2,0,0,2,0,19],[20,43,0.4651,0.6116,0.37667,0.28571,0.64286,1.0,0.0,1.0,4,12,0,4,0,3,0,0,2,0,0,5,0,0,2,0,0,1,0,0,3,0,12],[24,43,0.5581,0.50442,0.34252,0.28571,0.42857,0.78571,0.0,1.0,4,8,0,4,0,3,0,0,5,0,0,6,0,0,4,0,0,2,0,0,0,0,8],[28,43,0.6512,0.5,0.38132,0.14286,0.42857,1.0,0.0,1.0,6,9,0,6,0,3,0,0,5,0,0,5,0,0,0,0,0,3,0,0,1,0,9],[32,43,0.7442,0.60713,0.34442,0.28571,0.64286,1.0,0.0,1.0,2,11,0,2,0,4,0,0,3,0,0,4,0,0,3,0,0,5,0,0,0,0,11],[36,43,0.8372,0.45981,0.34392,0.14286,0.42857,0.75,0.0,1.0,5,5,0,5,0,5,0,0,4,0,0,5,0,0,3,0,0,2,0,0,3,0,5],[40,43,0.9302,0.33928,0.27373,0.14286,0.28571,0.4286,0.0,1.0,5,3,0,5,0,7,0,0,6,0,0,8,0,0,3,0,0,0,0,0,0,0,3],[43,43,1.0,0.29463,0.28556,0.10714,0.14288,0.4286,0.0,1.0,8,2,0,8,0,9,0,0,3,0,0,5,0,0,4,0,0,0,0,0,1,0,2]]},{"b":7,"e":0.571,"k":"flat","v":0.41515,"x":0.68747,"p":[[0,40,0.0,0.6517,0.41956,0.14286,1.0,1.0,0.0,1.0,4,18,0,4,0,6,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,18],[4,40,0.1,0.61606,0.35614,0.28571,0.71429,1.0,0.0,1.0,3,11,0,3,0,3,0,0,3,0,0,4,0,0,2,0,0,4,0,0,2,0,11],[8,40,0.2,0.68747,0.32624,0.39286,0.85707,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,4,0,0,2,0,0,4,0,0,1,0,0,4,0,13],[12,40,0.3,0.63838,0.33879,0.39286,0.64286,1.0,0.0,1.0,1,12,0,1,0,4,0,0,3,0,0,5,0,0,3,0,0,2,0,0,2,0,12],[16,40,0.4,0.66069,0.34581,0.39286,0.71429,1.0,0.0,1.0,1,14,0,1,0,4,0,0,3,0,0,4,0,0,3,0,0,2,0,0,1,0,14],[20,40,0.5,0.61603,0.2976,0.42857,0.57121,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,3,0,0,6,0,0,4,0,0,4,0,0,3,0,8],[24,40,0.6,0.56697,0.37026,0.25002,0.50001,1.0,0.0,1.0,2,11,0,2,0,6,0,0,5,0,0,3,0,0,1,0,0,3,0,0,1,0,11],[28,40,0.7,0.41515,0.31814,0.14289,0.35714,0.57111,0.0,1.0,4,5,0,4,0,6,0,0,6,0,0,6,0,0,4,0,0,0,0,0,1,0,5],[32,40,0.8,0.45094,0.30327,0.28571,0.42857,0.60712,0.0,1.0,3,4,0,3,0,4,0,0,6,0,0,10,0,0,1,0,0,1,0,0,3,0,4],[36,40,0.9,0.55342,0.27387,0.42857,0.571,0.71429,0.0,1.0,1,5,0,1,0,4,0,0,2,0,0,6,0,0,7,0,0,7,0,0,0,0,5],[40,40,1.0,0.62945,0.2392,0.42859,0.71429,0.74996,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,4,0,0,6,0,0,9,0,0,4,0,4]]}]},{"i":"5fcde3987174f082","q":"Find all positive integers  $n$  with the following property: the  $k$  positive divisors of  $n$  have a permutation  $(d_1,d_2,\\ldots,d_k)$  such that for  $i=1,2,\\ldots,k$ , the number  $d_1+d_2+\\cdots+d_i$  is a perfect square.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.06697,"x":0.67857,"p":[[0,106,0.0,0.0982,0.05283,0.07143,0.14286,0.14286,0.0,0.14286,5,0,0,5,10,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,106,0.0377,0.67857,0.29014,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,4,0,0,1,0,0,5,0,0,2,0,0,5,0,0,7,0,8],[8,106,0.0755,0.60714,0.31744,0.2857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,6,0,0,4,0,0,3,0,0,0,0,0,8,0,0,4,0,7],[12,106,0.1132,0.66068,0.29613,0.53539,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,2,0,0,1,0,0,4,0,0,8,0,0,4,0,8],[16,106,0.1509,0.58927,0.36553,0.14286,0.64286,1.0,0.0,1.0,1,10,0,1,0,8,0,0,3,0,0,2,0,0,2,0,0,2,0,0,4,0,10],[20,106,0.1887,0.51786,0.33217,0.14286,0.42857,0.85714,0.0,1.0,1,5,0,1,0,9,0,0,2,0,1,4,0,0,0,1,0,5,0,0,4,0,5],[24,106,0.2264,0.54686,0.35111,0.14286,0.4998,1.0,0.07143,1.0,0,9,0,0,1,9,0,0,2,0,0,4,0,0,2,0,0,4,0,0,1,0,9],[28,106,0.2642,0.60044,0.35619,0.14286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,9,0,0,2,0,1,2,0,0,0,0,0,5,0,0,3,0,10],[32,106,0.3019,0.48661,0.33666,0.14286,0.35714,0.75,0.14286,1.0,0,5,0,0,0,12,0,0,4,0,0,2,0,0,0,0,0,6,0,0,3,0,5],[36,106,0.3396,0.57363,0.34185,0.25,0.571,1.0,0.14286,1.0,0,9,0,0,0,8,0,0,4,0,0,2,0,1,2,0,0,4,0,0,2,0,9],[40,106,0.3774,0.58705,0.32229,0.28571,0.57143,0.89286,0.07143,1.0,0,8,0,0,1,6,0,0,2,0,0,4,0,0,4,0,0,5,0,0,2,0,8],[44,106,0.4151,0.45978,0.32286,0.14286,0.35714,0.74996,0.0,1.0,1,2,0,1,0,12,0,0,3,0,0,1,0,0,3,0,0,4,0,0,6,0,2],[48,106,0.4528,0.49107,0.32328,0.14286,0.42857,0.85714,0.14286,1.0,0,4,0,0,0,10,0,0,5,0,0,4,0,0,0,0,0,4,0,0,5,0,4],[52,106,0.4906,0.61593,0.32594,0.28571,0.71429,0.85714,0.07,1.0,0,7,0,0,2,4,0,0,3,0,0,4,0,0,0,0,0,6,0,0,6,0,7],[56,106,0.5283,0.5714,0.34442,0.2857,0.571,0.85714,0.0,1.0,2,7,0,2,0,5,0,0,4,0,0,4,0,0,2,0,0,2,0,0,6,0,7],[60,106,0.566,0.56697,0.32632,0.28571,0.50001,0.85714,0.14286,1.0,0,7,0,0,0,7,0,0,4,0,0,5,0,0,1,0,0,4,0,0,4,0,7],[64,106,0.6038,0.52444,0.34986,0.14286,0.4998,0.85714,0.071,1.0,0,7,0,0,1,10,0,0,3,0,0,2,0,0,2,0,0,4,0,0,3,0,7],[68,106,0.6415,0.52221,0.35832,0.14286,0.42859,1.0,0.14,1.0,0,9,0,0,0,11,0,0,4,0,0,2,0,0,2,0,0,3,0,0,1,0,9],[72,106,0.6792,0.48205,0.31903,0.14286,0.42857,0.75,0.14,1.0,0,4,0,0,0,11,0,0,3,0,0,5,0,0,1,0,0,4,0,0,4,0,4],[76,106,0.717,0.35258,0.27435,0.14286,0.2857,0.57111,0.0,1.0,1,2,0,1,0,14,0,0,6,0,0,2,0,0,3,0,0,3,0,0,1,0,2],[80,106,0.7547,0.40393,0.33348,0.14286,0.2857,0.71429,0.0,1.0,1,4,0,1,1,13,0,0,5,0,0,2,0,0,0,0,0,3,0,0,3,0,4],[84,106,0.7925,0.42186,0.31861,0.14286,0.28571,0.58929,0.0,1.0,1,4,0,1,0,12,0,0,5,0,0,2,0,0,4,1,0,0,0,0,3,0,4],[88,106,0.8302,0.52677,0.30813,0.1429,0.64264,0.71429,0.0,1.0,1,3,0,1,0,8,0,0,2,0,0,4,0,0,1,0,0,9,0,0,4,0,3],[92,106,0.8679,0.52442,0.36244,0.14286,0.42859,0.89286,0.07,1.0,0,8,0,0,1,10,0,0,4,0,0,2,0,0,2,0,0,1,0,0,4,0,8],[96,106,0.9057,0.46872,0.32188,0.14286,0.28571,0.74996,0.14286,1.0,0,4,0,0,0,11,0,0,6,0,0,1,0,0,3,0,0,3,0,0,4,0,4],[100,106,0.9434,0.36158,0.3364,0.14286,0.14286,0.60714,0.0,1.0,5,3,0,5,2,11,0,0,1,0,0,1,0,0,4,0,0,3,0,0,2,0,3],[104,106,0.9811,0.27682,0.29438,0.14286,0.14286,0.32143,0.0,1.0,3,3,0,3,3,16,0,1,1,0,0,2,0,0,2,0,0,0,0,0,1,0,3],[106,106,1.0,0.06697,0.0617,0.0,0.07143,0.14286,0.0,0.1429,13,0,0,13,8,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"rising","v":0.07585,"x":0.87051,"p":[[0,47,0.0,0.07585,0.0617,0.0,0.07143,0.14286,0.0,0.14286,11,0,1,11,8,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.51343,0.33091,0.2857,0.35714,0.85714,0.14286,1.0,0,6,0,0,0,7,0,0,9,0,0,3,0,0,1,0,0,1,0,0,5,0,6],[8,47,0.1702,0.61158,0.33165,0.25,0.71429,0.85714,0.0,1.0,2,4,0,2,0,6,0,0,1,0,0,1,0,0,3,0,0,4,0,0,11,0,4],[12,47,0.2553,0.69418,0.31906,0.49968,0.71429,1.0,0.07143,1.0,0,11,0,0,1,4,0,0,3,0,0,0,0,0,1,0,0,8,0,0,4,0,11],[16,47,0.3404,0.61155,0.29931,0.39286,0.57143,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,4,0,0,3,0,0,6,0,0,3,0,0,6,0,6],[20,47,0.4255,0.69196,0.31157,0.42857,0.78564,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,3,0,0,3,0,0,1,0,0,5,0,0,5,0,11],[24,47,0.5106,0.50884,0.33453,0.24999,0.42857,0.875,0.0,1.0,1,7,0,1,0,7,0,0,6,0,0,4,0,0,3,1,0,1,0,0,1,1,7],[28,47,0.5957,0.72089,0.29037,0.42857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,4,0,0,3,0,0,1,1,0,3,0,0,7,0,11],[32,47,0.6809,0.68081,0.32092,0.42859,0.78564,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,1,0,0,5,0,0,2,1,0,2,0,0,4,0,12],[36,47,0.766,0.72538,0.25146,0.53539,0.78564,0.89286,0.07,1.0,0,8,0,0,1,0,0,0,2,0,0,5,0,0,1,0,0,7,0,0,8,0,8],[40,47,0.8511,0.67188,0.26478,0.42859,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,1,2,0,0,4,0,0,3,0,0,7,1,0,5,1,6],[44,47,0.9362,0.84373,0.19353,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,8,0,15],[47,47,1.0,0.87051,0.14883,0.857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,11,0,14]]}]},{"i":"1cbb5f17a2f0d36e","q":"Let ABC be a triangle with O as the center of the circumscribed circle. Let $\\ell$ be a line perpendicular to the line $(A O)$. The line $(\\ell)$ intersects the sides $(A B)$ and $(A C)$ at points $D$ and $E$. Show that the points $B, C, E$ and $D$ are concyclic.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.24107,"x":0.37054,"p":[[0,25,0.0,0.24107,0.3163,0.0,0.0,0.42857,0.0,1.0,17,3,0,17,0,2,0,0,1,0,0,7,0,0,2,0,0,0,0,0,0,0,3],[4,25,0.16,0.3125,0.30813,0.0,0.42857,0.42858,0.0,1.0,12,2,0,12,0,3,0,0,0,0,0,11,0,0,1,0,0,2,0,0,1,0,2],[8,25,0.32,0.25445,0.27135,0.0,0.21428,0.42857,0.0,1.0,14,1,0,14,0,2,0,0,2,0,0,10,0,0,2,0,0,0,0,0,1,0,1],[12,25,0.48,0.29909,0.20933,0.10714,0.42857,0.42857,0.0,0.71429,8,0,0,8,0,4,0,0,1,0,0,16,0,0,2,0,0,1,0,0,0,0,0],[16,25,0.64,0.27679,0.2141,0.0,0.42857,0.42857,0.0,0.71429,10,0,0,10,0,3,0,0,1,0,0,16,0,0,1,0,0,1,0,0,0,0,0],[20,25,0.8,0.37054,0.1631,0.28571,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,3,0,0,3,0,0,18,0,0,5,0,0,0,0,0,0,0,0],[24,25,0.96,0.3214,0.18554,0.14289,0.42857,0.42857,0.0,0.57143,6,0,0,6,0,3,0,0,3,0,0,17,0,0,3,0,0,0,0,0,0,0,0],[25,25,1.0,0.28125,0.19061,0.0,0.42857,0.42857,0.0,0.4286,9,0,0,9,0,2,0,0,2,0,0,19,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"rising","v":0.16518,"x":0.48657,"p":[[0,22,0.0,0.16518,0.21756,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,4,0,0,1,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[4,22,0.1818,0.33036,0.25614,0.0,0.42857,0.57143,0.0,0.57143,11,0,0,11,0,1,0,0,1,0,0,5,0,0,14,0,0,0,0,0,0,0,0],[8,22,0.3636,0.44633,0.18485,0.39285,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,6,0,0,1,0,0,4,0,0,20,0,0,0,0,0,0,0,0],[12,22,0.5455,0.433,0.2097,0.24999,0.57143,0.57143,0.0,0.57143,3,0,0,3,0,5,0,0,1,0,0,2,0,0,21,0,0,0,0,0,0,0,0],[16,22,0.7273,0.48657,0.17073,0.5354,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,6,0,0,0,0,0,2,0,0,23,0,0,1,0,0,0,0,0],[20,22,0.9091,0.43302,0.19392,0.28571,0.57143,0.57143,0.0,0.57143,2,0,0,2,0,5,0,0,2,0,0,4,0,0,19,0,0,0,0,0,0,0,0],[22,22,1.0,0.4464,0.20437,0.39286,0.57143,0.57143,0.0,0.57143,3,0,0,3,0,4,0,0,1,0,0,2,0,0,22,0,0,0,0,0,0,0,0]]}]},{"i":"97f97acd542464b7","q":"A line through  $A$  intersects a circle at points  $B,C$  with  $B$  between  $A,C$ . The two tangents from  $A$  intersect the circle at  $S,T$ .  $ST$  and  $AC$  intersect at  $P$ . Show that  $\\frac{AP}{PC}=2\\frac{AB}{BC}$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.00893,"x":0.24551,"p":[[0,27,0.0,0.24551,0.22367,0.0,0.2857,0.42857,0.0,0.71429,12,0,0,12,0,2,0,0,7,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[4,27,0.1481,0.12054,0.22335,0.0,0.0,0.17857,0.0,0.85714,23,0,0,23,0,1,0,0,3,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[8,27,0.2963,0.11383,0.18715,0.0,0.0,0.2857,0.0,0.643,22,0,0,22,0,1,0,0,5,0,0,2,0,0,1,1,0,0,0,0,0,0,0],[12,27,0.4444,0.04465,0.10972,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,27,0.7407,0.04464,0.12079,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.03116,0.10546,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.28571,"p":[[0,20,0.0,0.12945,0.19997,0.0,0.0,0.28571,0.0,0.57143,22,0,0,22,0,0,0,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[4,20,0.2,0.28571,0.22867,0.0,0.28571,0.42857,0.0,0.857,10,0,0,10,0,1,0,0,6,0,0,12,0,0,1,0,0,1,0,0,1,0,0],[8,20,0.4,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"33d82e4ea56c84ca","q":"Consider the multiplicative group  $A=\\{z\\in\\mathbb{C}|z^{2006^k}=1, 0<k\\in\\mathbb{Z}\\}$  of all the roots of unity of degree  $2006^k$  for all positive integers  $k$ .\n\nFind the number of homomorphisms  $f:A\\to A$  that satisfy  $f(f(x))=f(x)$  for all elements  $x\\in A$ .","t":[{"b":4,"e":0.4286,"k":"flat","v":0.52232,"x":0.71429,"p":[[0,23,0.0,0.71429,0.16366,0.67857,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,14,0,0,7,0,3],[4,23,0.1739,0.58482,0.20934,0.42857,0.42857,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,6,0,0,2,0,4],[8,23,0.3478,0.54464,0.17655,0.42857,0.42857,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,21,0,0,1,0,0,7,0,0,1,0,2],[12,23,0.5217,0.57598,0.20966,0.42857,0.4286,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,18,0,0,1,0,0,7,0,0,1,0,4],[16,23,0.6957,0.57143,0.17857,0.42857,0.42857,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,10,0,0,1,0,2],[20,23,0.8696,0.52232,0.16213,0.42857,0.42857,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,22,0,0,3,0,0,5,0,0,0,0,2],[23,23,1.0,0.62503,0.198,0.42857,0.64286,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,14,0,0,2,0,0,9,0,0,4,0,3]]},{"b":6,"e":0.42857,"k":"flat","v":0.5625,"x":0.67411,"p":[[0,12,0.0,0.62054,0.16982,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,2,0,0,15,0,0,1,0,2],[4,12,0.3333,0.67411,0.25313,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,13,0,0,1,0,0,6,0,0,1,0,10],[8,12,0.6667,0.67396,0.25058,0.42857,0.71214,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,13,0,0,1,0,0,5,0,0,3,0,9],[12,12,1.0,0.5625,0.2141,0.42857,0.42857,0.60714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,21,0,0,3,0,0,2,0,0,1,0,5]]}]},{"i":"7538112859e9de3d","q":"$x,y,z$  are three real numbers inequal to zero satisfying  $x+y+z=xyz$ .\nProve that   $$  \\sum (\\frac{x^2-1}{x})^2 \\geq 4 $$  *Proposed by Amin Fathpour*","t":[{"b":0,"e":0.0,"k":"falling","v":0.0625,"x":0.94643,"p":[[0,106,0.0,0.79464,0.35881,0.82143,1.0,1.0,0.0,1.0,4,22,0,4,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,22],[4,106,0.0377,0.79018,0.3589,0.57143,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,23],[8,106,0.0755,0.84821,0.31326,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,24],[12,106,0.1132,0.79464,0.3272,0.57143,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,1,0,21],[16,106,0.1509,0.87054,0.27747,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,25],[20,106,0.1887,0.9375,0.15947,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,27],[24,106,0.2264,0.86161,0.30615,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,25],[28,106,0.2642,0.94196,0.20159,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[32,106,0.3019,0.86607,0.26949,0.85714,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,22],[36,106,0.3396,0.89732,0.26058,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,26],[40,106,0.3774,0.93304,0.21424,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[44,106,0.4151,0.91518,0.23381,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,28],[48,106,0.4528,0.90179,0.21852,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,5,0,23],[52,106,0.4906,0.8125,0.34522,0.82143,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,23],[56,106,0.5283,0.89732,0.2237,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,24],[60,106,0.566,0.94643,0.15047,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,28],[64,106,0.6038,0.24107,0.42021,0.0,0.0,0.17857,0.0,1.0,24,7,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,7],[68,106,0.6415,0.32143,0.46015,0.0,0.0,1.0,0.0,1.0,21,10,0,21,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[72,106,0.6792,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[76,106,0.717,0.12054,0.30117,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[80,106,0.7547,0.13839,0.33405,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[84,106,0.7925,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[88,106,0.8302,0.1875,0.36672,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,4],[92,106,0.8679,0.22322,0.37954,0.0,0.0,0.42858,0.0,1.0,23,5,0,23,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,5],[96,106,0.9057,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[100,106,0.9434,0.26339,0.42575,0.0,0.0,0.71429,0.0,1.0,23,7,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,7],[104,106,0.9811,0.09821,0.27302,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[106,106,1.0,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]},{"b":4,"e":0.42857,"k":"falling","v":0.43304,"x":0.95536,"p":[[0,48,0.0,0.70982,0.39526,0.42857,1.0,1.0,0.0,1.0,6,17,0,6,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,3,0,17],[4,48,0.0833,0.67411,0.44498,0.0,1.0,1.0,0.0,1.0,9,19,0,9,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,19],[8,48,0.1667,0.84375,0.31003,0.85714,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,23],[12,48,0.25,0.82589,0.34019,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,25],[16,48,0.3333,0.87054,0.31209,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,27],[20,48,0.4167,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[24,48,0.5,0.91964,0.23128,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,28],[28,48,0.5833,0.92411,0.2448,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[32,48,0.6667,0.90177,0.24076,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,27],[36,48,0.75,0.60714,0.31339,0.42857,0.42857,1.0,0.0,1.0,2,11,0,2,0,0,0,0,2,0,0,15,0,0,0,0,0,2,0,0,0,0,11],[40,48,0.8333,0.45089,0.14773,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,27,0,0,0,0,0,0,0,0,0,0,2],[44,48,0.9167,0.43304,0.05629,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,1,0,0,0,0,0],[48,48,1.0,0.44643,0.09942,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"b43d1111006b4427","q":"Given an acute triangle  $ABC$ , let  $l_a$  be the line passing  $A$  and perpendicular to  $AB$ ,  $l_b$  be the line passing  $B$  and perpendicular to  $BC$ , and  $l_c$  be the line passing  $C$  and perpendicular to  $CA$ . Let  $D$  be the intersection of  $l_b$  and  $l_c$ ,  $E$  be the intersection of  $l_c$  and  $l_a$ , and  $F$  be the intersection of  $l_a$  and  $l_b$ . Prove that the area of the triangle  $DEF$  is at least three times of the area of  $ABC$ 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7,0,0],[96,116,0.8276,0.64285,0.14285,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,9,0,0,6,0,0],[100,116,0.8621,0.59374,0.10779,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,22,0,0,5,0,0,2,0,0],[104,116,0.8966,0.60267,0.15458,0.57143,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,2,0,0,17,0,0,5,0,0,5,0,0],[108,116,0.931,0.59375,0.13415,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,4,0,0,15,0,0,9,0,0,2,0,0],[112,116,0.9655,0.65176,0.1126,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,6,0,0,6,0,0],[116,116,1.0,0.62496,0.14618,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,15,0,0,8,0,0,5,0,0]]}]},{"i":"3b2a8ab2d09331a8","q":"A sequence of real numbers  $ a_{0},\\ a_{1},\\ a_{2},\\dots$  is defined by the formula\n\\[ a_{i \\plus{} 1} \\equal{} \\left\\lfloor a_{i}\\right\\rfloor\\cdot \\left\\langle a_{i}\\right\\rangle\\qquad\\text{for}\\quad i\\geq 0;\n\\]here  $a_0$  is an arbitrary real number,  $\\lfloor a_i\\rfloor$  denotes the greatest integer not exceeding  $a_i$ , and  $\\left\\langle a_i\\right\\rangle=a_i-\\lfloor a_i\\rfloor$ .  Prove that  $a_i=a_{i+2}$  for  $i$  sufficiently large.\n\n*Proposed by Harmel Nestra, Estionia*","t":[{"b":4,"e":0.57143,"k":"falling","v":0.58929,"x":0.88844,"p":[[0,44,0.0,0.88844,0.18106,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,3,0,21],[4,44,0.0909,0.79469,0.20489,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,6,0,0,10,0,10],[8,44,0.1818,0.75,0.23146,0.57143,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,3,0,0,13,0,7],[12,44,0.2727,0.6964,0.20126,0.571,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,4,0,0,8,0,5],[16,44,0.3636,0.70979,0.16166,0.57143,0.78571,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,3,0,0,16,0,0],[20,44,0.4545,0.58929,0.23891,0.42857,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,8,0,0,5,0,0,3,0,0,8,0,2],[24,44,0.5455,0.65178,0.21705,0.42857,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,7,0,0,4,0,0,4,0,0,14,0,0],[28,44,0.6364,0.74106,0.1357,0.67836,0.78571,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,16,0,0],[32,44,0.7273,0.7321,0.17038,0.57143,0.85707,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,4,0,0,15,0,2],[36,44,0.8182,0.74998,0.14727,0.67836,0.78572,0.85714,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,14,0,2],[40,44,0.9091,0.66517,0.17716,0.57143,0.71429,0.857,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,10,0,0,8,0,1],[44,44,1.0,0.65175,0.1673,0.571,0.57143,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,11,0,0,5,0,0,10,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.05804,"x":0.90624,"p":[[0,120,0.0,0.90624,0.16214,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,3,0,22],[4,120,0.0333,0.875,0.20438,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,9,0,17],[8,120,0.0667,0.8125,0.19045,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,0,12,0,10],[12,120,0.1,0.80357,0.2165,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,2,0,0,10,0,12],[16,120,0.1333,0.76339,0.19759,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,13,0,0,4,0,9],[20,120,0.1667,0.70978,0.20669,0.571,0.78571,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,3,0,0,12,0,4],[24,120,0.2,0.87946,0.13882,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,12,0,14],[28,120,0.2333,0.81695,0.21201,0.67857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,4,0,0,5,0,15],[32,120,0.2667,0.86607,0.16727,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,7,0,16],[36,120,0.3,0.81694,0.18978,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,9,0,12],[40,120,0.3333,0.83034,0.15745,0.71429,0.857,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,6,0,12],[44,120,0.3667,0.78122,0.20514,0.57143,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,2,0,0,11,0,9],[48,120,0.4,0.81697,0.1931,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,7,0,0,8,0,12],[52,120,0.4333,0.87054,0.15303,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,4,0,17],[56,120,0.4667,0.89284,0.1557,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,19],[60,120,0.5,0.85268,0.16554,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,8,0,14],[64,120,0.5333,0.05804,0.22548,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[68,120,0.5667,0.26337,0.40579,0.0,0.0,0.57111,0.0,1.0,22,6,0,22,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,6],[72,120,0.6,0.16963,0.33394,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,3],[76,120,0.6333,0.25446,0.3908,0.0,0.0,0.57143,0.0,1.0,22,4,0,22,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,4],[80,120,0.6667,0.26339,0.40107,0.0,0.0,0.60714,0.0,1.0,22,4,0,22,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,4],[84,120,0.7,0.25891,0.39355,0.0,0.0,0.60714,0.0,1.0,22,3,0,22,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,3],[88,120,0.7333,0.29464,0.39276,0.0,0.0,0.71429,0.0,1.0,20,3,0,20,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,3],[92,120,0.7667,0.25447,0.41914,0.0,0.0,0.50002,0.0,1.0,23,7,0,23,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,7],[96,120,0.8,0.33481,0.44407,0.0,0.0,0.89286,0.0,1.0,20,8,0,20,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,8],[100,120,0.8333,0.20087,0.36219,0.0,0.0,0.14275,0.0,1.0,24,4,0,24,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,4],[104,120,0.8667,0.22767,0.37771,0.0,0.0,0.57111,0.0,1.0,23,4,0,23,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,4],[108,120,0.9,0.30804,0.43757,0.0,0.0,0.78571,0.0,1.0,21,8,0,21,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,8],[112,120,0.9333,0.3125,0.41869,0.0,0.0,0.71429,0.0,1.0,20,6,0,20,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,6],[116,120,0.9667,0.30802,0.43464,0.0,0.0,0.75,0.0,1.0,21,7,0,21,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,7],[120,120,1.0,0.7589,0.28892,0.71429,0.78571,1.0,0.0,1.0,3,13,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,3,0,13]]}]},{"i":"01a2576e5a4dbc90","q":"For positive integers $a$ and $b$, we define $a \\ominus b=\\frac{a-b}{\\operatorname{ggd}(a, b)}$. Prove that for any integer $n>1$: $n$ is a prime power (i.e., $n$ can be written as $n=p^{k}$ with $p$ a prime number and $k$ a positive integer) if and only if for all positive integers $m<n$ it holds that $\\operatorname{ggd}(n, n \\ominus m)=1$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":0.98214,"p":[[0,25,0.0,0.94195,0.14226,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,26],[4,25,0.16,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,25,0.32,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,25,0.48,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[16,25,0.64,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[20,25,0.8,0.95982,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[24,25,0.96,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[25,25,1.0,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":4,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,40,0.0,0.94642,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[4,40,0.1,0.91964,0.16728,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,2,0,25],[8,40,0.2,0.93304,0.18205,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[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],[20,40,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,40,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[36,40,0.9,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[40,40,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"033bc58252258df4","q":"Given is a triangle $ABC$. Let $\\Gamma_{1}$ be the circle through $B$ that is tangent to side $AC$ at $A$. Let $\\Gamma_{2}$ be the circle through $C$ that is tangent to side $AB$ at $A$. The second intersection point of $\\Gamma_{1}$ and $\\Gamma_{2}$ is called $D$. The line $AD$ intersects the circumcircle of $\\triangle ABC$ again at $E$. Prove that $D$ is the midpoint of $AE$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.13384,"p":[[0,97,0.0,0.06697,0.14719,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,97,0.0412,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,97,0.0825,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,97,0.1237,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,97,0.1649,0.02683,0.09081,0.0,0.0,0.0,0.0,0.43,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,97,0.2062,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,97,0.2474,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,97,0.2887,0.09375,0.15407,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,3,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,97,0.3299,0.10269,0.15664,0.0,0.0,0.28571,0.0,0.42857,22,0,0,22,0,0,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,97,0.3711,0.10714,0.15152,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,3,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,97,0.4124,0.04911,0.11633,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,97,0.4536,0.08482,0.15093,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,3,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,97,0.4948,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,97,0.5361,0.07589,0.13825,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,97,0.5773,0.12937,0.15302,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,4,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[60,97,0.6186,0.09821,0.15335,0.0,0.0,0.2857,0.0,0.42857,22,0,0,22,0,1,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[64,97,0.6598,0.09822,0.14032,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[68,97,0.701,0.0759,0.14719,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[72,97,0.7423,0.0625,0.12846,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,97,0.7835,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[80,97,0.8247,0.0758,0.11831,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,97,0.866,0.11161,0.17399,0.0,0.0,0.17857,0.0,0.57143,21,0,0,21,0,3,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[88,97,0.9072,0.13384,0.12845,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,97,0.9485,0.08482,0.13296,0.0,0.0,0.17857,0.0,0.42857,22,0,0,22,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,97,0.9897,0.12947,0.13997,0.0,0.14286,0.2857,0.0,0.4286,15,0,0,15,0,7,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[97,97,1.0,0.12498,0.17401,0.0,0.0,0.2857,0.0,0.571,20,0,0,20,0,1,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,81,0.0,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,81,0.0494,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,81,0.0988,0.04464,0.12596,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,81,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,81,0.1975,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,81,0.2469,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[28,81,0.3457,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,81,0.3951,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,81,0.4444,0.03125,0.09268,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,81,0.4938,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,81,0.5432,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,81,0.5926,0.03572,0.0945,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,81,0.642,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,81,0.6914,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,81,0.7407,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,81,0.7901,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,81,0.8395,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,81,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,81,0.9383,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,81,0.9877,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[81,81,1.0,0.00893,0.03459,0.0,0.0,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positive sequence  $\\{a_n\\}$  satisfies: $a_1=1$  and   $$ a_n=2+\\sqrt{a_{n-1}}-2 \\sqrt{1+\\sqrt{a_{n-1}}}(n\\geq 2) $$  Let  $S_n=\\sum\\limits_{k=1}^{n}{2^ka_k}$ . Find the value of  $S_{2023}$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.92411,"x":0.98214,"p":[[0,8,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,8,0.5,0.92411,0.07973,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[8,8,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]},{"b":5,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,54,0.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[4,54,0.0741,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,54,0.1481,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[20,54,0.3704,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,54,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"65b3e436d70f3324","q":"Find all pairs  $(a,b)$  of integers satisfying: there exists an integer  $d \\ge 2$  such that  $a^n + b^n +1$  is divisible by  $d$  for all positive integers  $n$ .","t":[{"b":0,"e":0.57143,"k":"flat","v":0.48658,"x":0.57138,"p":[[0,89,0.0,0.53566,0.10712,0.42857,0.5712,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,12,0,0,18,0,0,0,0,0,2,0,0],[4,89,0.0449,0.57138,0.15971,0.42857,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,9,0,0,17,0,0,1,0,0,2,0,2],[8,89,0.0899,0.48658,0.0787,0.42857,0.42859,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,0,1,0,0,0,0,0],[12,89,0.1348,0.49995,0.07137,0.42857,0.4998,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0],[16,89,0.1798,0.51781,0.10561,0.42859,0.57121,0.57143,0.1429,0.71429,0,0,0,0,0,1,0,0,0,0,0,11,0,0,18,0,0,2,0,0,0,0,0],[20,89,0.2247,0.54011,0.05902,0.571,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0],[24,89,0.2697,0.53566,0.07984,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,20,0,0,2,0,0,0,0,0],[28,89,0.3146,0.52671,0.07518,0.42857,0.571,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,20,0,0,1,0,0,0,0,0],[32,89,0.3596,0.50892,0.08701,0.42857,0.4998,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,14,0,0,2,0,0,0,0,0],[36,89,0.4045,0.52674,0.08325,0.42859,0.57121,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,12,0,0,18,0,0,2,0,0,0,0,0],[40,89,0.4494,0.55352,0.08563,0.5354,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,20,0,0,4,0,0,0,0,0],[44,89,0.4944,0.53568,0.101,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,12,0,0,17,0,0,2,0,0,1,0,0],[48,89,0.5393,0.53567,0.09447,0.42857,0.57141,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,11,0,0,19,0,0,1,0,0,1,0,0],[52,89,0.5843,0.51333,0.10008,0.42857,0.4286,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,11,0,0,4,0,0,0,0,0],[56,89,0.6292,0.52227,0.08455,0.42857,0.571,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,17,0,0,2,0,0,0,0,0],[60,89,0.6742,0.54459,0.0906,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,21,0,0,1,0,0,1,0,0],[64,89,0.7191,0.54011,0.09266,0.42857,0.57121,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,17,0,0,4,0,0,0,0,0],[68,89,0.764,0.5312,0.08168,0.42857,0.57141,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,19,0,0,2,0,0,0,0,0],[72,89,0.809,0.50001,0.07978,0.42857,0.4293,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,14,0,0,1,0,0,0,0,0],[76,89,0.8539,0.55351,0.09941,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,10,0,0,16,0,0,6,0,0,0,0,0],[80,89,0.8989,0.54908,0.09522,0.42859,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,20,0,0,2,0,0,1,0,0],[84,89,0.9438,0.51337,0.0787,0.4286,0.57141,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,11,0,0,20,0,0,0,0,0,0,0,0],[88,89,0.9888,0.53568,0.08746,0.4286,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,0,3,0,0,0,0,0],[89,89,1.0,0.54015,0.09931,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,0,2,0,0,1,0,0]]},{"b":1,"e":0.57143,"k":"flat","v":0.52227,"x":0.60714,"p":[[0,74,0.0,0.52227,0.1268,0.42857,0.571,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,14,0,0,14,0,0,2,0,0,0,0,1],[4,74,0.0541,0.53569,0.08747,0.4286,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,10,0,0,21,0,0,0,0,0,1,0,0],[8,74,0.1081,0.54012,0.0855,0.53539,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,7,0,0,22,0,0,2,0,0,0,0,0],[12,74,0.1622,0.53565,0.11843,0.42857,0.571,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,16,0,0,2,0,0,0,0,1],[16,74,0.2162,0.53121,0.11424,0.42857,0.5712,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,17,0,0,1,0,0,0,0,1],[20,74,0.2703,0.56247,0.13803,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,16,0,0,2,0,0,2,0,1],[24,74,0.3243,0.60708,0.14726,0.571,0.57143,0.71407,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,16,0,0,4,0,0,4,0,1],[28,74,0.3784,0.59817,0.16537,0.5354,0.57143,0.60714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,16,0,0,3,0,0,3,0,2],[32,74,0.4324,0.58925,0.13244,0.5354,0.57143,0.60714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,8,0,0,16,0,0,4,0,0,4,0,0],[36,74,0.4865,0.57142,0.12372,0.4286,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,8,0,0,14,0,0,8,0,0,1,0,0],[40,74,0.5405,0.53122,0.10247,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,13,0,0,16,0,0,2,0,0,1,0,0],[44,74,0.5946,0.55353,0.15046,0.42857,0.57121,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,13,0,0,15,0,0,1,0,0,1,0,2],[48,74,0.6486,0.52232,0.11633,0.42857,0.57141,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,15,0,0,15,0,0,1,0,0,0,0,1],[52,74,0.7027,0.5982,0.13092,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,21,0,0,2,0,0,3,0,1],[56,74,0.7568,0.56695,0.10402,0.5354,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,8,0,0,18,0,0,5,0,0,1,0,0],[60,74,0.8108,0.58474,0.13997,0.571,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,7,0,0,20,0,0,2,0,0,1,0,2],[64,74,0.8649,0.54908,0.11903,0.42857,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,17,0,0,3,0,0,0,0,1],[68,74,0.9189,0.60714,0.13832,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,17,0,0,5,0,0,3,0,1],[72,74,0.973,0.54905,0.08824,0.4286,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,9,0,0,19,0,0,4,0,0,0,0,0],[74,74,1.0,0.56697,0.19061,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,9,0,0,12,0,0,6,0,0,1,0,2]]}]},{"i":"da1b8fb8e329ae95","q":"Solve in real number the system  $x^3=\\frac{z}{y}-\\frac{2y}{z}, y^3=\\frac{x}{z}-\\frac{2z}{x}, z^3=\\frac{y}{x}-\\frac{2x}{y}$","t":[{"b":3,"e":1.0,"k":"rising","v":0.50893,"x":1.0,"p":[[0,43,0.0,0.72768,0.32803,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,7,0,0,1,0,0,1,0,0,3,0,0,1,0,17],[4,43,0.093,0.83482,0.28596,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,0,3,0,22],[8,43,0.186,0.70982,0.33021,0.28571,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,10,0,0,3,0,0,0,0,0,1,0,0,1,0,17],[12,43,0.2791,0.50893,0.32525,0.28571,0.28571,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,19,0,0,2,0,0,0,0,0,0,0,0,1,0,9],[16,43,0.3721,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,43,0.4651,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[28,43,0.6512,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,43,0.7442,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":1.0,"k":"rising","v":0.60714,"x":1.0,"p":[[0,24,0.0,0.80357,0.27607,0.64286,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,4,0,0,0,0,20],[4,24,0.1667,0.77679,0.3071,0.42857,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,5,0,0,3,0,0,1,0,0,2,0,0,0,0,20],[8,24,0.3333,0.60714,0.33693,0.28571,0.42857,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,15,0,0,2,0,0,1,0,0,1,0,0,0,0,13],[12,24,0.5,0.70089,0.32997,0.28571,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,9,0,0,2,0,0,1,0,0,2,0,0,1,0,16],[16,24,0.6667,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[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":"c258b95f1e317583","q":"Let $n$ be a positive integer. Let $V_{n}$ be the set of all sequences of 0 's and 1's of length $n$. Define $G_{n}$ to be the graph having vertex set $V_{n}$, such that two sequences are adjacent in $G_{n}$ if and only if they differ in either 1 or 2 places. For instance, if $n=3$, the sequences $(1,0,0)$, $(1,1,0)$, and $(1,1,1)$ are mutually adjacent, but $(1,0,0)$ is not adjacent to $(0,1,1)$.\nShow that, if $n+1$ is not a power of 2 , then the chromatic number of $G_{n}$ is at least $n+2$.","t":[{"b":2,"e":0.57143,"k":"flat","v":0.87945,"x":0.95536,"p":[[0,38,0.0,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[4,38,0.1053,0.89284,0.14289,0.85711,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,18],[8,38,0.2105,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[12,38,0.3158,0.91964,0.11259,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,20],[16,38,0.4211,0.88393,0.15746,0.85714,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,10,0,16],[20,38,0.5263,0.90625,0.14555,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,20],[24,38,0.6316,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[28,38,0.7368,0.9241,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[32,38,0.8421,0.9375,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[36,38,0.9474,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24],[38,38,1.0,0.87945,0.15615,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,5,0,18]]},{"b":4,"e":0.85714,"k":"flat","v":0.91071,"x":0.92857,"p":[[0,28,0.0,0.92411,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,20],[4,28,0.1429,0.91071,0.10564,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17],[8,28,0.2857,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19],[12,28,0.4286,0.92857,0.10714,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21],[16,28,0.5714,0.9241,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[20,28,0.7143,0.9241,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,20],[24,28,0.8571,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19],[28,28,1.0,0.9241,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,18]]}]},{"i":"96d56a959f4c91f0","q":"Point  $M$  is middle of side  $AB$  of equilateral triangle  $ABC$ . Points  $D$  and  $E$  lie on segments  $AC$  and  $BC$ , respectively and  $\\angle DME = 60 ^{\\circ}$ . Prove that,  $AD + BE = DE + \\frac{1}{2}AB$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.07143,"x":0.11161,"p":[[0,22,0.0,0.08036,0.17835,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,22,0.1818,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.11161,0.17399,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,22,0.9091,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.04911,"x":0.10706,"p":[[0,20,0.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.10706,0.17495,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,20,0.4,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"925c39c6a5d259db","q":"For a prime number  $p$  and a positive integer  $n$ , denote by  $f(p, n)$  the largest integer  $k$  such that  $p^k \\mid n!$ . Let  $p$  be a given prime number and let  $m$  and  $c$  be given positive integers. Prove that there exist infinitely many positive integers  $n$  such that  $f(p, n) \\equiv c \\pmod m$ .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.79463,"x":0.91964,"p":[[0,39,0.0,0.79463,0.20185,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,10,0,0,6,0,11],[4,39,0.1026,0.87946,0.10779,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,13,0,12],[8,39,0.2051,0.87054,0.08268,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,21,0,7],[12,39,0.3077,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[16,39,0.4103,0.90178,0.13092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,18],[20,39,0.5128,0.89284,0.11298,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,13,0,14],[24,39,0.6154,0.875,0.16269,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,13,0,14],[28,39,0.7179,0.90178,0.10374,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,15],[32,39,0.8205,0.87946,0.10779,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,19,0,10],[36,39,0.9231,0.87054,0.10326,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,9],[39,39,1.0,0.87946,0.10779,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,16,0,11]]},{"b":7,"e":0.85714,"k":"flat","v":0.75891,"x":0.91518,"p":[[0,46,0.0,0.75891,0.2096,0.67857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,9,0,0,6,0,9],[4,46,0.087,0.90179,0.12078,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,16],[8,46,0.1739,0.91071,0.12753,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,18],[12,46,0.2609,0.875,0.12242,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,16,0,11],[16,46,0.3478,0.87944,0.10783,0.85714,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,16,0,11],[20,46,0.4348,0.91071,0.08564,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[24,46,0.5217,0.89732,0.10853,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,14],[28,46,0.6087,0.91517,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,17],[32,46,0.6957,0.85714,0.13363,0.82143,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,13,0,11],[36,46,0.7826,0.91518,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[40,46,0.8696,0.86161,0.10999,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,16,0,9],[44,46,0.9565,0.83929,0.09279,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,24,0,3],[46,46,1.0,0.80357,0.14174,0.85714,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,23,0,2]]}]},{"i":"2c5e1e4fbb3a2f0f","q":"Let  $ K$  be a finite field of  $ p$  elements, where  $ p$  is a prime. For every polynomial\r $ f(x)\\equal{}\\sum_{i\\equal{}0}^na_ix^i$  ( $ \\in K[x]$ )\r\nput\r $ \\overline{f(x)}\\equal{}\\sum_{i\\equal{}0}^n a_ix^{p^i}$ .\r\nProve that for any pair of polynomials  $ f(x),g(x)\\in K[x]$ ,  $ \\overline{f(x)}|\\overline{g(x)}$  if and only if  $ f(x)|g(x)$ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.15179,"x":0.34375,"p":[[0,41,0.0,0.15179,0.20184,0.0,0.07143,0.17857,0.0,0.71429,16,0,0,16,0,8,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[4,41,0.0976,0.32588,0.29065,0.14286,0.14286,0.4286,0.0,1.0,5,3,0,5,0,12,0,0,1,0,0,7,0,0,3,0,0,1,0,0,0,0,3],[8,41,0.1951,0.34375,0.32115,0.0,0.21429,0.57143,0.0,1.0,9,2,0,9,0,7,0,0,1,0,0,4,0,0,5,0,0,2,0,0,2,0,2],[12,41,0.2927,0.23661,0.20079,0.14286,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,14,0,0,3,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[16,41,0.3902,0.16072,0.14617,0.0,0.14286,0.1786,0.0,0.57143,9,0,0,9,0,15,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,41,0.4878,0.23659,0.24381,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,9,0,0,3,0,0,6,0,0,2,0,0,1,0,0,0,0,1],[24,41,0.5854,0.26339,0.29474,0.0,0.14286,0.46431,0.0,1.0,11,1,0,11,0,9,0,0,2,0,0,2,0,0,4,0,0,1,0,0,2,0,1],[28,41,0.6829,0.24106,0.2214,0.0,0.14286,0.42857,0.0,0.57143,11,0,0,11,0,6,0,0,3,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[32,41,0.7805,0.32579,0.23489,0.14214,0.35714,0.57143,0.0,0.71429,7,0,0,7,0,5,0,0,4,0,0,6,0,0,8,0,0,2,0,0,0,0,0],[36,41,0.878,0.29911,0.24578,0.14286,0.28571,0.46431,0.0,1.0,7,1,0,7,0,8,0,0,3,0,0,6,0,0,7,0,0,0,0,0,0,0,1],[40,41,0.9756,0.26338,0.18933,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,8,0,0,7,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[41,41,1.0,0.15625,0.15714,0.0,0.14286,0.28571,0.0,0.42857,13,0,0,13,0,8,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.2857,"k":"rising","v":0.13393,"x":0.37947,"p":[[0,24,0.0,0.13393,0.14258,0.0,0.14286,0.14287,0.0,0.57143,12,0,0,12,0,14,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,24,0.1667,0.37947,0.32065,0.14286,0.28571,0.57143,0.0,1.0,5,4,0,5,0,9,0,0,3,0,0,5,0,0,4,0,0,1,0,0,1,0,4],[8,24,0.3333,0.16964,0.2126,0.0,0.14286,0.28571,0.0,1.0,14,1,1,14,0,7,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[12,24,0.5,0.25892,0.27065,0.0,0.14286,0.42857,0.0,1.0,11,1,0,11,0,8,0,0,1,0,0,5,0,0,4,0,0,2,0,0,0,0,1],[16,24,0.6667,0.26784,0.22515,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,4,0,0,9,0,0,4,0,0,5,0,0,0,0,0,1,0,0],[20,24,0.8333,0.22321,0.18189,0.0,0.21428,0.42857,0.0,0.57143,9,0,0,9,0,7,0,0,7,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[24,24,1.0,0.30357,0.21354,0.14286,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,6,0,0,10,0,0,5,0,0,4,0,0,1,0,0,1,0,0]]}]},{"i":"9cbe901d3cb85522","q":"In a middle school, each class has an odd number of students. Furthermore, each student has a best friend (possibly from a different class). Everyone is the best friend of their best friend. On the city trip, every student goes to Rome or Paris. Prove that the students can be distributed over the two destinations such that\n(i) every student goes on the city trip with their best friend;\n(ii) for every class, the absolute difference between the number of students going to Rome and the number of students going to Paris is equal to 1.","t":[{"b":2,"e":0.2857,"k":"rising","v":0.32357,"x":0.71427,"p":[[0,41,0.0,0.32357,0.33412,0.14286,0.14286,0.42858,0.0,1.0,4,5,0,4,0,16,0,0,3,0,0,2,0,1,0,0,0,0,0,0,1,0,5],[4,41,0.0976,0.71427,0.272,0.67846,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,2,0,0,2,0,0,1,0,0,10,0,0,5,0,9],[8,41,0.1951,0.68303,0.27833,0.57143,0.71429,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,1,0,0,2,0,0,6,0,0,7,0,0,6,0,7],[12,41,0.2927,0.54909,0.30327,0.28571,0.571,0.75,0.0,1.0,1,5,0,1,0,5,0,0,4,0,0,5,0,0,3,0,0,6,0,0,3,0,5],[16,41,0.3902,0.52231,0.35285,0.28571,0.4286,0.89286,0.0,1.0,5,8,0,5,0,2,0,0,3,0,0,8,0,0,3,0,0,1,0,0,2,0,8],[20,41,0.4878,0.62499,0.26183,0.39286,0.71429,0.85704,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,3,0,0,1,0,0,11,0,0,5,0,4],[24,41,0.5854,0.53124,0.36287,0.25,0.4998,0.89286,0.0,1.0,6,8,0,6,0,2,0,0,2,0,0,6,0,0,2,0,0,5,0,0,1,0,8],[28,41,0.6829,0.53124,0.3199,0.14286,0.57121,0.75,0.0,1.0,2,4,0,2,0,7,0,0,1,0,0,5,0,0,2,0,0,7,0,0,4,0,4],[32,41,0.7805,0.45982,0.35126,0.14286,0.35714,0.85714,0.0,1.0,4,5,0,4,0,7,0,0,5,0,0,3,0,0,2,0,0,2,0,0,4,0,5],[36,41,0.878,0.56692,0.35801,0.28571,0.571,1.0,0.0,1.0,5,9,0,5,0,0,0,0,6,0,0,3,0,0,5,0,0,1,0,0,3,0,9],[40,41,0.9756,0.63833,0.27429,0.5354,0.64286,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,3,0,0,2,0,0,8,0,0,4,0,0,7,0,5],[41,41,1.0,0.58485,0.2657,0.42857,0.57121,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,10,0,0,5,0,0,3,0,0,4,0,5]]},{"b":3,"e":0.85714,"k":"rising","v":0.30356,"x":0.85268,"p":[[0,42,0.0,0.30356,0.32877,0.14286,0.14286,0.32143,0.0,1.0,5,5,1,5,0,16,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,5],[4,42,0.0952,0.53123,0.29716,0.39286,0.57143,0.71429,0.0,1.0,3,4,0,3,0,4,0,0,1,0,0,5,0,0,6,0,0,8,0,0,1,0,4],[8,42,0.1905,0.76784,0.26367,0.67846,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,1,1,0,0,4,0,0,1,0,0,6,1,0,3,0,14],[12,42,0.2857,0.76783,0.27142,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,1,0,0,1,0,0,2,0,0,7,0,0,5,0,13],[16,42,0.381,0.79451,0.22291,0.71429,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,13,0,0,4,0,12],[20,42,0.4762,0.77678,0.24468,0.71429,0.85707,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,10,0,0,5,0,12],[24,42,0.5714,0.85268,0.16066,0.71429,0.85714,1.0,0.357,1.0,0,14,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,8,0,0,7,0,14],[28,42,0.6667,0.75893,0.23197,0.64296,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,3,0,0,2,2,0,8,0,0,4,0,11],[32,42,0.7619,0.70758,0.259,0.53572,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,1,3,0,0,1,0,0,8,0,0,8,0,7],[36,42,0.8571,0.75445,0.20278,0.71429,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,11,0,0,6,0,8],[40,42,0.9524,0.70536,0.2765,0.53571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,4,0,0,2,0,0,1,0,0,10,0,0,3,0,10],[42,42,1.0,0.61605,0.31019,0.42857,0.71429,0.85714,0.0,1.0,2,5,0,2,0,4,0,0,1,0,0,3,0,0,4,0,0,6,0,0,7,0,5]]}]},{"i":"74b24f9924265607","q":"Let  $T$  be the set of all triples  $(a,b,c)$  of positive integers for which there exist triangles with side lengths  $a,b,c.$  Express \\[\\sum_{(a,b,c)\\in T}\\frac{2^a}{3^b5^c}\\] as a rational number in lowest terms.","t":[{"b":2,"e":0.57143,"k":"rising","v":0.33036,"x":0.94196,"p":[[0,51,0.0,0.33036,0.40945,0.0,0.0,0.71429,0.0,1.0,17,7,0,17,0,0,0,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,7],[4,51,0.0784,0.52679,0.47439,0.0,0.71429,1.0,0.0,1.0,14,14,0,14,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,14],[8,51,0.1569,0.52232,0.47462,0.0,0.64286,1.0,0.0,1.0,13,15,0,13,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,15],[12,51,0.2353,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[16,51,0.3137,0.71875,0.27545,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,4,0,0,1,0,0,9,0,0,2,0,0,2,0,13],[20,51,0.3922,0.55357,0.21943,0.28571,0.57143,0.57143,0.28571,1.0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,18,0,0,0,0,0,1,0,4],[24,51,0.4706,0.49554,0.1636,0.28571,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,10,0,0,1,0,0,19,0,0,1,0,0,0,0,1],[28,51,0.549,0.49107,0.13803,0.39286,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,8,0,0,5,0,0,16,0,0,3,0,0,0,0,0],[32,51,0.6275,0.51339,0.12807,0.53571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,1,0,0,22,0,0,2,0,0,0,0,0],[36,51,0.7059,0.55356,0.06916,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0],[40,51,0.7843,0.52678,0.11538,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,2,0,0,25,0,0,1,0,0,0,0,0],[44,51,0.8627,0.51786,0.1171,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,6,0,0,1,0,0,24,0,0,1,0,0,0,0,0],[48,51,0.9412,0.50893,0.13333,0.57143,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,4,0,0,0,0,0,26,0,0,0,0,0,0,0,0],[51,51,1.0,0.52679,0.0974,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,4,0,0,2,0,0,26,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.48661,"x":0.99554,"p":[[0,73,0.0,0.48661,0.46956,0.0,0.35716,1.0,0.0,1.0,13,14,0,13,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,14],[4,73,0.0548,0.94196,0.17807,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[8,73,0.1096,0.91964,0.23402,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,27],[12,73,0.1644,0.89286,0.25754,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,25],[16,73,0.2192,0.81696,0.36287,0.96429,1.0,1.0,0.0,1.0,4,24,0,4,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,24],[20,73,0.274,0.84821,0.31731,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[24,73,0.3288,0.92857,0.22016,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,28],[28,73,0.3836,0.88393,0.23538,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,23],[32,73,0.4384,0.88393,0.23266,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,1,0,24],[36,73,0.4932,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[40,73,0.5479,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[44,73,0.6027,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[48,73,0.6575,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,73,0.7123,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,73,0.7671,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,73,0.8219,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,73,0.8767,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[68,73,0.9315,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,73,0.9863,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[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":"55010814d77e250f","q":"21. C1 (IND) Let $n \\geq 1$ be an integer. A path from $(0,0)$ to $(n, n)$ in the $x y$ plane is a chain of consecutive unit moves either to the right (move denoted by $E$ ) or upwards (move denoted by $N$ ), all the moves being made inside the half-plane $x \\geq y$. A step in a path is the occurrence of two consecutive moves of the form $E N$. Show that the number of paths from $(0,0)$ to $(n, n)$ that contain exactly $s$ steps $(n \\geq s \\geq 1)$ is  $$ \\frac{1}{s}\\binom{n-1}{s-1}\\binom{n}{s-1} $$","t":[{"b":0,"e":0.71429,"k":"rising","v":0.49992,"x":0.93304,"p":[[0,19,0.0,0.49992,0.16363,0.39286,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,7,0,0,1,0,0,21,0,0,1,0,0,1,0,0],[4,19,0.2105,0.93304,0.17122,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,26],[8,19,0.4211,0.79909,0.27863,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,3,0,0,2,0,0,4,0,0,1,0,0,2,0,19],[12,19,0.6316,0.76784,0.30879,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,4,0,0,1,0,0,2,0,0,3,0,0,3,0,17],[16,19,0.8421,0.76784,0.28066,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,3,0,0,2,0,0,4,0,0,3,0,0,4,0,15],[19,19,1.0,0.78566,0.21134,0.57143,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,6,0,0,5,0,12]]},{"b":1,"e":1.0,"k":"flat","v":0.56249,"x":0.8616,"p":[[0,19,0.0,0.56249,0.19865,0.57142,0.57143,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,5,0,0,0,0,0,19,0,0,4,0,0,1,0,2],[4,19,0.2105,0.85268,0.28231,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,23],[8,19,0.4211,0.83929,0.28065,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,5,0,20],[12,19,0.6316,0.8616,0.18029,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,7,0,16],[16,19,0.8421,0.72768,0.31816,0.39286,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,6,0,0,1,0,0,0,0,0,5,0,0,4,0,14],[19,19,1.0,0.6339,0.35163,0.28571,0.71429,1.0,0.0,1.0,3,10,0,3,0,2,0,0,5,0,0,1,0,0,3,0,0,3,0,0,5,0,10]]}]},{"i":"9f405912ea054899","q":"Show that if the sum of all positive divisors of an integer $n \\geqslant 1$ is a power of two, then the number of positive divisors of $n$ is a power of two.","t":[{"b":3,"e":1.0,"k":"rising","v":0.79911,"x":1.0,"p":[[0,22,0.0,0.79911,0.25218,0.53572,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,7,0,0,2,0,0,1,0,0,4,0,17],[4,22,0.1818,0.9107,0.18126,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,24],[8,22,0.3636,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,22,0.7273,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[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":5,"e":0.42857,"k":"flat","v":0.64732,"x":0.80802,"p":[[0,23,0.0,0.77677,0.25491,0.5354,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,3,0,0,4,0,15],[4,23,0.1739,0.64732,0.24739,0.42857,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,10,0,0,5,0,0,1,0,0,7,0,6],[8,23,0.3478,0.76339,0.23585,0.4286,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,4,0,0,6,0,12],[12,23,0.5217,0.80802,0.20396,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,9,0,12],[16,23,0.6957,0.80357,0.21354,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,2,0,0,8,0,13],[20,23,0.8696,0.71875,0.20039,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,8,0,0,5,0,7],[23,23,1.0,0.65177,0.19865,0.42857,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,12,0,0,1,0,5]]}]},{"i":"7a7914e52a376325","q":"Let  $BCDK$  be a convex quadrilateral such that  $BC=BK$  and  $DC=DK$ .  $A$  and  $E$  are points such that  $ABCDE$  is a convex pentagon such that  $AB=BC$  and  $DE=DC$  and  $K$  lies in the interior of the pentagon  $ABCDE$ . If  $\\angle ABC=120^{\\circ}$  and  $\\angle CDE=60^{\\circ}$  and  $BD=2$  then determine area of the pentagon  $ABCDE$ .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.07143,"x":0.24107,"p":[[0,195,0.0,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.42857,20,0,11,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,195,0.0205,0.08929,0.09942,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,195,0.041,0.08473,0.09348,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,195,0.0615,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,195,0.0821,0.08036,0.10677,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,195,0.1026,0.08928,0.10564,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,195,0.1231,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,195,0.1436,0.1875,0.1448,0.14286,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,13,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[32,195,0.1641,0.24107,0.13092,0.14286,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,8,0,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[36,195,0.1846,0.15179,0.11259,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,195,0.2051,0.15179,0.11258,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,195,0.2256,0.18751,0.1448,0.0,0.2857,0.28571,0.0,0.4286,11,0,0,11,0,2,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,195,0.2462,0.19196,0.12682,0.10714,0.2857,0.28571,0.0,0.42857,8,0,0,8,0,6,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,195,0.2667,0.15169,0.1234,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,11,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,195,0.2872,0.12054,0.12428,0.0,0.14286,0.28571,0.0,0.28571,15,0,0,15,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,195,0.3077,0.15179,0.12846,0.0,0.14286,0.28571,0.0,0.4286,11,0,0,11,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,195,0.3282,0.125,0.12752,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,9,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,195,0.3487,0.16518,0.13883,0.0,0.14286,0.2857,0.0,0.42857,11,0,0,11,0,7,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,195,0.3692,0.1429,0.12382,0.0,0.14286,0.17857,0.0,0.43,10,0,0,10,0,14,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,195,0.3897,0.16964,0.12595,0.0,0.21428,0.28571,0.0,0.28571,10,0,0,10,0,6,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,195,0.4103,0.16071,0.12242,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,195,0.4308,0.16963,0.14028,0.0,0.14286,0.28571,0.0,0.571,10,0,0,10,0,8,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,195,0.4513,0.1875,0.1448,0.0,0.2857,0.28571,0.0,0.4286,10,0,0,10,0,5,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[92,195,0.4718,0.18303,0.12492,0.10714,0.2143,0.28571,0.0,0.42857,8,0,0,8,0,8,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,195,0.4923,0.18303,0.14826,0.0,0.21428,0.28571,0.0,0.57143,10,0,0,10,0,6,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[100,195,0.5128,0.15625,0.13534,0.0,0.14286,0.28571,0.0,0.4286,11,0,0,11,0,9,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[104,195,0.5333,0.19195,0.14552,0.0,0.28571,0.28571,0.0,0.571,9,0,0,9,0,6,0,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[108,195,0.5538,0.17411,0.13236,0.0,0.21428,0.28571,0.0,0.42857,10,0,0,10,0,6,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[112,195,0.5744,0.16071,0.13716,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,8,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[116,195,0.5949,0.17857,0.12877,0.0,0.2143,0.28571,0.0,0.42857,9,0,0,9,0,7,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[120,195,0.6154,0.12946,0.13054,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,8,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[124,195,0.6359,0.19643,0.13716,0.0,0.2857,0.28571,0.0,0.4286,9,0,0,9,0,4,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[128,195,0.6564,0.15178,0.13803,0.0,0.14286,0.2857,0.0,0.4286,12,0,0,12,0,8,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[132,195,0.6769,0.15625,0.1488,0.0,0.14286,0.28571,0.0,0.4286,13,0,0,13,0,6,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[136,195,0.6974,0.11607,0.12595,0.0,0.07143,0.28571,0.0,0.28571,16,0,0,16,0,6,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,195,0.7179,0.1516,0.12341,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,11,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[144,195,0.7385,0.11607,0.12078,0.0,0.14286,0.14287,0.0,0.42857,14,0,0,14,0,11,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[148,195,0.759,0.19196,0.12169,0.14286,0.2857,0.28571,0.0,0.4286,7,0,0,7,0,8,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[152,195,0.7795,0.11152,0.13234,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,6,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[156,195,0.8,0.16509,0.12429,0.105,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,13,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[160,195,0.8205,0.16518,0.12931,0.0,0.14286,0.28571,0.0,0.4286,10,0,0,10,0,8,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[164,195,0.841,0.16071,0.13243,0.0,0.2143,0.28571,0.0,0.28571,12,0,0,12,0,4,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[168,195,0.8615,0.13393,0.13333,0.0,0.14286,0.28571,0.0,0.28571,15,0,0,15,0,4,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[172,195,0.8821,0.12054,0.11904,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,9,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[176,195,0.9026,0.1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$ABC$ be a triangle with acute angles. The altitude from $A$ intersects the circumcircle of triangle $ABC$ at a point $D$. The altitude from $B$ intersects the circumcircle of triangle $ABC$ at a point $E$. The line $(ED)$ intersects the sides $[AC]$ and $[BC]$ at points $P$ and $Q$ respectively. Show that the points $P, Q, B$ and $A$ are concyclic.","t":[{"b":2,"e":0.0,"k":"falling","v":0.16964,"x":0.59375,"p":[[0,74,0.0,0.52219,0.24133,0.28571,0.57121,0.71429,0.14,1.0,0,2,0,0,0,5,0,0,4,0,0,4,0,0,8,0,0,8,0,0,1,0,2],[4,74,0.0541,0.55356,0.39568,0.14286,0.50001,1.0,0.0,1.0,2,12,0,2,0,10,0,0,2,0,0,2,0,0,1,0,0,2,0,0,1,0,12],[8,74,0.1081,0.59375,0.40265,0.14286,0.71429,1.0,0.0,1.0,1,14,0,1,0,11,0,0,2,0,0,0,0,0,1,0,0,2,0,0,1,0,14],[12,74,0.1622,0.45524,0.33403,0.14286,0.49979,0.71429,0.0,1.0,5,5,0,5,0,5,0,0,5,0,0,1,0,0,6,0,0,5,0,0,0,0,5],[16,74,0.2162,0.44196,0.37519,0.14286,0.2857,0.75,0.0,1.0,4,7,0,4,0,11,0,0,3,0,0,1,0,0,1,0,0,4,0,0,1,0,7],[20,74,0.2703,0.48214,0.36202,0.14286,0.42857,0.75,0.0,1.0,4,7,0,4,0,7,0,0,4,0,0,3,0,0,1,0,0,5,0,0,1,0,7],[24,74,0.3243,0.34375,0.26692,0.14286,0.14288,0.57143,0.14286,1.0,0,1,0,0,0,18,0,0,3,0,0,1,0,0,3,0,0,5,0,0,1,0,1],[28,74,0.3784,0.40625,0.37983,0.14286,0.14286,0.71429,0.0,1.0,5,7,0,5,0,13,0,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,7],[32,74,0.4324,0.50882,0.37794,0.14286,0.49979,0.89286,0.0,1.0,3,8,0,3,0,10,0,0,1,0,0,2,0,0,3,0,0,2,0,0,3,0,8],[36,74,0.4865,0.4464,0.33454,0.14286,0.49979,0.71429,0.0,1.0,3,4,0,3,0,11,0,0,1,0,0,1,0,0,6,0,0,4,0,0,2,0,4],[40,74,0.5405,0.37036,0.36757,0.14214,0.14286,0.71429,0.0,1.0,6,6,0,6,0,13,0,0,1,0,0,1,0,0,2,0,0,3,0,0,0,0,6],[44,74,0.5946,0.41964,0.3008,0.14286,0.35714,0.71429,0.0,1.0,3,2,0,3,0,9,0,0,4,0,0,2,0,0,5,0,0,5,0,0,2,0,2],[48,74,0.6486,0.3125,0.32818,0.14286,0.14286,0.42857,0.0,1.0,5,4,0,5,0,15,0,0,3,0,0,3,0,0,0,0,0,0,0,0,2,0,4],[52,74,0.7027,0.375,0.28291,0.14286,0.28571,0.57143,0.0,1.0,4,2,0,4,0,8,0,0,6,0,0,3,0,0,4,0,0,5,0,0,0,0,2],[56,74,0.7568,0.41963,0.35343,0.14286,0.28571,0.71429,0.0,1.0,5,6,0,5,0,8,0,0,5,0,0,2,0,0,2,0,0,4,0,0,0,0,6],[60,74,0.8108,0.35713,0.27198,0.14286,0.28571,0.57141,0.0,1.0,6,1,0,6,0,4,0,0,9,0,0,3,0,0,4,0,0,4,0,0,1,0,1],[64,74,0.8649,0.26786,0.24936,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,15,0,0,6,0,0,2,0,0,2,0,0,0,0,0,2,0,1],[68,74,0.9189,0.37499,0.32683,0.14286,0.28571,0.57143,0.0,1.0,6,4,0,6,0,9,0,0,2,0,0,4,0,0,4,0,0,3,0,0,0,0,4],[72,74,0.973,0.23205,0.21946,0.14214,0.14286,0.2857,0.0,0.857,7,0,0,7,0,12,0,0,7,0,0,2,0,0,1,0,0,2,0,0,1,0,0],[74,74,1.0,0.16964,0.16536,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,5,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.33928,"x":0.5758,"p":[[0,69,0.0,0.49087,0.27866,0.24999,0.571,0.71429,0.0,1.0,1,3,0,1,0,7,0,0,3,0,0,4,0,0,7,0,0,6,0,0,1,0,3],[4,69,0.058,0.45079,0.34097,0.14286,0.28574,0.71429,0.14,1.0,0,6,0,0,0,16,0,0,0,0,0,2,0,0,3,0,0,5,0,0,0,0,6],[8,69,0.1159,0.53572,0.41802,0.14286,0.42859,1.0,0.0,1.0,5,13,0,5,0,7,0,0,3,0,0,2,0,0,0,0,0,2,0,0,0,0,13],[12,69,0.1739,0.42857,0.38465,0.14286,0.21429,0.78571,0.0,1.0,5,8,0,5,0,11,0,0,2,0,0,2,0,0,1,0,0,3,0,0,0,0,8],[16,69,0.2319,0.5758,0.36341,0.14289,0.64286,1.0,0.0,1.0,2,10,0,2,0,7,0,0,3,0,0,1,0,0,3,0,0,5,0,0,1,0,10],[20,69,0.2899,0.50884,0.3642,0.14286,0.42859,1.0,0.0,1.0,3,9,0,3,0,7,0,0,4,0,0,3,0,0,3,0,0,3,0,0,0,0,9],[24,69,0.3478,0.5,0.35535,0.14286,0.42859,0.85714,0.0,1.0,3,7,0,3,0,8,0,0,2,0,0,4,0,0,3,0,0,3,0,0,2,0,7],[28,69,0.4058,0.49098,0.36768,0.14286,0.49999,0.75,0.0,1.0,4,7,0,4,0,8,0,0,3,0,0,1,0,0,2,0,0,6,0,0,1,0,7],[32,69,0.4638,0.4598,0.38254,0.14286,0.2857,0.85714,0.0,1.0,5,7,0,5,0,8,0,0,5,0,0,0,0,0,2,0,0,2,0,0,3,0,7],[36,69,0.5217,0.44186,0.33579,0.14286,0.42857,0.71429,0.0,1.0,5,5,0,5,0,6,0,0,3,0,0,5,0,0,4,0,0,3,0,0,1,0,5],[40,69,0.5797,0.33928,0.3067,0.14286,0.14288,0.4642,0.0,1.0,5,3,0,5,0,12,0,0,2,0,0,5,0,0,1,0,0,4,0,0,0,0,3],[44,69,0.6377,0.558,0.33381,0.28571,0.57121,0.85714,0.0,1.0,3,7,0,3,0,4,0,0,3,0,0,3,0,0,5,0,0,5,0,0,2,0,7],[48,69,0.6957,0.50893,0.34799,0.2857,0.42857,1.0,0.0,1.0,2,9,0,2,0,4,0,0,9,0,0,6,0,0,0,0,0,1,0,0,1,0,9],[52,69,0.7536,0.47317,0.28444,0.14286,0.49979,0.71429,0.0,1.0,2,2,0,2,0,7,0,0,3,0,0,4,0,0,4,0,0,9,0,0,1,0,2],[56,69,0.8116,0.36604,0.19538,0.2857,0.28571,0.571,0.0,0.71429,2,0,0,2,0,5,0,0,10,0,0,6,0,0,6,0,0,3,0,0,0,0,0],[60,69,0.8696,0.43744,0.23937,0.2857,0.42859,0.57111,0.0,1.0,2,2,0,2,0,3,0,0,8,0,0,6,0,0,8,0,0,3,0,0,0,0,2],[64,69,0.9275,0.45981,0.2666,0.24999,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,8,0,0,5,0,0,6,0,0,4,0,0,6,0,0,0,0,3],[68,69,0.9855,0.51337,0.19515,0.39286,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,6,0,0,9,0,0,8,0,0,0,0,1],[69,69,1.0,0.40625,0.19597,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,10,0,0,6,0,0,6,0,0,5,0,0,0,0,0]]}]},{"i":"7cf3c4415d4533ba","q":"Let $A B C D$ be a parallelogram. The circle with diameter $A C$ intersects the line $B D$ at points $P$ and $Q$. The perpendicular to the line $A C$ passing through the point $C$ intersects the lines $A B$ and $A D$ at points $X$ and $Y$, respectively. Prove that the points $P, Q, X$ and $Y$ lie on the same circle.","t":[{"b":3,"e":0.14,"k":"flat","v":0.08465,"x":0.16062,"p":[[0,59,0.0,0.13367,0.13332,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,20,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,59,0.0678,0.14277,0.07143,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,59,0.1356,0.12929,0.07454,0.14214,0.14286,0.14286,0.0,0.2857,6,0,0,6,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,59,0.2034,0.12036,0.06292,0.14214,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,59,0.2712,0.12946,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,59,0.339,0.14722,0.12099,0.14214,0.14286,0.1429,0.0,0.571,7,0,0,7,0,20,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,59,0.4068,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,59,0.4746,0.1517,0.12847,0.105,0.14286,0.1429,0.0,0.57143,8,0,0,8,0,17,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,59,0.5424,0.12938,0.10925,0.0,0.14286,0.1429,0.0,0.42857,10,0,0,10,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,59,0.6102,0.15616,0.10327,0.14214,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.14278,0.10716,0.105,0.14286,0.14287,0.0,0.42857,8,0,0,8,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.13393,0.13803,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,17,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,59,0.8136,0.16062,0.11153,0.14286,0.14286,0.14287,0.0,0.57143,5,0,0,5,0,20,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,59,0.8814,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.08465,0.0786,0.0,0.14143,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[59,59,1.0,0.12929,0.18335,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":4,"e":0.14286,"k":"flat","v":0.08465,"x":0.17402,"p":[[0,78,0.0,0.17402,0.13238,0.14286,0.14286,0.1786,0.0,0.57143,5,0,0,5,0,19,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,78,0.0513,0.12036,0.07235,0.14,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,78,0.1026,0.12036,0.08068,0.105,0.14286,0.14287,0.0,0.28571,8,0,0,8,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,78,0.1538,0.14268,0.13363,0.105,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,19,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,78,0.2051,0.13357,0.06116,0.14214,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,78,0.2564,0.1383,0.10998,0.14214,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,21,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,78,0.3077,0.15607,0.13056,0.14214,0.14286,0.1429,0.0,0.71429,6,0,0,6,0,20,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,78,0.359,0.13366,0.13332,0.0,0.14286,0.14286,0.0,0.71429,9,0,0,9,0,19,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[32,78,0.4103,0.11599,0.06618,0.14214,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,78,0.4615,0.15608,0.14882,0.14,0.14286,0.1429,0.0,0.71429,7,0,0,7,0,20,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[40,78,0.5128,0.125,0.07784,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,78,0.5641,0.12947,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,78,0.6154,0.09804,0.07513,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,78,0.6667,0.12045,0.07237,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,78,0.7179,0.12464,0.07776,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,78,0.7692,0.16054,0.06921,0.14286,0.14286,0.14287,0.0,0.28571,2,0,0,2,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,78,0.8205,0.11161,0.07771,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,78,0.8718,0.10697,0.07134,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,78,0.9231,0.11152,0.09265,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,78,0.9744,0.08465,0.07002,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[78,78,1.0,0.09357,0.06773,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2de6b7fe455ef8e4","q":"The audience arranges  $n$  coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between  $1$  and  $n$  inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.\n[list]**(a)** Prove that if this is possible for some  $n$ , then it is also possible for  $2n$ .**(b)** Determine all  $n$  for which this is possible.[/list]","t":[{"b":1,"e":0.57143,"k":"flat","v":0.375,"x":0.5134,"p":[[0,32,0.0,0.375,0.13243,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,7,0,0,15,0,0,5,0,0,0,0,0,0,0,0],[4,32,0.125,0.4821,0.16267,0.42857,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,6,0,0,9,0,0,11,0,0,5,0,0,0,0,0],[8,32,0.25,0.473,0.17322,0.42857,0.4286,0.57143,0.14,0.71429,0,0,0,0,0,4,0,0,2,0,0,12,0,0,8,0,0,6,0,0,0,0,0],[12,32,0.375,0.5134,0.14664,0.42857,0.4286,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,14,0,0,5,0,0,9,0,0,0,0,0],[16,32,0.5,0.50442,0.16744,0.42857,0.57121,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,5,0,0,6,0,0,12,0,0,7,0,0,0,0,0],[20,32,0.625,0.47758,0.15833,0.42857,0.4286,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,15,0,0,8,0,0,5,0,0,0,0,0],[24,32,0.75,0.5,0.14725,0.42857,0.57141,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,8,0,0,11,0,0,6,0,0,0,0,0],[28,32,0.875,0.44633,0.16671,0.28571,0.42857,0.57143,0.14,0.71429,0,0,0,0,0,2,0,0,8,0,0,12,0,0,4,0,0,6,0,0,0,0,0],[32,32,1.0,0.41069,0.09275,0.39286,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,21,0,0,2,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.33035,"x":0.57587,"p":[[0,42,0.0,0.33035,0.14912,0.2857,0.42857,0.42857,0.0,0.57143,3,0,2,3,0,3,0,0,9,0,0,15,0,0,2,0,0,0,0,0,0,0,0],[4,42,0.0952,0.52225,0.16211,0.42857,0.5705,0.71429,0.1429,0.71429,0,0,0,0,0,1,0,0,4,0,0,10,0,0,7,0,0,10,0,0,0,0,0],[8,42,0.1905,0.51781,0.1587,0.42857,0.571,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,10,0,0,8,0,0,9,0,0,0,0,0],[12,42,0.2857,0.54459,0.14913,0.42857,0.571,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,9,0,0,8,0,0,11,0,0,0,0,0],[16,42,0.381,0.55802,0.17627,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,7,0,0,6,0,0,13,0,0,1,0,0],[20,42,0.4762,0.50444,0.17851,0.28571,0.571,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,6,0,0,7,0,0,10,0,0,0,0,0],[24,42,0.5714,0.50449,0.16743,0.42857,0.5005,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,9,0,0,7,0,0,9,0,0,0,0,0],[28,42,0.6667,0.49553,0.16745,0.42857,0.4286,0.71429,0.1429,0.71429,0,0,0,0,0,1,0,0,6,0,0,11,0,0,5,0,0,9,0,0,0,0,0],[32,42,0.7619,0.4732,0.16535,0.42857,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,4,0,0,10,0,0,10,0,0,5,0,0,0,0,0],[36,42,0.8571,0.56242,0.16341,0.42859,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,9,0,0,9,0,0,11,0,0,1,0,0],[40,42,0.9524,0.46651,0.1694,0.33918,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,5,0,1,8,0,0,10,0,0,5,0,0,0,0,0],[42,42,1.0,0.57587,0.14499,0.4286,0.57143,0.71429,0.14286,0.857,0,0,0,0,0,1,0,0,0,0,0,9,0,0,10,0,0,11,0,0,1,0,0]]}]},{"i":"7d1dae5bef6aaaa2","q":"Let  $ ABC$  be an acute-angled triangle such that there exist points  $ D,E,F$  on side  $ BC,CA,AB$ , respectively such that the inradii of triangle  $ AEF,BDF,CDE$  are all equal to  $ r_0$ . If the inradii of triangle  $ DEF$  and  $ ABC$  are  $ r$  and  $ R$ , respectively, prove that \\[ r\\plus{}r_0\\equal{}R.\\]\r\n*Soewono, Bandung*","t":[{"b":6,"e":0.0,"k":"falling","v":0.07589,"x":0.39284,"p":[[0,112,0.0,0.39284,0.21128,0.28571,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,17,0,0,2,0,0,4,0,0,5,0,0,1,0,0],[4,112,0.0357,0.31249,0.26106,0.14286,0.28571,0.42858,0.0,1.0,7,1,0,7,0,4,0,0,11,0,0,4,0,0,1,0,0,3,0,0,1,0,1],[8,112,0.0714,0.3258,0.24291,0.14286,0.28571,0.42857,0.0,1.0,6,1,0,6,0,4,0,0,9,0,0,7,0,0,2,0,0,3,0,0,0,0,1],[12,112,0.1071,0.28124,0.26117,0.0,0.21428,0.42858,0.0,0.85714,10,0,0,10,0,6,0,0,3,0,0,6,0,0,3,0,0,3,0,0,1,0,0],[16,112,0.1429,0.34821,0.32721,0.0,0.2857,0.57143,0.0,1.0,9,3,0,9,0,5,0,0,4,0,0,5,0,0,3,0,0,1,0,0,2,0,3],[20,112,0.1786,0.24554,0.2321,0.0,0.14286,0.42857,0.0,1.0,9,1,0,9,0,8,0,0,4,0,0,9,0,0,0,0,0,1,0,0,0,0,1],[24,112,0.2143,0.23214,0.23623,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,9,0,0,7,0,0,4,0,0,0,0,0,2,0,0,0,0,1],[28,112,0.25,0.28124,0.26601,0.0,0.2857,0.42857,0.0,1.0,10,1,0,10,0,4,0,0,8,0,0,3,0,0,3,0,0,3,0,0,0,0,1],[32,112,0.2857,0.29911,0.25595,0.0,0.28571,0.42857,0.0,1.0,9,1,0,9,0,4,0,0,5,0,0,8,0,0,3,0,0,2,0,0,0,0,1],[36,112,0.3214,0.21875,0.22583,0.0,0.14288,0.28571,0.0,0.71429,12,0,0,12,0,5,0,0,8,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[40,112,0.3571,0.21429,0.22588,0.0,0.14286,0.32143,0.0,0.71429,13,0,0,13,0,5,0,0,6,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[44,112,0.3929,0.23661,0.25407,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,4,0,0,6,0,0,4,0,0,0,0,0,5,0,0,0,0,0],[48,112,0.4286,0.25893,0.23538,0.0,0.2857,0.42857,0.0,0.85714,9,0,0,9,0,6,0,0,8,0,0,4,0,0,2,0,0,2,0,0,1,0,0],[52,112,0.4643,0.20982,0.24218,0.0,0.14286,0.28571,0.0,0.85714,14,0,0,14,0,4,0,0,7,0,0,3,0,0,1,0,0,2,0,0,1,0,0],[56,112,0.5,0.21874,0.25995,0.0,0.07143,0.32143,0.0,0.85714,16,0,0,16,0,1,0,0,7,0,0,2,0,0,3,0,0,2,0,0,1,0,0],[60,112,0.5357,0.22321,0.23402,0.0,0.21428,0.32143,0.0,1.0,12,1,0,12,0,4,0,0,8,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[64,112,0.5714,0.27232,0.26812,0.0,0.2857,0.42857,0.0,1.0,11,1,0,11,0,3,0,0,9,0,0,2,0,0,3,0,0,3,0,0,0,0,1],[68,112,0.6071,0.15177,0.19538,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,7,0,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[72,112,0.6429,0.22321,0.2111,0.0,0.2857,0.32143,0.0,0.71429,12,0,0,12,0,3,0,0,9,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[76,112,0.6786,0.28125,0.19719,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,4,0,0,13,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[80,112,0.7143,0.24098,0.26354,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,5,0,0,5,0,0,2,0,0,2,0,0,5,0,0,0,0,0],[84,112,0.75,0.1875,0.24856,0.0,0.07143,0.28571,0.0,0.85714,16,0,0,16,0,5,0,0,5,0,0,1,0,0,2,0,0,2,0,0,1,0,0],[88,112,0.7857,0.31695,0.28286,0.0,0.28571,0.42857,0.0,0.85714,10,0,0,10,0,1,0,0,9,0,0,5,0,0,2,0,0,1,0,0,4,0,0],[92,112,0.8214,0.21875,0.25501,0.0,0.14286,0.42857,0.0,1.0,14,1,0,14,0,4,0,0,5,0,0,6,0,0,0,0,0,2,0,0,0,0,1],[96,112,0.8571,0.21418,0.25506,0.0,0.07,0.42857,0.0,0.71429,16,0,0,16,0,3,0,0,3,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[100,112,0.8929,0.27231,0.25343,0.0,0.28571,0.42857,0.0,0.85714,12,0,0,12,0,1,0,0,7,0,0,6,0,0,3,0,0,2,0,0,1,0,0],[104,112,0.9286,0.15625,0.19351,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,3,0,0,6,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[108,112,0.9643,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[112,112,1.0,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.14286,"x":0.50444,"p":[[0,89,0.0,0.37053,0.24448,0.28571,0.28571,0.60714,0.0,0.85714,5,0,0,5,0,0,0,0,16,0,0,2,0,0,1,0,0,7,0,0,1,0,0],[4,89,0.0449,0.31697,0.2618,0.10714,0.28571,0.42857,0.0,0.85714,8,0,0,8,0,3,0,0,8,0,0,8,0,0,0,0,0,2,0,0,3,0,0],[8,89,0.0899,0.43304,0.30823,0.28571,0.35714,0.71429,0.0,1.0,5,4,0,5,0,1,0,0,10,0,0,6,0,0,1,0,0,4,0,0,1,0,4],[12,89,0.1348,0.42411,0.30196,0.24999,0.42857,0.71429,0.0,1.0,6,2,0,6,0,2,0,0,7,0,0,4,0,0,4,0,0,5,0,0,2,0,2],[16,89,0.1798,0.36161,0.27195,0.14289,0.28571,0.46431,0.0,1.0,4,2,0,4,0,5,0,0,12,0,0,3,0,0,2,0,0,3,0,0,1,0,2],[20,89,0.2247,0.37499,0.27606,0.14286,0.28571,0.60714,0.0,1.0,4,1,0,4,0,7,0,0,8,0,0,2,0,0,3,0,0,6,0,0,1,0,1],[24,89,0.2697,0.34374,0.24184,0.14286,0.28571,0.42857,0.0,1.0,3,2,0,3,0,7,0,0,9,0,0,7,0,0,3,0,0,1,0,0,0,0,2],[28,89,0.3146,0.39732,0.33832,0.14286,0.28571,0.71429,0.0,1.0,7,4,0,7,0,6,0,0,4,0,0,3,0,0,2,0,0,6,0,0,0,0,4],[32,89,0.3596,0.43303,0.32632,0.14286,0.42857,0.71429,0.0,1.0,4,5,0,4,0,7,0,0,4,0,0,5,0,0,3,0,0,4,0,0,0,0,5],[36,89,0.4045,0.43302,0.289,0.14286,0.28571,0.71429,0.0,1.0,2,2,0,2,0,7,0,0,8,0,0,2,0,0,3,0,0,6,0,0,2,0,2],[40,89,0.4494,0.50444,0.29662,0.2857,0.571,0.71429,0.0,1.0,2,3,0,2,0,5,0,0,5,0,0,3,0,0,5,0,0,6,0,0,3,0,3],[44,89,0.4944,0.46874,0.31589,0.28571,0.28571,0.71429,0.0,1.0,2,6,0,2,0,4,0,0,11,0,0,3,0,0,2,0,0,4,0,0,0,0,6],[48,89,0.5393,0.35268,0.26483,0.14286,0.28571,0.57143,0.0,1.0,2,2,0,2,0,12,0,0,5,0,0,4,0,0,4,0,0,3,0,0,0,0,2],[52,89,0.5843,0.29909,0.21534,0.14286,0.28571,0.4286,0.0,0.71429,5,0,0,5,0,7,0,0,10,0,0,3,0,0,4,0,0,3,0,0,0,0,0],[56,89,0.6292,0.25447,0.15865,0.14286,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,9,0,0,12,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[60,89,0.6742,0.25,0.20516,0.14286,0.14286,0.28571,0.0,1.0,3,1,0,3,0,16,0,0,6,0,0,3,0,0,3,0,0,0,0,0,0,0,1],[64,89,0.7191,0.17411,0.11143,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,22,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[68,89,0.764,0.23214,0.15872,0.14286,0.14286,0.42857,0.0,0.57143,4,0,0,4,0,15,0,0,3,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[72,89,0.809,0.19197,0.11633,0.14286,0.14286,0.2857,0.0,0.42857,3,0,0,3,0,19,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[76,89,0.8539,0.16518,0.11355,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,23,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.17857,0.11845,0.14286,0.14286,0.17857,0.0,0.42857,4,0,0,4,0,20,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.22768,0.11769,0.14286,0.21428,0.28571,0.0,0.42857,2,0,0,2,0,14,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[88,89,0.9888,0.14286,0.11294,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,23,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[89,89,1.0,0.16072,0.11152,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,23,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"66d681e7d4264c40","q":"The set  $A{}$  of positive integers satisfies the following conditions:\n[list=1]\n[*]If a positive integer  $n{}$  belongs to  $A{}$ , then  $2n$  also belongs to  $A{}$ ;\n[*]For any positive integer  $n{}$  there exists an element of  $A{}$  divisible by  $n{}$ ;\n[*]There exist finite subsets of  $A{}$  with arbitrarily large sums of reciprocals of elements. \n[/list]Prove that for any positive rational number  $r{}$  there exists a finite subset  $B\\subset A$  such that \\[\\sum_{x\\in B}\\frac{1}{x}=r.\\]","t":[{"b":3,"e":0.0,"k":"flat","v":0.06697,"x":0.29463,"p":[[0,66,0.0,0.06697,0.07974,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.28571,0.21724,0.14286,0.2857,0.42857,0.0,0.71429,5,0,0,5,0,10,0,0,6,0,0,5,0,0,3,0,0,3,0,0,0,0,0],[8,66,0.1212,0.29463,0.24467,0.14286,0.21428,0.4286,0.0,0.85714,6,0,0,6,0,10,0,0,4,0,0,5,0,0,3,0,0,3,0,0,1,0,0],[12,66,0.1818,0.22767,0.24706,0.0,0.14286,0.28571,0.0,0.85714,10,0,0,10,0,10,0,0,5,0,0,2,0,0,2,0,0,1,0,0,2,0,0],[16,66,0.2424,0.27009,0.24597,0.14286,0.14286,0.44643,0.0,0.71429,7,0,0,7,0,11,0,0,5,0,0,1,0,1,2,0,0,5,0,0,0,0,0],[20,66,0.303,0.19186,0.14555,0.14286,0.14286,0.1786,0.0,0.57143,4,0,0,4,0,20,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[24,66,0.3636,0.24999,0.21127,0.14286,0.2143,0.32143,0.0,0.71429,7,0,0,7,0,9,0,0,8,0,0,4,0,0,1,0,0,3,0,0,0,0,0],[28,66,0.4242,0.2009,0.19515,0.0,0.14288,0.28571,0.0,0.71429,10,0,0,10,0,10,0,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[32,66,0.4848,0.19197,0.19759,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,12,0,0,3,0,0,5,0,0,0,0,0,2,0,0,0,0,0],[36,66,0.5455,0.17858,0.18558,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,11,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[40,66,0.6061,0.2365,0.26634,0.0,0.14286,0.32143,0.0,0.85714,10,0,0,10,0,11,0,0,3,0,0,3,0,0,1,0,0,1,0,0,3,0,0],[44,66,0.6667,0.15178,0.14694,0.0,0.14286,0.2857,0.0,0.571,11,0,0,11,0,12,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[48,66,0.7273,0.19642,0.20121,0.0,0.14286,0.2857,0.0,0.71429,9,0,0,9,0,13,0,0,5,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[52,66,0.7879,0.20536,0.2257,0.0,0.14286,0.32143,0.0,0.71429,11,0,0,11,0,11,0,0,2,0,0,4,0,0,1,0,0,3,0,0,0,0,0],[56,66,0.8485,0.12938,0.19678,0.0,0.0,0.14287,0.0,0.71429,17,0,0,17,0,9,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[60,66,0.9091,0.12929,0.15713,0.0,0.14143,0.14286,0.0,0.57143,14,0,0,14,0,12,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[64,66,0.9697,0.14732,0.19059,0.0,0.14286,0.14286,0.0,0.857,13,0,0,13,0,13,0,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[66,66,1.0,0.10268,0.11425,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,12,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.03572,"x":0.31253,"p":[[0,63,0.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,63,0.0635,0.26116,0.25915,0.0,0.21428,0.42857,0.0,0.85714,11,0,0,11,0,5,0,0,5,0,1,5,0,0,0,0,0,4,0,0,1,0,0],[8,63,0.127,0.23214,0.20746,0.10714,0.14288,0.42857,0.0,0.71429,8,0,0,8,0,11,0,0,4,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[12,63,0.1905,0.25447,0.24152,0.10714,0.14288,0.42857,0.0,0.71429,8,0,0,8,0,11,0,0,3,0,0,5,0,0,0,0,0,5,0,0,0,0,0],[16,63,0.254,0.20534,0.20802,0.0,0.14286,0.32143,0.0,0.857,11,0,0,11,0,8,0,0,5,0,0,6,0,0,1,0,0,0,0,0,1,0,0],[20,63,0.3175,0.2723,0.26328,0.0,0.14288,0.57111,0.0,0.71429,9,0,0,9,0,11,0,0,0,0,0,2,0,0,6,0,0,4,0,0,0,0,0],[24,63,0.381,0.23294,0.25816,0.0,0.14286,0.32143,0.0,0.85714,11,0,0,11,0,9,0,0,4,0,0,2,0,0,2,1,0,1,0,0,2,0,0],[28,63,0.4444,0.30802,0.26988,0.14286,0.14288,0.57143,0.0,0.85714,6,0,0,6,0,12,0,0,2,0,0,2,0,0,4,0,0,5,0,0,1,0,0],[32,63,0.5079,0.24999,0.20822,0.14286,0.14295,0.42857,0.0,0.85714,6,0,0,6,0,11,0,0,6,0,0,6,0,0,1,0,0,1,0,0,1,0,0],[36,63,0.5714,0.27899,0.23304,0.14286,0.14286,0.46418,0.0,0.71429,6,0,0,6,0,11,0,1,3,0,0,3,0,0,5,0,0,3,0,0,0,0,0],[40,63,0.6349,0.2589,0.23535,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,11,0,0,8,0,0,2,0,0,2,0,0,2,0,0,0,0,1],[44,63,0.6984,0.21875,0.17851,0.10714,0.14288,0.28571,0.0,0.57143,8,0,0,8,0,9,0,0,8,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[48,63,0.7619,0.31253,0.25112,0.14286,0.2857,0.42893,0.0,1.0,5,1,0,5,0,9,0,0,6,0,0,5,0,0,4,0,0,1,0,0,1,0,1],[52,63,0.8254,0.28125,0.24087,0.14286,0.2857,0.42857,0.0,0.85714,7,0,0,7,0,8,0,0,7,0,0,4,0,0,2,0,0,3,0,0,1,0,0],[56,63,0.8889,0.16518,0.17169,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,9,0,0,7,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[60,63,0.9524,0.14285,0.1923,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,11,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[63,63,1.0,0.1317,0.11588,0.0,0.14286,0.16075,0.0,0.4286,11,0,0,11,0,13,0,1,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4a7f12d5ee1cb034","q":"Consider a graph with  $n$  vertices and  $\\frac{7n}{4}$  edges.\n(a) Prove that there are two cycles of equal length.\n(25 points)\n(b) Can you give a smaller function than  $\\frac{7n}{4}$  that still fits in part (a)? Prove your claim.\nWe say function  $a(n)$  is smaller than  $b(n)$  if there exists an  $N$  such that for each  $n>N$  , $a(n)<b(n)$ (At most 5 points)\n*Proposed by Afrooz Jabal'ameli*","t":[{"b":1,"e":0.71429,"k":"rising","v":0.05804,"x":0.40177,"p":[[0,37,0.0,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.2857,21,0,1,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.34368,0.22541,0.14286,0.28571,0.571,0.0,0.71429,6,0,0,6,0,3,0,0,8,0,0,4,0,0,9,0,0,2,0,0,0,0,0],[8,37,0.2162,0.2723,0.23514,0.10714,0.2857,0.42857,0.0,0.85714,8,0,0,8,0,7,0,0,6,0,0,6,0,0,2,0,0,2,0,0,1,0,0],[12,37,0.3243,0.2857,0.25252,0.14286,0.2857,0.42857,0.0,0.85714,7,0,0,7,0,8,0,0,8,0,0,2,0,0,3,0,0,2,0,0,2,0,0],[16,37,0.4324,0.26785,0.2223,0.14286,0.2143,0.42857,0.0,0.71429,7,0,0,7,0,9,0,0,5,0,0,6,0,0,2,0,0,3,0,0,0,0,0],[20,37,0.5405,0.2857,0.26485,0.10714,0.14288,0.4642,0.0,0.85714,8,0,0,8,0,10,0,0,2,0,0,4,0,0,3,0,0,4,0,0,1,0,0],[24,37,0.6486,0.3437,0.21381,0.14286,0.28571,0.571,0.0,0.71429,3,0,0,3,0,8,0,0,6,0,0,6,0,0,6,0,0,3,0,0,0,0,0],[28,37,0.7568,0.36606,0.25984,0.14286,0.28571,0.60714,0.0,0.85714,2,0,0,2,0,11,0,0,6,0,0,3,0,0,2,0,0,6,0,0,2,0,0],[32,37,0.8649,0.31697,0.21939,0.14286,0.28571,0.4286,0.0,0.85714,3,0,0,3,0,11,0,0,4,0,0,8,0,0,3,0,0,2,0,0,1,0,0],[36,37,0.973,0.34372,0.15909,0.14289,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,10,0,0,5,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[37,37,1.0,0.40177,0.2185,0.14289,0.42859,0.57111,0.14286,0.85714,0,0,0,0,0,10,0,0,3,0,0,9,0,0,4,0,0,5,0,0,1,0,0]]},{"b":7,"e":0.1429,"k":"rising","v":0.10714,"x":0.41719,"p":[[0,28,0.0,0.10714,0.17128,0.0,0.0,0.1429,0.0,0.71429,20,0,0,20,0,5,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,28,0.1429,0.3839,0.26105,0.14289,0.28571,0.57143,0.0,1.0,4,1,0,4,0,5,0,0,8,0,0,3,0,0,8,0,0,1,0,0,2,0,1],[8,28,0.2857,0.37048,0.27162,0.14286,0.28571,0.57111,0.0,1.0,5,1,0,5,0,6,0,0,6,0,0,3,0,0,8,0,0,1,0,0,2,0,1],[12,28,0.4286,0.29461,0.22846,0.10714,0.28571,0.4286,0.0,0.71429,8,0,0,8,0,5,0,0,5,0,0,7,0,0,5,0,0,2,0,0,0,0,0],[16,28,0.5714,0.41719,0.21898,0.2857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,9,0,0,3,0,1,9,0,0,3,0,0,0,0,1],[20,28,0.7143,0.33032,0.20953,0.1429,0.28571,0.42858,0.0,0.85714,4,0,0,4,0,5,0,0,10,0,0,6,0,0,5,0,0,1,0,0,1,0,0],[24,28,0.8571,0.27676,0.24724,0.14286,0.14286,0.4286,0.0,0.71429,7,0,0,7,0,11,0,0,3,0,0,4,0,0,2,0,0,5,0,0,0,0,0],[28,28,1.0,0.29007,0.21575,0.14286,0.14286,0.42857,0.0,0.857,1,0,0,1,0,18,0,0,2,0,0,5,0,0,3,0,0,2,0,0,1,0,0]]}]},{"i":"5c6047d6b1868020","q":"27. (GBR 4) The triangle $A B C$ is acute-angled. Let $L$ be any line in the plane of the triangle and let $u, v, w$ be the lengths of the perpendiculars from $A, B, C$ respectively to $L$. Prove that  $$ u^{2} \\tan A+v^{2} \\tan B+w^{2} \\tan C \\geq 2 \\Delta $$  where $\\Delta$ is the area of the triangle, and determine the lines $L$ for which equality holds.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.60711,"x":0.91518,"p":[[0,56,0.0,0.60711,0.27894,0.5354,0.64286,0.74996,0.0,1.0,3,4,3,3,0,1,0,0,1,0,0,3,0,0,8,0,0,8,0,0,4,0,4],[4,56,0.0714,0.89732,0.16065,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,16],[8,56,0.1429,0.91071,0.09943,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,15,0,15],[12,56,0.2143,0.875,0.14617,0.85714,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,13,0,13],[16,56,0.2857,0.87052,0.1204,0.85714,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,15,0,11],[20,56,0.3571,0.90623,0.12686,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,18],[24,56,0.4286,0.91518,0.11214,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,18],[28,56,0.5,0.875,0.18472,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,7,0,18],[32,56,0.5714,0.83481,0.18597,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,11,0,12],[36,56,0.6429,0.85713,0.14728,0.82132,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,12,0,12],[40,56,0.7143,0.87052,0.19021,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,8,0,17],[44,56,0.7857,0.83035,0.19377,0.82132,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,0,14,0,10],[48,56,0.8571,0.78568,0.21726,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,10,0,9],[52,56,0.9286,0.77232,0.21086,0.71429,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,0,12,0,7],[56,56,1.0,0.75445,0.17583,0.71429,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,10,0,0,15,0,2]]},{"b":2,"e":1.0,"k":"flat","v":0.66072,"x":0.89283,"p":[[0,40,0.0,0.66072,0.25692,0.57143,0.71429,0.85714,0.0,1.0,2,3,2,2,0,1,0,0,0,0,0,4,0,0,5,0,0,8,0,0,9,0,3],[4,40,0.1,0.88839,0.1665,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,10,0,17],[8,40,0.2,0.89283,0.11298,0.85714,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,16,0,13],[12,40,0.3,0.87946,0.14335,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,13,0,14],[16,40,0.4,0.88839,0.1504,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,13,0,15],[20,40,0.5,0.85713,0.13832,0.857,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,18,0,9],[24,40,0.6,0.84153,0.15946,0.82132,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,13,1,10],[28,40,0.7,0.82588,0.20119,0.82132,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,13,0,11],[32,40,0.8,0.83036,0.16146,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,8,0,0,13,0,9],[36,40,0.9,0.76335,0.23588,0.57132,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,4,0,0,7,0,11],[40,40,1.0,0.741,0.17661,0.67857,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,6,0,0,10,0,0,10,0,4]]}]},{"i":"acf1d5b404ff483d","q":"Let  $\\mathcal C$  be a circle centered at  $O$  and  $A\\ne O$  be a point in its interior. The perpendicular bisector of the segment  $OA$  meets  $\\mathcal C$  at the points  $B$  and  $C$ , and the lines  $AB$  and  $AC$  meet  $\\mathcal C$  again at  $D$  and  $E$ , respectively. Show that the circles  $(OBC)$  and  $(ADE)$  have the same centre.\n\n*Ion P\u0103tra\u0219cu, Ion Cotoi*","t":[{"b":1,"e":0.71429,"k":"rising","v":0.06697,"x":0.29464,"p":[[0,31,0.0,0.06697,0.17852,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,31,0.129,0.27231,0.38524,0.0,0.07143,0.35704,0.0,1.0,16,6,0,16,0,6,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,6],[8,31,0.2581,0.21429,0.35535,0.0,0.0,0.14287,0.0,1.0,18,5,0,18,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[12,31,0.3871,0.12947,0.27516,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[16,31,0.5161,0.22322,0.34798,0.0,0.14286,0.1429,0.0,1.0,15,5,0,15,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[20,31,0.6452,0.13839,0.25123,0.0,0.0,0.14286,0.0,1.0,18,2,0,18,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[24,31,0.7742,0.15625,0.29529,0.0,0.0,0.14286,0.0,1.0,20,3,0,20,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[28,31,0.9032,0.17411,0.29609,0.0,0.0,0.1429,0.0,1.0,18,3,0,18,0,7,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[31,31,1.0,0.29464,0.40079,0.0,0.14286,0.50002,0.0,1.0,15,7,0,15,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,7]]},{"b":5,"e":0.1429,"k":"flat","v":0.07143,"x":0.24107,"p":[[0,28,0.0,0.08036,0.18877,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,28,0.1429,0.12947,0.26088,0.0,0.0,0.14287,0.0,1.0,19,2,0,19,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[8,28,0.2857,0.10714,0.25,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,28,0.4286,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.15625,0.29093,0.0,0.0,0.14286,0.0,1.0,19,3,0,19,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[24,28,0.8571,0.24107,0.32229,0.0,0.14286,0.2857,0.0,1.0,11,4,0,11,0,12,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[28,28,1.0,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"865dbe4e06403525","q":"A tournament is held among  $n$  teams, following such rules:\na) every team plays all others once at home and once away.(i.e. double round-robin schedule)\nb) each team may participate in several away games in a week(from Sunday to Saturday).\nc) there is no away game arrangement for a team, if it has a home game in the same week.\nIf the tournament finishes in 4 weeks, determine the maximum value of  $n$ .","t":[{"b":4,"e":1.0,"k":"rising","v":0.61607,"x":1.0,"p":[[0,38,0.0,0.76786,0.40994,0.82143,1.0,1.0,0.0,1.0,7,23,4,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,23],[4,38,0.1053,0.76786,0.40681,0.82143,1.0,1.0,0.0,1.0,6,24,0,6,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[8,38,0.2105,0.61607,0.46898,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[12,38,0.3158,0.62054,0.48129,0.0,1.0,1.0,0.0,1.0,12,19,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,19],[16,38,0.4211,0.78348,0.37007,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,1,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[20,38,0.5263,0.6383,0.46846,0.0,1.0,1.0,0.0,1.0,9,20,0,9,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[24,38,0.6316,0.76339,0.41282,0.82143,1.0,1.0,0.0,1.0,6,24,0,6,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[28,38,0.7368,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":7,"e":0.0,"k":"falling","v":0.0,"x":0.8125,"p":[[0,56,0.0,0.8125,0.39031,1.0,1.0,1.0,0.0,1.0,6,26,3,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[4,56,0.0714,0.46874,0.46598,0.0,0.28571,1.0,0.0,1.0,14,12,0,14,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,12],[8,56,0.1429,0.41071,0.46805,0.0,0.0,1.0,0.0,1.0,17,11,1,17,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,11],[12,56,0.2143,0.42856,0.45456,0.0,0.2143,1.0,0.0,1.0,15,10,0,15,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,10],[16,56,0.2857,0.37053,0.44872,0.0,0.0,0.89286,0.0,1.0,17,8,0,17,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,8],[20,56,0.3571,0.19643,0.34022,0.0,0.0,0.17857,0.0,1.0,20,4,0,20,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[24,56,0.4286,0.28125,0.43373,0.0,0.0,0.78571,0.0,1.0,21,8,0,21,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[28,56,0.5,0.30357,0.43704,0.0,0.0,0.78571,0.0,1.0,20,8,0,20,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,8],[32,56,0.5714,0.1875,0.36672,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,4],[36,56,0.6429,0.29018,0.41724,0.0,0.0,0.75,0.0,1.0,20,6,0,20,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,6],[40,56,0.7143,0.10268,0.27254,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[44,56,0.7857,0.23214,0.41304,0.0,0.0,0.10714,0.0,1.0,24,7,0,24,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[48,56,0.8571,0.08034,0.25736,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[52,56,0.9286,0.08929,0.27837,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[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]]}]},{"i":"a82294c2698057cf","q":"Find all positive integers  $a$  and  $b$  such that \\[\\frac{a^{2}+b}{b^{2}-a}\\text{ and }\\frac{b^{2}+a}{a^{2}-b}\\] are both integers.","t":[{"b":6,"e":1.0,"k":"flat","v":0.87053,"x":0.94643,"p":[[0,18,0.0,0.9375,0.09407,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[4,18,0.2222,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[8,18,0.4444,0.89286,0.07986,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,10],[12,18,0.6667,0.89285,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[16,18,0.8889,0.87053,0.08268,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,21,0,7],[18,18,1.0,0.88839,0.09268,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,20,0,10]]},{"b":7,"e":1.0,"k":"flat","v":0.93304,"x":0.99554,"p":[[0,68,0.0,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,25],[4,68,0.0588,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[8,68,0.1176,0.93304,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,6,0,23],[12,68,0.1765,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[16,68,0.2353,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,68,0.2941,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,68,0.3529,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,68,0.4118,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,68,0.4706,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,68,0.5294,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,68,0.5882,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[44,68,0.6471,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[48,68,0.7059,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,68,0.7647,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[56,68,0.8235,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[60,68,0.8824,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[64,68,0.9412,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[68,68,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"7d2d3041f5f8ed7e","q":"Prove that\n\n$$\n\\frac{1995}{2}-\\frac{1994}{3}+\\frac{1993}{4}-\\cdots-\\frac{2}{1995}+\\frac{1}{1996}=\\frac{1}{999}+\\frac{3}{1000}+\\cdots+\\frac{1995}{1996} .\n$$","t":[{"b":0,"e":0.85714,"k":"flat","v":0.96429,"x":0.99107,"p":[[0,9,0.0,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,9,0.4444,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[8,9,0.8889,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[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":7,"e":1.0,"k":"flat","v":0.95982,"x":0.99107,"p":[[0,14,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,14,0.2857,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,14,0.5714,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,14,0.8571,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[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":"f4e9fdb50df45fba","q":"Show that among the square roots of the first  $ 2015 $  natural numbers, we cannot choose an arithmetic sequence composed of  $ 45 $  elements.","t":[{"b":3,"e":1.0,"k":"rising","v":0.54016,"x":1.0,"p":[[0,26,0.0,0.74988,0.32945,0.42857,1.0,1.0,0.14,1.0,0,19,0,0,0,3,0,0,4,0,0,2,0,0,3,0,0,0,0,0,1,0,19],[4,26,0.1538,0.77679,0.30917,0.53571,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,6,0,0,1,0,0,2,0,0,2,0,0,0,0,20],[8,26,0.3077,0.54016,0.33261,0.24999,0.57143,1.0,0.14286,1.0,0,9,0,0,0,8,0,0,5,0,0,1,0,0,8,0,0,1,0,0,0,0,9],[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":6,"e":1.0,"k":"rising","v":0.73213,"x":0.99554,"p":[[0,11,0.0,0.73213,0.32488,0.42859,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,3,0,0,2,0,0,1,0,0,4,0,0,2,0,16],[4,11,0.3636,0.76786,0.28738,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,5,0,0,1,0,0,2,0,0,4,0,0,3,0,16],[8,11,0.7273,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[11,11,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"d9d43851830ec3f4","q":"Freddy writes down numbers  $1, 2,\\ldots ,n$  in some order. Then he makes a list of all pairs  $(i, j)$  such that  $1\\le i<j\\le n$  and the  $i$ -th number is bigger than the  $j$ -th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair  $(i, j)$  from the current list, interchange the  $i$ -th and the  $j$ -th number in the current permutation, and delete  $(i, j)$  from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.","t":[{"b":1,"e":1.0,"k":"rising","v":0.1875,"x":0.625,"p":[[0,29,0.0,0.30804,0.36089,0.0,0.14286,0.71429,0.0,1.0,15,2,7,15,0,2,0,0,5,0,0,0,0,0,0,0,0,5,0,0,3,0,2],[4,29,0.1379,0.29009,0.4001,0.0,0.0,0.71429,0.0,1.0,18,5,0,18,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,5],[8,29,0.2759,0.33482,0.44264,0.0,0.0,0.85714,0.0,1.0,20,7,0,20,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,7],[12,29,0.4138,0.1875,0.36672,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,4],[16,29,0.5517,0.25893,0.40317,0.0,0.0,0.60714,0.0,1.0,21,5,0,21,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,5],[20,29,0.6897,0.36607,0.45448,0.0,0.0,0.89286,0.0,1.0,19,8,0,19,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,8],[24,29,0.8276,0.25,0.40564,0.0,0.0,0.71429,0.0,1.0,23,5,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,5],[28,29,0.9655,0.625,0.43411,0.0,0.85714,1.0,0.0,1.0,9,13,0,9,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,13],[29,29,1.0,0.61161,0.46185,0.0,0.92857,1.0,0.0,1.0,11,16,0,11,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,16]]},{"b":7,"e":0.57143,"k":"rising","v":0.27232,"x":0.73214,"p":[[0,30,0.0,0.27232,0.39182,0.0,0.0,0.57143,0.0,1.0,20,5,4,20,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,5],[4,30,0.1333,0.29018,0.41876,0.0,0.0,0.75,0.0,1.0,21,5,0,21,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,5],[8,30,0.2667,0.36607,0.40238,0.0,0.21429,0.85714,0.0,1.0,15,4,0,15,0,1,0,0,3,0,0,1,0,0,1,0,0,2,0,0,5,0,4],[12,30,0.4,0.43304,0.42631,0.0,0.35714,0.85714,0.0,1.0,14,5,0,14,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,5],[16,30,0.5333,0.64286,0.43006,0.14286,0.85714,1.0,0.0,1.0,7,15,0,7,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,15],[20,30,0.6667,0.39732,0.43409,0.0,0.14285,0.85714,0.0,1.0,16,7,0,16,0,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,7],[24,30,0.8,0.63392,0.41332,0.10714,0.85705,1.0,0.0,1.0,8,13,0,8,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,13],[28,30,0.9333,0.67411,0.34299,0.39286,0.85714,1.0,0.0,1.0,3,9,0,3,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,0,10,0,9],[30,30,1.0,0.73214,0.28959,0.53571,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,4,0,0,2,0,0,3,0,0,1,0,0,9,0,11]]}]},{"i":"df3dde1a309ab7cc","q":"Let  $ n \\geq 3$  and consider a set  $ E$  of  $ 2n \\minus{} 1$  distinct points on a circle. Suppose that exactly  $ k$  of these points are to be colored black.  Such a coloring is **good** if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly  $ n$  points from  $ E$ .  Find the smallest value of  $ k$  so that every such coloring of  $ k$  points of  $ E$  is good.","t":[{"b":5,"e":0.28571,"k":"falling","v":0.37054,"x":1.0,"p":[[0,19,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,19,0.2105,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,19,0.4211,0.94188,0.2258,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[12,19,0.6316,0.80357,0.34947,0.85714,1.0,1.0,0.0,1.0,2,24,0,2,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,24],[16,19,0.8421,0.77232,0.39747,0.82143,1.0,1.0,0.0,1.0,4,24,0,4,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[19,19,1.0,0.37054,0.38608,0.0,0.28571,0.57143,0.0,1.0,9,8,0,9,0,6,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,8]]},{"b":6,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,18,0.0,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[4,18,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"2775d357b7602cd0","q":"Let $M$ be a finite set of lattice points and $n$ be a positive integer. A mine-avoiding path is a path of lattice points with length $n$, beginning at $(0,0)$ and ending at a point on the line $x+y=n$, that does not contain any point in $M$. Prove that if there exists a mine-avoiding path, then there exist at least $2^{n-|M|}$ mine-avoiding paths.","t":[{"b":0,"e":0.0,"k":"falling","v":0.01339,"x":0.56696,"p":[[0,28,0.0,0.56696,0.47311,0.0,1.0,1.0,0.0,1.0,11,17,0,11,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,17],[4,28,0.1429,0.36161,0.37962,0.0,0.14286,0.60714,0.0,1.0,10,6,0,10,0,7,0,0,3,0,0,1,0,0,3,0,0,1,0,0,1,0,6],[8,28,0.2857,0.15625,0.31816,0.0,0.0,0.14287,0.0,1.0,23,3,0,23,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[12,28,0.4286,0.17856,0.30092,0.0,0.0,0.17857,0.0,1.0,20,2,0,20,0,4,0,0,2,0,0,0,0,0,2,0,0,2,0,0,0,0,2],[16,28,0.5714,0.08929,0.24419,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,28,0.7143,0.10714,0.25,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[24,28,0.8571,0.06249,0.16724,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[28,28,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"falling","v":0.04,"x":0.5982,"p":[[0,18,0.0,0.5982,0.43659,0.10714,0.85714,1.0,0.0,1.0,8,16,0,8,0,2,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,16],[4,18,0.2222,0.33482,0.41589,0.0,0.0,0.74996,0.0,1.0,17,7,0,17,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,7],[8,18,0.4444,0.30357,0.42371,0.0,0.0,0.71429,0.0,1.0,20,7,0,20,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,7],[12,18,0.6667,0.08036,0.2257,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[16,18,0.8889,0.04,0.09587,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cfa9b5b6ccb0f73f","q":"Find all functions  $f: R \\to  R$  such that for all real numbers  $x, y$  holds  $f(y - xy) = f(x)y + (x - 1)^2 f(y)$","t":[{"b":6,"e":0.28571,"k":"flat","v":0.28125,"x":0.29019,"p":[[0,69,0.0,0.28125,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,69,0.058,0.28125,0.04352,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,69,0.1159,0.28125,0.08364,0.2857,0.28571,0.28571,0.0,0.57143,1,0,1,1,0,2,0,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,69,0.1739,0.28571,1e-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],[16,69,0.2319,0.28571,1e-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],[20,69,0.2899,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.29017,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.28571,1e-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],[36,69,0.5217,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.28571,1e-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],[44,69,0.6377,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.28571,1e-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],[52,69,0.7536,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.29019,0.02485,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[69,69,1.0,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.27678,"x":0.29018,"p":[[0,14,0.0,0.27678,0.04971,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.28571,0.05051,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.28571,0.03571,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.28571,1e-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],[14,14,1.0,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ef4a2210810ea8a2","q":"Suppose that points  $X, Y,Z$  are located on sides  $BC, CA$ , and  $AB$ , respectively, of triangle  $ABC$  in such a way that triangle  $XY Z$  is similar to triangle  $ABC$ . Prove that the orthocenter of triangle  $XY Z$  is the circumcenter of triangle  $ABC.$","t":[{"b":4,"e":0.2857,"k":"flat","v":0.04911,"x":0.11161,"p":[[0,60,0.0,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.1025,0.0734,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,60,0.1333,0.11161,0.09268,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.09821,0.10374,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,60,0.2667,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.08911,0.08553,0.0,0.14143,0.14286,0.0,0.28571,14,0,0,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.10705,0.09446,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.08036,0.08702,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.08929,0.10564,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.09375,0.11633,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.0759,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.2857,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.07588,0.11831,0.0,0.0,0.14286,0.0,0.571,19,0,0,19,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,60,1.0,0.05348,0.07774,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.02679,"x":0.16071,"p":[[0,59,0.0,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.06697,0.07973,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,59,0.1356,0.07125,0.08734,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,59,0.2034,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,59,0.2712,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,59,0.339,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.2857,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,59,0.4068,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,59,0.4746,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,59,0.5424,0.05348,0.10557,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,59,0.6102,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.05795,0.10005,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.1384,0.08364,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,24,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,59,0.8136,0.16071,0.12242,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,23,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[52,59,0.8814,0.1384,0.07563,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.12482,0.06912,0.14214,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[59,59,1.0,0.1383,0.0435,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"19ecaf1c1fb23a05","q":"An acute triangle  $ABC$  is given. Let us denote  $X$  for each of its inner points  $X_a, X_b, X_c$  its images in axial symmetries sequentially along the lines  $BC, CA, AB$ . Prove that all  $X_aX_bX_c$  triangles have a common interior point. \n\n(Josef Tkadlec)","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.125,"p":[[0,38,0.0,0.11598,0.06618,0.14214,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.12482,0.13241,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,19,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,38,0.2105,0.10705,0.12875,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,19,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,38,0.3158,0.12054,0.12931,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,20,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,38,0.4211,0.125,0.12242,0.10714,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,23,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,38,0.5263,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.16947,"p":[[0,22,0.0,0.09348,0.07651,0.0,0.14286,0.14286,0.0,0.28571,12,0,2,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.16947,0.10379,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,22,0.3636,0.10714,0.08748,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0625,0.13333,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,22,0.7273,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"60a25be7ac20a2e5","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}$ be strictly positive integers. For all $k=1,2, \\ldots, n$, we denote\n\n$$\nm_{k}=\\max _{1 \\leq \\ell \\leq k} \\frac{a_{k-\\ell+1}+a_{k-\\ell+2}+\\cdots+a_{k}}{\\ell} .\n$$\n\nShow that for any $\\alpha>0$, the number of integers $k$ such that $m_{k}>\\alpha$ is strictly less than $\\frac{a_{1}+a_{2}+\\cdots+a_{n}}{\\alpha}$.","t":[{"b":6,"e":1.0,"k":"rising","v":0.38838,"x":0.89732,"p":[[0,45,0.0,0.38838,0.39967,0.0,0.14286,0.85714,0.0,1.0,10,7,1,10,0,7,0,0,2,0,0,1,0,0,3,0,0,0,0,0,2,0,7],[4,45,0.0889,0.86161,0.24868,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,0,0,23],[8,45,0.1778,0.74553,0.28512,0.67857,0.85707,1.0,0.0,1.0,2,11,0,2,0,1,0,0,1,0,0,0,0,0,4,0,0,7,0,0,6,0,11],[12,45,0.2667,0.78572,0.25505,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,3,0,15],[16,45,0.3556,0.84821,0.2141,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,4,0,0,4,0,18],[20,45,0.4444,0.82141,0.26488,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,6,0,0,0,0,20],[24,45,0.5333,0.79015,0.24221,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,3,0,0,2,0,16],[28,45,0.6222,0.84373,0.25596,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,4,0,20],[32,45,0.7111,0.8482,0.23675,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,8,0,17],[36,45,0.8,0.89732,0.1684,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,4,0,21],[40,45,0.8889,0.85712,0.21728,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,2,0,0,3,0,20],[44,45,0.9778,0.82584,0.2075,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,2,0,0,3,0,17],[45,45,1.0,0.87942,0.18256,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,5,0,20]]},{"b":7,"e":1.0,"k":"rising","v":0.40179,"x":1.0,"p":[[0,26,0.0,0.40179,0.39518,0.14286,0.14286,0.89286,0.0,1.0,5,8,0,5,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,8],[4,26,0.1538,0.83929,0.29178,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,22],[8,26,0.3077,0.86606,0.22853,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,2,0,21],[12,26,0.4615,0.73661,0.35734,0.5,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,5,0,0,0,0,0,4,0,0,0,0,0,1,0,19],[16,26,0.6154,0.82143,0.27433,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,1,0,0,1,0,21],[20,26,0.7692,0.9375,0.13803,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[24,26,0.9231,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[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":"6f825150b83d0b1b","q":"For any strictly positive integer $x$, we denote $S(x)$ as the sum of the digits of its decimal representation.\nLet $k>0$ be an integer. We define the sequence $\\left(x_{n}\\right)$ by $x_{1}=1$ and $x_{n+1}=S\\left(k x_{n}\\right)$ for all $n>0$.\nProve that $x_{n}<27 \\sqrt{k}$, for all $n>0$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.04018,"x":0.06696,"p":[[0,33,0.0,0.06696,0.09439,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.143,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.06687,0.0712,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.06687,0.09432,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.05804,"x":0.08473,"p":[[0,15,0.0,0.07125,0.0797,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.07589,0.11837,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.08473,0.09348,0.0,0.14143,0.14286,0.0,0.42857,15,0,0,15,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e33aaa8e78dcafe6","q":"Suppose that real numbers  $x,y$  and  $z$  satisfy the following equations:\n\n\\begin{align*}\nx+\\frac{y}{z} &=2,\ny+\\frac{z}{x} &=2,\nz+\\frac{x}{y} &=2.\n\\end{align*}\n\nShow that  $s=x+y+z$  must be equal to  $3$  or  $7$ .\n\n*Note:* It is not required to show the existence of such numbers  $x,y,z$ .","t":[{"b":3,"e":0.14286,"k":"falling","v":0.22768,"x":0.52679,"p":[[0,149,0.0,0.46427,0.29233,0.2857,0.42859,0.60714,0.0,1.0,3,4,0,3,0,4,0,0,6,0,0,4,0,0,7,0,0,4,0,0,0,0,4],[4,149,0.0268,0.52679,0.22142,0.42857,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,6,0,0,15,0,0,0,0,0,0,0,4],[8,149,0.0537,0.49999,0.21428,0.42857,0.57143,0.57143,0.0,1.0,1,2,0,1,0,4,0,0,1,0,0,5,0,0,18,0,0,1,0,0,0,0,2],[12,149,0.0805,0.44196,0.17627,0.28571,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,7,0,0,3,0,0,17,0,0,1,0,0,0,0,0],[16,149,0.1074,0.36161,0.22583,0.14286,0.35714,0.57143,0.0,0.71429,3,0,0,3,0,9,0,0,4,0,0,2,0,0,12,0,0,2,0,0,0,0,0],[20,149,0.1342,0.44196,0.22968,0.28571,0.57143,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,5,0,0,3,0,0,15,0,0,2,0,0,0,0,1],[24,149,0.1611,0.45087,0.23174,0.28571,0.571,0.57143,0.0,1.0,1,2,0,1,0,5,0,0,6,0,0,3,0,0,14,0,0,1,0,0,0,0,2],[28,149,0.1879,0.41518,0.23787,0.25,0.42857,0.57143,0.0,1.0,1,2,0,1,0,7,0,0,6,0,0,5,0,0,10,0,0,1,0,0,0,0,2],[32,149,0.2148,0.5089,0.27879,0.28571,0.57121,0.57143,0.0,1.0,1,5,0,1,0,4,0,0,6,0,0,3,0,0,12,0,0,0,0,0,1,0,5],[36,149,0.2416,0.40177,0.25363,0.14286,0.28571,0.57143,0.0,1.0,1,2,0,1,0,8,0,0,8,0,0,3,0,0,8,0,0,1,0,0,1,0,2],[40,149,0.2685,0.44643,0.26426,0.1429,0.57143,0.57143,0.0,1.0,1,3,0,1,0,8,0,0,4,0,0,2,0,0,13,0,0,1,0,0,0,0,3],[44,149,0.2953,0.45982,0.25439,0.25,0.50001,0.57143,0.14286,1.0,0,2,0,0,0,8,0,0,5,0,0,3,0,0,9,0,0,4,0,0,1,0,2],[48,149,0.3221,0.38393,0.29329,0.14286,0.28571,0.57143,0.0,1.0,6,2,0,6,0,5,0,0,6,0,0,1,0,0,9,0,0,2,0,0,1,0,2],[52,149,0.349,0.46873,0.30977,0.14286,0.42857,0.60714,0.0,1.0,1,5,0,1,0,9,0,0,4,0,0,3,0,0,7,0,0,2,0,0,1,0,5],[56,149,0.3758,0.39732,0.26422,0.14286,0.28571,0.57143,0.0,1.0,1,2,0,1,0,8,0,0,11,0,0,0,0,0,6,0,0,3,0,0,1,0,2],[60,149,0.4027,0.29911,0.19678,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,12,0,0,10,0,0,4,0,0,4,0,0,0,0,0,0,0,1],[64,149,0.4295,0.41071,0.25691,0.14289,0.42857,0.57143,0.0,0.85714,4,0,0,4,0,5,0,0,4,0,0,5,0,0,9,0,0,2,0,0,3,0,0],[68,149,0.4564,0.42411,0.2382,0.28571,0.5,0.57143,0.0,1.0,3,1,0,3,0,4,0,0,6,0,0,3,0,0,12,0,0,3,0,0,0,0,1],[72,149,0.4832,0.48661,0.29851,0.28571,0.57143,0.60714,0.0,1.0,3,4,0,3,0,3,0,0,7,0,0,2,0,0,9,0,0,2,0,0,2,0,4],[76,149,0.5101,0.45981,0.25438,0.28571,0.49979,0.60714,0.0,1.0,1,2,0,1,0,6,0,0,6,0,0,3,0,0,8,0,0,6,0,0,0,0,2],[80,149,0.5369,0.35268,0.24996,0.14286,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,9,0,0,1,0,0,3,0,0,12,0,0,1,0,0,1,0,0],[84,149,0.5638,0.43304,0.28456,0.14286,0.5,0.57143,0.0,1.0,2,3,0,2,0,8,0,0,5,0,0,1,0,0,11,0,0,1,0,0,1,0,3],[88,149,0.5906,0.36159,0.25249,0.14286,0.35714,0.57143,0.0,0.71429,5,0,0,5,0,8,0,0,3,0,0,1,0,0,11,0,0,4,0,0,0,0,0],[92,149,0.6174,0.37946,0.27804,0.14286,0.28571,0.57143,0.0,1.0,1,2,0,1,0,12,0,0,6,0,0,3,0,0,4,0,0,2,0,0,2,0,2],[96,149,0.6443,0.41518,0.31209,0.14286,0.28571,0.57143,0.0,1.0,4,4,0,4,0,7,0,0,6,0,0,1,0,0,7,0,0,3,0,0,0,0,4],[100,149,0.6711,0.39286,0.25505,0.24999,0.28571,0.57143,0.0,1.0,2,2,0,2,0,6,0,0,10,0,0,3,0,0,7,0,0,1,0,0,1,0,2],[104,149,0.698,0.31695,0.19798,0.14286,0.28571,0.32143,0.0,0.85714,2,0,0,2,0,7,0,0,15,0,0,2,0,0,3,0,0,2,0,0,1,0,0],[108,149,0.7248,0.29464,0.22851,0.14286,0.21428,0.57143,0.0,1.0,3,1,0,3,0,13,0,0,7,0,0,0,0,0,8,0,0,0,0,0,0,0,1],[112,149,0.7517,0.37054,0.20782,0.25,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,8,0,0,12,0,0,2,0,0,7,0,0,1,0,0,2,0,0],[116,149,0.7785,0.32143,0.23146,0.14286,0.28571,0.46429,0.0,1.0,3,1,0,3,0,9,0,0,10,0,0,2,0,0,5,0,0,2,0,0,0,0,1],[120,149,0.8054,0.41071,0.28738,0.24999,0.28571,0.57143,0.0,1.0,1,4,0,1,0,7,0,0,12,0,0,2,0,0,4,0,0,1,0,0,1,0,4],[124,149,0.8322,0.27232,0.17627,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,14,0,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[128,149,0.8591,0.25446,0.08552,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,10,0,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[132,149,0.8859,0.25446,0.08552,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,10,0,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[136,149,0.9128,0.26339,0.08828,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,9,0,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[140,149,0.9396,0.24545,0.06437,0.14289,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,149,0.9664,0.22768,0.07873,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,14,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[148,149,0.9933,0.25446,0.11701,0.14286,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,12,0,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[149,149,1.0,0.22768,0.10012,0.14286,0.2857,0.28571,0.0,0.4286,1,0,0,1,0,14,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.33481,"x":0.86607,"p":[[0,169,0.0,0.4017,0.27076,0.14286,0.28571,0.57143,0.0,1.0,2,2,0,2,0,8,0,0,7,0,0,3,0,0,6,0,0,3,0,0,1,0,2],[4,169,0.0237,0.44196,0.27049,0.14286,0.57143,0.57143,0.0,1.0,1,3,0,1,0,8,0,0,6,0,0,0,0,0,12,0,0,2,0,0,0,0,3],[8,169,0.0473,0.51338,0.1885,0.42857,0.57143,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,4,0,0,5,0,0,17,0,0,2,0,0,1,0,1],[12,169,0.071,0.56695,0.25123,0.57132,0.57143,0.60714,0.14286,1.0,0,4,0,0,0,5,0,0,2,0,0,0,0,0,17,0,0,2,0,0,2,0,4],[16,169,0.0947,0.53124,0.21498,0.42857,0.57143,0.57143,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,4,0,0,14,0,0,4,0,0,1,0,2],[20,169,0.1183,0.45981,0.17398,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,2,0,0,6,0,0,18,0,0,1,0,0,0,0,0],[24,169,0.142,0.47319,0.19044,0.39286,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,3,0,0,4,0,0,16,0,0,4,0,0,0,0,0],[28,169,0.1657,0.47765,0.17352,0.42857,0.57141,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,2,0,0,7,0,0,16,0,0,3,0,0,0,0,0],[32,169,0.1893,0.51784,0.24156,0.42857,0.57143,0.57143,0.0,1.0,2,3,0,2,0,2,0,0,3,0,0,4,0,0,15,0,0,3,0,0,0,0,3],[36,169,0.213,0.49107,0.2141,0.42857,0.57143,0.57143,0.0,1.0,2,1,0,2,0,2,0,0,3,0,0,5,0,0,16,0,0,2,0,0,1,0,1],[40,169,0.2367,0.52231,0.22759,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,2,0,0,4,0,0,15,0,0,2,0,0,2,0,2],[44,169,0.2604,0.48214,0.22798,0.42857,0.57143,0.57143,0.0,1.0,2,2,0,2,0,3,0,0,2,0,0,7,0,0,14,0,0,2,0,0,0,0,2],[48,169,0.284,0.45088,0.21161,0.28571,0.57121,0.57143,0.0,0.85714,1,0,0,1,0,6,0,0,2,0,0,6,0,0,14,0,0,1,0,0,2,0,0],[52,169,0.3077,0.41964,0.25489,0.24999,0.49999,0.57143,0.0,1.0,3,2,0,3,0,5,0,0,6,0,0,2,0,0,13,0,0,1,0,0,0,0,2],[56,169,0.3314,0.39732,0.24675,0.24999,0.42857,0.57143,0.0,1.0,3,2,0,3,0,5,0,0,7,0,0,4,0,0,11,0,0,0,0,0,0,0,2],[60,169,0.355,0.4107,0.21052,0.14286,0.49979,0.57143,0.0,0.71429,2,0,0,2,0,7,0,0,2,0,0,5,0,0,14,0,0,2,0,0,0,0,0],[64,169,0.3787,0.37938,0.1977,0.25001,0.35714,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,8,0,0,5,0,0,8,0,0,3,0,0,0,0,0],[68,169,0.4024,0.37054,0.24185,0.14286,0.28571,0.57143,0.0,1.0,1,1,0,1,0,12,0,0,5,0,0,0,0,0,11,0,0,2,0,0,0,0,1],[72,169,0.426,0.34372,0.2047,0.14286,0.28571,0.57143,0.0,0.57143,1,0,0,1,0,13,0,0,3,0,0,2,0,0,13,0,0,0,0,0,0,0,0],[76,169,0.4497,0.39283,0.22013,0.14286,0.35714,0.57143,0.0,1.0,1,1,0,1,0,8,0,0,7,0,0,1,0,0,14,0,0,0,0,0,0,0,1],[80,169,0.4734,0.40177,0.26591,0.14286,0.42857,0.57143,0.0,1.0,3,2,0,3,0,8,0,0,3,0,0,4,0,0,10,0,0,2,0,0,0,0,2],[84,169,0.497,0.40178,0.22711,0.14286,0.57141,0.57143,0.0,0.71429,3,0,0,3,0,6,0,0,5,0,0,1,0,0,14,0,0,3,0,0,0,0,0],[88,169,0.5207,0.43304,0.24868,0.14286,0.5,0.57143,0.0,1.0,1,2,0,1,0,8,0,0,4,0,0,3,0,0,12,0,0,2,0,0,0,0,2],[92,169,0.5444,0.34373,0.22546,0.14286,0.28571,0.57143,0.0,0.857,3,0,0,3,0,8,0,0,8,0,0,3,0,0,7,0,0,2,0,0,1,0,0],[96,169,0.568,0.35713,0.22867,0.14286,0.28571,0.57143,0.0,1.0,2,1,0,2,0,9,0,0,7,0,0,3,0,0,9,0,0,1,0,0,0,0,1],[100,169,0.5917,0.42409,0.24609,0.14286,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,10,0,0,3,0,0,5,0,0,11,0,0,0,0,0,1,0,2],[104,169,0.6154,0.4732,0.18012,0.39286,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,4,0,0,4,0,0,17,0,0,3,0,0,0,0,0],[108,169,0.6391,0.52232,0.21902,0.28571,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,4,0,0,5,0,0,0,0,0,16,0,0,5,0,0,0,0,2],[112,169,0.6627,0.49999,0.27199,0.28571,0.57143,0.60707,0.0,1.0,1,3,0,1,0,5,0,0,6,0,0,1,0,0,11,0,0,3,0,0,2,0,3],[116,169,0.6864,0.35714,0.26486,0.14286,0.28571,0.57143,0.0,1.0,3,2,0,3,0,9,0,0,8,0,0,1,0,0,7,0,0,2,0,0,0,0,2],[120,169,0.7101,0.35268,0.31336,0.14286,0.14286,0.57143,0.0,1.0,6,3,0,6,0,11,0,0,1,0,0,1,0,0,8,0,0,2,0,0,0,0,3],[124,169,0.7337,0.424,0.23289,0.14286,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,7,0,0,1,0,0,1,0,0,17,0,0,3,0,0,0,0,0],[128,169,0.7574,0.43749,0.20805,0.2857,0.57121,0.57143,0.14286,1.0,0,1,0,0,0,7,0,0,5,0,0,3,0,0,15,0,0,1,0,0,0,0,1],[132,169,0.7811,0.33481,0.25406,0.10714,0.42857,0.57143,0.0,0.71429,8,0,0,8,0,6,0,0,1,0,0,2,0,0,14,0,0,1,0,0,0,0,0],[136,169,0.8047,0.36606,0.24205,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,9,0,0,3,0,0,6,0,0,8,0,0,2,0,0,0,0,1],[140,169,0.8284,0.50445,0.23953,0.28571,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,7,0,0,2,0,0,1,0,0,16,0,0,3,0,0,1,0,2],[144,169,0.8521,0.47766,0.26632,0.28571,0.57143,0.57143,0.0,1.0,2,3,0,2,0,5,0,0,4,0,0,2,0,0,13,0,0,3,0,0,0,0,3],[148,169,0.8757,0.4464,0.27835,0.14289,0.42857,0.57143,0.0,1.0,2,3,0,2,0,7,0,0,4,0,0,4,0,0,8,0,0,4,0,0,0,0,3],[152,169,0.8994,0.63393,0.28333,0.53571,0.71429,0.78571,0.0,1.0,1,8,0,1,0,2,0,0,4,0,0,1,0,0,7,0,0,9,0,0,0,0,8],[156,169,0.9231,0.70982,0.29339,0.57143,0.71429,1.0,0.0,1.0,1,12,0,1,0,2,0,0,2,0,0,2,0,0,3,0,0,9,0,0,1,0,12],[160,169,0.9467,0.81697,0.30771,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,22],[164,169,0.9704,0.75,0.30093,0.71429,0.71429,1.0,0.0,1.0,2,15,0,2,0,1,0,0,1,0,0,2,0,0,1,0,0,10,0,0,0,0,15],[168,169,0.9941,0.86607,0.20183,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,2,0,20],[169,169,1.0,0.83929,0.21354,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,10,0,0,2,0,17]]}]},{"i":"dc473f84cf6b8bd8","q":"$56$  lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly  $594$  points, what is the maximum number of them that could have the same slope?","t":[{"b":5,"e":1.0,"k":"flat","v":0.86607,"x":0.96875,"p":[[0,59,0.0,0.86607,0.14698,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,6,0,15],[4,59,0.0678,0.91517,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,17],[8,59,0.1356,0.91518,0.14664,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,9,0,20],[12,59,0.2034,0.87497,0.17039,0.82143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,7,0,17],[16,59,0.2712,0.89284,0.17499,0.85711,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,4,0,21],[20,59,0.339,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[24,59,0.4068,0.91071,0.12753,0.85711,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,20],[28,59,0.4746,0.9241,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[32,59,0.5424,0.92411,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[36,59,0.6102,0.91518,0.13296,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,21],[40,59,0.678,0.92411,0.15146,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,4,0,23],[44,59,0.7458,0.9241,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19],[48,59,0.8136,0.91518,0.13767,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,20],[52,59,0.8814,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[56,59,0.9492,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[59,59,1.0,0.91071,0.14174,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,20]]},{"b":7,"e":0.85714,"k":"falling","v":0.74551,"x":0.91964,"p":[[0,40,0.0,0.91964,0.14258,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,22],[4,40,0.1,0.79016,0.12366,0.71429,0.857,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,13,0,4],[8,40,0.2,0.74551,0.15041,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,16,0,0,7,0,4],[12,40,0.3,0.82589,0.15458,0.71429,0.78564,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,4,0,12],[16,40,0.4,0.77678,0.15947,0.71429,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,15,0,0,8,0,6],[20,40,0.5,0.80804,0.19434,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,9,0,0,8,0,11],[24,40,0.6,0.82142,0.13363,0.71429,0.85707,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,10,0,8],[28,40,0.7,0.76338,0.14112,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,18,0,0,6,0,5],[32,40,0.8,0.75,0.17128,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,17,0,0,3,0,7],[36,40,0.9,0.75893,0.21261,0.71429,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,17,0,0,1,0,10],[40,40,1.0,0.76785,0.13716,0.71429,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,21,0,0,5,0,5]]}]},{"i":"9f8e49949fd9769f","q":"Let  $f:\\mathbb{Z}\\rightarrow\\{ 1,2,\\ldots ,n\\}$  be a function such that  $f(x)\\not= f(y)$ , for all  $x,y\\in\\mathbb{Z}$  such that  $|x-y|\\in\\{2,3,5\\}$ . Prove that  $n\\ge 4$ .\n\n*Ioan Tomescu*","t":[{"b":0,"e":0.85714,"k":"flat","v":0.72767,"x":0.85267,"p":[[0,16,0.0,0.7857,0.22587,0.857,0.85714,0.85714,0.0,1.0,2,3,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,23,0,3],[4,16,0.25,0.85267,0.17307,0.85711,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,12,0,13],[8,16,0.5,0.72767,0.32015,0.67857,0.85714,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,0,15,0,8],[12,16,0.75,0.79908,0.17079,0.71429,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,18,0,5],[16,16,1.0,0.79907,0.14227,0.85708,0.85714,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,25,0,1]]},{"b":7,"e":0.6667,"k":"falling","v":0.47024,"x":0.79911,"p":[[0,16,0.0,0.7589,0.2381,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,0,0,0,6,0,0,0,0,0,15,0,7],[4,16,0.25,0.75446,0.28175,0.82143,0.85714,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,17,0,7],[8,16,0.5,0.79911,0.2392,0.82143,0.85714,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,16,0,8],[12,16,0.75,0.61977,0.30378,0.4286,0.71429,0.85714,0.0,1.0,3,1,0,3,0,3,0,0,0,0,0,4,0,0,2,0,0,5,0,1,13,0,1],[16,16,1.0,0.47024,0.31106,0.14289,0.42857,0.71429,0.0,1.0,2,3,0,2,0,7,0,0,5,1,0,3,0,0,3,0,0,4,0,0,4,0,3]]}]},{"i":"afdd117ae7a0f616","q":"Find all positive integers  $n$  for such the following condition holds:\n\"If  $a$ ,  $b$  and  $c$  are positive integers such are all numbers \\[ a^2+2ab+b^2,\\  b^2+2bc+c^2, \\ c^2+2ca+a^2 \\] are divisible by  $n$ , then  $(a+b+c)^2$  is also divisible by  $n$ .\"\n*G.M.Sharafetdinova*","t":[{"b":2,"e":0.0,"k":"falling","v":0.31696,"x":0.92856,"p":[[0,47,0.0,0.92409,0.07975,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[4,47,0.0851,0.92856,0.07987,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[8,47,0.1702,0.9107,0.07785,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[12,47,0.2553,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[16,47,0.3404,0.88838,0.06902,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,8],[20,47,0.4255,0.7991,0.16311,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,15,0,6],[24,47,0.5106,0.8125,0.14032,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,17,0,5],[28,47,0.5957,0.8616,0.13592,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,17,0,10],[32,47,0.6809,0.74553,0.18465,0.57143,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,4,0,0,16,0,3],[36,47,0.766,0.69196,0.21461,0.42859,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,6,0,0,10,0,4],[40,47,0.8511,0.7232,0.1954,0.42859,0.857,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,5,0,0,16,0,2],[44,47,0.9362,0.72321,0.17105,0.67857,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,9,0,0,14,0,1],[47,47,1.0,0.31696,0.30459,0.0,0.28571,0.46429,0.0,1.0,12,1,0,12,0,2,0,0,3,0,0,7,0,0,1,0,0,5,0,0,1,0,1]]},{"b":4,"e":1.0,"k":"flat","v":0.92409,"x":0.98214,"p":[[0,40,0.0,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[4,40,0.1,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[8,40,0.2,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,40,0.3,0.92409,0.07975,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[16,40,0.4,0.92857,0.10715,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[20,40,0.5,0.94642,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[24,40,0.6,0.94197,0.11769,0.85714,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,23],[28,40,0.7,0.92856,0.0945,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,18],[32,40,0.8,0.93749,0.09408,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[36,40,0.9,0.95088,0.07669,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[40,40,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"a49d3eec5016cbdd","q":"$\\mathrm{ABC}$ is an equilateral triangle with side 2 . Show that any point $\\mathrm{P}$ on the incircle satisfies $\\mathrm{PA}^{2}$ $+\\mathrm{PB}^{2}+\\mathrm{PC}^{2}=5$. Show also that the triangle with side lengths $\\mathrm{PA}, \\mathrm{PB}, \\mathrm{PC}$ has area $(\\sqrt{3}) / 4$.","t":[{"b":4,"e":0.4286,"k":"flat","v":0.37054,"x":0.51786,"p":[[0,73,0.0,0.51786,0.1948,0.42857,0.42859,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,11,0,0,6,0,0,4,0,0,3,0,1],[4,73,0.0548,0.41515,0.13994,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,13,0,0,2,0,0,4,0,0,0,0,0],[8,73,0.1096,0.40625,0.11904,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,18,0,0,0,0,0,3,0,0,0,0,0],[12,73,0.1644,0.38393,0.11538,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,14,0,0,1,0,0,2,0,0,0,0,0],[16,73,0.2192,0.40183,0.12596,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,15,0,0,1,0,0,3,0,0,0,0,0],[20,73,0.274,0.38393,0.10972,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,16,0,0,0,0,0,2,0,0,0,0,0],[24,73,0.3288,0.39732,0.10555,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,19,0,0,0,0,0,2,0,0,0,0,0],[28,73,0.3836,0.40177,0.12594,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,15,0,0,1,0,0,3,0,0,0,0,0],[32,73,0.4384,0.37945,0.13174,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,18,0,0,10,0,0,1,0,0,3,0,0,0,0,0],[36,73,0.4932,0.43302,0.12618,0.39286,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,19,0,0,1,0,0,4,0,0,0,0,0],[40,73,0.5479,0.37945,0.11631,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,13,0,0,1,0,0,2,0,0,0,0,0],[44,73,0.6027,0.43304,0.14054,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,16,0,0,1,0,0,5,0,0,0,0,0],[48,73,0.6575,0.42411,0.14054,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,16,0,0,0,0,0,5,0,0,0,0,0],[52,73,0.7123,0.39286,0.10715,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,18,0,0,0,0,0,2,0,0,0,0,0],[56,73,0.7671,0.39732,0.12234,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,16,0,0,0,0,0,3,0,0,0,0,0],[60,73,0.8219,0.40625,0.11904,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,18,0,0,0,0,0,3,0,0,0,0,0],[64,73,0.8767,0.375,0.09279,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,17,0,0,0,0,0,1,0,0,0,0,0],[68,73,0.9315,0.37054,0.11214,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,13,0,0,0,0,0,2,0,0,0,0,0],[72,73,0.9863,0.41071,0.12243,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,17,0,0,1,0,0,3,0,0,0,0,0],[73,73,1.0,0.38838,0.10246,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,16,0,0,2,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.38393,"x":0.46875,"p":[[0,36,0.0,0.45532,0.16915,0.28571,0.42857,0.57111,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,10,0,0,7,0,0,3,0,0,0,0,1],[4,36,0.1111,0.38839,0.11426,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,15,0,0,1,0,0,2,0,0,0,0,0],[8,36,0.2222,0.40178,0.12593,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,15,0,0,1,0,0,3,0,0,0,0,0],[12,36,0.3333,0.38393,0.11539,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,14,0,0,1,0,0,2,0,0,0,0,0],[16,36,0.4444,0.42862,0.12877,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,18,0,0,1,0,0,4,0,0,0,0,0],[20,36,0.5556,0.4107,0.1417,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,12,0,0,2,0,0,4,0,0,0,0,0],[24,36,0.6667,0.40625,0.13883,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,13,0,0,1,0,0,4,0,0,0,0,0],[28,36,0.7778,0.40625,0.14773,0.28571,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,12,0,0,0,0,0,5,0,0,0,0,0],[32,36,0.8889,0.40625,0.13415,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,15,0,0,0,0,0,4,0,0,0,0,0],[36,36,1.0,0.46875,0.16458,0.28571,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,14,0,0,0,0,0,9,0,0,0,0,0]]}]},{"i":"110e3da4f4a6ceb3","q":"Find all functions  $f$  from the reals to the reals such that \\[ \\left(f(x)+f(z)\\right)\\left(f(y)+f(t)\\right)=f(xy-zt)+f(xt+yz)  \\] for all real  $x,y,z,t$ .","t":[{"b":2,"e":0.42857,"k":"flat","v":0.40399,"x":0.73652,"p":[[0,71,0.0,0.40399,0.22769,0.28571,0.28571,0.4286,0.0,1.0,1,1,0,1,0,2,0,1,14,0,0,7,0,0,2,0,0,1,0,0,3,0,1],[4,71,0.0563,0.6339,0.25239,0.42857,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,9,0,0,6,0,0,2,0,0,4,0,7],[8,71,0.1127,0.64953,0.24703,0.42857,0.57143,0.87501,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,9,0,0,8,0,0,2,0,0,3,1,7],[12,71,0.169,0.73652,0.29922,0.57142,0.85714,1.0,0.14,1.0,0,15,0,0,0,3,0,0,2,0,0,2,0,0,5,0,0,3,0,0,2,0,15],[16,71,0.2254,0.63838,0.25996,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,9,0,0,8,0,0,2,0,0,1,0,9],[20,71,0.2817,0.58702,0.25488,0.42857,0.57121,0.85704,0.14286,1.0,0,5,0,0,0,1,0,0,6,0,0,6,0,1,7,0,0,2,0,0,4,0,5],[24,71,0.338,0.5625,0.23673,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,3,0,0,8,0,0,9,0,0,4,0,0,3,0,3],[28,71,0.3944,0.73212,0.23893,0.42859,0.78571,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,2,0,0,5,0,11],[32,71,0.4507,0.64061,0.23385,0.42857,0.57143,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,5,0,0,9,1,0,5,0,0,1,0,7],[36,71,0.507,0.68301,0.2544,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,4,0,0,6,0,0,4,0,0,6,0,0,3,0,9],[40,71,0.5634,0.58927,0.25442,0.42857,0.57143,0.75,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,6,0,0,7,0,0,5,0,0,4,0,4],[44,71,0.6197,0.55356,0.20438,0.42857,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,11,0,0,8,0,0,4,0,0,3,0,2],[48,71,0.6761,0.58034,0.23941,0.42857,0.57143,0.75,0.2857,1.0,0,4,0,0,0,0,0,0,7,0,0,7,0,0,7,0,0,3,0,0,4,0,4],[52,71,0.7324,0.56694,0.26362,0.42857,0.49979,0.75,0.0,1.0,1,5,0,1,0,0,0,0,6,0,0,9,0,0,5,0,0,3,0,0,3,0,5],[56,71,0.7887,0.62942,0.23108,0.42857,0.57143,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,4,0,0,6,0,0,9,0,0,5,0,0,2,0,6],[60,71,0.8451,0.5223,0.21311,0.42857,0.4998,0.57143,0.1429,1.0,0,4,0,0,0,1,0,0,5,0,0,10,0,0,12,0,0,0,0,0,0,0,4],[64,71,0.9014,0.53122,0.22653,0.42857,0.4286,0.60714,0.14286,1.0,0,3,0,0,0,2,0,0,4,0,0,11,0,0,7,0,0,3,0,0,2,0,3],[68,71,0.9577,0.4598,0.15456,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,13,0,0,9,0,0,2,0,0,1,0,0],[71,71,1.0,0.46875,0.16457,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,14,0,0,7,0,0,3,0,0,0,0,1]]},{"b":3,"e":0.4286,"k":"flat","v":0.46875,"x":0.68525,"p":[[0,89,0.0,0.46875,0.25313,0.28571,0.42857,0.46431,0.14286,1.0,0,4,0,0,0,2,0,0,11,0,0,11,0,0,2,0,0,0,0,0,2,0,4],[4,89,0.0449,0.62052,0.26633,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,8,0,0,9,0,0,1,0,0,2,0,8],[8,89,0.0899,0.65848,0.27007,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,7,0,1,5,0,0,4,0,0,1,0,10],[12,89,0.1348,0.68525,0.26115,0.42857,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,1,7,0,0,4,0,0,4,0,0,3,0,10],[16,89,0.1798,0.60255,0.23066,0.42857,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,12,0,0,6,0,0,6,0,0,0,0,6],[20,89,0.2247,0.64732,0.27429,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,2,0,0,8,0,0,5,0,0,4,0,0,2,0,9],[24,89,0.2697,0.63392,0.26711,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,8,0,0,6,0,0,4,0,0,2,0,8],[28,89,0.3146,0.61607,0.27534,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,5,0,0,5,0,0,6,0,0,4,0,0,3,0,7],[32,89,0.3596,0.60267,0.23072,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,5,0,0,8,0,0,7,0,0,4,0,3],[36,89,0.4045,0.61606,0.25862,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,3,0,0,6,0,0,9,0,0,2,0,0,4,0,6],[40,89,0.4494,0.62942,0.23653,0.42859,0.57143,0.85704,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,4,0,0,11,0,0,3,0,0,3,0,6],[44,89,0.4944,0.54911,0.24513,0.42857,0.5,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,9,0,0,6,0,0,4,0,0,2,0,4],[48,89,0.5393,0.63396,0.21993,0.4286,0.57143,0.74996,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,8,0,0,10,0,0,4,0,0,2,0,6],[52,89,0.5843,0.60267,0.22794,0.53539,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,0,0,0,5,0,0,14,0,0,3,0,0,3,0,4],[56,89,0.6292,0.65399,0.23899,0.53539,0.57143,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,4,0,0,4,0,0,10,1,0,4,0,0,1,0,8],[60,89,0.6742,0.66963,0.26351,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,8,0,0,6,0,0,2,0,0,4,0,9],[64,89,0.7191,0.56692,0.22155,0.42857,0.57121,0.57143,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,8,0,0,13,0,0,0,0,0,1,0,5],[68,89,0.764,0.58034,0.25739,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,4,0,0,4,0,0,10,0,0,4,0,0,2,0,5],[72,89,0.809,0.64284,0.25754,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,4,0,0,5,0,0,8,0,0,3,0,0,4,0,7],[76,89,0.8539,0.49105,0.21997,0.42857,0.42859,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,2,0,0,10,0,0,8,0,0,5,0,0,0,0,2],[80,89,0.8989,0.58476,0.23243,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,1,0,0,7,0,0,10,0,0,4,0,0,4,0,3],[84,89,0.9438,0.59374,0.23176,0.42857,0.57143,0.60714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,9,0,0,12,0,0,1,0,0,1,0,6],[88,89,0.9888,0.56246,0.16728,0.42857,0.57141,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,9,0,0,16,0,0,1,0,0,2,0,2],[89,89,1.0,0.55802,0.16888,0.42857,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,6,0,0,15,0,0,5,0,0,0,0,2]]}]},{"i":"793eb904ff8f2730","q":"Let  $p$  be a prime number and let  $X$  be a finite set containing at least  $p$  elements. A collection of pairwise mutually disjoint  $p$ -element subsets of  $X$  is called a  $p$ -family. (In particular, the empty collection is a  $p$ -family.) Let  $A$ (respectively,  $B$ ) denote the number of  $p$ -families having an even (respectively, odd) number of  $p$ -element subsets of  $X$ . Prove that  $A$  and  $B$  differ by a multiple of  $p$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.37499,"p":[[0,10,0.0,0.37499,0.33455,0.0,0.57143,0.71429,0.0,0.71429,14,0,0,14,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,0,0,0],[4,10,0.4,0.36161,0.34807,0.0,0.57143,0.71429,0.0,1.0,15,1,0,15,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,0,0,1],[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],[10,10,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.41963,"p":[[0,16,0.0,0.41963,0.32914,0.0,0.57143,0.71429,0.0,0.71429,12,0,0,12,0,0,0,0,0,0,0,0,0,0,6,0,0,14,0,0,0,0,0],[4,16,0.25,0.41518,0.35058,0.0,0.57143,0.71429,0.0,1.0,13,1,0,13,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,0,0,1],[8,16,0.5,0.17411,0.30249,0.0,0.0,0.14286,0.0,0.71429,24,0,0,24,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"1b10e11025b7de8d","q":"Define a function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ by $f(1)=1, f(n+1)=f(n)+2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \\ldots, f\\left(3^{2013}\\right)$ leave distinct remainders when divided by $3^{2013}$.","t":[{"b":0,"e":0.71429,"k":"rising","v":0.32125,"x":0.54683,"p":[[0,34,0.0,0.32125,0.27676,0.14286,0.21428,0.42858,0.0,1.0,4,1,1,4,0,12,0,0,5,0,0,4,0,0,2,0,0,1,0,0,3,0,1],[4,34,0.1176,0.35713,0.21427,0.14286,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,8,0,0,8,0,0,4,0,0,6,0,0,4,0,0,0,0,0],[8,34,0.2353,0.4375,0.19865,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,8,0,0,12,0,0,3,0,0,4,0,0,2,0,0],[12,34,0.3529,0.42408,0.22154,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,10,0,0,7,0,0,6,0,0,2,0,0,3,0,0],[16,34,0.4706,0.42853,0.22866,0.28571,0.42857,0.57111,0.0,0.85714,1,0,0,1,0,6,0,0,5,0,0,9,0,0,5,0,0,3,0,0,3,0,0],[20,34,0.5882,0.44641,0.21052,0.28571,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,6,0,0,7,0,0,9,0,0,3,0,0,2,0,0],[24,34,0.7059,0.42622,0.18265,0.28571,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,8,0,1,7,0,0,10,0,0,3,0,0,0,0,0],[28,34,0.8235,0.42411,0.18724,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,9,0,0,11,0,0,4,0,0,4,0,0,1,0,0],[32,34,0.9412,0.49552,0.20511,0.42857,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,13,0,0,3,0,0,8,0,0,2,0,0],[34,34,1.0,0.54683,0.13902,0.42857,0.571,0.71429,0.2857,0.78571,0,0,0,0,0,0,0,0,2,0,0,12,0,0,8,0,0,9,1,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.19643,"x":0.51326,"p":[[0,42,0.0,0.37052,0.36745,0.10714,0.2143,0.64286,0.0,1.0,8,5,0,8,0,8,0,0,4,0,0,2,0,0,2,0,0,0,0,0,3,0,5],[4,42,0.0952,0.37054,0.2392,0.14286,0.35714,0.57143,0.0,1.0,3,1,0,3,0,7,0,0,6,0,0,6,0,0,6,0,0,3,0,0,0,0,1],[8,42,0.1905,0.19643,0.18123,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,15,0,0,5,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[12,42,0.2857,0.34596,0.24941,0.14286,0.2857,0.46418,0.0,0.85714,3,0,0,3,1,8,0,0,7,0,0,5,0,0,2,0,0,4,0,0,2,0,0],[16,42,0.381,0.29911,0.16114,0.14297,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,6,0,0,11,0,0,10,0,0,1,0,0,1,0,0,0,0,0],[20,42,0.4762,0.3683,0.25946,0.14286,0.28571,0.42857,0.0,1.0,2,2,0,2,0,8,0,0,9,0,0,6,0,0,1,1,0,2,0,0,1,0,2],[24,42,0.5714,0.3973,0.23885,0.14289,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,6,0,0,6,0,0,4,0,0,6,0,0,7,0,0,0,0,0],[28,42,0.6667,0.42853,0.25998,0.25,0.42857,0.60714,0.0,1.0,4,1,0,4,0,4,0,0,4,0,0,6,0,0,6,0,0,7,0,0,0,0,1],[32,42,0.7619,0.41962,0.1921,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,9,0,0,7,0,0,5,0,0,6,0,0,0,0,0],[36,42,0.8571,0.43304,0.21572,0.2857,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,7,0,0,5,0,0,9,0,0,3,0,0,7,0,0,1,0,0],[40,42,0.9524,0.51326,0.17824,0.42857,0.4998,0.71429,0.14,0.71429,0,0,0,0,0,2,0,0,4,0,0,10,0,0,5,0,0,11,0,0,0,0,0],[42,42,1.0,0.35712,0.21426,0.14289,0.28571,0.571,0.0,0.71429,2,0,0,2,0,7,0,0,10,0,0,4,0,0,4,0,0,5,0,0,0,0,0]]}]},{"i":"014a37d52896dbf0","q":"Let  $a$  and  $b$  be positive integers. Prove that the numbers  $an^2+b$  and  $a(n+1)^2+b$  are both perfect squares only for finitely many integers  $n$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.79017,"x":0.99107,"p":[[0,11,0.0,0.85267,0.3103,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[4,11,0.3636,0.8125,0.35072,0.85714,1.0,1.0,0.0,1.0,4,22,0,4,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,22],[8,11,0.7273,0.79017,0.37963,0.82143,1.0,1.0,0.0,1.0,5,23,0,5,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,23],[11,11,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":6,"e":1.0,"k":"rising","v":0.82142,"x":1.0,"p":[[0,41,0.0,0.82142,0.33881,0.85714,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,21],[4,41,0.0976,0.83482,0.32948,0.85714,1.0,1.0,0.0,1.0,2,23,0,2,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,23],[8,41,0.1951,0.90178,0.27765,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,27],[12,41,0.2927,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[20,41,0.4878,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,41,0.5854,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,41,0.6829,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,41,0.7805,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[36,41,0.878,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[40,41,0.9756,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"263c268addca1191","q":"Prove that $2^{n}+3^{n}$ is not a perfect cube for any positive integer $n$.","t":[{"b":4,"e":1.0,"k":"rising","v":0.60268,"x":0.97768,"p":[[0,20,0.0,0.61161,0.27947,0.42857,0.4286,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,15,0,0,5,0,0,0,0,0,0,0,10],[4,20,0.2,0.61159,0.28175,0.42857,0.42857,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,16,0,0,2,0,0,0,0,0,1,0,10],[8,20,0.4,0.60268,0.25935,0.42857,0.42857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,17,0,0,5,0,0,0,0,0,0,0,9],[12,20,0.6,0.71875,0.27545,0.42857,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,14,0,0,2,0,0,0,0,0,1,0,15],[16,20,0.8,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[20,20,1.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30]]},{"b":6,"e":0.71429,"k":"flat","v":0.56249,"x":0.71865,"p":[[0,28,0.0,0.71865,0.28919,0.42857,0.78571,1.0,0.14,1.0,0,16,0,0,0,1,0,0,0,0,0,12,0,0,3,0,0,0,0,0,0,0,16],[4,28,0.1429,0.67857,0.26964,0.42857,0.57143,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,15,0,0,4,0,0,0,0,0,0,0,13],[8,28,0.2857,0.61157,0.23211,0.42857,0.571,0.67857,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,15,0,0,9,0,0,0,0,0,0,0,8],[12,28,0.4286,0.56249,0.18189,0.42857,0.57121,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,15,0,0,12,0,0,1,0,0,0,0,4],[16,28,0.5714,0.60261,0.21939,0.42857,0.571,0.64286,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,15,0,0,9,0,0,0,0,0,2,0,6],[20,28,0.7143,0.58029,0.19212,0.42857,0.571,0.57143,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,13,0,0,14,0,0,0,0,0,0,0,5],[24,28,0.8571,0.65177,0.22571,0.42857,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,11,0,0,9,0,0,3,0,0,1,0,8],[28,28,1.0,0.62498,0.22799,0.42857,0.571,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,15,0,0,5,0,0,4,0,0,1,0,7]]}]},{"i":"34ef08a8fcf1e4b4","q":"Let $a, b, c$ be the sides of a triangle, with $a+b+c=1$, and let $n \\geq 2$ be an integer. Show that\n\n$$\n\\sqrt[n]{a^{n}+b^{n}}+\\sqrt[n]{b^{n}+c^{n}}+\\sqrt[n]{c^{n}+a^{n}}<1+\\frac{\\sqrt[n]{2}}{2}\n$$","t":[{"b":4,"e":0.14286,"k":"flat","v":0.14732,"x":0.19196,"p":[[0,26,0.0,0.17393,0.13713,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,26,0.1538,0.14732,0.07563,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.19196,0.11071,0.14286,0.14286,0.17857,0.14286,0.71429,0,0,0,0,0,24,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,26,0.4615,0.17848,0.06191,0.14286,0.14286,0.17857,0.14,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.19196,0.07667,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.14732,"x":0.1875,"p":[[0,23,0.0,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.16072,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.16071,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.1875,0.06621,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.1875,0.06622,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.18304,0.06423,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,23,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"afbfc5493aa81ed3","q":"Let  $ABC$  be a triangle with  $\\angle ABC $  as the largest angle. Let  $R$  be its circumcenter.  Let the circumcircle of triangle  $ARB$  cut  $AC$  again at  $X$ . Prove that  $RX$  is perpendicular to  $BC$ .","t":[{"b":1,"e":0.85714,"k":"rising","v":0.35268,"x":0.84375,"p":[[0,65,0.0,0.56683,0.29977,0.28571,0.42857,0.85714,0.0,1.0,2,6,1,2,0,0,0,0,7,0,0,8,0,0,2,0,0,3,0,0,4,0,6],[4,65,0.0615,0.55354,0.32093,0.28571,0.42857,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,10,0,0,6,0,0,3,0,0,0,0,0,3,0,8],[8,65,0.1231,0.54464,0.27994,0.28571,0.42857,0.75,0.14286,1.0,0,6,0,0,0,1,0,0,11,0,0,6,0,0,3,0,0,3,0,0,2,0,6],[12,65,0.1846,0.63391,0.31731,0.28571,0.4998,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,8,0,0,7,0,0,2,0,0,0,0,0,2,0,12],[16,65,0.2462,0.49999,0.30722,0.28571,0.28571,0.85704,0.0,1.0,1,7,0,1,0,0,0,0,16,0,0,5,0,0,1,0,0,0,0,0,2,0,7],[20,65,0.3077,0.42857,0.22868,0.28571,0.28571,0.46431,0.14286,1.0,0,2,0,0,0,1,0,0,18,0,0,5,0,0,2,0,0,2,0,0,2,0,2],[24,65,0.3692,0.40178,0.23538,0.28571,0.28571,0.42857,0.0,1.0,2,2,0,2,0,0,0,0,16,0,0,9,0,0,0,0,0,1,0,0,2,0,2],[28,65,0.4308,0.41963,0.26471,0.28571,0.42857,0.4642,0.0,1.0,3,4,0,3,0,1,0,0,11,0,0,9,0,0,4,0,0,0,0,0,0,0,4],[32,65,0.4923,0.46875,0.27254,0.28571,0.42857,0.57143,0.0,1.0,1,5,0,1,0,2,0,0,10,0,0,10,0,0,3,0,0,0,0,0,1,0,5],[36,65,0.5538,0.40178,0.20652,0.28571,0.42857,0.42857,0.0,1.0,2,1,0,2,0,1,0,0,10,0,0,14,0,0,2,0,0,0,0,0,2,0,1],[40,65,0.6154,0.39284,0.1675,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,18,0,0,10,0,0,1,0,0,0,0,0,3,0,0],[44,65,0.6769,0.35268,0.2789,0.28571,0.28571,0.42857,0.0,1.0,5,4,0,5,0,1,0,0,17,0,0,4,0,0,1,0,0,0,0,0,0,0,4],[48,65,0.7385,0.375,0.23077,0.28571,0.35714,0.42857,0.0,1.0,3,2,0,3,0,2,0,0,11,0,0,12,0,0,1,0,0,0,0,0,1,0,2],[52,65,0.8,0.41072,0.12753,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,23,0,0,0,0,0,1,0,0,1,0,0],[56,65,0.8615,0.41517,0.19999,0.28571,0.42857,0.42857,0.0,1.0,2,1,0,2,0,1,0,0,6,0,0,19,0,0,1,0,0,0,0,0,2,0,1],[60,65,0.9231,0.82587,0.12747,0.71429,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,17,0,6],[64,65,0.9846,0.83034,0.12598,0.85714,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,21,0,5],[65,65,1.0,0.84375,0.13054,0.85711,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,17,0,8]]},{"b":3,"e":0.28571,"k":"falling","v":0.30803,"x":0.63839,"p":[[0,59,0.0,0.63393,0.29868,0.42857,0.42857,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,5,0,0,11,0,0,0,0,0,3,0,0,1,0,11],[4,59,0.0678,0.45091,0.33713,0.28571,0.28571,0.75,0.0,1.0,4,7,0,4,0,0,0,0,17,0,0,1,0,0,1,0,0,1,0,0,1,0,7],[8,59,0.1356,0.57589,0.34159,0.28571,0.42859,1.0,0.0,1.0,2,11,0,2,0,0,0,0,11,0,0,4,0,0,3,0,0,0,0,0,1,0,11],[12,59,0.2034,0.51339,0.36221,0.2857,0.28571,1.0,0.0,1.0,3,10,0,3,0,0,0,0,17,0,0,0,0,0,0,0,0,1,0,0,1,0,10],[16,59,0.2712,0.49999,0.30093,0.28571,0.28571,0.74996,0.0,1.0,1,6,0,1,0,0,0,0,17,0,0,2,0,0,2,0,0,2,0,0,2,0,6],[20,59,0.339,0.53569,0.34626,0.2857,0.42857,1.0,0.0,1.0,3,9,0,3,0,0,0,0,12,0,0,4,0,0,1,0,0,1,0,0,2,0,9],[24,59,0.4068,0.63839,0.3589,0.28571,0.78571,1.0,0.0,1.0,3,12,0,3,0,1,0,0,6,0,0,3,0,0,2,0,0,1,0,0,4,0,12],[28,59,0.4746,0.58927,0.3458,0.28571,0.49979,1.0,0.0,1.0,3,10,0,3,0,0,0,0,8,0,0,5,0,0,2,0,0,1,0,0,3,0,10],[32,59,0.5424,0.51785,0.3229,0.28571,0.35714,0.85704,0.0,1.0,2,7,0,2,0,0,0,0,14,0,0,4,0,0,1,0,0,1,0,0,3,0,7],[36,59,0.6102,0.40178,0.25111,0.28571,0.28571,0.32143,0.14286,1.0,0,4,0,0,0,1,0,0,23,0,0,3,0,0,0,0,0,0,0,0,1,0,4],[40,59,0.678,0.36161,0.21424,0.28571,0.28571,0.28571,0.0,1.0,1,2,0,1,0,1,0,0,23,0,0,2,0,0,2,0,0,0,0,0,1,0,2],[44,59,0.7458,0.37052,0.19838,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,0,0,0,22,0,0,3,0,0,4,0,0,0,0,0,0,0,2],[48,59,0.8136,0.30803,0.11355,0.28571,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,1,0,0,25,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[52,59,0.8814,0.34374,0.17805,0.28571,0.28571,0.42857,0.0,1.0,2,1,0,2,0,0,0,0,21,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[56,59,0.9492,0.33034,0.09737,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,25,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[59,59,1.0,0.38393,0.15335,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,20,0,0,6,0,0,3,0,0,2,0,0,1,0,0]]}]},{"i":"3c3cc3be58d8adaa","q":"Let  $N{}$  be the number of positive integers with  $10$  digits  $\\overline{d_9d_8\\cdots d_0}$  in base  $10$  (where  $0\\le d_i\\le9$  for all  $i$  and  $d_9>0$ ) such that the polynomial\n\\[d_9x^9+d_8x^8+\\cdots+d_1x+d_0\\]\nis irreducible in  $\\Bbb Q$ . Prove that  $N$  is even.\n\n(A polynomial is irreducible in  $\\Bbb Q$  if it cannot be factored into two non-constant polynomials with rational coefficients.)","t":[{"b":1,"e":1.0,"k":"flat","v":0.95089,"x":0.99554,"p":[[0,9,0.0,0.95089,0.19759,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[4,9,0.4444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,9,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[9,9,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.91518,"x":0.99554,"p":[[0,12,0.0,0.91518,0.25218,1.0,1.0,1.0,0.0,1.0,2,28,1,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[4,12,0.3333,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,12,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]}]},{"i":"c978f9755f4bc029","q":"It is known that each two of the 12 competitors, that participated in the finals of the competition \u201cMathematical duels\u201d, have a common friend among the other 10. Prove that there is one of them that has at least 5 friends among the group.","t":[{"b":2,"e":0.42857,"k":"rising","v":0.24107,"x":0.50452,"p":[[0,27,0.0,0.24107,0.26351,0.0,0.14286,0.42857,0.0,1.0,13,1,2,13,0,5,0,0,2,0,0,8,0,0,1,0,0,2,0,0,0,0,1],[4,27,0.1481,0.32589,0.27019,0.14286,0.28571,0.42857,0.0,1.0,5,2,0,5,0,10,0,0,2,0,0,9,0,0,3,0,0,0,0,0,1,0,2],[8,27,0.2963,0.37502,0.30461,0.10714,0.42857,0.57143,0.0,1.0,8,2,0,8,0,4,0,0,2,0,0,7,0,0,6,0,0,1,0,0,2,0,2],[12,27,0.4444,0.33481,0.25154,0.14286,0.35714,0.4286,0.0,0.85714,7,0,0,7,0,4,0,0,5,0,0,9,0,0,3,0,0,2,0,0,2,0,0],[16,27,0.5926,0.34375,0.32704,0.0,0.28586,0.42857,0.0,1.0,9,4,0,9,0,5,0,0,3,0,0,8,0,0,2,0,0,0,0,0,1,0,4],[20,27,0.7407,0.50452,0.2672,0.42857,0.4286,0.57143,0.0,1.0,2,3,0,2,0,3,0,0,1,0,0,13,0,0,6,0,0,0,0,0,4,0,3],[24,27,0.8889,0.46866,0.24815,0.39286,0.42857,0.60714,0.0,1.0,1,2,0,1,0,5,0,0,2,0,0,14,0,0,2,0,0,4,0,0,2,0,2],[27,27,1.0,0.42856,0.23418,0.39285,0.42857,0.57143,0.0,1.0,3,1,0,3,0,4,0,0,1,0,0,14,0,0,5,0,0,3,0,0,1,0,1]]},{"b":3,"e":0.4286,"k":"flat","v":0.08929,"x":0.42409,"p":[[0,25,0.0,0.16517,0.2146,0.0,0.14286,0.17857,0.0,0.857,14,0,6,14,0,10,0,0,2,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[4,25,0.16,0.3929,0.29234,0.14286,0.42857,0.57143,0.0,1.0,4,3,0,4,0,7,0,0,4,0,0,8,0,0,3,0,0,2,0,0,1,0,3],[8,25,0.32,0.40167,0.29335,0.14286,0.42857,0.571,0.0,1.0,4,2,0,4,0,7,0,0,3,0,0,9,0,0,2,0,0,2,0,0,3,0,2],[12,25,0.48,0.42409,0.28455,0.14289,0.42857,0.60714,0.0,1.0,4,2,0,4,0,5,0,0,3,0,0,10,0,0,2,0,0,4,0,0,2,0,2],[16,25,0.64,0.37051,0.20156,0.1429,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,6,0,0,2,0,0,14,0,0,4,0,0,3,0,0,0,0,0],[20,25,0.8,0.25,0.25254,0.0,0.14288,0.42857,0.0,1.0,10,1,0,10,0,8,0,0,2,0,0,9,0,0,1,0,0,0,0,0,1,0,1],[24,25,0.96,0.125,0.18472,0.0,0.0,0.1786,0.0,0.71429,19,0,0,19,0,5,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[25,25,1.0,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,10,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"6b851f648ae099d6","q":"Let $a<b<c$ be three positive integers. Prove that among any $2 c$ consecutive positive integers there exist three different numbers $x, y, z$ such that $a b c$ divides $x y z$.","t":[{"b":1,"e":0.14,"k":"flat","v":0.21427,"x":0.5089,"p":[[0,40,0.0,0.21427,0.18896,0.0,0.21428,0.28571,0.0,0.57143,10,0,0,10,0,6,0,0,10,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[4,40,0.1,0.5089,0.21409,0.42857,0.57143,0.60714,0.0,1.0,2,1,0,2,0,1,0,0,4,0,0,5,0,0,12,0,0,7,0,0,0,0,1],[8,40,0.2,0.39286,0.23958,0.28571,0.42857,0.57143,0.0,1.0,4,1,0,4,0,3,0,0,7,0,0,8,0,0,5,0,0,4,0,0,0,0,1],[12,40,0.3,0.41514,0.21532,0.28571,0.4998,0.57143,0.0,0.71429,5,0,0,5,0,0,0,0,6,0,0,5,0,0,14,0,0,2,0,0,0,0,0],[16,40,0.4,0.29018,0.23279,0.0,0.28571,0.57141,0.0,0.71429,10,0,0,10,0,1,0,0,9,0,0,3,0,0,8,0,0,1,0,0,0,0,0],[20,40,0.5,0.34375,0.2392,0.0,0.42857,0.57143,0.0,0.71429,9,0,0,9,0,0,0,0,5,0,0,6,0,0,11,0,0,1,0,0,0,0,0],[24,40,0.6,0.33033,0.23805,0.10714,0.28571,0.57141,0.0,0.71429,8,0,0,8,0,2,0,0,7,0,0,4,0,0,9,0,0,2,0,0,0,0,0],[28,40,0.7,0.30347,0.22522,0.14214,0.28571,0.42858,0.0,0.85714,7,0,0,7,0,3,0,0,11,0,0,4,0,0,5,0,0,1,0,0,1,0,0],[32,40,0.8,0.35706,0.21733,0.28571,0.42857,0.57143,0.0,0.85714,6,0,0,6,0,1,0,0,7,0,0,9,0,0,8,0,0,0,0,0,1,0,0],[36,40,0.9,0.3482,0.20182,0.2857,0.35714,0.57111,0.0,0.71429,5,0,0,5,0,2,0,0,9,0,0,7,0,0,8,0,0,1,0,0,0,0,0],[40,40,1.0,0.35267,0.15559,0.28571,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,2,0,0,8,0,0,15,0,0,4,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.20536,"x":0.53566,"p":[[0,37,0.0,0.20536,0.22286,0.0,0.14288,0.32143,0.0,0.71429,14,0,0,14,0,4,0,0,6,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[4,37,0.1081,0.4508,0.21474,0.39286,0.57143,0.57143,0.0,0.71429,4,0,0,4,0,1,0,0,3,0,0,6,0,0,14,0,0,4,0,0,0,0,0],[8,37,0.2162,0.49997,0.22868,0.28571,0.57143,0.60714,0.0,1.0,2,1,0,2,0,1,0,0,7,0,0,2,0,0,12,0,0,6,0,0,1,0,1],[12,37,0.3243,0.45088,0.22047,0.28571,0.57121,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,5,0,0,4,0,0,10,0,0,7,0,0,0,0,0],[16,37,0.4324,0.52676,0.2126,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,4,0,0,2,0,0,14,0,0,7,0,0,2,0,0],[20,37,0.5405,0.43302,0.23002,0.2857,0.42857,0.60714,0.0,0.71429,3,0,0,3,0,3,0,0,6,0,0,6,0,0,6,0,0,8,0,0,0,0,0],[24,37,0.6486,0.53566,0.1821,0.42857,0.57143,0.60714,0.0,1.0,1,1,0,1,0,0,0,0,4,0,0,6,0,0,13,0,0,7,0,0,0,0,1],[28,37,0.7568,0.47765,0.21311,0.28571,0.57143,0.60714,0.0,0.71429,2,0,0,2,0,2,0,0,6,0,0,3,0,0,11,0,0,8,0,0,0,0,0],[32,37,0.8649,0.45088,0.2146,0.28571,0.42857,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,8,0,0,9,0,0,8,0,0,2,0,0,0,0,2],[36,37,0.973,0.38839,0.21498,0.28571,0.42857,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,8,0,0,8,0,0,8,0,0,1,0,0,0,0,1],[37,37,1.0,0.32585,0.22365,0.10717,0.28571,0.57143,0.0,0.57143,8,0,0,8,0,1,0,0,8,0,0,4,0,0,11,0,0,0,0,0,0,0,0]]}]},{"i":"e4b7864002db3250","q":"Let $A B C$ be a triangle and let $\\omega$ be its incircle. Denote by $D_{1}$ and $E_{1}$ the points where $\\omega$ is tangent to sides $B C$ and $A C$, respectively. Denote by $D_{2}$ and $E_{2}$ the points on sides $B C$ and $A C$, respectively, such that $C D_{2}=B D_{1}$ and $C E_{2}=A E_{1}$, and denote by $P$ the point of intersection of segments $A D_{2}$ and $B E_{2}$. Circle $\\omega$ intersects segment $A D_{2}$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $A Q=D_{2} P$.","t":[{"b":1,"e":0.0,"k":"falling","v":0.0625,"x":0.71875,"p":[[0,91,0.0,0.56696,0.23278,0.42857,0.4286,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,12,0,0,5,0,0,4,0,0,1,0,5],[4,91,0.044,0.71875,0.16935,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,13,0,0,4,0,5],[8,91,0.0879,0.70981,0.16165,0.57143,0.71429,0.75,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,15,0,0,5,0,3],[12,91,0.1319,0.60713,0.25754,0.57132,0.71429,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,3,0,0,1,0,0,8,0,0,12,0,0,1,0,4],[16,91,0.1758,0.5982,0.23538,0.57143,0.57143,0.71429,0.0,1.0,3,2,0,3,0,0,0,0,1,0,0,0,0,0,14,0,0,10,0,0,2,0,2],[20,91,0.2198,0.62945,0.23381,0.57143,0.71429,0.71429,0.0,1.0,3,2,0,3,0,0,0,0,0,0,0,0,0,0,11,0,0,13,0,0,3,0,2],[24,91,0.2637,0.60267,0.24415,0.57143,0.64286,0.71429,0.0,1.0,3,2,0,3,0,1,0,0,0,0,0,0,0,0,12,0,0,12,0,0,2,0,2],[28,91,0.3077,0.67409,0.24546,0.57143,0.71429,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,1,0,0,0,0,0,12,0,0,6,0,0,6,0,5],[32,91,0.3516,0.60714,0.22588,0.57143,0.64286,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,3,0,0,0,0,0,11,0,0,12,0,0,2,0,2],[36,91,0.3956,0.58482,0.28652,0.57143,0.57143,0.71429,0.0,1.0,3,5,0,3,0,2,0,0,2,0,0,0,0,0,10,0,0,10,0,0,0,0,5],[40,91,0.4396,0.56695,0.27078,0.57132,0.57143,0.71429,0.0,1.0,4,4,0,4,0,0,0,0,1,0,0,2,0,0,15,0,0,5,0,0,1,0,4],[44,91,0.4835,0.55354,0.26184,0.57132,0.57143,0.71429,0.0,1.0,4,3,0,4,0,0,0,0,2,0,0,0,0,0,17,0,0,5,0,0,1,0,3],[48,91,0.5275,0.5446,0.30396,0.49967,0.57143,0.71429,0.0,1.0,5,4,0,5,0,1,0,0,2,0,0,0,0,0,13,0,0,5,0,0,2,0,4],[52,91,0.5714,0.52232,0.31463,0.28571,0.57143,0.71429,0.0,1.0,6,3,0,6,0,1,0,0,2,0,0,0,0,0,13,0,0,3,0,0,4,0,3],[56,91,0.6154,0.57141,0.22868,0.57143,0.57143,0.71429,0.0,1.0,3,1,0,3,0,0,0,0,2,0,0,0,0,0,15,0,0,9,0,0,2,0,1],[60,91,0.6593,0.54911,0.27225,0.53571,0.57143,0.71429,0.0,1.0,4,2,0,4,0,1,0,0,2,0,0,1,0,0,11,0,0,9,0,0,2,0,2],[64,91,0.7033,0.50445,0.29877,0.42857,0.57143,0.71429,0.0,1.0,7,3,0,7,0,0,0,0,0,0,0,2,0,0,14,0,0,6,0,0,0,0,3],[68,91,0.7473,0.52232,0.27342,0.57143,0.57143,0.71429,0.0,1.0,5,1,0,5,0,2,0,0,0,0,0,0,0,0,13,0,0,10,0,0,1,0,1],[72,91,0.7912,0.52229,0.26392,0.57132,0.57143,0.71429,0.0,0.85714,6,0,0,6,0,0,0,0,0,0,0,0,0,0,15,0,0,9,0,0,2,0,0],[76,91,0.8352,0.56249,0.23402,0.57143,0.57143,0.71429,0.0,0.85714,3,0,0,3,0,1,0,0,1,0,0,1,0,0,14,0,0,8,0,0,4,0,0],[80,91,0.8791,0.46874,0.30353,0.10714,0.57143,0.71429,0.0,1.0,8,1,0,8,0,1,0,0,0,0,0,2,0,0,10,0,0,9,0,0,1,0,1],[84,91,0.9231,0.52676,0.2889,0.42857,0.57143,0.71429,0.0,1.0,5,2,0,5,0,2,0,0,0,0,0,2,0,0,11,0,0,8,0,0,2,0,2],[88,91,0.967,0.45533,0.28668,0.10714,0.57143,0.71429,0.0,0.85714,8,0,0,8,0,1,0,0,0,0,0,1,0,0,13,0,0,8,0,0,1,0,0],[91,91,1.0,0.0625,0.14698,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.02678,"x":0.55357,"p":[[0,69,0.0,0.5357,0.27664,0.28571,0.57143,0.71429,0.0,1.0,2,5,1,2,0,2,0,0,5,0,0,3,0,0,11,0,0,4,0,0,0,0,5],[4,69,0.058,0.55357,0.27836,0.42857,0.71429,0.71429,0.0,1.0,4,1,0,4,0,2,0,0,1,0,0,2,0,0,6,0,0,13,0,0,3,0,1],[8,69,0.1159,0.45535,0.3203,0.14286,0.57143,0.71429,0.0,1.0,6,2,0,6,0,5,0,0,2,0,0,0,0,0,8,0,0,7,0,0,2,0,2],[12,69,0.1739,0.33481,0.31664,0.0,0.28571,0.71429,0.0,0.71429,13,0,0,13,0,3,0,0,0,0,0,1,0,0,6,0,0,9,0,0,0,0,0],[16,69,0.2319,0.25892,0.26349,0.0,0.14286,0.57143,0.0,0.71429,13,0,0,13,0,5,0,0,1,0,0,3,0,0,8,0,0,2,0,0,0,0,0],[20,69,0.2899,0.30356,0.28958,0.0,0.21428,0.57143,0.0,1.0,10,1,0,10,0,6,0,0,4,0,0,1,0,0,6,0,0,4,0,0,0,0,1],[24,69,0.3478,0.48214,0.22232,0.28571,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,2,0,0,4,0,0,1,0,0,15,0,0,7,0,0,0,0,0],[28,69,0.4058,0.31696,0.29609,0.0,0.28571,0.57143,0.0,1.0,12,1,0,12,0,2,0,0,3,0,0,4,0,0,6,0,0,4,0,0,0,0,1],[32,69,0.4638,0.23211,0.28288,0.0,0.14286,0.46418,0.0,1.0,15,1,0,15,0,5,0,0,2,0,0,2,0,0,5,0,0,2,0,0,0,0,1],[36,69,0.5217,0.30802,0.28817,0.0,0.2857,0.57143,0.0,1.0,11,1,0,11,0,4,0,0,4,0,0,0,0,0,10,0,0,2,0,0,0,0,1],[40,69,0.5797,0.23434,0.25639,0.0,0.14286,0.571,0.0,0.71429,15,0,0,15,0,3,0,0,3,0,0,1,0,1,8,0,0,1,0,0,0,0,0],[44,69,0.6377,0.21429,0.28571,0.0,0.0,0.46431,0.0,0.85714,18,0,0,18,0,3,0,0,1,0,0,2,0,0,4,0,0,3,0,0,1,0,0],[48,69,0.6957,0.23213,0.29611,0.0,0.0,0.57111,0.0,1.0,17,1,0,17,0,2,0,0,4,0,0,0,0,0,5,0,0,3,0,0,0,0,1],[52,69,0.7536,0.20982,0.26483,0.0,0.14286,0.57143,0.0,0.71429,15,0,0,15,0,8,0,0,0,0,0,0,0,0,6,0,0,3,0,0,0,0,0],[56,69,0.8116,0.28124,0.32826,0.0,0.14286,0.57143,0.0,1.0,15,2,0,15,0,4,0,0,1,0,0,0,0,0,7,0,0,3,0,0,0,0,2],[60,69,0.8696,0.20089,0.2854,0.0,0.0,0.57143,0.0,0.71429,20,0,0,20,0,2,0,0,0,0,0,1,0,0,5,0,0,4,0,0,0,0,0],[64,69,0.9275,0.29464,0.3071,0.0,0.21428,0.57143,0.0,0.71429,15,0,0,15,0,1,0,0,3,0,0,0,0,0,6,0,0,7,0,0,0,0,0],[68,69,0.9855,0.02678,0.10971,0.0,0.0,0.0,0.0,0.5714,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[69,69,1.0,0.03125,0.12745,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"3bdea53bf75b2d88","q":"Solve in N  $a^2 = 2^b3^c + 1$ .","t":[{"b":1,"e":0.14286,"k":"falling","v":0.33025,"x":0.99107,"p":[[0,72,0.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,72,0.0556,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,72,0.1111,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,72,0.1667,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[16,72,0.2222,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[20,72,0.2778,0.95089,0.1411,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[24,72,0.3333,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,72,0.3889,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,72,0.4444,0.97321,0.05578,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,72,0.5,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[40,72,0.5556,0.92857,0.14726,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,23],[44,72,0.6111,0.94642,0.10566,0.96429,1.0,1.0,0.5714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[48,72,0.6667,0.9375,0.13333,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,25],[52,72,0.7222,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[56,72,0.7778,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[60,72,0.8333,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[64,72,0.8889,0.8213,0.32737,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,21],[68,72,0.9444,0.71866,0.36694,0.39285,0.92857,1.0,0.0,1.0,2,16,0,2,0,5,0,0,1,0,0,1,0,0,0,0,0,3,0,0,4,0,16],[72,72,1.0,0.33025,0.36676,0.105,0.14286,0.60682,0.0,1.0,8,5,0,8,0,12,0,0,3,0,0,0,0,0,1,0,0,1,0,0,2,0,5]]},{"b":5,"e":1.0,"k":"flat","v":0.91964,"x":0.99553,"p":[[0,60,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,60,0.0667,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[8,60,0.1333,0.96427,0.11299,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[12,60,0.2,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[16,60,0.2667,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,60,0.3333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,60,0.4,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[28,60,0.4667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,60,0.5333,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,60,0.6,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[40,60,0.6667,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,60,0.7333,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,60,0.8,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[52,60,0.8667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,60,0.9333,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[60,60,1.0,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,22]]}]},{"i":"35131aa4b599b1a8","q":"Some of the $n$ cities are connected by air routes (all routes are bidirectional). There are exactly $m$ routes. Let $d_{i}$ be the number of routes departing from city $i$, for $i=1,2, \\ldots, n$. If $1 \\leqslant d_{i} \\leqslant 2010$, for every $i=1,2, \\ldots, n$, prove that\n\n$$\n\\sum_{i=1}^{n} d_{i}^{2} \\leqslant 4022 m-2010 n\n$$\n\nDetermine all $n$ for which equality can be achieved.\n\n(Alexander Ili\u0107)","t":[{"b":1,"e":0.85714,"k":"flat","v":0.95089,"x":0.99553,"p":[[0,29,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,29,0.1379,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,29,0.2759,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[12,29,0.4138,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[16,29,0.5517,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,29,0.6897,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[24,29,0.8276,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[28,29,0.9655,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[29,29,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]},{"b":4,"e":1.0,"k":"flat","v":0.97321,"x":0.98661,"p":[[0,6,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,6,0.6667,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[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]]}]},{"i":"56bd5142d8eaec23","q":"It is given isosceles triangle  $ABC$  with  $AB = AC$ .  $AD$  is diameter of circumcircle of triangle  $ABC$ . On the side  $BC$  is chosen point  $E$ . On the sides  $AC, AB$  there are points  $F, G$  respectively such that  $AFEG$  is parallelogram. Prove that  $DE$  is perpendicular to  $FG$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.04911,"x":0.09357,"p":[[0,22,0.0,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.143,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.09357,0.14549,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,22,0.3636,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.06687,0.0712,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0134,"x":0.0758,"p":[[0,32,0.0,0.0758,0.07966,0.0,0.07,0.14286,0.0,0.2857,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.0134,0.04165,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.04893,0.06761,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f6eda2e1d1d40075","q":"Let  $N=\\overline{abcd}$  be a positive integer with four digits. We name *pl\u00e1tano power* to the smallest positive integer  $p(N)=\\overline{\\alpha_1\\alpha_2\\ldots\\alpha_k}$  that can be inserted between  the numbers  $\\overline{ab}$  and  $\\overline{cd}$  in such a way the new number  $\\overline{ab\\alpha_1\\alpha_2\\ldots\\alpha_kcd}$  is divisible by  $N$ . Determine the value of  $p(2025)$ .","t":[{"b":4,"e":0.85714,"k":"flat","v":0.88838,"x":0.94642,"p":[[0,81,0.0,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[4,81,0.0494,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[8,81,0.0988,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[12,81,0.1481,0.91517,0.07874,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[16,81,0.1975,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[20,81,0.2469,0.93302,0.0713,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[24,81,0.2963,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[28,81,0.3457,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[32,81,0.3951,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[36,81,0.4444,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[40,81,0.4938,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[44,81,0.5432,0.88838,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[48,81,0.5926,0.88839,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[52,81,0.642,0.91517,0.07874,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[56,81,0.6914,0.89732,0.07349,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[60,81,0.7407,0.89732,0.07349,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[64,81,0.7901,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[68,81,0.8395,0.89732,0.06423,0.85714,0.85714,1.0,0.85714,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[72,81,0.8889,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[76,81,0.9383,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[80,81,0.9877,0.90624,0.06786,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[81,81,1.0,0.90177,0.06623,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10]]},{"b":5,"e":1.0,"k":"flat","v":0.91071,"x":0.92857,"p":[[0,10,0.0,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[4,10,0.4,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[8,10,0.8,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[10,10,1.0,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12]]}]},{"i":"561d92c7d0a09909","q":"Two circles $\\mathcal{C}$ and $\\mathcal{C}^{\\prime}$ with centers $O$ and $O^{\\prime}$ are externally tangent at $B$. A common external tangent touches $\\mathcal{C}$ at $M$ and $\\mathcal{C}^{\\prime}$ at $N$. The common tangent to $\\mathcal{C}$ and $\\mathcal{C}^{\\prime}$ at $B$ intersects $(M N)$ at $A$. We denote $C$ as the intersection of $(O A)$ and $(B M)$, and $D$ as the intersection of $\\left(O^{\\prime} A\\right)$ and $(B N)$.\nShow that $(C D)$ is parallel to $(M N)$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,20,0.0,0.92857,0.18558,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,27],[4,20,0.2,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,20,0.4,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,20,0.6,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,20,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":1.0,"k":"flat","v":0.87054,"x":1.0,"p":[[0,35,0.0,0.94196,0.21086,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[4,35,0.1143,0.91071,0.22799,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,27],[8,35,0.2286,0.87054,0.24317,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,23],[12,35,0.3429,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,35,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,35,0.6857,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,35,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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]]}]},{"i":"8edc414b06d19f2a","q":"Solve the system\n\\[(4x)_5+7y=14  (2y)_5 -(3x)_7=74\\]\nin the domain of integers, where  $(n)_k$  stands for the multiple of the number  $k$  closest to the number  $n$ .","t":[{"b":1,"e":0.5,"k":"falling","v":0.45533,"x":0.77232,"p":[[0,12,0.0,0.77232,0.2621,0.57143,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,3,0,0,4,0,0,3,0,0,8,0,12],[4,12,0.3333,0.77232,0.24964,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,8,0,0,7,0,11],[8,12,0.6667,0.57143,0.3093,0.28571,0.71429,0.85714,0.0,1.0,1,3,0,1,0,6,0,0,3,0,0,3,0,0,2,0,0,6,0,0,8,0,3],[12,12,1.0,0.45533,0.24597,0.28571,0.42857,0.60714,0.0,1.0,1,1,0,1,0,5,0,0,5,0,0,10,0,0,3,0,0,4,0,0,3,0,1]]},{"b":2,"e":0.71429,"k":"flat","v":0.79016,"x":0.80357,"p":[[0,15,0.0,0.80357,0.22517,0.71429,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,12,0,10],[4,15,0.2667,0.79911,0.23381,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,5,0,0,9,0,12],[8,15,0.5333,0.79464,0.23673,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,8,0,12],[12,15,0.8,0.79911,0.10012,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,17,0,2],[15,15,1.0,0.79016,0.12871,0.71429,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,16,0,3]]}]},{"i":"f9074c5c0016dac3","q":"The integer  $k$  is a *good number*, if we can divide a square into  $k$  squares. How many good numbers not greater than  $2006$  are there? $ \n\\textbf{(A)}\\ 1003\n\\qquad\\textbf{(B)}\\ 1026\n\\qquad\\textbf{(C)}\\ 2000\n\\qquad\\textbf{(D)}\\ 2003\n\\qquad\\textbf{(E)}\\ 2004\n$","t":[{"b":3,"e":1.0,"k":"volatile","v":0.45982,"x":0.90625,"p":[[0,12,0.0,0.45982,0.19475,0.28571,0.42857,0.57143,0.0,0.85714,2,0,1,2,0,0,0,0,7,0,0,10,0,0,7,0,0,5,0,0,1,0,0],[4,12,0.3333,0.90625,0.10479,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,16],[8,12,0.6667,0.90625,0.10479,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,16],[12,12,1.0,0.83482,0.13882,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,10,0,10]]},{"b":4,"e":1.0,"k":"rising","v":0.48214,"x":0.96875,"p":[[0,30,0.0,0.48214,0.15465,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,10,0,0,11,0,0,3,0,0,1,0,0],[4,30,0.1333,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[8,30,0.2667,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[12,30,0.4,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[16,30,0.5333,0.90179,0.13092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,18],[20,30,0.6667,0.9375,0.08703,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20],[24,30,0.8,0.90179,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,14],[28,30,0.9333,0.88839,0.12234,0.85714,0.85714,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,15,0,13],[30,30,1.0,0.90624,0.10483,0.85714,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,15]]}]},{"i":"0ee141ecdda8c43e","q":"Find the number of different numbers of the form  $\\left\\lfloor\\frac{i^2}{2015} \\right\\rfloor$ , with  $i = 1,2, ..., 2015$ .","t":[{"b":1,"e":0.57143,"k":"falling","v":0.39286,"x":0.72321,"p":[[0,38,0.0,0.64732,0.4103,0.32143,0.78571,1.0,0.0,1.0,8,15,5,8,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,1,0,15],[4,38,0.1053,0.62945,0.38109,0.2857,0.71429,1.0,0.0,1.0,4,13,0,4,0,3,0,0,4,0,0,0,0,0,2,0,0,5,0,0,1,0,13],[8,38,0.2105,0.51344,0.36745,0.24999,0.57143,0.78571,0.0,1.0,6,8,0,6,0,2,0,0,6,0,0,1,0,0,3,0,0,6,0,0,0,0,8],[12,38,0.3158,0.53563,0.38969,0.14286,0.50001,1.0,0.0,1.0,5,10,0,5,0,5,0,0,3,0,0,3,0,0,2,0,0,2,0,0,2,0,10],[16,38,0.4211,0.39286,0.32927,0.0,0.35714,0.71429,0.0,1.0,10,1,0,10,0,1,0,0,5,0,0,3,0,0,2,0,0,7,0,0,3,0,1],[20,38,0.5263,0.58482,0.32411,0.42857,0.64286,0.89286,0.0,1.0,3,8,0,3,0,2,0,0,2,0,0,8,0,0,1,0,0,7,0,0,1,0,8],[24,38,0.6316,0.64286,0.27894,0.53572,0.57143,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,2,0,0,3,0,0,9,0,0,5,0,0,2,0,8],[28,38,0.7368,0.72321,0.27418,0.53571,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,4,0,0,5,0,0,4,0,0,2,0,13],[32,38,0.8421,0.63838,0.25997,0.53539,0.64286,0.75,0.14286,1.0,0,7,0,0,0,3,0,0,2,0,0,3,0,0,8,0,0,8,0,0,1,0,7],[36,38,0.9474,0.54906,0.17535,0.4286,0.57143,0.60714,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,5,0,0,14,0,0,6,0,0,1,0,1],[38,38,1.0,0.39731,0.19473,0.14289,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,9,0,0,3,0,0,10,0,0,6,0,0,4,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"falling","v":0.44643,"x":0.74106,"p":[[0,114,0.0,0.74106,0.37362,0.53539,1.0,1.0,0.0,1.0,4,20,3,4,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,20],[4,114,0.0351,0.70089,0.37177,0.39286,1.0,1.0,0.0,1.0,3,17,0,3,0,3,0,0,2,0,0,2,0,0,0,0,0,5,0,0,0,0,17],[8,114,0.0702,0.56696,0.35978,0.24999,0.57143,1.0,0.0,1.0,2,10,0,2,0,6,0,0,4,0,0,3,0,0,2,0,0,4,0,0,1,0,10],[12,114,0.1053,0.53125,0.35217,0.28571,0.5,0.89275,0.0,1.0,4,8,0,4,0,2,0,0,8,0,0,2,0,0,2,0,0,5,0,0,1,0,8],[16,114,0.1404,0.47321,0.38206,0.14286,0.28571,0.85714,0.0,1.0,6,7,0,6,0,5,0,0,6,0,0,1,0,0,1,0,0,3,0,0,3,0,7],[20,114,0.1754,0.45534,0.36323,0.14286,0.42857,0.75,0.0,1.0,5,7,0,5,0,7,0,0,3,0,0,4,0,0,3,0,0,2,0,0,1,0,7],[24,114,0.2105,0.44643,0.3567,0.10714,0.42857,0.71429,0.0,1.0,8,6,0,8,0,3,0,0,1,0,0,7,0,0,3,0,0,4,0,0,0,0,6],[28,114,0.2456,0.45536,0.29974,0.2857,0.42857,0.71429,0.0,1.0,3,4,0,3,0,4,0,0,8,0,0,4,0,0,3,0,0,6,0,0,0,0,4],[32,114,0.2807,0.66964,0.3597,0.28571,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,5,0,0,4,0,0,1,0,0,1,0,0,2,0,15],[36,114,0.3158,0.52678,0.36498,0.28571,0.42857,0.89286,0.0,1.0,5,8,0,5,0,1,0,0,9,0,0,2,0,0,0,0,0,5,0,0,2,0,8],[40,114,0.3509,0.5625,0.30291,0.39286,0.57143,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,4,0,0,6,0,0,8,0,0,1,0,0,2,0,7],[44,114,0.386,0.60713,0.33693,0.28571,0.57121,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,5,0,0,5,0,0,3,0,0,2,0,0,0,0,12],[48,114,0.4211,0.53125,0.31589,0.28571,0.57143,0.71429,0.0,1.0,2,7,0,2,0,4,0,0,5,0,0,4,0,0,6,0,0,4,0,0,0,0,7],[52,114,0.4561,0.4641,0.36261,0.14286,0.42857,0.74996,0.0,1.0,5,6,0,5,0,7,0,0,3,0,0,3,0,0,2,0,0,4,0,0,2,0,6],[56,114,0.4912,0.54016,0.31689,0.28571,0.4998,0.74996,0.0,1.0,3,7,0,3,0,2,0,0,4,0,0,7,0,0,5,0,0,3,0,0,1,0,7],[60,114,0.5263,0.52675,0.2911,0.28571,0.57121,0.71429,0.0,1.0,3,3,0,3,0,3,0,0,3,0,0,4,0,0,7,0,0,6,0,0,3,0,3],[64,114,0.5614,0.50447,0.26483,0.28571,0.57143,0.71429,0.0,1.0,2,2,0,2,0,3,0,0,6,0,0,4,0,0,3,0,0,12,0,0,0,0,2],[68,114,0.5965,0.52678,0.32623,0.2857,0.5,0.85714,0.0,1.0,1,6,0,1,0,6,0,0,7,0,0,2,0,0,3,0,0,4,0,0,3,0,6],[72,114,0.6316,0.6317,0.30152,0.42857,0.71429,0.85714,0.0,1.0,3,7,0,3,0,0,0,0,4,0,0,2,0,1,2,0,0,11,0,0,2,0,7],[76,114,0.6667,0.63838,0.24996,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,6,0,0,4,0,0,8,0,0,3,0,6],[80,114,0.7018,0.6607,0.26667,0.42859,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,2,0,0,4,0,0,3,0,0,10,0,0,3,0,7],[84,114,0.7368,0.47757,0.32074,0.2857,0.42836,0.71429,0.0,1.0,4,4,0,4,0,3,0,0,9,0,0,0,0,0,3,0,0,8,0,0,1,0,4],[88,114,0.7719,0.53125,0.32583,0.28571,0.42859,0.78571,0.14286,1.0,0,8,0,0,0,7,0,0,6,0,0,5,0,0,1,0,0,5,0,0,0,0,8],[92,114,0.807,0.60268,0.27833,0.42857,0.64286,0.75,0.0,1.0,1,5,0,1,0,3,0,0,3,0,0,3,0,0,6,0,0,8,0,0,3,0,5],[96,114,0.8421,0.60267,0.26901,0.42857,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,5,0,0,6,0,0,2,0,0,8,0,0,5,0,4],[100,114,0.8772,0.69643,0.24936,0.4286,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,8,0,0,3,0,9],[104,114,0.9123,0.57367,0.18939,0.42857,0.60721,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,6,0,0,4,1,0,13,0,0,1,0,1],[108,114,0.9474,0.54464,0.2635,0.28571,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,10,0,0,3,0,0,3,0,0,9,0,0,1,0,4],[112,114,0.9825,0.62054,0.20395,0.42859,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,5,0,0,3,0,0,14,0,0,3,0,2],[114,114,1.0,0.46874,0.19637,0.28571,0.42857,0.60714,0.14286,1.0,0,1,0,0,0,2,0,0,8,0,0,11,0,0,3,0,0,7,0,0,0,0,1]]}]},{"i":"ffff26a3806d40f1","q":"Solve the inequation:   $$ 5\\mid x\\mid \\leq x(3x+2-2\\sqrt{8-2x-x^2}) $$","t":[{"b":3,"e":1.0,"k":"flat","v":0.99553,"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],[4,51,0.0784,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,51,0.1569,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,51,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,51,0.3137,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,51,0.3922,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,51,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,51,0.549,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,51,0.6275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[40,51,0.7843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,51,0.8627,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,51,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[51,51,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.42857,"k":"falling","v":0.65625,"x":1.0,"p":[[0,47,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,47,0.0851,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,47,0.1702,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,47,0.2553,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,47,0.3404,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,47,0.5106,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,47,0.6809,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[44,47,0.9362,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[47,47,1.0,0.65625,0.24707,0.42857,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,14,0,0,6,0,0,0,0,0,3,0,9]]}]},{"i":"04d3b13a59caf2ea","q":"It is said that an integer $k$ is olympic if there exist four integers $a, b, c$ and $d$, all coprime with $k$, such that $k$ divides $a^{4}+b^{4}+c^{4}+d^{4}$. Let $n$ be any integer.\nShow that $n^{2}-2$ is olympic.","t":[{"b":4,"e":0.0,"k":"falling","v":0.0625,"x":0.84374,"p":[[0,106,0.0,0.66517,0.23037,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,5,0,0,2,0,0,8,0,0,6,0,0,6,0,5],[4,106,0.0377,0.78124,0.23416,0.67857,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,0,9,0,11],[8,106,0.0755,0.84374,0.18683,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,8,0,15],[12,106,0.1132,0.74999,0.25255,0.57143,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,2,0,0,4,0,13],[16,106,0.1509,0.75446,0.1996,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,6,0,0,6,0,9],[20,106,0.1887,0.67856,0.28122,0.57132,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,2,0,0,2,0,0,6,0,0,6,0,0,5,0,8],[24,106,0.2264,0.63838,0.27196,0.42857,0.64286,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,5,0,0,5,0,0,5,0,0,4,0,0,6,0,6],[28,106,0.2642,0.61161,0.24545,0.42857,0.64286,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,5,0,0,3,0,0,6,0,0,6,0,0,8,0,2],[32,106,0.3019,0.52907,0.32081,0.2857,0.571,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,6,0,1,2,0,0,6,0,0,2,0,0,3,0,6],[36,106,0.3396,0.56694,0.29121,0.42857,0.57143,0.71429,0.0,1.0,3,4,0,3,0,2,0,0,2,0,0,4,0,0,7,0,0,7,0,0,3,0,4],[40,106,0.3774,0.59821,0.21558,0.42857,0.57143,0.75,0.28571,1.0,0,2,0,0,0,0,0,0,5,0,0,7,0,0,7,0,0,5,0,0,6,0,2],[44,106,0.4151,0.59823,0.21555,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,6,0,0,9,0,0,5,0,0,4,0,3],[48,106,0.4528,0.58471,0.2536,0.42857,0.57143,0.85714,0.14,1.0,0,2,0,0,0,3,0,0,4,0,0,5,0,0,7,0,0,3,0,0,8,0,2],[52,106,0.4906,0.57584,0.26362,0.42857,0.57143,0.71429,0.0,1.0,3,4,0,3,0,0,0,0,1,0,0,7,0,0,9,0,0,6,0,0,2,0,4],[56,106,0.5283,0.58036,0.23941,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,4,0,0,6,0,0,5,0,0,9,0,0,4,0,2],[60,106,0.566,0.50446,0.24996,0.28571,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,2,0,0,6,0,0,5,0,0,6,0,0,6,0,0,5,0,0],[64,106,0.6038,0.48658,0.22262,0.42857,0.571,0.57143,0.0,1.0,3,1,0,3,0,0,0,0,4,0,0,8,0,0,11,0,0,4,0,0,1,0,1],[68,106,0.6415,0.35263,0.22578,0.14286,0.35729,0.571,0.0,0.85714,5,0,0,5,0,4,0,0,7,0,0,6,0,0,8,0,0,1,0,0,1,0,0],[72,106,0.6792,0.41068,0.23074,0.28571,0.42857,0.57111,0.0,0.85714,2,0,0,2,0,4,0,0,9,0,0,6,0,0,6,0,0,2,0,0,3,0,0],[76,106,0.717,0.28125,0.18029,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,9,0,0,8,0,0,6,0,0,5,0,0,0,0,0,0,0,0],[80,106,0.7547,0.41964,0.23673,0.2857,0.35714,0.57143,0.0,1.0,1,1,0,1,0,5,0,0,10,0,0,5,0,0,4,0,0,5,0,0,1,0,1],[84,106,0.7925,0.32579,0.23759,0.14286,0.28571,0.57143,0.0,0.71429,4,0,0,4,0,10,0,0,6,0,0,1,0,0,7,0,0,4,0,0,0,0,0],[88,106,0.8302,0.23659,0.23036,0.0,0.14286,0.42857,0.0,0.71429,11,0,0,11,0,7,0,0,4,0,0,3,0,0,6,0,0,1,0,0,0,0,0],[92,106,0.8679,0.24554,0.18977,0.14286,0.21428,0.28571,0.0,0.85714,5,0,0,5,0,11,0,0,9,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[96,106,0.9057,0.2857,0.22866,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,10,0,0,8,0,0,2,0,0,4,0,0,2,0,0,1,0,0],[100,106,0.9434,0.10714,0.15152,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,6,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[104,106,0.9811,0.09821,0.1448,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[106,106,1.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.38168,"x":0.70981,"p":[[0,82,0.0,0.67857,0.26486,0.42859,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,6,0,0,3,0,0,9,0,0,3,0,8],[4,82,0.0488,0.66071,0.29179,0.42857,0.71429,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,4,0,0,2,0,0,0,0,0,10,0,0,8,0,5],[8,82,0.0976,0.70981,0.2328,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,2,0,0,6,0,0,6,0,0,7,0,7],[12,82,0.1463,0.58481,0.32214,0.42857,0.57143,1.0,0.0,1.0,3,9,0,3,0,1,0,0,3,0,0,7,0,0,6,0,0,2,0,0,1,0,9],[16,82,0.1951,0.53122,0.24803,0.39286,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,5,0,0,5,0,0,9,0,0,4,0,0,5,0,1],[20,82,0.2439,0.53569,0.22868,0.39286,0.57143,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,6,0,0,5,0,0,9,0,0,5,0,0,4,0,1],[24,82,0.2927,0.43749,0.24205,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,3,0,0,8,0,0,8,0,0,4,0,0,4,0,0,2,0,1],[28,82,0.3415,0.52674,0.24337,0.28571,0.57121,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,6,0,0,4,0,0,10,0,0,4,0,0,3,0,2],[32,82,0.3902,0.44631,0.19492,0.28571,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,8,0,0,4,0,0,11,0,0,5,0,0,0,0,0],[36,82,0.439,0.46872,0.22082,0.2857,0.42857,0.60714,0.14286,1.0,0,1,0,0,0,4,0,0,8,0,0,6,0,0,6,0,0,6,0,0,1,0,1],[40,82,0.4878,0.49551,0.22863,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,7,0,0,7,0,0,8,0,0,3,0,0,2,0,2],[44,82,0.5366,0.51783,0.22516,0.39286,0.571,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,7,0,0,10,0,0,1,0,0,5,0,1],[48,82,0.5854,0.57141,0.22587,0.42857,0.57143,0.74996,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,7,0,0,8,0,0,4,0,0,8,0,0],[52,82,0.6341,0.5357,0.20516,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,8,0,0,9,0,0,5,0,0,3,0,1],[56,82,0.6829,0.46871,0.17213,0.39286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,12,0,0,9,0,0,1,0,0,1,0,1],[60,82,0.7317,0.49554,0.17122,0.42857,0.4286,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,1,0,0,16,0,0,6,0,0,6,0,0,1,0,0],[64,82,0.7805,0.44641,0.17404,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,6,0,0,12,0,0,8,0,0,1,0,0,2,0,0],[68,82,0.8293,0.4687,0.20586,0.28571,0.42859,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,6,0,0,8,0,0,12,0,0,0,0,0,2,0,1],[72,82,0.878,0.45087,0.20549,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,4,0,0,8,0,0,7,0,0,8,0,0,2,0,0,3,0,0],[76,82,0.9268,0.46429,0.16752,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,12,0,0,8,0,0,1,0,0,1,0,1],[80,82,0.9756,0.41514,0.16884,0.28571,0.42857,0.571,0.0,0.71429,1,0,0,1,0,0,0,0,14,0,0,7,0,0,6,0,0,4,0,0,0,0,0],[82,82,1.0,0.38168,0.12575,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,9,0,1,15,0,0,3,0,0,1,0,0,0,0,0]]}]},{"i":"57b9ef69b2e64e41","q":"Let  $n$  and  $m$  be natural numbers such that  $m+\u0006 i=a_ib_i^2$  for  $i=1,2, \\cdots\u0002 n$ \u000e where  $a_i$  and  $b_i$  are natural numbers and  $a_i$  is not divisible by a square of a prime number.\nFind all  $n$  for which there exists an  $m$  such that  $\\sum_{i=1}^{n}a_i=12$","t":[{"b":0,"e":0.14286,"k":"flat","v":0.52227,"x":0.87945,"p":[[0,142,0.0,0.56693,0.27312,0.42857,0.57143,0.85714,0.0,1.0,3,1,1,3,0,1,0,0,1,0,0,8,0,0,6,0,0,3,0,0,9,0,1],[4,142,0.0282,0.87945,0.12428,0.857,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,13,0,13],[8,142,0.0563,0.84372,0.18684,0.71429,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,5,0,16],[12,142,0.0845,0.8348,0.17171,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,8,0,0,7,0,13],[16,142,0.1127,0.82142,0.20517,0.71429,0.85714,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,8,0,12],[20,142,0.1408,0.79909,0.20783,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,8,0,0,6,0,12],[24,142,0.169,0.81245,0.18368,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,7,0,12],[28,142,0.1972,0.81695,0.16843,0.67857,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,9,0,11],[32,142,0.2254,0.74996,0.17861,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,7,0,0,9,0,6],[36,142,0.2535,0.7321,0.21944,0.571,0.85707,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,5,0,0,4,0,0,2,0,0,14,0,5],[40,142,0.2817,0.69193,0.21162,0.571,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,5,0,0,13,0,2],[44,142,0.3099,0.67408,0.21794,0.571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,6,0,0,11,0,0,0,0,0,10,0,4],[48,142,0.338,0.79461,0.17837,0.67857,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,15,0,7],[52,142,0.3662,0.71873,0.20973,0.57143,0.857,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,1,0,0,12,0,5],[56,142,0.3944,0.67851,0.21431,0.571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,2,0,0,10,0,0,4,0,0,10,0,3],[60,142,0.4225,0.75889,0.20655,0.57143,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,0,0,0,13,0,7],[64,142,0.4507,0.66514,0.18423,0.571,0.71429,0.857,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,7,0,0,8,0,2],[68,142,0.4789,0.68746,0.20343,0.571,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,3,0,0,13,0,2],[72,142,0.507,0.73213,0.21054,0.57143,0.78564,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,5,0,0,3,0,0,7,0,0,11,0,5],[76,142,0.5352,0.69639,0.24681,0.5354,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,5,0,0,6,0,0,3,0,0,8,0,7],[80,142,0.5634,0.71413,0.18898,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,3,0,0,11,0,4],[84,142,0.5915,0.74104,0.18709,0.57143,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,3,0,0,14,0,4],[88,142,0.6197,0.73655,0.23452,0.57132,0.85707,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,3,0,0,9,0,8],[92,142,0.6479,0.73659,0.21462,0.57143,0.85707,0.85714,0.0,1.0,1,4,1,1,0,0,0,0,0,0,0,3,0,0,6,0,0,4,0,0,14,0,4],[96,142,0.6761,0.70532,0.21111,0.5354,0.78564,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,3,0,0,12,0,4],[100,142,0.7042,0.7589,0.19378,0.57143,0.85707,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,4,0,0,14,0,5],[104,142,0.7324,0.67856,0.2342,0.53539,0.71429,0.85714,0.1429,1.0,0,3,0,0,0,1,0,0,3,0,0,4,0,0,5,0,0,4,0,0,12,0,3],[108,142,0.7606,0.58926,0.20438,0.4286,0.57143,0.71429,0.1429,1.0,0,1,0,0,0,2,0,0,2,0,0,5,0,0,11,0,0,6,0,0,5,0,1],[112,142,0.7887,0.66961,0.18364,0.57132,0.64286,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,5,0,0,9,0,2],[116,142,0.8169,0.70087,0.21237,0.57143,0.78564,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,3,0,0,9,0,0,2,0,0,13,0,3],[120,142,0.8451,0.66512,0.21904,0.4286,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,4,0,0,10,0,3],[124,142,0.8732,0.66071,0.22517,0.42859,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,6,0,0,3,0,0,7,0,0,12,0,1],[128,142,0.9014,0.70979,0.20668,0.57132,0.85707,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,2,0,0,14,0,3],[132,142,0.9296,0.60713,0.19233,0.42857,0.57143,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,10,0,0,7,0,0,5,0,0,7,0,1],[136,142,0.9577,0.5803,0.20805,0.42857,0.571,0.85704,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,8,0,0,9,0,0,0,0,0,10,0,0],[140,142,0.9859,0.52227,0.14984,0.42857,0.4286,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,18,0,0,8,0,0,1,0,0,4,0,0],[142,142,1.0,0.57585,0.19059,0.42859,0.4998,0.74996,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,14,0,0,5,0,0,3,0,0,8,0,0]]},{"b":2,"e":0.2857,"k":"flat","v":0.54018,"x":0.86604,"p":[[0,91,0.0,0.54018,0.28735,0.42857,0.57143,0.71429,0.0,1.0,4,3,2,4,0,1,0,0,2,0,0,5,0,0,8,0,0,6,0,0,3,0,3],[4,91,0.044,0.86604,0.16347,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,5,0,17],[8,91,0.0879,0.81249,0.18363,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,11,0,10],[12,91,0.1319,0.82138,0.19567,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,9,0,13],[16,91,0.1758,0.78568,0.19235,0.67857,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,5,0,0,10,0,9],[20,91,0.2198,0.81697,0.1996,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,6,0,0,7,0,13],[24,91,0.2637,0.75439,0.21205,0.57132,0.857,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,4,0,0,9,0,8],[28,91,0.3077,0.65174,0.19866,0.5354,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,6,0,0,9,0,0,4,0,0,9,0,2],[32,91,0.3516,0.72318,0.18191,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,6,0,0,10,0,4],[36,91,0.3956,0.70981,0.22156,0.57132,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,4,0,0,9,0,6],[40,91,0.4396,0.75442,0.18981,0.57143,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,5,0,0,13,0,5],[44,91,0.4835,0.69195,0.20858,0.4286,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,2,0,0,11,0,4],[48,91,0.5275,0.76784,0.20748,0.67857,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,2,0,0,17,0,5],[52,91,0.5714,0.65622,0.1984,0.5354,0.57143,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,6,0,0,9,0,0,2,0,0,12,0,1],[56,91,0.6154,0.70979,0.16934,0.57143,0.78564,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,2,0,0,15,0,1],[60,91,0.6593,0.63391,0.21409,0.42857,0.57143,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,8,0,0,6,0,0,4,0,0,9,0,2],[64,91,0.7033,0.66511,0.21903,0.571,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,5,0,0,10,0,0,3,0,0,8,0,4],[68,91,0.7473,0.7455,0.23073,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,3,0,0,7,0,0,0,0,0,13,0,7],[72,91,0.7912,0.75446,0.21199,0.67857,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,6,0,0,12,0,6],[76,91,0.8352,0.7455,0.19147,0.57143,0.85707,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,5,0,0,12,0,5],[80,91,0.8791,0.67851,0.22869,0.571,0.857,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,2,0,0,6,0,0,1,0,0,18,0,0],[84,91,0.9231,0.71426,0.15569,0.57143,0.71429,0.85714,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,6,0,0,13,0,1],[88,91,0.967,0.68301,0.19145,0.5354,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,5,0,0,13,0,1],[91,91,1.0,0.60262,0.19144,0.42859,0.571,0.85704,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,11,0,0,6,0,0,4,0,0,9,0,0]]}]},{"i":"65f12b8cd089f24d","q":"Show that for any integer  $n \\ge 2$  the sum of the fractions  $\\frac{1}{ab}$ , where  $a$  and  $b$  are relatively prime positive integers such that  $a < b \\le n$  and  $a+b > n$ , equals  $\\frac{1}{2}$ .\n(Integers  $a$  and  $b$  are called relatively prime if the greatest common divisor of  $a$  and  $b$  is  $1$ .)","t":[{"b":0,"e":0.42857,"k":"falling","v":0.3884,"x":1.0,"p":[[0,38,0.0,0.70089,0.42161,0.42857,1.0,1.0,0.0,1.0,8,19,0,8,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,19],[4,38,0.1053,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,38,0.2105,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[12,38,0.3158,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[16,38,0.4211,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[20,38,0.5263,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,38,0.6316,0.96427,0.12376,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[28,38,0.7368,0.85714,0.30929,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,26],[32,38,0.8421,0.62054,0.37561,0.39286,0.4286,1.0,0.0,1.0,2,15,0,2,0,5,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,15],[36,38,0.9474,0.44197,0.31412,0.14286,0.42857,0.71429,0.0,1.0,3,5,0,3,0,8,0,0,0,0,0,12,0,0,0,0,0,4,0,0,0,0,5],[38,38,1.0,0.3884,0.28846,0.14286,0.42857,0.4286,0.0,1.0,4,3,0,4,0,8,0,0,1,0,0,12,0,0,0,0,0,4,0,0,0,0,3]]},{"b":7,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,47,0.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[4,47,0.0851,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,47,0.1702,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[12,47,0.2553,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,47,0.3404,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[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],[24,47,0.5106,0.94196,0.19187,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[28,47,0.5957,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,47,0.6809,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,47,0.766,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,47,0.8511,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,47,0.9362,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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":"a5e4e824c7a5911e","q":"Let  $n\\geq 2$  be an even integer. Find the greatest integer  $m\\geq 2^{n-2}+1$  such that there exist  $m$  distinct subsets of  $\\{1,2,\\dots ,n\\}$ , any  $2^{n-2}+1$  of them having empty intersection.\n\n*Cristi S\u0103vescu*","t":[{"b":1,"e":0.14,"k":"falling","v":0.14286,"x":0.99554,"p":[[0,58,0.0,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[4,58,0.069,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,58,0.1379,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,58,0.2069,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,58,0.2759,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[20,58,0.3448,0.87946,0.28372,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,26],[24,58,0.4138,0.89732,0.24284,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,26],[28,58,0.4828,0.75446,0.38338,0.14286,1.0,1.0,0.14286,1.0,0,22,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,22],[32,58,0.5517,0.78795,0.35602,0.75,1.0,1.0,0.14286,1.0,0,22,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,22],[36,58,0.6207,0.90179,0.26351,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[40,58,0.6897,0.53125,0.40913,0.14286,0.28571,1.0,0.14286,1.0,0,13,0,0,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,13],[44,58,0.7586,0.37054,0.35149,0.14286,0.14286,0.5,0.14286,1.0,0,7,0,0,0,20,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,7],[48,58,0.8276,0.3125,0.3223,0.14286,0.14286,0.28571,0.14286,1.0,0,5,0,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[52,58,0.8966,0.25,0.25254,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,25,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[56,58,0.9655,0.18304,0.15663,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[58,58,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.2009,"x":1.0,"p":[[0,61,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,61,0.0656,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,61,0.1311,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,61,0.1967,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,61,0.2623,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,61,0.3279,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,61,0.3934,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,61,0.459,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,61,0.5246,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,61,0.5902,0.9375,0.20806,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[40,61,0.6557,0.91071,0.25191,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[44,61,0.7213,0.81696,0.34669,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[48,61,0.7869,0.70973,0.40179,0.14286,1.0,1.0,0.14,1.0,0,21,0,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[52,61,0.8525,0.67858,0.39929,0.14286,1.0,1.0,0.14286,1.0,0,18,0,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,18],[56,61,0.918,0.59821,0.42773,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[60,61,0.9836,0.21429,0.21724,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[61,61,1.0,0.2009,0.20782,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]}]},{"i":"0ff2d77b76f22bc6","q":"Let  $n$  be an even positive integer. A sequence of  $n$  real numbers is called complete if for every integer  $m$  with  $1 \\leq  m \\leq  n$  either the sum of the first  $m$  terms of the sum or the sum of the last  $m$  terms is integral. Determine\nthe minimum number of integers in a complete sequence of  $n$  numbers.","t":[{"b":4,"e":0.0,"k":"falling","v":0.02679,"x":0.71873,"p":[[0,72,0.0,0.53125,0.35577,0.14286,0.57143,0.85714,0.0,1.0,6,4,5,6,0,3,0,0,2,0,0,2,0,0,4,0,0,4,0,0,7,0,4],[4,72,0.0556,0.71873,0.38711,0.49968,1.0,1.0,0.0,1.0,5,18,0,5,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,0,2,0,18],[8,72,0.1111,0.49551,0.42104,0.0,0.42857,1.0,0.0,1.0,9,10,0,9,0,3,0,0,3,0,0,2,0,0,2,0,0,0,0,0,3,0,10],[12,72,0.1667,0.61158,0.38172,0.28571,0.71429,1.0,0.0,1.0,7,9,0,7,0,0,0,0,3,0,0,0,0,0,2,0,0,6,0,0,5,0,9],[16,72,0.2222,0.23213,0.33263,0.0,0.0,0.4642,0.0,1.0,19,2,0,19,0,2,0,0,2,0,0,1,0,0,1,0,0,5,0,0,0,0,2],[20,72,0.2778,0.31248,0.35968,0.0,0.14286,0.60714,0.0,1.0,15,2,0,15,0,2,0,0,3,0,0,2,0,0,2,0,0,2,0,0,4,0,2],[24,72,0.3333,0.28124,0.37197,0.0,0.0,0.46418,0.0,1.0,19,3,0,19,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,3,0,3],[28,72,0.3889,0.35714,0.40406,0.0,0.14286,0.71429,0.0,1.0,15,5,0,15,0,2,0,0,3,0,0,0,0,0,0,0,0,5,0,0,2,0,5],[32,72,0.4444,0.29017,0.35621,0.0,0.07143,0.60682,0.0,1.0,16,3,0,16,0,2,0,0,3,0,0,2,0,0,1,0,0,4,0,0,1,0,3],[36,72,0.5,0.22768,0.31715,0.0,0.0,0.32144,0.0,1.0,18,1,0,18,0,2,0,0,4,0,0,1,0,0,1,0,0,3,0,0,2,0,1],[40,72,0.5556,0.21872,0.35709,0.0,0.0,0.42858,0.0,1.0,22,4,0,22,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,4],[44,72,0.6111,0.20536,0.31931,0.0,0.0,0.32144,0.0,1.0,20,1,0,20,0,2,0,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,1],[48,72,0.6667,0.12945,0.27046,0.0,0.0,0.03572,0.0,1.0,24,2,0,24,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[52,72,0.7222,0.12946,0.25091,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[56,72,0.7778,0.07589,0.21124,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[60,72,0.8333,0.13839,0.29555,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[64,72,0.8889,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.09374,0.1943,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[72,72,1.0,0.04464,0.14033,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.125,"x":0.79015,"p":[[0,72,0.0,0.48211,0.27138,0.28571,0.42859,0.71429,0.0,0.85714,4,0,2,4,0,0,0,0,8,0,0,5,0,0,3,0,0,7,0,0,5,0,0],[4,72,0.0556,0.53571,0.43153,0.0,0.71429,1.0,0.0,1.0,11,9,0,11,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,9],[8,72,0.1111,0.79015,0.26243,0.67857,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,0,4,0,15],[12,72,0.1667,0.77232,0.2854,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,4,0,0,1,0,0,2,0,0,6,0,0,2,0,16],[16,72,0.2222,0.69192,0.2882,0.5354,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,6,0,0,1,0,0,4,0,0,6,0,0,4,0,10],[20,72,0.2778,0.68749,0.33964,0.57132,0.71429,1.0,0.0,1.0,4,12,0,4,0,1,0,0,1,0,0,1,0,0,3,0,0,8,0,0,2,0,12],[24,72,0.3333,0.63839,0.3694,0.28571,0.71429,1.0,0.0,1.0,5,11,0,5,0,1,0,0,3,0,0,1,0,0,3,0,0,4,0,0,4,0,11],[28,72,0.3889,0.63392,0.31731,0.42857,0.71429,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,3,0,0,4,0,0,1,0,0,8,0,0,5,0,7],[32,72,0.4444,0.4375,0.37276,0.0,0.35714,0.75,0.0,1.0,9,4,0,9,0,3,0,0,4,0,0,2,0,0,1,0,0,5,0,0,4,0,4],[36,72,0.5,0.53121,0.35933,0.14286,0.64264,0.85704,0.0,1.0,7,5,0,7,0,2,0,0,1,0,0,3,0,0,3,0,0,7,0,0,4,0,5],[40,72,0.5556,0.74997,0.27895,0.71429,0.71429,1.0,0.0,1.0,2,12,0,2,0,1,0,0,0,0,0,1,0,0,3,0,0,10,0,0,3,0,12],[44,72,0.6111,0.6205,0.32851,0.39285,0.71429,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,3,0,0,1,0,0,4,0,0,8,0,0,4,0,7],[48,72,0.6667,0.57588,0.33405,0.28571,0.64286,0.85714,0.0,1.0,4,7,0,4,0,0,0,0,7,0,0,2,0,0,3,0,0,6,0,0,3,0,7],[52,72,0.7222,0.61158,0.29502,0.53539,0.71429,0.71429,0.0,1.0,4,4,0,4,0,0,0,0,3,0,0,1,0,0,3,0,0,14,0,0,3,0,4],[56,72,0.7778,0.42854,0.33311,0.0,0.49979,0.71429,0.0,1.0,9,1,0,9,0,2,0,0,3,0,0,2,0,0,3,0,0,9,0,0,3,0,1],[60,72,0.8333,0.54014,0.39241,0.0,0.71429,0.85714,0.0,1.0,10,5,0,10,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,7,0,5],[64,72,0.8889,0.19643,0.30671,0.0,0.0,0.28571,0.0,1.0,20,2,0,20,0,1,0,0,4,0,0,2,0,0,0,0,0,3,0,0,0,0,2],[68,72,0.9444,0.15625,0.25843,0.0,0.0,0.28571,0.0,0.85714,22,0,0,22,0,0,0,0,4,0,0,1,0,0,2,0,0,2,0,0,1,0,0],[72,72,1.0,0.125,0.26904,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,1]]}]},{"i":"8d229c74deb909ce","q":"40. (SWE 1) The numbers $1,2,3, \\ldots, 64$ are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size $2 \\times 2$. Prove that there are at least three such squares for which the sum of the 4 numbers contained exceeds 100.","t":[{"b":3,"e":1.0,"k":"volatile","v":0.17411,"x":0.97768,"p":[[0,7,0.0,0.17411,0.11143,0.14286,0.14286,0.17857,0.0,0.42857,4,0,0,4,0,20,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.35268,0.26483,0.1429,0.28571,0.42857,0.0,1.0,1,4,0,1,0,8,0,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,4],[7,7,1.0,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":4,"e":0.14286,"k":"flat","v":0.16063,"x":0.3125,"p":[[0,11,0.0,0.20071,0.16318,0.14286,0.14286,0.1786,0.0,1.0,1,1,0,1,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,11,0.3636,0.3125,0.21852,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,12,0,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,2],[8,11,0.7273,0.28116,0.27317,0.14286,0.14286,0.32143,0.0,1.0,2,2,0,2,0,20,0,0,2,0,0,3,0,0,1,0,0,0,0,0,2,0,2],[11,11,1.0,0.16063,0.09944,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b11eed7821105328","q":"Let  $f(x)$  be a quadratic polynomial with two real roots in the interval  $[-1,1]$ . Prove that if the maximum value of  $|f(x)|$  in the interval  $[-1,1]$  is equal to  $1$ , then the maximum value of  $|f'(x)|$  in the interval  $[-1,1]$  is not less than  $1$ .","t":[{"b":5,"e":1.0,"k":"flat","v":0.91072,"x":0.99554,"p":[[0,37,0.0,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[4,37,0.1081,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,37,0.2162,0.91072,0.16656,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,23],[12,37,0.3243,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[16,37,0.4324,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,37,0.5405,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,37,0.6486,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[28,37,0.7568,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,37,0.8649,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,37,0.973,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[37,37,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":0.85714,"k":"flat","v":0.9241,"x":0.99107,"p":[[0,40,0.0,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27],[4,40,0.1,0.94641,0.10569,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[8,40,0.2,0.9241,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,0,21],[12,40,0.3,0.94643,0.09942,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[16,40,0.4,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,40,0.5,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,40,0.6,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[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],[32,40,0.8,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,40,0.9,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[40,40,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]}]},{"i":"cbe181941d5d6763","q":"Find all positive integers  $n$  for which there exists an integer multiple of  $2022$  such that the sum of the squares of its digits is equal to  $n$ .","t":[{"b":3,"e":0.14286,"k":"falling","v":0.2142,"x":0.68293,"p":[[0,71,0.0,0.40173,0.1653,0.28571,0.42857,0.571,0.0,0.71429,2,0,2,2,0,1,0,0,9,0,0,10,0,0,9,0,0,1,0,0,0,0,0],[4,71,0.0563,0.68293,0.33467,0.53539,0.71429,1.0,0.0,1.0,1,13,0,1,0,5,0,0,1,0,0,1,0,0,5,0,0,4,0,0,2,0,13],[8,71,0.1127,0.37501,0.24936,0.14286,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,13,0,0,2,0,0,10,0,0,3,0,0,1,0,0,1,0,2],[12,71,0.169,0.42847,0.31142,0.14286,0.28571,0.71429,0.14,1.0,0,4,0,0,0,13,0,0,4,0,0,4,0,0,2,0,0,3,0,0,2,0,4],[16,71,0.2254,0.46866,0.318,0.14286,0.42857,0.71429,0.14,1.0,0,7,0,0,0,9,0,0,5,0,0,9,0,0,0,0,0,2,0,0,0,0,7],[20,71,0.2817,0.40624,0.31361,0.14286,0.35714,0.60714,0.0,1.0,2,4,1,2,0,12,0,0,2,0,0,6,0,0,2,0,0,3,0,0,1,0,4],[24,71,0.338,0.31697,0.23347,0.14286,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,15,0,0,7,0,0,5,0,0,2,0,0,1,0,0,0,0,2],[28,71,0.3944,0.41964,0.2765,0.14286,0.42857,0.42858,0.14286,1.0,0,4,0,0,0,10,0,0,4,0,0,11,0,0,0,0,0,3,0,0,0,0,4],[32,71,0.4507,0.32581,0.2726,0.14286,0.14286,0.42857,0.14,1.0,0,3,0,0,0,18,0,0,4,0,0,4,0,0,1,0,0,2,0,0,0,0,3],[36,71,0.507,0.41963,0.27183,0.14286,0.42857,0.60682,0.14286,1.0,0,2,0,0,0,12,0,0,1,0,0,10,0,0,1,0,0,4,0,0,2,0,2],[40,71,0.5634,0.3525,0.2288,0.14286,0.28571,0.42857,0.14,1.0,0,1,0,0,0,12,0,0,6,0,0,9,0,0,0,0,0,3,0,0,1,0,1],[44,71,0.6197,0.4732,0.33011,0.14286,0.42857,0.75,0.14286,1.0,0,6,0,0,0,12,0,0,2,0,0,6,0,0,2,0,0,2,0,0,2,0,6],[48,71,0.6761,0.30348,0.24163,0.14286,0.14286,0.42857,0.14,1.0,0,2,0,0,0,18,0,0,3,0,0,8,0,0,0,0,0,0,0,0,1,0,2],[52,71,0.7324,0.28572,0.20203,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,18,0,0,3,0,0,8,0,0,1,0,0,1,0,0,0,0,1],[56,71,0.7887,0.26791,0.12758,0.14286,0.2857,0.42857,0.14286,0.43,0,0,0,0,0,15,0,0,6,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.29452,0.21083,0.14286,0.1429,0.42857,0.14286,1.0,0,1,0,0,0,18,0,0,2,0,0,9,0,0,0,0,0,2,0,0,0,0,1],[64,71,0.9014,0.24089,0.25118,0.14286,0.14286,0.14286,0.14,1.0,0,3,0,0,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[68,71,0.9577,0.23214,0.1171,0.14286,0.14286,0.28571,0.14286,0.4286,0,0,0,0,0,19,0,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.2142,0.11851,0.14286,0.14286,0.28571,0.14,0.57143,0,0,0,0,0,22,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"falling","v":0.17857,"x":0.70979,"p":[[0,44,0.0,0.41517,0.21236,0.28571,0.42857,0.57111,0.0,1.0,2,1,1,2,0,2,0,0,9,0,0,10,0,0,4,0,0,4,0,0,0,0,1],[4,44,0.0909,0.70979,0.2934,0.42857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,5,0,0,4,0,0,2,0,0,6,0,0,0,0,14],[8,44,0.1818,0.55802,0.31208,0.39286,0.57121,0.75,0.0,1.0,1,7,1,1,0,6,0,0,1,0,0,7,0,0,4,0,0,5,0,0,1,0,7],[12,44,0.2727,0.48213,0.27836,0.25,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,2,0,0,10,0,0,3,0,0,4,0,0,1,0,4],[16,44,0.3636,0.37491,0.25947,0.14286,0.28571,0.46431,0.14,1.0,0,2,0,0,0,12,0,0,7,0,0,5,0,0,2,0,0,3,0,0,1,0,2],[20,44,0.4545,0.37045,0.25727,0.14286,0.28571,0.42857,0.14,1.0,0,3,0,0,0,12,0,0,5,0,0,10,0,0,0,0,0,2,0,0,0,0,3],[24,44,0.5455,0.38393,0.27994,0.14286,0.35714,0.60714,0.0,1.0,2,2,0,2,0,12,0,0,2,0,0,6,0,0,2,0,0,6,0,0,0,0,2],[28,44,0.6364,0.33482,0.22192,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,13,0,0,7,0,0,7,0,0,0,0,0,4,0,0,0,0,1],[32,44,0.7273,0.52214,0.31887,0.14286,0.42859,0.71429,0.14,1.0,0,6,0,0,0,9,0,0,3,0,0,5,0,0,1,0,0,7,0,0,1,0,6],[36,44,0.8182,0.44642,0.30461,0.14286,0.28571,0.71429,0.14286,1.0,0,4,0,0,0,11,0,0,6,0,0,2,0,0,3,0,0,5,0,0,1,0,4],[40,44,0.9091,0.27232,0.19352,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,17,0,0,8,0,0,4,0,0,1,0,0,1,0,0,0,0,1],[44,44,1.0,0.17857,0.06185,0.14286,0.14286,0.1786,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f8c4d893bb9bea0d","q":"Suppose  $S= \\{x|x=a^2+ab+b^2,a,b \\in Z\\}$ . Prove that:\n\n(1) If  $m \\in S$ ,   $3|m$  , then  $\\frac{m}{3} \\in S$ (2) If  $m,n \\in S$  , then  $mn\\in S$ .","t":[{"b":1,"e":0.85714,"k":"flat","v":0.83481,"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],[4,29,0.1379,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,29,0.2759,0.92856,0.13363,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,22],[12,29,0.4138,0.87051,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[16,29,0.5517,0.85267,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,1],[20,29,0.6897,0.83481,0.07239,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0],[24,29,0.8276,0.85267,0.02485,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[28,29,0.9655,0.85267,0.05629,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,1],[29,29,1.0,0.85265,0.05629,0.85714,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,1]]},{"b":2,"e":1.0,"k":"flat","v":0.94196,"x":0.99107,"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],[4,22,0.1818,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[8,22,0.3636,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[12,22,0.5455,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[16,22,0.7273,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[20,22,0.9091,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[22,22,1.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"a8a09b41d6bd4f3b","q":"Let $a, b$ and $c$ be real numbers such that\n\n$$\n|a-b| \\geq|c|,|b-c| \\geq|a| \\text { and }|c-a| \\geq|b| .\n$$\n\nProve that one of the three numbers $a, b$ and $c$ is the sum of the other two.","t":[{"b":4,"e":0.42857,"k":"rising","v":0.17857,"x":0.76784,"p":[[0,23,0.0,0.29462,0.34613,0.0,0.28571,0.46418,0.0,1.0,15,4,0,15,0,0,0,0,7,0,0,2,0,0,2,0,0,2,0,0,0,0,4],[4,23,0.1739,0.17857,0.28571,0.0,0.0,0.2857,0.0,1.0,20,2,0,20,0,1,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[8,23,0.3478,0.20982,0.31132,0.0,0.0,0.28571,0.0,1.0,18,3,0,18,0,1,0,0,8,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[12,23,0.5217,0.23661,0.33619,0.0,0.0,0.28571,0.0,1.0,17,4,0,17,0,2,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,4],[16,23,0.6957,0.76784,0.26906,0.67846,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,5,0,0,2,0,0,1,0,0,8,0,0,0,0,16],[20,23,0.8696,0.74554,0.2618,0.42859,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,7,0,0,3,0,0,5,0,0,0,0,15],[23,23,1.0,0.68308,0.30664,0.42857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,5,0,0,8,0,0,0,0,0,4,0,0,0,0,14]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.41964,"p":[[0,33,0.0,0.38839,0.38998,0.0,0.28571,0.71429,0.0,1.0,13,6,0,13,0,0,0,0,5,0,0,2,0,0,2,0,0,3,0,0,1,0,6],[4,33,0.1212,0.29463,0.28332,0.0,0.28571,0.28571,0.0,1.0,9,3,0,9,0,1,0,0,16,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[8,33,0.2424,0.33035,0.34151,0.0,0.2857,0.57143,0.0,1.0,12,4,0,12,0,0,0,0,10,0,0,1,0,0,3,0,0,1,0,0,1,0,4],[12,33,0.3636,0.34375,0.39907,0.0,0.28571,0.71429,0.0,1.0,15,7,0,15,0,0,0,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,7],[16,33,0.4848,0.41964,0.33491,0.2857,0.28571,0.71429,0.0,1.0,7,5,0,7,0,0,0,0,12,0,0,2,0,0,1,0,0,5,0,0,0,0,5],[20,33,0.6061,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ecc6387f4fb20bd7","q":"Find all integers  $n=2k+1>1$  so that there exists a permutation  $a_0, a_1,\\ldots,a_{k}$  of  $0, 1, \\ldots, k$  such that \n\\[a_1^2-a_0^2\\equiv a_2^2-a_1^2\\equiv \\cdots\\equiv a_{k}^2-a_{k-1}^2\\pmod n.\\]\n\n*Proposed by usjl*","t":[{"b":3,"e":0.0,"k":"rising","v":0.0,"x":0.73206,"p":[[0,112,0.0,0.20536,0.1234,0.14286,0.14286,0.17857,0.0,0.42857,1,0,0,1,0,23,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,112,0.0357,0.37494,0.23887,0.24999,0.28571,0.571,0.0,1.0,3,1,0,3,0,5,0,0,9,0,0,6,0,0,5,0,0,2,0,0,1,0,1],[8,112,0.0714,0.49995,0.2988,0.28571,0.42859,0.60714,0.0,1.0,3,5,1,3,0,1,0,0,8,0,0,5,0,0,7,0,0,1,0,0,2,0,5],[12,112,0.1071,0.43298,0.26839,0.28571,0.42857,0.57143,0.0,1.0,2,3,0,2,0,5,0,0,8,0,0,4,0,0,7,0,0,3,0,0,0,0,3],[16,112,0.1429,0.54015,0.28061,0.39286,0.4286,0.75,0.14286,1.0,0,5,0,0,0,5,0,0,3,0,0,9,0,0,5,0,0,2,0,0,3,0,5],[20,112,0.1786,0.56246,0.2671,0.42857,0.4998,0.74996,0.14286,1.0,0,5,0,0,0,3,0,0,4,0,0,9,0,0,5,0,0,3,0,0,3,0,5],[24,112,0.2143,0.46425,0.28346,0.14289,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,9,0,0,4,0,0,6,0,0,4,0,0,5,0,0,0,0,4],[28,112,0.25,0.59823,0.28888,0.39286,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,5,0,0,3,0,0,4,0,0,4,0,0,6,0,0,5,0,5],[32,112,0.2857,0.51779,0.24678,0.39286,0.4998,0.60714,0.14286,1.0,0,4,0,0,0,4,0,0,4,0,0,8,0,0,8,0,0,4,0,0,0,0,4],[36,112,0.3214,0.53566,0.24221,0.39286,0.571,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,5,0,0,7,0,0,7,0,0,5,0,0,2,0,3],[40,112,0.3571,0.5267,0.27532,0.28571,0.571,0.71429,0.14286,1.0,0,4,0,0,0,6,0,0,4,0,0,4,0,0,8,0,0,4,0,0,2,0,4],[44,112,0.3929,0.49551,0.22862,0.39286,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,3,0,0,10,0,0,4,0,0,7,0,0,2,0,1],[48,112,0.4286,0.41068,0.22229,0.14289,0.42857,0.571,0.14286,1.0,0,1,0,0,0,9,0,0,3,0,0,10,0,0,6,0,0,2,0,0,1,0,1],[52,112,0.4643,0.51335,0.25217,0.39286,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,4,0,0,11,0,0,4,0,0,3,0,0,3,0,3],[56,112,0.5,0.73206,0.25697,0.571,0.85707,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,2,0,0,8,0,0,2,0,0,7,0,10],[60,112,0.5357,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,112,0.5714,0.08482,0.26453,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[68,112,0.6071,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[72,112,0.6429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,112,0.6786,0.04017,0.15659,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[80,112,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,112,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,112,0.7857,0.05804,0.22548,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[92,112,0.8214,0.05357,0.20748,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[96,112,0.8571,0.07588,0.21717,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[100,112,0.8929,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[104,112,0.9286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,112,0.9643,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[112,112,1.0,0.6428,0.3093,0.571,0.71429,0.85714,0.0,1.0,3,6,0,3,0,2,0,0,2,0,0,0,0,0,4,0,0,10,0,0,5,0,6]]},{"b":6,"e":0.71429,"k":"rising","v":0.19197,"x":0.61601,"p":[[0,62,0.0,0.19197,0.11071,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,24,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,62,0.0645,0.53563,0.24221,0.42857,0.571,0.60714,0.14286,1.0,0,4,0,0,0,3,0,0,4,0,0,8,0,0,9,0,0,3,0,0,1,0,4],[8,62,0.129,0.55352,0.31083,0.25,0.571,0.75,0.14286,1.0,0,7,0,0,0,8,0,0,1,0,0,5,0,0,6,0,0,4,0,0,1,0,7],[12,62,0.1935,0.51781,0.2389,0.42857,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,10,0,0,6,0,0,5,0,0,1,0,3],[16,62,0.2581,0.55357,0.28515,0.42857,0.42859,0.85704,0.14286,1.0,0,6,0,0,0,4,0,0,3,0,0,12,0,0,2,0,0,2,0,0,3,0,6],[20,62,0.3226,0.53567,0.26725,0.28571,0.4998,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,5,0,0,7,0,0,6,0,0,3,0,0,3,0,4],[24,62,0.3871,0.55799,0.25842,0.42857,0.571,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,1,0,0,8,0,0,6,0,0,5,0,0,4,0,3],[28,62,0.4516,0.5714,0.25505,0.42857,0.4286,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,6,0,0,10,0,0,2,0,0,6,0,0,2,0,5],[32,62,0.5161,0.59373,0.27689,0.39286,0.57121,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,6,0,0,7,0,0,3,0,0,3,0,0,6,0,5],[36,62,0.5806,0.60712,0.28122,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,4,0,0,5,0,0,7,0,0,3,0,0,3,0,7],[40,62,0.6452,0.57136,0.26964,0.42857,0.571,0.71429,0.14286,1.0,0,5,0,0,0,5,0,0,2,0,0,5,0,0,7,0,0,7,0,0,1,0,5],[44,62,0.7097,0.61601,0.23538,0.42857,0.57143,0.857,0.14286,1.0,0,3,0,0,0,2,0,0,2,0,0,7,0,0,6,0,0,6,0,0,6,0,3],[48,62,0.7742,0.59148,0.23639,0.42857,0.57121,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,9,0,0,6,1,0,5,0,0,3,0,4],[52,62,0.8387,0.54905,0.26512,0.28571,0.571,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,7,0,0,4,0,0,7,0,0,4,0,0,3,0,4],[56,62,0.9032,0.58034,0.27184,0.39285,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,5,0,0,3,0,0,4,0,0,3,0,0,11,0,0,2,0,4],[60,62,0.9677,0.52228,0.25154,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,6,0,0,9,0,0,4,0,0,4,0,0,3,0,3],[62,62,1.0,0.61151,0.21499,0.42857,0.571,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,9,0,0,7,0,0,6,0,0,3,0,4]]}]},{"i":"e08dc3c77bd7b513","q":"25. (GBR 1) A positive integer is called a double number if its decimal representation consists of a block of digits, not commencing with 0 , followed immediately by an identical block. For instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers that are perfect squares.","t":[{"b":1,"e":0.42857,"k":"falling","v":0.76784,"x":0.98661,"p":[[0,66,0.0,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[4,66,0.0606,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,66,0.1212,0.95535,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[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],[16,66,0.2424,0.95536,0.18013,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[20,66,0.303,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,66,0.3636,0.95982,0.15664,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[28,66,0.4242,0.95089,0.18423,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[32,66,0.4848,0.97321,0.06623,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[36,66,0.5455,0.94196,0.18161,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,28],[40,66,0.6061,0.91963,0.22,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[44,66,0.6667,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[48,66,0.7273,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[52,66,0.7879,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[56,66,0.8485,0.96874,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[60,66,0.9091,0.88839,0.19145,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,22],[64,66,0.9697,0.88391,0.20342,0.857,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4,0,0,4,0,21],[66,66,1.0,0.76784,0.27837,0.57132,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,4,0,0,3,0,0,4,0,0,2,0,16]]},{"b":7,"e":1.0,"k":"flat","v":0.81696,"x":0.98661,"p":[[0,78,0.0,0.86607,0.27879,0.96425,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,24],[4,78,0.0513,0.93304,0.18893,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[8,78,0.1026,0.98437,0.04277,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,1,28],[12,78,0.1538,0.91518,0.22829,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,27],[16,78,0.2051,0.95982,0.11426,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[20,78,0.2564,0.95535,0.15746,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[24,78,0.3077,0.83013,0.32254,0.82143,1.0,1.0,0.0,1.0,2,23,2,2,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,23],[28,78,0.359,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,78,0.4103,0.91518,0.22548,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[36,78,0.4615,0.92857,0.18558,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[40,78,0.5128,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,78,0.5641,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[48,78,0.6154,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[52,78,0.6667,0.81696,0.32972,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,22],[56,78,0.7179,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[60,78,0.7692,0.82143,0.29014,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,20],[64,78,0.8205,0.94643,0.20748,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[68,78,0.8718,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,78,0.9231,0.96875,0.12745,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[76,78,0.9744,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[78,78,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"13833e3ea6133857","q":"Let  $ABC$  be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is  $10$ , what is the largest possible area of the triangle? $ \n\\textbf{(A)}\\ 20\n\\qquad\\textbf{(B)}\\ 25\n\\qquad\\textbf{(C)}\\ 30 \n\\qquad\\textbf{(D)}\\ 40\n\\qquad\\textbf{(E)}\\ 50\n$","t":[{"b":0,"e":0.71429,"k":"flat","v":0.59375,"x":0.71429,"p":[[0,47,0.0,0.59375,0.11355,0.53572,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,11,0,0,13,0,0,0,0,0],[4,47,0.0851,0.67411,0.13475,0.71429,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,28,0,0,0,0,0],[8,47,0.1702,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[12,47,0.2553,0.70536,0.10062,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,27,0,0,1,0,1],[16,47,0.3404,0.66964,0.12595,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,27,0,0,0,0,0],[20,47,0.4255,0.66963,0.14479,0.71429,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,28,0,0,0,0,0],[24,47,0.5106,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[28,47,0.5957,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[32,47,0.6809,0.69195,0.0519,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0],[36,47,0.766,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[40,47,0.8511,0.69643,0.06916,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,27,0,0,1,0,0],[44,47,0.9362,0.69197,0.06297,0.71429,0.71429,0.71429,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[47,47,1.0,0.65624,0.07018,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.55804,"x":0.72321,"p":[[0,70,0.0,0.55804,0.16506,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,8,0,0,8,0,0,13,0,0,0,0,0],[4,70,0.0571,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[8,70,0.1143,0.68302,0.06904,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,26,0,0,0,0,0],[12,70,0.1714,0.64732,0.18893,0.71429,0.71429,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,28,0,0,0,0,0],[16,70,0.2286,0.69196,0.06298,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[20,70,0.2857,0.67411,0.09606,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,26,0,0,0,0,0],[24,70,0.3429,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[28,70,0.4,0.69196,0.08073,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,29,0,0,0,0,0],[32,70,0.4571,0.70534,0.07938,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,25,0,0,1,0,1],[36,70,0.5143,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[40,70,0.5714,0.69643,0.04725,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[44,70,0.6286,0.69643,0.04725,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[48,70,0.6857,0.67857,0.12877,0.71429,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0],[52,70,0.7429,0.70982,0.07563,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,26,0,0,1,0,1],[56,70,0.8,0.70982,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,1,0,0],[60,70,0.8571,0.69196,0.10779,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,28,0,0,1,0,0],[64,70,0.9143,0.69196,0.05187,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0],[68,70,0.9714,0.69643,0.04725,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0],[70,70,1.0,0.67857,0.11294,0.71429,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,25,0,0,1,0,0]]}]},{"i":"b01c6c95733d3fb6","q":"For how many  $x \\in \\{1,2,3,\\dots, 2024\\}$  is it possible that *Bekhzod* summed  $2024$  non-negative consecutive integers, *Ozod* summed  $2024+x$  non-negative consecutive integers and they got the same result? \n\n*Proposed by Marek Maruin, Slovakia*","t":[{"b":4,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,87,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,87,0.046,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,87,0.092,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,87,0.1379,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,87,0.1839,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[20,87,0.2299,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,87,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,87,0.3218,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,87,0.3678,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,87,0.4138,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[40,87,0.4598,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[44,87,0.5057,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,87,0.5517,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,87,0.5977,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[56,87,0.6437,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,87,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,87,0.7356,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,87,0.7816,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,87,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,87,0.8736,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,87,0.9195,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,87,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[87,87,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.92409,"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],[4,58,0.069,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,58,0.1379,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,58,0.2069,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,58,0.2759,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[20,58,0.3448,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,58,0.4138,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[28,58,0.4828,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[32,58,0.5517,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[36,58,0.6207,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[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],[44,58,0.7586,0.92409,0.18208,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,27],[48,58,0.8276,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[52,58,0.8966,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"edc8a4b5267ac140","q":"$\\boxed{N1}$ Let  $n$  be a positive integer, $g(n)$  be the number of positive divisors of  $n$  of the form  $6k+1$  and  $h(n)$  be the number of positive divisors of  $n$  of the form  $6k-1,$ where  $k$  is a nonnegative integer.Find all positive integers  $n$  such that  $g(n)$  and  $h(n)$  have different parity.","t":[{"b":6,"e":1.0,"k":"flat","v":0.93749,"x":1.0,"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],[4,47,0.0851,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,47,0.1702,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[12,47,0.2553,0.95088,0.11637,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[16,47,0.3404,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,47,0.4255,0.94196,0.09355,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,21],[24,47,0.5106,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[28,47,0.5957,0.93749,0.10063,0.85714,1.0,1.0,0.5714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[32,47,0.6809,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[36,47,0.766,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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],[44,47,0.9362,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,58,0.0,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[4,58,0.069,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[8,58,0.1379,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[12,58,0.2069,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,58,0.2759,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,58,0.3448,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,58,0.4138,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[28,58,0.4828,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,58,0.5517,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[36,58,0.6207,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,58,0.6897,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,58,0.7586,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[48,58,0.8276,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[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],[56,58,0.9655,0.93304,0.24218,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[58,58,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"0d31d6db6d694982","q":"Determine all positive real numbers  $x$  for which   $$ \\left [x\\right ]+\\left [\\sqrt{1996x}\\right ]=1996 $$   is verified\n\nClarification:The brackets indicate the integer part of the number they enclose.","t":[{"b":4,"e":0.85714,"k":"flat","v":0.94196,"x":0.98214,"p":[[0,9,0.0,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[4,9,0.4444,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,9,0.8889,0.94642,0.09943,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[9,9,1.0,0.94196,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22]]},{"b":6,"e":1.0,"k":"flat","v":0.90625,"x":0.97321,"p":[[0,6,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,6,0.6667,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[6,6,1.0,0.90625,0.13175,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,7,0,19]]}]},{"i":"529bb46812937e63","q":"Let $b \\geq 2$ be a fixed integer, and let $s_{b}(n)$ denote the sum of the base- $b$ digits of $n$. Show that there are infinitely many positive integers that cannot be represented in the from $n+s_{b}(n)$ where $n$ is a positive integer.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.16054,"x":0.34375,"p":[[0,36,0.0,0.16054,0.12245,0.14214,0.14286,0.14286,0.0,0.4286,6,0,1,6,0,20,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.24545,0.16071,0.14286,0.14288,0.42857,0.0,0.57143,2,0,0,2,0,17,0,0,4,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[8,36,0.2222,0.26784,0.23621,0.14286,0.14286,0.32143,0.0,1.0,4,1,0,4,0,14,0,0,6,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[12,36,0.3333,0.34375,0.22545,0.14286,0.28571,0.46525,0.0,1.0,1,1,0,1,0,11,0,0,7,0,0,5,0,0,6,0,0,0,0,0,1,0,1],[16,36,0.4444,0.21429,0.21129,0.14286,0.14286,0.1786,0.0,1.0,4,1,0,4,0,20,0,0,4,0,0,0,0,0,2,0,0,1,0,0,0,0,1],[20,36,0.5556,0.24105,0.19699,0.14286,0.14286,0.32142,0.0,0.71429,3,0,0,3,0,19,0,0,2,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[24,36,0.6667,0.23214,0.17035,0.14286,0.14286,0.28571,0.0,0.85714,2,0,0,2,0,18,0,0,6,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[28,36,0.7778,0.19197,0.12168,0.14286,0.14286,0.2857,0.0,0.57143,3,0,0,3,0,19,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[32,36,0.8889,0.20981,0.17849,0.14286,0.14286,0.2857,0.0,0.71429,5,0,0,5,0,16,0,0,7,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[36,36,1.0,0.16509,0.1243,0.14286,0.14286,0.14287,0.0,0.4286,6,0,0,6,0,19,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.15839,"x":0.35268,"p":[[0,38,0.0,0.15839,0.11258,0.14286,0.14286,0.14286,0.0,0.71429,2,0,1,2,1,26,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,38,0.1053,0.26785,0.19147,0.14286,0.2143,0.42857,0.0,1.0,2,1,0,2,0,14,0,0,7,0,0,7,0,0,1,0,0,0,0,0,0,0,1],[8,38,0.2105,0.29464,0.22569,0.14286,0.1429,0.42857,0.0,0.857,2,0,0,2,0,16,0,0,3,0,0,5,0,0,2,0,0,3,0,0,1,0,0],[12,38,0.3158,0.20972,0.15966,0.14286,0.14286,0.2857,0.0,0.71429,3,0,0,3,0,20,0,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[16,38,0.4211,0.29902,0.17635,0.14286,0.2857,0.32143,0.14,0.85714,0,0,0,0,0,12,0,0,12,0,0,4,0,0,2,0,0,1,0,0,1,0,0],[20,38,0.5263,0.29017,0.20353,0.14286,0.2143,0.42857,0.0,0.85714,2,0,0,2,0,14,0,0,5,0,0,6,0,0,3,0,0,1,0,0,1,0,0],[24,38,0.6316,0.29465,0.21706,0.14286,0.14288,0.42858,0.0,1.0,2,1,0,2,0,15,0,0,3,0,0,6,0,0,5,0,0,0,0,0,0,0,1],[28,38,0.7368,0.33918,0.21657,0.14286,0.28571,0.42858,0.0,1.0,1,1,0,1,0,11,0,0,6,0,0,8,0,0,3,0,0,2,0,0,0,0,1],[32,38,0.8421,0.34821,0.19541,0.2857,0.42857,0.42857,0.0,1.0,3,1,0,3,0,4,0,0,8,0,0,13,0,0,3,0,0,0,0,0,0,0,1],[36,38,0.9474,0.24107,0.18708,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,6,0,0,4,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.35268,0.18205,0.24999,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,4,0,0,3,0,0,15,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"8af5c751a4c304e3","q":"Let $ABC$ be an acute triangle (all angles are acute) with $BA \\neq BC$. Let $O$ be the center of its circumcircle. The line $(AB)$ intersects the circumcircle of $BOC$ a second time at $P \\neq B$. Show that $PA = PC$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.6875,"x":0.96429,"p":[[0,68,0.0,0.6875,0.30605,0.42859,0.85705,1.0,0.0,1.0,1,10,1,1,0,2,0,0,2,0,0,6,0,0,2,0,0,2,0,0,7,0,10],[4,68,0.0588,0.85267,0.22443,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,0,5,0,19],[8,68,0.1176,0.87945,0.22336,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,2,0,22],[12,68,0.1765,0.85711,0.25257,0.82132,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,21],[16,68,0.2353,0.87947,0.25532,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,24],[20,68,0.2941,0.83034,0.26831,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,2,0,0,3,0,20],[24,68,0.3529,0.79463,0.27418,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,3,0,0,1,0,0,3,0,0,7,0,15],[28,68,0.4118,0.91518,0.1984,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,24],[32,68,0.4706,0.95089,0.12682,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,26],[36,68,0.5294,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[40,68,0.5882,0.875,0.20124,0.85711,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,5,0,20],[44,68,0.6471,0.87946,0.20858,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,8,0,19],[48,68,0.7059,0.91071,0.18814,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,5,0,23],[52,68,0.7647,0.82589,0.25688,0.67857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,4,0,0,3,0,0,2,0,0,3,0,19],[56,68,0.8235,0.95088,0.09187,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[60,68,0.8824,0.94643,0.13717,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[64,68,0.9412,0.90625,0.14555,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,10,0,18],[68,68,1.0,0.93747,0.14703,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,24]]},{"b":4,"e":1.0,"k":"rising","v":0.66963,"x":0.96873,"p":[[0,31,0.0,0.66963,0.38205,0.39285,0.85714,1.0,0.0,1.0,4,13,4,4,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,0,7,0,13],[4,31,0.129,0.91072,0.19479,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,24],[8,31,0.2581,0.90165,0.21863,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,25],[12,31,0.3871,0.78125,0.2766,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,1,0,0,5,0,0,2,0,0,4,0,16],[16,31,0.5161,0.82143,0.26244,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,3,0,0,3,0,0,2,0,0,3,0,19],[20,31,0.6452,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[24,31,0.7742,0.91964,0.20806,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,26],[28,31,0.9032,0.96873,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[31,31,1.0,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27]]}]},{"i":"f28b02659ccd5cd0","q":"Positive integers  $a_1, a_2, \\ldots, a_{101}$  are such that  $a_i+1$  is divisible by  $a_{i+1}$  for all  $1 \\le i \\le 101$ , where  $a_{102} = a_1$ . What is the largest possible value of  $\\max(a_1, a_2, \\ldots, a_{101})$ ?\n\n*Proposed by Oleksiy Masalitin*","t":[{"b":1,"e":0.42857,"k":"falling","v":0.1875,"x":0.58927,"p":[[0,51,0.0,0.58927,0.36025,0.14286,0.57143,1.0,0.14286,1.0,0,10,0,0,0,10,0,0,1,0,0,3,0,0,3,0,0,1,0,0,4,0,10],[4,51,0.0784,0.38839,0.31791,0.14286,0.14286,0.71429,0.14286,1.0,0,3,0,0,0,19,0,0,0,0,0,1,0,0,3,0,0,4,0,0,2,0,3],[8,51,0.1569,0.37945,0.31663,0.14286,0.14288,0.57143,0.0,1.0,2,4,0,2,0,15,0,0,1,0,0,3,0,0,5,0,0,1,0,0,1,0,4],[12,51,0.2353,0.54911,0.31564,0.42857,0.5,0.75,0.0,1.0,2,7,0,2,0,5,0,0,0,0,0,9,0,0,4,0,0,4,0,0,1,0,7],[16,51,0.3137,0.32589,0.30143,0.14286,0.14286,0.57143,0.0,1.0,4,1,0,4,0,16,0,0,1,0,0,1,0,0,3,0,0,3,0,0,3,0,1],[20,51,0.3922,0.31249,0.29973,0.10714,0.14286,0.57143,0.0,1.0,8,1,0,8,0,10,0,0,2,0,0,2,0,0,3,0,0,5,0,0,1,0,1],[24,51,0.4706,0.32143,0.36246,0.0,0.14286,0.46431,0.0,1.0,12,5,0,12,0,6,0,0,1,0,0,5,0,0,2,0,0,0,0,0,1,0,5],[28,51,0.549,0.1875,0.2126,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,9,0,0,6,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[32,51,0.6275,0.23661,0.23585,0.0,0.14286,0.42857,0.0,0.71429,10,0,0,10,0,9,0,0,4,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[36,51,0.7059,0.26329,0.35015,0.0,0.14286,0.46418,0.0,1.0,14,4,0,14,0,8,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,4],[40,51,0.7843,0.29017,0.3453,0.0,0.14286,0.46418,0.0,1.0,13,3,0,13,0,7,0,0,0,0,0,4,0,0,2,0,0,1,0,0,2,0,3],[44,51,0.8627,0.26786,0.25939,0.14286,0.1429,0.42857,0.0,1.0,7,2,0,7,0,10,0,0,6,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[48,51,0.9412,0.21875,0.18552,0.0,0.14286,0.42857,0.0,0.42857,10,0,0,10,0,8,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.20982,0.18552,0.0,0.14286,0.42857,0.0,0.4286,11,0,0,11,0,7,0,0,2,0,0,12,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.04464,"x":0.64284,"p":[[0,70,0.0,0.64284,0.33313,0.39286,0.64286,1.0,0.0,1.0,1,12,0,1,0,3,0,0,4,0,0,5,0,0,3,0,0,2,0,0,2,0,12],[4,70,0.0571,0.39286,0.3312,0.14286,0.14286,0.60714,0.14286,1.0,0,5,0,0,0,17,0,0,3,0,0,3,0,0,1,0,0,1,0,0,2,0,5],[8,70,0.1143,0.29018,0.28901,0.14286,0.14286,0.21429,0.0,1.0,1,3,0,1,0,23,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,3],[12,70,0.1714,0.32147,0.29016,0.14286,0.14286,0.42857,0.14286,1.0,0,3,0,0,0,21,0,0,1,0,0,4,0,0,0,0,0,2,0,0,1,0,3],[16,70,0.2286,0.33479,0.28256,0.14286,0.14286,0.571,0.14286,1.0,0,2,0,0,0,20,0,0,1,0,0,2,0,0,4,0,0,1,0,0,2,0,2],[20,70,0.2857,0.29463,0.2671,0.14286,0.14286,0.4642,0.0,1.0,3,1,0,3,0,18,0,0,1,0,0,2,0,0,3,0,0,3,0,0,1,0,1],[24,70,0.3429,0.43749,0.34798,0.14286,0.28571,0.71429,0.0,1.0,2,6,0,2,0,13,0,0,2,0,0,1,0,0,5,0,0,2,0,0,1,0,6],[28,70,0.4,0.3616,0.33499,0.14286,0.14286,0.60714,0.0,1.0,3,3,0,3,0,16,0,0,2,0,0,1,0,0,2,0,0,1,0,0,4,0,3],[32,70,0.4571,0.28572,0.30305,0.14286,0.14286,0.4286,0.0,1.0,6,3,0,6,0,16,0,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,3],[36,70,0.5143,0.36606,0.33107,0.14286,0.14286,0.64286,0.0,1.0,2,2,0,2,0,18,0,0,0,0,0,2,0,0,2,0,0,0,0,0,6,0,2],[40,70,0.5714,0.24554,0.26058,0.14286,0.14286,0.14286,0.0,1.0,4,2,0,4,0,21,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,2],[44,70,0.6286,0.31696,0.34577,0.14286,0.14286,0.57143,0.0,1.0,6,4,0,6,0,16,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,4],[48,70,0.6857,0.29911,0.29957,0.14286,0.14286,0.42857,0.0,1.0,5,2,0,5,0,15,0,0,3,0,0,2,0,0,1,0,0,2,0,0,2,0,2],[52,70,0.7429,0.40625,0.34646,0.14286,0.28571,0.71429,0.0,1.0,6,3,0,6,0,10,0,0,0,0,0,2,0,0,4,0,0,4,0,0,3,0,3],[56,70,0.8,0.2633,0.33905,0.0,0.14286,0.21429,0.0,1.0,10,4,0,10,0,14,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,4],[60,70,0.8571,0.20089,0.31106,0.0,0.07143,0.14286,0.0,1.0,16,3,0,16,0,9,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,3],[64,70,0.9143,0.10714,0.27664,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[68,70,0.9714,0.09821,0.25364,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,1],[70,70,1.0,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"2ec727be8dc19bef","q":"How many positive integers which divide  $5n^{11}-2n^5-3n$  for all positive integers  $n$  are there? $ \n\\textbf{(A)}\\ 2\n\\qquad\\textbf{(B)}\\ 5\n\\qquad\\textbf{(C)}\\ 6\n\\qquad\\textbf{(D)}\\ 12\n\\qquad\\textbf{(E)}\\ 18\n$","t":[{"b":0,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,9,0.0,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[4,9,0.4444,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":7,"e":1.0,"k":"flat","v":0.91964,"x":0.99107,"p":[[0,54,0.0,0.91964,0.17835,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,24],[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],[8,54,0.1481,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[12,54,0.2222,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[16,54,0.2963,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[20,54,0.3704,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[24,54,0.4444,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,54,0.5185,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[32,54,0.5926,0.92409,0.13359,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,23],[36,54,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,54,0.7407,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,54,0.8148,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[48,54,0.8889,0.95089,0.11633,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[52,54,0.963,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[54,54,1.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28]]}]},{"i":"83fffa80b3390ebd","q":"Consider the sequence  $u_0, u_1, u_2, ...$  defined by  $u_0 = 0, u_1 = 1,$  and  $u_n = 6u_{n - 1} + 7u_{n - 2}$  for  $n \\ge 2$ . Show that there are no non-negative integers  $a, b, c, n$  such that  $$ ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n. $$","t":[{"b":4,"e":1.0,"k":"rising","v":0.62054,"x":0.99554,"p":[[0,40,0.0,0.62054,0.31866,0.53572,0.57143,1.0,0.0,1.0,4,9,0,4,0,0,0,0,2,0,0,2,0,0,10,0,0,4,0,0,1,0,9],[4,40,0.1,0.64732,0.31132,0.42859,0.57143,1.0,0.0,1.0,3,11,0,3,0,0,0,0,1,0,0,6,0,0,7,0,0,4,0,0,0,0,11],[8,40,0.2,0.90624,0.19436,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,24],[12,40,0.3,0.84375,0.25595,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,3,0,0,2,0,21],[16,40,0.4,0.91518,0.19516,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,26],[20,40,0.5,0.86158,0.20976,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,4,0,0,4,0,19],[24,40,0.6,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[28,40,0.7,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[32,40,0.8,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[36,40,0.9,0.95089,0.18423,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[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]]},{"b":5,"e":0.4286,"k":"flat","v":0.52677,"x":0.69644,"p":[[0,61,0.0,0.63825,0.26719,0.5354,0.57143,0.85714,0.0,1.0,1,7,1,1,0,1,0,0,3,0,0,3,0,0,10,0,0,4,0,0,3,0,7],[4,61,0.0656,0.69644,0.21942,0.57143,0.57143,1.0,0.286,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,13,0,0,3,0,0,2,0,9],[8,61,0.1311,0.53571,0.07143,0.57143,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,6,0,0,25,0,0,0,0,0,0,0,0],[12,61,0.1967,0.56249,0.03458,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0],[16,61,0.2623,0.55802,0.04163,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0],[20,61,0.3279,0.56247,0.03457,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0],[24,61,0.3934,0.54911,0.05187,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0],[28,61,0.459,0.55357,0.04725,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[32,61,0.5246,0.55357,0.04725,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[36,61,0.5902,0.52677,0.09061,0.57132,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,4,0,0,25,0,0,0,0,0,0,0,0],[40,61,0.6557,0.54909,0.05186,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0],[44,61,0.7213,0.54014,0.05903,0.571,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0],[48,61,0.7869,0.55353,0.04723,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[52,61,0.8525,0.56246,0.03457,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0],[56,61,0.918,0.54018,0.06901,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,5,0,0,26,0,0,0,0,0,0,0,0],[60,61,0.9836,0.54016,0.06901,0.5714,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,5,0,0,26,0,0,0,0,0,0,0,0],[61,61,1.0,0.56695,0.02485,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0]]}]},{"i":"e1b33b15edc2ebe8","q":"Six real numbers  $x_1<x_2<x_3<x_4<x_5<x_6$  are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the  $20$  sums are pairwise distinct; denote those sums by   $$ s_1<s_2<s_3<\\cdots<s_{19}<s_{20}. $$   It is known that  $x_2+x_3+x_4=s_{11}$ ,  $x_2+x_3+x_6=s_{15}$  and  $x_1+x_2+x_6=s_{m}$ . Find all possible values of  $m$ .","t":[{"b":1,"e":0.0,"k":"rising","v":0.35714,"x":0.88393,"p":[[0,259,0.0,0.44642,0.3809,0.0,0.5,0.85714,0.0,1.0,11,1,8,11,0,1,0,0,3,0,0,1,0,0,3,0,0,1,0,0,11,0,1],[4,259,0.0154,0.73661,0.33713,0.57143,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,3,0,0,1,0,0,4,0,0,3,0,0,1,0,17],[8,259,0.0309,0.57143,0.36246,0.28571,0.57144,0.89286,0.0,1.0,5,8,1,5,0,1,0,0,4,0,0,6,0,0,0,0,0,3,0,0,5,0,8],[12,259,0.0463,0.53571,0.38795,0.24999,0.64286,0.89286,0.0,1.0,7,8,0,7,0,1,0,0,6,0,0,1,0,0,1,0,0,4,0,0,4,0,8],[16,259,0.0618,0.59821,0.3597,0.28571,0.71429,1.0,0.0,1.0,5,9,0,5,0,0,0,0,5,0,0,3,0,0,2,0,0,4,0,0,4,0,9],[20,259,0.0772,0.59375,0.38317,0.25,0.71429,1.0,0.0,1.0,5,11,0,5,0,3,0,0,1,0,0,6,0,0,0,0,0,3,0,0,3,0,11],[24,259,0.0927,0.45526,0.4233,0.0,0.28571,1.0,0.0,1.0,11,9,0,11,0,2,0,0,4,0,0,1,0,0,1,0,0,2,0,0,2,0,9],[28,259,0.1081,0.424,0.36338,0.0,0.35714,0.71429,0.0,1.0,9,5,0,9,0,2,0,0,5,0,0,3,0,0,3,0,0,3,0,0,2,0,5],[32,259,0.1236,0.50893,0.37785,0.25,0.42859,0.85714,0.0,1.0,7,7,0,7,0,1,0,0,6,0,0,3,0,0,1,0,0,3,0,0,4,0,7],[36,259,0.139,0.35714,0.30929,0.0,0.28571,0.60714,0.0,1.0,9,2,0,9,0,3,0,0,6,0,0,3,0,0,3,0,0,6,0,0,0,0,2],[40,259,0.1544,0.50445,0.37455,0.21427,0.42857,0.85714,0.0,1.0,8,7,0,8,0,0,0,0,4,0,0,5,0,0,2,0,0,3,0,0,3,0,7],[44,259,0.1699,0.5,0.42558,0.0,0.42859,1.0,0.0,1.0,11,10,0,11,0,0,0,0,3,0,0,3,0,0,0,0,0,3,0,0,2,0,10],[48,259,0.1853,0.52679,0.43366,0.0,0.64286,1.0,0.0,1.0,10,10,0,10,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,5,0,10],[52,259,0.2008,0.71428,0.34069,0.42857,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,3,0,0,2,0,0,2,0,0,3,0,0,3,0,15],[56,259,0.2162,0.70089,0.33381,0.4286,0.85714,1.0,0.0,1.0,3,13,0,3,0,1,0,0,1,0,0,4,0,0,3,0,0,3,0,0,4,0,13],[60,259,0.2317,0.70535,0.35344,0.57143,0.85714,1.0,0.0,1.0,4,13,1,4,0,1,0,0,2,0,0,0,0,0,4,0,0,2,0,0,6,0,13],[64,259,0.2471,0.71427,0.36944,0.5354,0.85714,1.0,0.0,1.0,5,15,0,5,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,5,0,15],[68,259,0.2625,0.85714,0.21129,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,0,2,0,20],[72,259,0.278,0.75,0.34442,0.57143,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,2,0,0,1,0,0,3,0,0,1,0,0,4,0,17],[76,259,0.2934,0.80357,0.28291,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,3,0,0,2,0,0,2,0,0,2,0,0,4,0,18],[80,259,0.3089,0.76784,0.2896,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,1,0,0,0,0,0,2,0,0,2,0,0,7,0,0,4,0,14],[84,259,0.3243,0.52228,0.35644,0.28571,0.571,0.85714,0.0,1.0,7,6,0,7,0,0,0,0,4,0,0,4,0,0,3,0,0,5,0,0,3,0,6],[88,259,0.3398,0.72321,0.3213,0.57143,0.85714,1.0,0.0,1.0,3,12,0,3,0,1,0,0,1,0,0,1,0,0,4,0,0,4,0,0,6,0,12],[92,259,0.3552,0.64286,0.3896,0.25001,0.71429,1.0,0.0,1.0,5,14,0,5,0,3,0,0,1,0,0,1,0,0,3,0,0,4,0,0,1,0,14],[96,259,0.3707,0.71425,0.32734,0.53539,0.85714,1.0,0.0,1.0,3,12,0,3,0,0,0,0,3,0,0,2,0,0,3,0,0,2,0,0,7,0,12],[100,259,0.3861,0.75892,0.2933,0.67857,0.85714,1.0,0.0,1.0,3,12,0,3,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,7,0,12],[104,259,0.4015,0.88393,0.22711,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,6,0,21],[108,259,0.417,0.70533,0.348,0.42857,0.85714,1.0,0.0,1.0,3,15,0,3,0,1,0,0,3,0,0,2,0,0,2,0,0,4,0,0,2,0,15],[112,259,0.4324,0.70088,0.35956,0.42857,0.85714,1.0,0.0,1.0,4,14,0,4,0,0,0,0,3,0,0,3,0,0,2,0,0,0,0,0,6,0,14],[116,259,0.4479,0.73661,0.30537,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,1,0,0,1,0,0,3,0,0,4,0,0,1,0,0,8,0,12],[120,259,0.4633,0.77678,0.32525,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,3,0,0,1,0,0,0,0,0,4,0,0,3,0,18],[124,259,0.4788,0.72766,0.31004,0.57142,0.85707,1.0,0.0,1.0,2,14,0,2,0,0,0,0,4,0,0,0,0,0,6,0,0,3,0,0,3,0,14],[128,259,0.4942,0.83036,0.24856,0.67857,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,1,0,0,4,0,19],[132,259,0.5097,0.74107,0.34523,0.57143,0.92857,1.0,0.0,1.0,3,16,0,3,0,1,0,0,3,0,0,0,0,0,2,0,0,3,0,0,4,0,16],[136,259,0.5251,0.73213,0.36026,0.28571,1.0,1.0,0.0,1.0,2,18,0,2,0,2,0,0,5,0,0,0,0,0,2,0,0,0,0,0,3,0,18],[140,259,0.5405,0.72768,0.36133,0.53571,1.0,1.0,0.0,1.0,3,18,1,3,0,3,0,0,0,0,0,2,0,0,2,0,0,4,0,0,0,0,18],[144,259,0.556,0.84375,0.29744,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,21],[148,259,0.5714,0.77679,0.3443,0.42857,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,22],[152,259,0.5869,0.77232,0.33093,0.53571,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,4,0,0,1,0,0,1,0,0,1,0,0,3,0,19],[156,259,0.6023,0.75884,0.36513,0.71429,1.0,1.0,0.0,1.0,3,18,0,3,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,6,0,18],[160,259,0.6178,0.79018,0.2789,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,2,0,0,2,0,0,3,0,0,5,0,16],[164,259,0.6332,0.83034,0.246,0.85711,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,10,0,15],[168,259,0.6486,0.64731,0.38629,0.28571,0.85714,1.0,0.0,1.0,5,13,0,5,0,0,0,0,6,0,0,1,0,0,1,0,0,1,0,0,5,0,13],[172,259,0.6641,0.76785,0.34209,0.71429,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,7,0,16],[176,259,0.6795,0.73661,0.36089,0.28571,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,6,0,0,0,0,0,1,0,0,1,0,0,3,0,18],[180,259,0.695,0.74107,0.33396,0.53571,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,4,0,0,1,0,0,2,0,0,2,0,0,4,0,16],[184,259,0.7104,0.80804,0.27574,0.64286,1.0,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$\\triangle A B C$ be a triangle with $|A C|=2|A B|$, and let $O$ be the center of the circumscribed circle. Let $D$ be the intersection of the angle bisector of $\\angle A$ with $B C$. Let $E$ be the orthogonal projection of $O$ on $A D$, and let $F \\neq D$ be the point on $A D$ for which $|C D|=|C F|$. Prove that $\\angle E B F=\\angle E C 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$A B$ be a diameter of a circle $S$, and let $L$ be the tangent at $A$. Furthermore, let $c$ be a fixed, positive real, and consider all pairs of points $X$ and $Y$ lying on $L$, on opposite sides of $A$, such that $|A X| \\cdot|A Y|=c$. The lines $B X$ and $B Y$ intersect $S$ at points $P$ and $Q$, respectively. Show that all the lines $P Q$ pass through a common point.","t":[{"b":2,"e":0.0,"k":"flat","v":0.01786,"x":0.10258,"p":[[0,34,0.0,0.07134,0.13358,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.08036,0.15542,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,6,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,34,0.2353,0.10258,0.24017,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[12,34,0.3529,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.06697,0.21124,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[20,34,0.5882,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.07589,0.19228,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[28,34,0.8235,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.08482,0.20782,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[34,34,1.0,0.04902,0.11626,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04464,"x":0.0625,"p":[[0,11,0.0,0.0625,0.18877,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,11,0.3636,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.05357,0.12242,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[11,11,1.0,0.04464,0.11538,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"075d6e810543c24c","q":"The vertices of a regular  $n$ -gon are initially marked with one of the signs  $+$  or  $-$  . A *move* consists in choosing three consecutive vertices and changing the signs from the vertices , from  $+$  to  $-$  and from  $-$  to  $+$ .**a)** Prove that if  $n=2015$  then for any initial configuration of signs  , there exists a sequence of *moves* such that we'll arrive at a configuration with only  $+$  signs.**b)** Prove that if  $n=2016$  , then there exists an initial configuration of signs such that no matter how we make the *moves* we'll never arrive at a configuration with only  $+$  signs.","t":[{"b":1,"e":1.0,"k":"flat","v":0.96428,"x":0.99554,"p":[[0,36,0.0,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[4,36,0.1111,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,36,0.2222,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[12,36,0.3333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,36,0.4444,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,36,0.5556,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[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],[28,36,0.7778,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,36,0.8889,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,36,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,8,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,8,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"8292345072766957","q":"25. (USS 3) Prove that every partition of 3-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \\in \\mathbb{R}_{+}$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,62,0.0,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,62,0.0645,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,62,0.129,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,62,0.1935,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,62,0.2581,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,62,0.3226,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,62,0.3871,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,62,0.4516,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,62,0.5161,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,62,0.5806,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,62,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[56,62,0.9032,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01562,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.01562,0.04276,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"57d066e8f9f4caea","q":"Let  $ABC$  be a triangle such that  $|AC|=1$  and  $|AB|=\\sqrt 2$ . Let  $M$  be a point such that  $|MA|=|AB|$ ,  $m(\\widehat{MAB}) = 90^\\circ$ , and  $C$  and  $M$  are on the opposite sides of  $AB$ . Let  $N$  be a point such that  $|NA|=|AX|$ ,  $m(\\widehat{NAC}) = 90^\\circ$ , and  $B$  and  $N$  are on the opposite sides of  $AC$ . If the line passing throung  $A$  and the circumcenter of triangle  $MAN$  meets  $[BC]$  at  $F$ , what is  $\\dfrac {|BF|}{|FC|}$ ? $ \n\\textbf{(A)}\\ 2\\sqrt 2\n\\qquad\\textbf{(B)}\\ 2\\sqrt 3\n\\qquad\\textbf{(C)}\\ 2\n\\qquad\\textbf{(D)}\\ 3\n\\qquad\\textbf{(E)}\\ 3\\sqrt 2\n$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.14286,"x":0.22768,"p":[[0,60,0.0,0.18295,0.12995,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,27,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[4,60,0.0667,0.2142,0.1473,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,25,0,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[8,60,0.1333,0.17856,0.09443,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,27,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,60,0.2,0.16963,0.10369,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,28,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,60,0.2667,0.18302,0.12988,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,26,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,60,0.3333,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.16965,0.10972,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,60,0.4667,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.16063,0.09281,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,29,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,60,0.6,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,60,0.6667,0.16518,0.06298,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.16062,0.06919,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.16955,0.09064,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,29,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,60,0.9333,0.15608,0.10328,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[60,60,1.0,0.22768,0.18162,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,24,0,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.1384,"x":0.20536,"p":[[0,61,0.0,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,61,0.0656,0.17411,0.09268,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,25,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,61,0.1311,0.16964,0.07523,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,61,0.1967,0.16963,0.10369,0.14286,0.14286,0.14286,0.0,0.571,1,0,0,1,0,28,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,61,0.2623,0.19197,0.13175,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[20,61,0.3279,0.1384,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,61,0.3934,0.16947,0.10379,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,30,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,61,0.459,0.16964,0.12079,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,27,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,61,0.5246,0.17384,0.10563,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,27,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,61,0.5902,0.15625,0.08268,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,29,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,61,0.6557,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,61,0.7213,0.14286,1e-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],[48,61,0.7869,0.16964,0.09061,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,27,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,61,0.8525,0.17857,0.13363,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,28,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[56,61,0.918,0.20536,0.15947,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[60,61,0.9836,0.19633,0.14174,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,27,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[61,61,1.0,0.19196,0.13175,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"069b341fbd85ee17","q":"Given  $a_i\\ge 0,x_i\\in\\mathbb{R},(i=1,2,\\ldots,n)$ . Prove that  $$ ((1-\\sum_{i=1}^n a_i\\cos x_i)^2+(1-\\sum_{i=1}^n a_i\\sin x_i)^2)^2\\ge 4(1-\\sum_{i=1}^n a_i)^3 $$","t":[{"b":2,"e":1.0,"k":"flat","v":0.8616,"x":0.99554,"p":[[0,46,0.0,0.9241,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[4,46,0.087,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,46,0.1739,0.89732,0.25812,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,25],[12,46,0.2609,0.8616,0.25376,0.85714,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,19],[16,46,0.3478,0.90179,0.20958,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,22],[20,46,0.4348,0.88393,0.23538,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,21],[24,46,0.5217,0.87054,0.23244,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,6,0,20],[28,46,0.6087,0.88393,0.23808,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,22],[32,46,0.6957,0.92857,0.15567,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,24],[36,46,0.7826,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[40,46,0.8696,0.96874,0.09274,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[44,46,0.9565,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[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]]},{"b":5,"e":0.85714,"k":"flat","v":0.79017,"x":0.97321,"p":[[0,72,0.0,0.92856,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[4,72,0.0556,0.94643,0.09942,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[8,72,0.1111,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[12,72,0.1667,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[16,72,0.2222,0.95088,0.11076,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[20,72,0.2778,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,72,0.3333,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[28,72,0.3889,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[32,72,0.4444,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[36,72,0.5,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[40,72,0.5556,0.95088,0.09183,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[44,72,0.6111,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[48,72,0.6667,0.91071,0.14617,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,8,0,20],[52,72,0.7222,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[56,72,0.7778,0.91964,0.08702,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,16],[60,72,0.8333,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[64,72,0.8889,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[68,72,0.9444,0.87051,0.16117,0.82132,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,16],[72,72,1.0,0.79017,0.19558,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,8,0,10]]}]},{"i":"61f5cb03c97283d9","q":"Find all pairs of prime numbers  $p$  and  $q$  that satisfy the equation  $3p^{q}-2q^{p-1}=19$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.75446,"x":0.96429,"p":[[0,37,0.0,0.76339,0.29148,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,4,0,0,1,0,0,2,0,0,2,0,0,7,0,14],[4,37,0.1081,0.92411,0.14719,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,6,0,22],[8,37,0.2162,0.75446,0.27487,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,5,0,0,1,0,0,1,0,0,5,0,0,7,0,12],[12,37,0.3243,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[16,37,0.4324,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[20,37,0.5405,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[24,37,0.6486,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[28,37,0.7568,0.90179,0.13092,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,11,0,17],[32,37,0.8649,0.93301,0.11841,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,22],[36,37,0.973,0.93304,0.14279,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,5,0,24],[37,37,1.0,0.93304,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22]]},{"b":2,"e":1.0,"k":"rising","v":0.75,"x":1.0,"p":[[0,39,0.0,0.80348,0.26926,0.78561,0.85714,1.0,0.14,1.0,0,15,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,0,0,0,9,0,15],[4,39,0.1026,0.75,0.29451,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,2,0,0,2,0,0,3,0,0,3,0,0,5,0,14],[8,39,0.2051,0.7991,0.26692,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,3,0,0,7,0,15],[12,39,0.3077,0.88839,0.16263,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,8,0,18],[16,39,0.4103,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,39,0.5128,0.9733,0.05557,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,39,0.6154,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,39,0.7179,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[32,39,0.8205,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"9d2f1e001c5c1e0a","q":"Let $n, k$ be given positive integers with $n>k$. Prove that\n\n$$\n\\frac{1}{n+1} \\cdot \\frac{n^{n}}{k^{k}(n-k)^{n-k}}<\\frac{n!}{k!(n-k)!}<\\frac{n^{n}}{k^{k}(n-k)^{n-k}} .\n$$","t":[{"b":0,"e":1.0,"k":"rising","v":0.37053,"x":1.0,"p":[[0,33,0.0,0.37053,0.38442,0.0,0.2857,0.57143,0.0,1.0,9,8,0,9,0,5,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,8],[4,33,0.1212,0.76339,0.34738,0.28571,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,21],[8,33,0.2424,0.77232,0.28316,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,5,0,0,2,0,0,5,0,0,1,0,0,1,0,18],[12,33,0.3636,0.79464,0.29437,0.53571,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,6,0,0,2,0,0,2,0,0,0,0,0,2,0,20],[16,33,0.4848,0.92857,0.20825,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[20,33,0.6061,0.80356,0.28961,0.57132,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0,2,0,0,0,0,21],[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],[32,33,0.9697,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[33,33,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]},{"b":3,"e":1.0,"k":"rising","v":0.47768,"x":0.99554,"p":[[0,24,0.0,0.47768,0.40344,0.14286,0.28571,1.0,0.0,1.0,6,11,0,6,0,5,0,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,11],[4,24,0.1667,0.72321,0.35524,0.28571,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,8,0,0,0,0,0,2,0,0,0,0,0,2,0,18],[8,24,0.3333,0.62054,0.34921,0.28571,0.35714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,0,0,1,0,14],[12,24,0.5,0.65625,0.34784,0.28571,0.78571,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,16],[16,24,0.6667,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[20,24,0.8333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,24,1.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29]]}]},{"i":"2e7e5b6eccc1d249","q":"The numbers  $2^0, 2^1, \\dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$  are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.11607,"p":[[0,25,0.0,0.11607,0.29329,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,25,0.16,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,25,0.32,0.04464,0.17655,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,25,0.8,0.04464,0.1729,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[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],[25,25,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.25893,"p":[[0,30,0.0,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,30,0.1333,0.05357,0.21053,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,30,0.2667,0.25893,0.41255,0.0,0.0,0.46429,0.0,1.0,22,7,0,22,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,7],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9e1e626c9527b027","q":"There are two prime numbers  $p$  so that  $5p$  can be expressed in the form  $\\left\\lfloor \\dfrac{n^2}{5}\\right\\rfloor$  for some positive integer  $n.$  What is the sum of these two prime numbers?","t":[{"b":0,"e":1.0,"k":"flat","v":0.85714,"x":0.98201,"p":[[0,64,0.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[4,64,0.0625,0.88839,0.12745,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,11,0,15],[8,64,0.125,0.92411,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[12,64,0.1875,0.93304,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,19],[16,64,0.25,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,64,0.3125,0.98201,0.05983,1.0,1.0,1.0,0.71,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,64,0.375,0.9375,0.11812,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[28,64,0.4375,0.94196,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[32,64,0.5,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[36,64,0.5625,0.93304,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,8,0,22],[40,64,0.625,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[44,64,0.6875,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[48,64,0.75,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[52,64,0.8125,0.92409,0.11841,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[56,64,0.875,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[60,64,0.9375,0.90179,0.14032,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,20],[64,64,1.0,0.85714,0.12877,0.82132,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,14,0,10]]},{"b":2,"e":0.2857,"k":"falling","v":0.46416,"x":0.9375,"p":[[0,185,0.0,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[4,185,0.0216,0.9375,0.09407,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,20],[8,185,0.0432,0.9241,0.11286,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[12,185,0.0649,0.92411,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20],[16,185,0.0865,0.9375,0.10677,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[20,185,0.1081,0.88839,0.13709,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,10,0,16],[24,185,0.1297,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[28,185,0.1514,0.91071,0.11152,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,17],[32,185,0.173,0.91071,0.12242,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,7,0,19],[36,185,0.1946,0.90625,0.12169,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,18],[40,185,0.2162,0.875,0.12242,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,16,0,11],[44,185,0.2378,0.89286,0.10714,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,14],[48,185,0.2595,0.84375,0.16887,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,9,0,13],[52,185,0.2811,0.89286,0.12877,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16],[56,185,0.3027,0.89286,0.17128,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,8,0,19],[60,185,0.3243,0.87945,0.13418,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,12,0,14],[64,185,0.3459,0.86161,0.16164,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,5,0,16],[68,185,0.3676,0.90179,0.12078,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,16],[72,185,0.3892,0.92411,0.11837,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,11,0,19],[76,185,0.4108,0.88392,0.11538,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,13],[80,185,0.4324,0.9241,0.10092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,18],[84,185,0.4541,0.84821,0.15542,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,11,0,12],[88,185,0.4757,0.89285,0.11294,0.85711,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,15],[92,185,0.4973,0.91071,0.14617,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,9,0,19],[96,185,0.5189,0.90178,0.12595,0.85711,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,7,0,18],[100,185,0.5405,0.91071,0.11152,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,17],[104,185,0.5622,0.875,0.14174,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,16],[108,185,0.5838,0.90625,0.11633,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,17],[112,185,0.6054,0.87054,0.13997,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,7,0,15],[116,185,0.627,0.87945,0.12934,0.85711,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,11,0,14],[120,185,0.6486,0.87946,0.11904,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,12,0,13],[124,185,0.6703,0.89732,0.10853,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,14],[128,185,0.6919,0.91071,0.10564,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17],[132,185,0.7135,0.88839,0.12745,0.82132,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,16],[136,185,0.7351,0.87053,0.13997,0.85711,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,12,0,13],[140,185,0.7568,0.86161,0.13115,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,8,0,13],[144,185,0.7784,0.91518,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,17],[148,185,0.8,0.88839,0.13236,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,9,0,16],[152,185,0.8216,0.85714,0.14725,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,7,0,14],[156,185,0.8432,0.87054,0.14445,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,13,0,13],[160,185,0.8649,0.87499,0.13716,0.85711,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,13,0,13],[164,185,0.8865,0.88839,0.12234,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,13,0,14],[168,185,0.9081,0.85714,0.16751,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,7,0,15],[172,185,0.9297,0.875,0.13716,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,11,0,14],[176,185,0.9514,0.88839,0.11701,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,14],[180,185,0.973,0.89286,0.11845,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,11,0,15],[184,185,0.9946,0.87946,0.21162,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,6,0,20],[185,185,1.0,0.46416,0.26972,0.28571,0.28571,0.57143,0.14,1.0,0,4,0,0,0,3,0,0,14,0,0,4,0,0,4,0,0,1,0,0,2,0,4]]}]},{"i":"72bd666bd23f44a7","q":"The function  $f(n)$  satisfies  $f(0)=0$ ,  $f(n)=n-f \\left( f(n-1) \\right)$ ,  $n=1,2,3 \\cdots$ . Find all polynomials  $g(x)$  with real coefficient such that\n\\[ f(n)= [ g(n) ], \\qquad n=0,1,2 \\cdots \\]\nWhere  $[ g(n) ]$  denote the greatest integer that does not exceed  $g(n)$ .","t":[{"b":6,"e":1.0,"k":"rising","v":0.66072,"x":0.97321,"p":[[0,36,0.0,0.66072,0.16656,0.53572,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,17,0,0,1,0,3],[4,36,0.1111,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[8,36,0.2222,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[12,36,0.3333,0.91518,0.13296,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,22],[16,36,0.4444,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[20,36,0.5556,0.90179,0.25364,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,26],[24,36,0.6667,0.84375,0.19019,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,12,0,0,2,0,16],[28,36,0.7778,0.89286,0.14725,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,4,0,19],[32,36,0.8889,0.84821,0.14698,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,4,0,14],[36,36,1.0,0.83036,0.16917,0.71429,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,3,0,14]]},{"b":7,"e":0.14286,"k":"falling","v":0.14286,"x":0.92411,"p":[[0,68,0.0,0.57588,0.14054,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,12,0,0,4,0,0,15,0,0,0,0,0],[4,68,0.0588,0.89732,0.1394,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,20],[8,68,0.1176,0.92411,0.14719,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,24],[12,68,0.1765,0.92411,0.11837,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,22],[16,68,0.2353,0.88393,0.20341,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,21],[20,68,0.2941,0.86159,0.21868,0.82132,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,4,0,20],[24,68,0.3529,0.88393,0.16146,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,1,0,20],[28,68,0.4118,0.58481,0.38855,0.14286,0.71429,1.0,0.0,1.0,5,9,0,5,0,6,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,9],[32,68,0.4706,0.89285,0.21429,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,22],[36,68,0.5294,0.76338,0.25156,0.71429,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,10,0,0,3,0,12],[40,68,0.5882,0.76338,0.25658,0.67857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,9,0,0,2,0,13],[44,68,0.6471,0.80341,0.23899,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,8,0,0,3,0,15],[48,68,0.7059,0.60266,0.35307,0.14286,0.71429,1.0,0.0,1.0,1,9,0,1,0,9,0,0,0,0,0,1,0,0,3,0,0,6,0,0,3,0,9],[52,68,0.7647,0.64732,0.34807,0.2857,0.71429,1.0,0.0,1.0,1,11,0,1,0,6,0,0,2,0,0,2,0,0,2,0,0,4,0,0,4,0,11],[56,68,0.8235,0.56249,0.33491,0.14286,0.71429,0.71429,0.0,1.0,1,6,0,1,0,10,0,0,0,0,0,0,0,0,2,0,0,12,0,0,1,0,6],[60,68,0.8824,0.50223,0.3769,0.14286,0.35714,0.94643,0.14286,1.0,0,8,0,0,0,16,0,0,0,0,0,0,0,0,2,0,0,4,0,0,1,1,8],[64,68,0.9412,0.22322,0.25238,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[68,68,1.0,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ad7d9e6cbfa549d9","q":"Let $\\triangle A B C$ be a right triangle with $\\angle A=90^{\\circ}$ and circumcircle $\\Gamma$. The incircle touches $B C$ at a point $D$. Let $E$ be the midpoint of the arc $A B$ of $\\Gamma$ that does not contain $C$ and let $F$ be the midpoint of the arc $A C$ of $\\Gamma$ that does not contain $B$.\na) Prove that $\\triangle A B C \\sim \\triangle D E F$.\nb) Prove that $E F$ passes through the points where the incircle touches $A B$ and $A C$.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.06249,"x":0.27231,"p":[[0,75,0.0,0.17857,0.3481,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,3],[4,75,0.0533,0.27231,0.39017,0.0,0.0,0.60712,0.0,1.0,20,4,0,20,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,4],[8,75,0.1067,0.21428,0.35355,0.0,0.0,0.32144,0.0,1.0,22,3,0,22,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,3],[12,75,0.16,0.20981,0.33688,0.0,0.0,0.28571,0.0,1.0,21,3,0,21,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,1,0,3],[16,75,0.2133,0.2142,0.31341,0.0,0.0,0.35714,0.0,1.0,19,2,0,19,0,2,0,0,3,0,0,0,0,0,4,0,0,2,0,0,0,0,2],[20,75,0.2667,0.06249,0.20802,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[24,75,0.32,0.15624,0.29955,0.0,0.0,0.03572,0.0,1.0,24,1,0,24,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,2,0,1],[28,75,0.3733,0.21424,0.35709,0.0,0.0,0.35704,0.0,1.0,22,3,0,22,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,3],[32,75,0.4267,0.21874,0.35888,0.0,0.0,0.28571,0.0,1.0,21,4,0,21,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,4],[36,75,0.48,0.14284,0.26484,0.0,0.0,0.2857,0.0,1.0,23,1,0,23,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[40,75,0.5333,0.23214,0.36202,0.0,0.0,0.46429,0.0,1.0,21,3,0,21,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,3],[44,75,0.5867,0.21425,0.32138,0.0,0.0,0.42858,0.0,1.0,20,2,0,20,0,1,0,0,2,0,0,2,0,0,3,0,0,1,0,0,1,0,2],[48,75,0.64,0.24105,0.36843,0.0,0.0,0.4642,0.0,1.0,21,3,0,21,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,3,0,3],[52,75,0.6933,0.13383,0.24467,0.0,0.0,0.1786,0.0,0.85714,22,0,0,22,0,2,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0],[56,75,0.7467,0.17411,0.30875,0.0,0.0,0.1786,0.0,1.0,22,2,0,22,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,2],[60,75,0.8,0.07588,0.20508,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[64,75,0.8533,0.16071,0.29826,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,1],[68,75,0.9067,0.13837,0.26838,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,0,1,0,1],[72,75,0.96,0.25445,0.3326,0.0,0.07143,0.46418,0.0,1.0,16,3,0,16,0,3,0,0,4,0,0,1,0,0,4,0,0,0,0,0,1,0,3],[75,75,1.0,0.09822,0.21558,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.14286,"k":"falling","v":0.02223,"x":0.29463,"p":[[0,79,0.0,0.23658,0.34919,0.0,0.0,0.4286,0.0,1.0,19,3,0,19,0,2,0,0,2,0,0,2,0,0,2,0,0,0,0,0,2,0,3],[4,79,0.0506,0.12937,0.30169,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[8,79,0.1013,0.1741,0.34206,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[12,79,0.1519,0.25,0.39123,0.0,0.0,0.60714,0.0,1.0,22,4,0,22,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,4],[16,79,0.2025,0.1875,0.36672,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[20,79,0.2532,0.20982,0.37962,0.0,0.0,0.17857,0.0,1.0,23,5,0,23,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,5],[24,79,0.3038,0.13838,0.31027,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3],[28,79,0.3544,0.12944,0.27044,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,1,0,1],[32,79,0.4051,0.14731,0.28229,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,1],[36,79,0.4557,0.29463,0.37275,0.0,0.07145,0.57111,0.0,1.0,16,5,0,16,0,1,0,0,6,0,0,0,0,0,3,0,0,0,0,0,1,0,5],[40,79,0.5063,0.13837,0.31636,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,2],[44,79,0.557,0.14286,0.30723,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,2],[48,79,0.6076,0.15178,0.28333,0.0,0.0,0.2857,0.0,1.0,22,2,0,22,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[52,79,0.6582,0.11607,0.24855,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,0],[56,79,0.7089,0.08034,0.20492,0.0,0.0,0.0,0.0,0.857,27,0,0,27,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[60,79,0.7595,0.20089,0.34784,0.0,0.0,0.28571,0.0,1.0,22,3,0,22,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,3],[64,79,0.8101,0.15625,0.29093,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,2,0,0,1,0,0,2,0,0,0,0,0,2,0,1],[68,79,0.8608,0.22765,0.36569,0.0,0.0,0.28571,0.0,1.0,20,5,0,20,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[72,79,0.9114,0.23658,0.35282,0.0,0.0,0.35704,0.0,1.0,20,3,0,20,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,2,0,3],[76,79,0.962,0.12945,0.27745,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,4,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[79,79,1.0,0.02223,0.06281,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b88ea5d4795b444c","q":"In a triangle  $ABC$ , a point  $P$  in the interior of  $ABC$  is such that   $$  \\angle BPC - \\angle BAC = \\angle CPA - \\angle CBA = \\angle APB - \\angle ACB. $$   Suppose  $\\angle BAC = 30^{\\circ}$  and  $AP = 12$ . Let  $D,E,F$  be the feet of perpendiculars from  $P$  on to  $BC,CA,AB$  respectively. If  $m \\sqrt{n}$  is the area of the triangle DEF where  $m,n$  are integers with  $n$  prime, then what is the value of the product  $mn$ 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,0,0,0,0,0,0,0,0,0,0],[280,305,0.918,0.24553,0.06423,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[284,305,0.9311,0.25,0.06186,0.24999,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[288,305,0.9443,0.25447,0.05906,0.2857,0.28571,0.28571,0.14286,0.286,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[292,305,0.9574,0.25446,0.10555,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,10,0,0,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[296,305,0.9705,0.24107,0.07523,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[300,305,0.9836,0.27232,0.05486,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[304,305,0.9967,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[305,305,1.0,0.27233,0.04164,0.28571,0.28571,0.28571,0.14286,0.286,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"04c5cae8b6782efa","q":"Given is an equilateral triangle $A B C$. On the line through $B$ parallel to $A C$ lies a point $D$, such that $D$ and $C$ are on the same side of line $A B$. The perpendicular bisector of $C D$ intersects the line $A B$ at $E$. Prove that triangle $C D E$ is equilateral.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.14723,"x":0.23205,"p":[[0,26,0.0,0.23205,0.11159,0.14286,0.21428,0.28571,0.0,0.42857,1,0,0,1,0,15,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.2188,0.10714,0.14286,0.14286,0.28571,0.14286,0.43,0,0,0,0,0,20,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.19187,0.08464,0.14286,0.14286,0.2857,0.14,0.4286,0,0,0,0,0,23,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.14723,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.14733,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.4286,"k":"flat","v":0.19634,"x":0.36161,"p":[[0,27,0.0,0.21428,0.10714,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,21,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.19634,0.12246,0.14286,0.14286,0.28571,0.0,0.4286,3,0,0,3,0,19,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.20094,0.12815,0.14286,0.14286,0.28571,0.0,0.43,4,0,0,4,0,16,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.25,0.13363,0.14286,0.2857,0.42857,0.0,0.4286,2,0,0,2,0,13,0,0,8,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.31688,0.1057,0.2857,0.28571,0.42857,0.14,0.4286,0,0,0,0,0,6,0,0,13,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.35714,0.09449,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,13,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[24,27,0.8889,0.36161,0.10705,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,11,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[27,27,1.0,0.35714,0.07144,0.28571,0.35714,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0c2b78159bf110b4","q":"In a tennis tournament,  $n$  players want to make  $2$  vs  $2$  matches such that each player has each of the other players as opponents exactly once. Find all possible values of  $n$ .","t":[{"b":3,"e":0.42857,"k":"flat","v":0.41964,"x":0.44643,"p":[[0,37,0.0,0.41964,0.07936,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0],[4,37,0.1081,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.43304,0.10403,0.42857,0.42857,0.42857,0.0,0.71429,1,0,1,1,0,0,0,0,0,0,0,29,0,0,0,0,0,2,0,0,0,0,0],[12,37,0.3243,0.4375,0.04971,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0],[16,37,0.4324,0.44643,0.06916,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0],[20,37,0.5405,0.42857,1e-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],[24,37,0.6486,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.40625,"x":0.44197,"p":[[0,24,0.0,0.40625,0.10779,0.42857,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,0,0,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0],[4,24,0.1667,0.44197,0.05486,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0],[8,24,0.3333,0.4375,0.03458,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0],[12,24,0.5,0.43304,0.02486,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0],[16,24,0.6667,0.41518,0.07457,0.42857,0.42857,0.42857,0.0,0.4286,1,0,1,1,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.43304,0.02486,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0],[24,24,1.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c854b04dac36884b","q":"Find the pairs of integers  $(a, b)$  such that  $a^2 + 2b^2 + 2a +1$  is a divisor of  $2ab$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.7589,"x":0.89731,"p":[[0,9,0.0,0.7589,0.26109,0.78571,0.85714,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,19,0,5],[4,9,0.4444,0.83928,0.11152,0.85714,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,23,0,4],[8,9,0.8889,0.89286,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[9,9,1.0,0.89731,0.08918,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,12]]},{"b":7,"e":0.85714,"k":"flat","v":0.8125,"x":0.85713,"p":[[0,67,0.0,0.83928,0.09279,0.85714,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,26,0,2],[4,67,0.0597,0.84821,0.10062,0.85714,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,24,0,4],[8,67,0.1194,0.81694,0.08921,0.85711,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,26,0,0],[12,67,0.1791,0.83481,0.16795,0.85714,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,25,0,4],[16,67,0.2388,0.84821,0.08702,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,26,0,3],[20,67,0.2985,0.82587,0.17401,0.85714,0.85714,0.85714,0.0,1.0,1,4,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,24,0,4],[24,67,0.3582,0.8348,0.10174,0.85714,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,26,0,2],[28,67,0.4179,0.83034,0.10377,0.85714,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,28,0,1],[32,67,0.4776,0.83926,0.08568,0.85714,0.85714,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,26,0,2],[36,67,0.5373,0.8348,0.09526,0.85714,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,23,0,3],[40,67,0.597,0.84375,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,1],[44,67,0.6567,0.84821,0.10062,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,22,0,5],[48,67,0.7164,0.85713,0.07991,0.85714,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,25,0,4],[52,67,0.7761,0.84375,0.08268,0.85714,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,27,0,2],[56,67,0.8358,0.84375,0.09689,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,23,0,4],[60,67,0.8955,0.83479,0.10786,0.85714,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,25,0,3],[64,67,0.9552,0.82142,0.10717,0.85714,0.85714,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,24,0,2],[67,67,1.0,0.8125,0.12595,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,19,0,4]]}]},{"i":"7aa11f253372964a","q":"Determine all functions  $f:(0,\\infty)\\to\\mathbb{R}$  satisfying   $$ \\left(x+\\frac{1}{x}\\right)f(y)=f(xy)+f\\left(\\frac{y}{x}\\right) $$   for all  $x,y>0$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.17856,"x":0.21875,"p":[[0,8,0.0,0.20982,0.09438,0.14286,0.21428,0.28571,0.0,0.42857,2,0,0,2,0,14,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.17856,0.13359,0.10714,0.14286,0.28571,0.0,0.571,8,0,0,8,0,10,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,8,1.0,0.21875,0.10091,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,18,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.16062,"x":0.24107,"p":[[0,20,0.0,0.1875,0.23808,0.0,0.14286,0.2857,0.0,1.0,11,2,0,11,0,10,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,20,0.2,0.16062,0.12244,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.1875,0.15746,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,8,0,0,13,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,20,0.6,0.16964,0.18013,0.0,0.14286,0.28571,0.0,0.85714,10,0,0,10,0,12,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[16,20,0.8,0.17411,0.17762,0.14286,0.14286,0.2857,0.0,1.0,7,1,0,7,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,20,1.0,0.24107,0.12078,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,13,0,0,15,0,0,1,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"6bf3560cff0e8291","q":"Two different  $3$  digit numbers are picked and then for every of them is calculated sum of all  $5$  numbers which are getting when digits of picked number change place (etc. if one of the number is  $707$ , the sum is   $2401=770+77+77+770+707$ ). Do the given results must be different?","t":[{"b":2,"e":1.0,"k":"rising","v":0.64285,"x":1.0,"p":[[0,31,0.0,0.64285,0.34256,0.28571,0.57144,1.0,0.0,1.0,1,14,0,1,0,0,0,0,10,0,0,5,0,0,0,0,0,1,0,0,1,0,14],[4,31,0.129,0.9375,0.19541,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[8,31,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[31,31,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"rising","v":0.65625,"x":0.99554,"p":[[0,47,0.0,0.65625,0.35149,0.28571,0.71429,1.0,0.0,1.0,2,15,0,2,0,0,0,0,7,0,0,6,0,0,1,0,0,0,0,0,1,0,15],[4,47,0.0851,0.66964,0.31831,0.42857,0.57143,1.0,0.0,1.0,1,14,0,1,0,0,0,0,5,0,0,8,0,0,3,0,0,0,0,0,1,0,14],[8,47,0.1702,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,47,0.2553,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,47,0.3404,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,47,0.4255,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,47,0.5106,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,47,0.5957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,47,0.6809,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,47,0.766,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[40,47,0.8511,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,47,0.9362,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[47,47,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"b4105e42da6f231e","q":"The numbers  $3$ ,  $4$  , $...$ ,  $2019$  are written on a blackboard. David and Edgardo play alternately, starting with David. On their turn, each player must erase a number from the board and write two positive integers whose sum is\nequal to the number just deleted. The winner is the one who makes all the numbers on the board equal. Determine who has a winning strategy and describe it.","t":[{"b":1,"e":0.71429,"k":"rising","v":0.54007,"x":0.77679,"p":[[0,5,0.0,0.54007,0.31092,0.14286,0.71429,0.75,0.14,1.0,0,3,0,0,0,9,0,0,4,0,0,0,0,0,2,0,0,9,0,0,5,0,3],[4,5,0.8,0.77679,0.09407,0.71429,0.71429,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,8,0,3],[5,5,1.0,0.77678,0.07935,0.71429,0.71429,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,12,0,1]]},{"b":2,"e":0.85714,"k":"rising","v":0.62053,"x":0.91071,"p":[[0,55,0.0,0.62053,0.31259,0.28571,0.71429,0.85714,0.0,1.0,2,4,0,2,0,3,0,0,4,0,0,2,0,0,0,0,0,8,0,0,9,0,4],[4,55,0.0727,0.77232,0.07873,0.71429,0.71429,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,1],[8,55,0.1455,0.72308,0.15127,0.71429,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,6,0,1],[12,55,0.2182,0.76338,0.06784,0.71429,0.71429,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,11,0,0],[16,55,0.2909,0.78125,0.09439,0.71429,0.71429,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,9,0,3],[20,55,0.3636,0.81249,0.08328,0.71429,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,18,0,2],[24,55,0.4364,0.79464,0.10062,0.71429,0.71429,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,10,0,4],[28,55,0.5091,0.83929,0.11152,0.71429,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,12,0,8],[32,55,0.5818,0.85254,0.09115,0.85711,0.85714,0.85714,0.71,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,19,0,6],[36,55,0.6545,0.85268,0.10999,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,13,0,9],[40,55,0.7273,0.87946,0.1017,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,15,0,11],[44,55,0.8,0.83929,0.09942,0.71429,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,16,0,6],[48,55,0.8727,0.85714,0.10714,0.85711,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,17,0,8],[52,55,0.9455,0.87052,0.08268,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,21,0,7],[55,55,1.0,0.91071,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15]]}]},{"i":"0c2c83720626e4ae","q":"Points  $N$  and  $M$  are on the sides  $CD$  and  $BC$  of square  $ABCD$ , respectively. The perimeter of triangle  $MCN$  is equal to the double of the length of the square's side. Find  $\\angle MAN$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.73652,"x":1.0,"p":[[0,19,0.0,0.73652,0.29275,0.42859,0.85714,1.0,0.14,1.0,0,14,0,0,0,2,0,0,3,0,0,4,0,0,1,0,0,5,0,0,3,0,14],[4,19,0.2105,0.9375,0.16342,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[8,19,0.4211,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,19,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,19,0.8421,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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":6,"e":0.42857,"k":"falling","v":0.69641,"x":0.87491,"p":[[0,24,0.0,0.87491,0.28314,1.0,1.0,1.0,0.14,1.0,0,25,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[4,24,0.1667,0.78572,0.29666,0.64286,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,1,0,0,4,0,0,0,0,0,2,0,0,5,0,17],[8,24,0.3333,0.8348,0.18597,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,7,0,14],[12,24,0.5,0.73215,0.2829,0.42857,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,4,0,0,1,0,0,2,0,0,10,0,10],[16,24,0.6667,0.80357,0.25442,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,6,0,0,4,0,16],[20,24,0.8333,0.70982,0.31234,0.39286,0.85714,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,5,0,0,1,0,0,3,0,0,0,0,0,9,0,11],[24,24,1.0,0.69641,0.24419,0.5354,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,5,0,0,3,0,0,6,0,0,10,0,5]]}]},{"i":"795c4c7624b65d88","q":"Zeroes are written in all cells of a  $5 \\times 5$  board.  We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it.  Is it possible to obtain the number 2012 in all cells simultaneously?","t":[{"b":1,"e":1.0,"k":"rising","v":0.18304,"x":0.49101,"p":[[0,16,0.0,0.18304,0.34113,0.0,0.0,0.14289,0.0,1.0,22,4,0,22,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,4],[4,16,0.25,0.4464,0.4252,0.0,0.35714,1.0,0.0,1.0,11,10,0,11,0,4,0,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,10],[8,16,0.5,0.30804,0.40894,0.0,0.0,0.71429,0.0,1.0,18,6,0,18,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,6],[12,16,0.75,0.45087,0.39946,0.0,0.35714,0.85714,0.0,1.0,11,7,0,11,0,0,0,0,5,0,0,1,0,0,4,0,0,1,0,0,3,0,7],[16,16,1.0,0.49101,0.35343,0.10714,0.571,0.71429,0.0,1.0,8,6,0,8,0,1,0,0,1,0,0,4,0,0,8,0,0,3,0,0,1,0,6]]},{"b":2,"e":0.85714,"k":"rising","v":0.16072,"x":0.96875,"p":[[0,108,0.0,0.16072,0.34022,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,4],[4,108,0.037,0.58481,0.41705,0.10714,0.71429,1.0,0.0,1.0,8,12,0,8,0,1,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,12],[8,108,0.0741,0.56686,0.43086,0.105,0.64264,1.0,0.0,1.0,8,14,0,8,0,2,0,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,14],[12,108,0.1111,0.56253,0.41177,0.10714,0.64264,1.0,0.0,1.0,8,12,0,8,0,2,0,0,0,0,0,5,0,0,1,0,0,3,0,0,1,0,12],[16,108,0.1481,0.52231,0.42047,0.0,0.49979,1.0,0.0,1.0,9,11,0,9,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,0,2,0,11],[20,108,0.1852,0.625,0.42068,0.14286,0.85714,1.0,0.0,1.0,7,13,0,7,0,3,0,0,1,0,0,0,0,0,1,0,0,2,0,0,5,0,13],[24,108,0.2222,0.7366,0.35012,0.42859,1.0,1.0,0.0,1.0,3,17,0,3,0,2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,2,0,17],[28,108,0.2593,0.82143,0.30094,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,3,0,21],[32,108,0.2963,0.73658,0.37136,0.49967,1.0,1.0,0.0,1.0,3,20,0,3,0,2,0,0,3,0,0,0,0,0,3,0,0,1,0,0,0,0,20],[36,108,0.3333,0.79464,0.29868,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,4,0,0,0,0,0,0,0,0,4,0,0,5,0,17],[40,108,0.3704,0.81695,0.29286,0.78571,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,5,0,19],[44,108,0.4074,0.81696,0.30978,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,0,1,0,22],[48,108,0.4444,0.72321,0.36585,0.28571,1.0,1.0,0.0,1.0,2,17,0,2,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,0,4,0,17],[52,108,0.4815,0.74105,0.37191,0.53539,1.0,1.0,0.0,1.0,5,18,0,5,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,3,0,18],[56,108,0.5185,0.8125,0.33396,0.82132,1.0,1.0,0.0,1.0,2,22,0,2,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,22],[60,108,0.5556,0.78569,0.32342,0.67857,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,6,0,17],[64,108,0.5926,0.79016,0.35173,0.82143,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,4,0,20],[68,108,0.6296,0.79463,0.32526,0.5354,1.0,1.0,0.0,1.0,1,22,0,1,0,2,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,22],[72,108,0.6667,0.8348,0.27692,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,1,0,0,1,0,0,3,0,0,2,0,0,2,0,21],[76,108,0.7037,0.8348,0.26272,0.67857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,1,0,0,2,0,21],[80,108,0.7407,0.94195,0.133,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[84,108,0.7778,0.90624,0.21314,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,25],[88,108,0.8148,0.90179,0.26592,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,26],[92,108,0.8519,0.84821,0.23941,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,8,0,17],[96,108,0.8889,0.92855,0.15158,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,25],[100,108,0.9259,0.78571,0.32927,0.75,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,5,0,19],[104,108,0.963,0.75448,0.35033,0.53575,1.0,1.0,0.0,1.0,3,18,0,3,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,0,4,0,18],[108,108,1.0,0.96875,0.10555,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28]]}]},{"i":"ea38c73c68360d00","q":"Let  $ABCDE$  be a convex pentagon such that  $AB = BC = CD$  and  $\\angle BDE =\n\\angle EAC = 30 ^{\\circ}$ . Find the possible values of  $\\angle BEC$ .\n\n*Proposed by Josef Tkadlec (Czech Republic)*","t":[{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.45086,"p":[[0,98,0.0,0.45086,0.37982,0.0,0.42857,0.85714,0.0,1.0,10,5,2,10,0,0,0,0,5,0,0,3,0,0,3,0,0,1,0,0,5,0,5],[4,98,0.0408,0.22768,0.309,0.0,0.14286,0.2857,0.0,1.0,13,3,0,13,0,9,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,3],[8,98,0.0816,0.18749,0.25612,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,13,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[12,98,0.1224,0.24098,0.27537,0.0,0.14286,0.32143,0.0,1.0,10,2,0,10,0,10,0,0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,2],[16,98,0.1633,0.23214,0.29613,0.0,0.14286,0.2857,0.0,1.0,11,3,0,11,0,11,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,3],[20,98,0.2041,0.16963,0.2214,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,9,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[24,98,0.2449,0.16518,0.24513,0.0,0.14286,0.14286,0.0,1.0,13,2,0,13,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[28,98,0.2857,0.12947,0.28652,0.0,0.0,0.14286,0.0,1.0,21,3,0,21,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[32,98,0.3265,0.12946,0.19678,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[36,98,0.3673,0.08927,0.20121,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,98,0.4082,0.09821,0.19704,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[44,98,0.449,0.19634,0.29829,0.0,0.0,0.2857,0.0,1.0,17,2,0,17,0,6,0,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,2],[48,98,0.4898,0.09375,0.18073,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,98,0.5306,0.21432,0.26246,0.0,0.14286,0.32143,0.0,1.0,12,1,0,12,0,10,0,0,2,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[56,98,0.5714,0.17634,0.29068,0.0,0.0,0.14286,0.0,1.0,18,2,0,18,1,6,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,2],[60,98,0.6122,0.41518,0.4062,0.0,0.2857,1.0,0.0,1.0,9,9,0,9,0,5,0,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,9],[64,98,0.6531,0.17857,0.26,0.0,0.14286,0.14286,0.0,1.0,13,2,0,13,0,12,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[68,98,0.6939,0.19195,0.25655,0.0,0.14286,0.1786,0.0,1.0,12,2,0,12,0,12,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[72,98,0.7347,0.35714,0.37287,0.10714,0.14286,0.46431,0.0,1.0,8,7,0,8,0,10,0,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,7],[76,98,0.7755,0.32589,0.41532,0.0,0.14286,0.78571,0.0,1.0,14,8,0,14,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,8],[80,98,0.8163,0.3125,0.39031,0.0,0.14286,0.53571,0.0,1.0,12,6,0,12,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,6],[84,98,0.8571,0.31686,0.33265,0.14214,0.14286,0.42857,0.0,1.0,7,5,0,7,0,12,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,5],[88,98,0.898,0.17854,0.2879,0.0,0.0,0.28571,0.0,1.0,19,2,0,19,0,4,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,2],[92,98,0.9388,0.07589,0.20198,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[96,98,0.9796,0.07143,0.22868,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[98,98,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.59587,"p":[[0,111,0.0,0.59587,0.38666,0.14214,0.857,0.85714,0.0,1.0,7,5,1,7,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,12,1,5],[4,111,0.036,0.29008,0.32635,0.14286,0.14286,0.28571,0.0,1.0,5,5,0,5,0,17,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,5],[8,111,0.0721,0.19197,0.25904,0.0,0.14286,0.1429,0.0,1.0,11,2,0,11,0,14,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[12,111,0.1081,0.20981,0.30509,0.0,0.14286,0.17857,0.0,1.0,14,3,0,14,0,10,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[16,111,0.1441,0.06915,0.1002,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,1,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,111,0.1802,0.13384,0.17835,0.0,0.14286,0.14286,0.0,0.85714,14,0,0,14,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[24,111,0.2162,0.11151,0.16257,0.0,0.07,0.14286,0.0,0.71429,16,0,0,16,0,12,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[28,111,0.2523,0.12482,0.24678,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[32,111,0.2883,0.14285,0.1923,0.0,0.07143,0.1429,0.0,0.71429,16,0,0,16,0,9,0,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[36,111,0.3243,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[40,111,0.3604,0.15179,0.21706,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,14,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[44,111,0.3964,0.05804,0.14665,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,111,0.4324,0.07143,0.24223,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[52,111,0.4685,0.04464,0.13092,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[56,111,0.5045,0.10714,0.24744,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[60,111,0.5405,0.03125,0.08552,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,111,0.5766,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[68,111,0.6126,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,111,0.6486,0.04911,0.15815,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[76,111,0.6847,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[80,111,0.7207,0.05357,0.18123,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[84,111,0.7568,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,111,0.7928,0.10268,0.29066,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[92,111,0.8288,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,111,0.8649,0.03795,0.08373,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,111,0.9009,0.06696,0.22583,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[104,111,0.9369,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,111,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[111,111,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2227c4d4ae64d774","q":"In a scalene triangle  $ABC$  with centroid  $G$  and circumcircle  $\\omega$  centred at  $O$ , the extension of  $AG$  meets  $\\omega$  at  $M$ ; lines  $AB$  and  $CM$  intersect at  $P$ ; and lines  $AC$  and  $BM$  intersect at  $Q$ . Suppose the circumcentre  $S$  of the triangle  $APQ$  lies on  $\\omega$  and  $A, O, S$  are collinear. Prove that  $\\angle AGO = 90^{o}$ .","t":[{"b":2,"e":0.14286,"k":"falling","v":0.17393,"x":0.69186,"p":[[0,79,0.0,0.3258,0.31187,0.14286,0.14286,0.42858,0.0,1.0,3,4,2,3,0,15,0,0,5,0,0,3,0,0,0,0,0,1,0,0,1,0,4],[4,79,0.0506,0.69186,0.32378,0.42857,0.85707,1.0,0.14,1.0,0,14,0,0,0,3,0,0,4,0,0,5,0,0,2,0,0,1,0,0,3,0,14],[8,79,0.1013,0.63392,0.33108,0.28571,0.64286,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,7,0,0,5,0,0,1,0,0,2,0,0,2,0,12],[12,79,0.1519,0.58928,0.34395,0.28571,0.5,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,8,0,0,3,0,0,1,0,0,3,0,0,1,0,11],[16,79,0.2025,0.60268,0.35307,0.28571,0.71429,1.0,0.0,1.0,1,10,0,1,0,5,0,0,7,0,0,1,0,0,0,0,0,5,0,0,3,0,10],[20,79,0.2532,0.60704,0.34638,0.2857,0.64286,1.0,0.14,1.0,0,11,0,0,0,6,0,0,6,0,0,2,0,0,2,0,0,3,0,0,2,0,11],[24,79,0.3038,0.5267,0.33215,0.2857,0.42857,1.0,0.14,1.0,0,9,0,0,0,6,0,0,9,0,0,3,0,0,3,0,0,2,0,0,0,0,9],[28,79,0.3544,0.58033,0.3443,0.24999,0.57141,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,4,0,0,1,0,0,5,0,0,3,0,0,1,0,10],[32,79,0.4051,0.66067,0.2874,0.42859,0.64286,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,3,0,0,3,0,0,7,0,0,3,0,0,4,0,9],[36,79,0.4557,0.54463,0.3183,0.28571,0.42859,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,9,0,0,5,0,0,0,0,0,5,0,0,2,0,7],[40,79,0.5063,0.49999,0.32926,0.2857,0.35714,0.78571,0.14286,1.0,0,8,0,0,0,7,0,0,9,0,0,4,0,0,1,0,0,3,0,0,0,0,8],[44,79,0.557,0.45981,0.30037,0.24999,0.28571,0.71429,0.14286,1.0,0,4,0,0,0,8,0,0,9,0,0,3,0,0,3,0,0,2,0,0,3,0,4],[48,79,0.6076,0.40625,0.36441,0.14286,0.21428,0.71429,0.0,1.0,3,7,0,3,0,13,0,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,7],[52,79,0.6582,0.42854,0.28119,0.24999,0.28571,0.5711,0.14286,1.0,0,3,0,0,0,8,0,0,10,0,0,4,0,0,3,0,0,1,0,0,3,0,3],[56,79,0.7089,0.44194,0.31814,0.14286,0.35714,0.71429,0.14286,1.0,0,5,0,0,0,13,0,0,3,0,0,4,0,0,3,0,0,3,0,0,1,0,5],[60,79,0.7595,0.41964,0.3272,0.14286,0.2857,0.71429,0.14286,1.0,0,5,0,0,0,14,0,0,6,0,0,1,0,0,1,0,0,4,0,0,1,0,5],[64,79,0.8101,0.37947,0.28259,0.14286,0.28571,0.46431,0.14286,1.0,0,3,0,0,0,13,0,0,7,0,0,4,0,0,1,0,0,3,0,0,1,0,3],[68,79,0.8608,0.3214,0.24741,0.14286,0.1429,0.42857,0.14286,1.0,0,2,0,0,0,17,0,0,4,0,0,5,0,0,2,0,0,2,0,0,0,0,2],[72,79,0.9114,0.18295,0.15253,0.14286,0.14286,0.14286,0.14,1.0,0,1,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[76,79,0.962,0.17411,0.07771,0.14286,0.14286,0.1429,0.0,0.42857,1,0,0,1,0,24,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[79,79,1.0,0.17393,0.15462,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.28571,"k":"flat","v":0.22322,"x":0.73214,"p":[[0,133,0.0,0.41936,0.33512,0.14286,0.2857,0.74996,0.14,1.0,0,5,0,0,0,13,0,0,9,0,0,0,0,0,0,0,0,2,0,0,3,0,5],[4,133,0.0301,0.59821,0.34151,0.28571,0.5,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,11,0,0,2,0,0,1,0,0,3,0,0,0,0,12],[8,133,0.0602,0.73214,0.36025,0.28571,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,6,0,0,0,0,0,0,0,0,2,0,0,1,0,19],[12,133,0.0902,0.59375,0.35555,0.14286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,9,0,0,4,0,0,0,0,0,1,0,0,5,0,0,4,0,9],[16,133,0.1203,0.63391,0.35344,0.2857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,7,0,0,2,0,0,1,0,0,2,0,0,2,0,13],[20,133,0.1504,0.59811,0.38216,0.14286,0.64286,1.0,0.0,1.0,1,13,0,1,0,9,0,0,2,0,0,2,0,0,2,0,0,2,0,0,1,0,13],[24,133,0.1805,0.62045,0.36886,0.2857,0.71429,1.0,0.14,1.0,0,14,0,0,0,7,0,0,6,0,0,1,0,0,1,0,0,3,0,0,0,0,14],[28,133,0.2105,0.43747,0.31528,0.14286,0.28571,0.57141,0.14286,1.0,0,6,0,0,0,9,0,0,11,0,0,1,0,0,4,0,0,0,0,0,1,0,6],[32,133,0.2406,0.47757,0.33627,0.24999,0.28571,0.78571,0.14,1.0,0,8,0,0,0,8,0,0,11,0,0,1,0,0,2,0,0,2,0,0,0,0,8],[36,133,0.2707,0.49991,0.36952,0.14286,0.28571,1.0,0.14,1.0,0,10,0,0,0,11,0,0,7,0,0,1,0,0,2,0,0,0,0,0,1,0,10],[40,133,0.3008,0.5357,0.37115,0.24999,0.35716,1.0,0.0,1.0,2,10,0,2,0,6,0,0,8,0,0,1,0,0,2,0,0,1,0,0,2,0,10],[44,133,0.3308,0.49552,0.3369,0.2857,0.28571,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,14,0,0,1,0,0,1,0,0,2,0,0,1,0,8],[48,133,0.3609,0.47768,0.36177,0.14286,0.35714,0.89286,0.0,1.0,2,8,0,2,0,10,0,0,4,0,0,3,0,0,2,0,0,2,0,0,1,0,8],[52,133,0.391,0.53562,0.37636,0.14286,0.28571,1.0,0.14,1.0,0,11,0,0,0,10,0,0,7,0,0,1,0,0,0,0,0,2,0,0,1,0,11],[56,133,0.4211,0.51325,0.34592,0.14289,0.35714,1.0,0.14286,1.0,0,9,0,0,0,9,0,0,7,0,0,1,0,0,4,0,0,2,0,0,0,0,9],[60,133,0.4511,0.61597,0.37202,0.25,0.71429,1.0,0.14,1.0,0,13,0,0,0,8,0,0,5,0,0,1,0,0,1,0,0,2,0,0,2,0,13],[64,133,0.4812,0.28562,0.2551,0.14286,0.14286,0.28571,0.0,1.0,1,2,0,1,0,18,0,0,7,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[68,133,0.5113,0.22322,0.21997,0.14286,0.14286,0.1429,0.14286,1.0,0,2,0,0,0,27,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[72,133,0.5414,0.28097,0.22171,0.14286,0.14286,0.32143,0.14,1.0,0,1,0,0,0,19,0,0,5,0,0,4,0,0,1,0,0,1,0,0,1,0,1],[76,133,0.5714,0.62053,0.33237,0.28571,0.42857,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,9,0,0,6,0,0,0,0,0,1,0,0,2,0,12],[80,133,0.6015,0.52678,0.30813,0.28571,0.35714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,16,0,0,6,0,0,0,0,0,1,0,0,0,0,9],[84,133,0.6316,0.38837,0.22652,0.2857,0.28571,0.42858,0.0,1.0,1,2,0,1,0,2,0,0,18,0,0,4,0,0,3,0,0,1,0,0,1,0,2],[88,133,0.6617,0.45982,0.25935,0.2857,0.28571,0.42858,0.2857,1.0,0,5,0,0,0,0,0,0,17,0,0,8,0,0,1,0,0,0,0,0,1,0,5],[92,133,0.6917,0.46873,0.30563,0.2857,0.28571,0.60682,0.14286,1.0,0,7,0,0,0,5,0,0,12,0,0,6,0,0,1,0,0,1,0,0,0,0,7],[96,133,0.7218,0.41962,0.23127,0.2857,0.28571,0.4286,0.2857,1.0,0,3,0,0,0,0,0,0,21,0,0,4,0,0,2,0,0,1,0,0,1,0,3],[100,133,0.7519,0.36607,0.24728,0.2857,0.28571,0.28571,0.0,1.0,1,4,0,1,0,1,0,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[104,133,0.782,0.41071,0.26184,0.2857,0.28571,0.32143,0.14286,1.0,0,5,0,0,0,1,0,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,5],[108,133,0.812,0.41518,0.26573,0.2857,0.28571,0.42858,0.0,1.0,1,4,0,1,0,2,0,0,18,0,0,4,0,0,0,0,0,3,0,0,0,0,4],[112,133,0.8421,0.33929,0.15047,0.28571,0.28571,0.42857,0.0,1.0,1,1,0,1,0,0,0,0,22,0,0,7,0,0,1,0,0,0,0,0,0,0,1],[116,133,0.8722,0.32142,0.10102,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,25,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[120,133,0.9023,0.40623,0.26989,0.2857,0.28571,0.4286,0.0,1.0,1,4,0,1,0,2,0,0,20,0,0,2,0,0,1,0,0,1,0,0,1,0,4],[124,133,0.9323,0.35704,0.17506,0.2857,0.28571,0.42857,0.14,1.0,0,1,0,0,0,3,0,0,19,0,0,6,0,0,1,0,0,2,0,0,0,0,1],[128,133,0.9624,0.34375,0.22829,0.2857,0.28571,0.28571,0.0,1.0,1,3,0,1,0,4,0,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,3],[132,133,0.9925,0.24545,0.11979,0.2857,0.28571,0.28571,0.0,0.4286,5,0,0,5,0,2,0,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[133,133,1.0,0.28571,0.16751,0.2857,0.28571,0.28571,0.0,1.0,3,1,0,3,0,3,0,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"e471b3eddc34e48a","q":"Determine all pairs of positive integers  $(m, n)$  for which there exists a bijective function \\[f : \\mathbb{Z}_m \\times \\mathbb{Z}_n \\to \\mathbb{Z}_m \\times \\mathbb{Z}_n\\]such that the vectors  $f(\\mathbf{v}) + \\mathbf{v}$ , as  $\\mathbf{v}$  runs through all of  $\\mathbb{Z}_m \\times \\mathbb{Z}_n$ , are pairwise distinct. \n\n(For any integers  $a$  and  $b$ , the vectors  $[a, b], [a + m, b]$  and  $[a, b + n]$  are treated as equal.)\n\n*Poland, Wojciech Nadara*","t":[{"b":5,"e":1.0,"k":"rising","v":0.50889,"x":0.89731,"p":[[0,35,0.0,0.50889,0.1554,0.42857,0.4286,0.57111,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,20,0,0,7,0,0,2,0,0,0,0,2],[4,35,0.1143,0.87053,0.23787,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,5,0,21],[8,35,0.2286,0.80802,0.23314,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,5,0,0,0,0,18],[12,35,0.3429,0.79014,0.24998,0.67836,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,4,0,0,3,0,0,6,0,0,3,0,15],[16,35,0.4571,0.87499,0.18472,0.82132,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,5,0,19],[20,35,0.5714,0.83925,0.17772,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,7,0,14],[24,35,0.6857,0.82588,0.22794,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,2,0,0,7,0,16],[28,35,0.8,0.89731,0.18638,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,6,0,21],[32,35,0.9143,0.84374,0.21831,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,3,0,0,3,0,19],[35,35,1.0,0.70526,0.25258,0.42859,0.71429,1.0,0.14,1.0,0,9,0,0,0,1,0,0,1,0,0,8,0,0,2,0,0,6,0,0,5,0,9]]},{"b":6,"e":1.0,"k":"rising","v":0.45533,"x":0.91071,"p":[[0,34,0.0,0.45533,0.09058,0.42857,0.42857,0.4286,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,24,0,0,4,0,0,2,0,0,0,0,0],[4,34,0.1176,0.88835,0.18125,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,3,0,21],[8,34,0.2353,0.91071,0.16656,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,23],[12,34,0.3529,0.89283,0.16754,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,8,0,19],[16,34,0.4706,0.86604,0.15952,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,6,0,16],[20,34,0.5882,0.71874,0.29556,0.5354,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,2,0,0,4,0,0,2,0,0,7,0,0,3,0,12],[24,34,0.7059,0.81247,0.22145,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,5,0,0,5,0,15],[28,34,0.8235,0.79004,0.20542,0.71429,0.857,1.0,0.14,1.0,0,10,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,8,0,0,8,0,10],[32,34,0.9412,0.83033,0.22145,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,4,0,0,4,0,17],[34,34,1.0,0.6428,0.23705,0.42859,0.71429,0.85714,0.14,1.0,0,5,0,0,0,1,0,0,2,0,0,9,0,0,2,0,0,9,0,0,4,0,5]]}]},{"i":"2f42b2ff69a0869d","q":"We have an infinite list of cells, the cells being numbered by $1, 2, \\ldots$. Initially, all cells contain the number 1. At each step, we choose a number $a \\in \\mathbb{N}^{*}$ such that:\n\n- either all the cells numbered by a multiple of $a$ contain 1, in which case we replace these 1s with 0s,\n- or all the cells numbered by a multiple of $a$ contain 0, in which case we replace these 0s with 1s.\nShow that, for all $n \\geqslant 2$, we can ensure that all the cells numbered at most $n$ contain a 0 except the first cell which contains a 1.","t":[{"b":1,"e":0.14,"k":"flat","v":0.02232,"x":0.13384,"p":[[0,36,0.0,0.06241,0.20801,0.0,0.0,0.0,0.0,0.85714,28,0,5,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[4,36,0.1111,0.05804,0.14223,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,36,0.2222,0.13384,0.2111,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,6,0,0,2,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[12,36,0.3333,0.0758,0.15557,0.0,0.0,0.14071,0.0,0.71429,23,0,0,23,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,36,0.4444,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.05357,0.09943,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.02679,"x":0.1875,"p":[[0,63,0.0,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,2,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,63,0.0635,0.1875,0.29974,0.0,0.0,0.14286,0.0,1.0,18,2,0,18,0,7,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,2],[8,63,0.127,0.12044,0.15609,0.0,0.14286,0.14286,0.0,0.857,12,0,0,12,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,63,0.1905,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,63,0.254,0.09374,0.13171,0.0,0.0,0.14286,0.0,0.571,17,0,0,17,0,12,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,63,0.3175,0.0625,0.08703,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,63,0.381,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,63,0.4444,0.11607,0.19704,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,63,0.5079,0.09822,0.18707,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,63,0.5714,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,63,0.6349,0.08036,0.11811,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,63,0.6984,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,63,0.7619,0.06241,0.094,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,63,0.8254,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,63,0.8889,0.06688,0.07965,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,63,0.9524,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[63,63,1.0,0.07134,0.08741,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"618d039b65ce6890","q":"Let  $P$  be an odd-degree integer-coefficient polynomial. Suppose that  $xP(x)=yP(y)$  for infinitely many pairs  $x,y$  of integers with  $x\\ne y$ . Prove that the equation  $P(x)=0$  has an integer root.\n\n*Victor Wang*","t":[{"b":0,"e":0.14286,"k":"falling","v":0.13393,"x":0.77231,"p":[[0,120,0.0,0.70534,0.29653,0.57143,0.71429,1.0,0.0,1.0,2,11,0,2,0,1,0,0,1,0,0,3,0,0,4,0,0,7,0,0,3,0,11],[4,120,0.0333,0.77231,0.26452,0.82132,0.85714,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,1,0,0,1,0,0,0,0,0,3,0,0,17,0,7],[8,120,0.0667,0.66515,0.25408,0.571,0.71429,0.85714,0.0,1.0,1,1,0,1,0,2,0,0,2,0,0,2,0,0,4,0,0,6,0,0,14,0,1],[12,120,0.1,0.67408,0.30564,0.57132,0.71429,0.85714,0.0,1.0,3,5,0,3,0,2,0,0,1,0,0,0,0,0,3,0,0,8,0,0,10,0,5],[16,120,0.1333,0.70087,0.2705,0.571,0.85714,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,4,0,0,1,0,0,2,0,0,6,0,0,12,0,5],[20,120,0.1667,0.56248,0.29,0.39286,0.57143,0.85714,0.0,1.0,2,1,0,2,0,4,0,0,2,0,0,4,0,0,6,0,0,3,0,0,10,0,1],[24,120,0.2,0.72766,0.25092,0.57143,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,1,0,0,3,0,0,2,0,0,17,0,4],[28,120,0.2333,0.65623,0.26452,0.53539,0.71429,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,2,0,0,3,0,0,2,0,0,8,0,0,13,0,1],[32,120,0.2667,0.56697,0.28456,0.39286,0.57143,0.85714,0.0,1.0,2,2,0,2,0,2,0,0,4,0,0,6,0,0,3,0,0,5,0,0,8,0,2],[36,120,0.3,0.5714,0.29451,0.28571,0.57143,0.85714,0.0,1.0,1,4,0,1,0,4,0,0,5,0,0,2,0,0,5,0,0,6,0,0,5,0,4],[40,120,0.3333,0.62945,0.30275,0.42857,0.71429,0.85714,0.0,1.0,2,5,0,2,0,2,0,0,2,0,0,6,0,0,2,0,0,4,0,0,9,0,5],[44,120,0.3667,0.5357,0.30093,0.28571,0.4998,0.85714,0.0,1.0,1,2,0,1,0,5,0,0,6,0,0,4,0,0,1,0,0,5,0,0,8,0,2],[48,120,0.4,0.50888,0.32914,0.24999,0.571,0.85714,0.0,1.0,4,3,0,4,0,4,0,0,4,0,0,3,0,0,5,0,0,2,0,0,7,0,3],[52,120,0.4333,0.58482,0.32015,0.28571,0.71429,0.85714,0.0,1.0,3,3,0,3,0,3,0,0,4,0,0,2,0,0,0,0,0,9,0,0,8,0,3],[56,120,0.4667,0.62052,0.30223,0.39286,0.71429,0.85714,0.0,1.0,1,4,0,1,0,5,0,0,2,0,0,2,0,0,2,0,0,8,0,0,8,0,4],[60,120,0.5,0.54911,0.31766,0.28571,0.64286,0.85714,0.0,1.0,1,3,0,1,0,6,0,0,6,0,0,1,0,0,2,0,0,5,0,0,8,0,3],[64,120,0.5333,0.66071,0.29827,0.42859,0.85707,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,4,0,0,3,0,0,1,0,0,4,0,0,13,0,4],[68,120,0.5667,0.48658,0.33475,0.25,0.42857,0.85704,0.0,1.0,6,2,0,6,0,2,0,0,4,0,0,5,0,0,3,0,0,2,0,0,8,0,2],[72,120,0.6,0.60266,0.35487,0.28571,0.78564,0.89286,0.0,1.0,2,8,0,2,0,4,0,0,6,0,0,2,0,0,1,0,0,1,0,0,8,0,8],[76,120,0.6333,0.51784,0.33645,0.14286,0.57121,0.85714,0.0,1.0,2,6,0,2,0,7,0,0,4,0,0,2,0,0,5,0,0,3,0,0,3,0,6],[80,120,0.6667,0.3125,0.29545,0.14286,0.14286,0.60714,0.0,1.0,4,1,0,4,0,16,0,0,3,0,0,0,0,0,1,0,0,5,0,0,2,0,1],[84,120,0.7,0.21875,0.22583,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,21,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[88,120,0.7333,0.24107,0.26351,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,21,0,0,1,0,0,2,0,0,0,0,0,0,0,0,3,0,1],[92,120,0.7667,0.21875,0.20198,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,18,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[96,120,0.8,0.25892,0.25613,0.14286,0.14286,0.28571,0.0,1.0,6,2,0,6,0,13,0,0,6,0,0,1,0,0,4,0,0,0,0,0,0,0,2],[100,120,0.8333,0.25,0.29233,0.14286,0.14286,0.28571,0.0,1.0,7,2,0,7,0,16,0,0,3,0,0,1,0,0,0,0,0,1,0,0,2,0,2],[104,120,0.8667,0.19643,0.24419,0.14286,0.14286,0.14286,0.0,1.0,7,1,0,7,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,1],[108,120,0.9,0.19196,0.18766,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,24,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[112,120,0.9333,0.13393,0.07087,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,120,0.9667,0.18304,0.11971,0.14286,0.14286,0.1786,0.0,0.71429,2,0,0,2,0,22,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[120,120,1.0,0.15625,0.10326,0.14286,0.14286,0.14286,0.0,0.71429,1,0,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"falling","v":0.29465,"x":0.72321,"p":[[0,28,0.0,0.71429,0.24744,0.53571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,5,0,0,2,0,0,9,0,0,4,0,9],[4,28,0.1429,0.72321,0.3071,0.71429,0.85714,0.85714,0.0,1.0,3,6,0,3,0,1,0,0,2,0,0,0,0,0,0,0,0,5,0,0,15,0,6],[8,28,0.2857,0.70087,0.25345,0.57143,0.78564,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,3,0,0,0,0,0,5,0,0,6,0,0,12,0,4],[12,28,0.4286,0.6249,0.35503,0.2857,0.85714,0.85714,0.0,1.0,3,7,0,3,0,4,0,0,3,0,0,1,0,0,2,0,0,2,0,0,10,0,7],[16,28,0.5714,0.68748,0.25364,0.57143,0.85707,0.85714,0.14286,1.0,0,3,0,0,0,3,0,0,2,0,0,1,0,0,6,0,0,3,0,0,14,0,3],[20,28,0.7143,0.52679,0.31831,0.14286,0.57143,0.85714,0.0,1.0,1,1,0,1,0,8,0,0,4,0,0,2,0,0,2,0,0,3,0,0,11,0,1],[24,28,0.8571,0.43303,0.32436,0.14286,0.28571,0.71429,0.14286,1.0,0,5,0,0,0,12,0,0,7,0,0,3,0,0,0,0,0,3,0,0,2,0,5],[28,28,1.0,0.29465,0.20497,0.14286,0.21431,0.42857,0.14286,1.0,0,1,0,0,0,16,0,0,5,0,0,9,0,0,0,0,0,0,0,0,1,0,1]]}]},{"i":"a6f6fe02cd17ee02","q":"Determine all prime numbers  $p$  and all positive integers  $x$  and  $y$  satisfying   $$ x^3+y^3=p(xy+p). $$","t":[{"b":2,"e":0.28571,"k":"falling","v":0.12938,"x":0.66964,"p":[[0,151,0.0,0.33034,0.31018,0.0,0.2857,0.60714,0.0,1.0,9,1,0,9,0,6,0,0,5,0,0,2,0,0,2,0,0,5,0,0,2,0,1],[4,151,0.0265,0.50222,0.32216,0.2857,0.49979,0.71429,0.0,1.0,4,3,0,4,0,3,0,0,6,0,0,3,0,0,2,0,0,7,0,0,3,1,3],[8,151,0.053,0.57142,0.27432,0.39286,0.71429,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,4,0,0,5,0,0,0,0,0,13,0,0,4,0,2],[12,151,0.0795,0.58704,0.29866,0.28571,0.71429,0.74996,0.0,1.0,2,5,0,2,0,2,0,1,4,0,0,2,0,0,4,0,0,9,0,0,3,0,5],[16,151,0.106,0.63828,0.27445,0.42857,0.71429,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,4,0,0,2,0,0,3,0,0,9,0,0,7,0,4],[20,151,0.1325,0.62277,0.25394,0.42859,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,2,0,0,3,0,0,3,0,0,13,0,0,3,1,3],[24,151,0.1589,0.65177,0.22286,0.5713,0.71429,0.75,0.0,1.0,1,2,0,1,0,1,0,0,1,0,0,4,0,0,4,0,0,13,0,0,6,0,2],[28,151,0.1854,0.6183,0.26099,0.5,0.71429,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,6,0,0,0,0,0,4,1,0,9,0,0,9,0,1],[32,151,0.2119,0.62035,0.30429,0.571,0.71429,0.85704,0.0,1.0,2,5,0,2,0,5,0,0,0,0,0,0,0,0,5,0,0,11,0,0,4,0,5],[36,151,0.2384,0.59821,0.29545,0.28571,0.71429,0.85714,0.0,1.0,3,2,0,3,0,1,0,0,5,0,0,1,0,0,2,0,0,10,0,0,8,0,2],[40,151,0.2649,0.58036,0.25738,0.42857,0.71429,0.74996,0.0,1.0,1,1,0,1,0,3,0,0,2,0,0,7,0,0,2,0,0,9,0,0,7,0,1],[44,151,0.2914,0.5267,0.27004,0.28571,0.57141,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,4,0,0,5,0,0,4,1,0,6,0,0,3,1,2],[48,151,0.3179,0.61827,0.30389,0.39288,0.71429,0.85714,0.0,1.0,2,4,0,2,1,2,0,0,3,0,0,1,0,0,3,0,0,9,0,0,7,0,4],[52,151,0.3444,0.55802,0.29957,0.39286,0.71429,0.71429,0.0,1.0,3,3,0,3,0,4,0,0,1,0,0,3,0,0,4,0,0,11,0,0,3,0,3],[56,151,0.3709,0.56027,0.28465,0.39286,0.67857,0.74996,0.0,1.0,2,2,0,2,0,3,0,0,3,0,0,6,0,0,1,1,0,8,0,0,6,0,2],[60,151,0.3974,0.61606,0.28669,0.4286,0.71429,0.75,0.0,1.0,4,3,0,4,0,0,0,0,1,0,0,4,0,0,2,0,0,13,0,0,5,0,3],[64,151,0.4238,0.65177,0.23942,0.57132,0.71429,0.75,0.0,1.0,1,3,0,1,0,1,0,0,3,0,0,2,0,0,3,0,0,14,0,0,5,0,3],[68,151,0.4503,0.66515,0.22761,0.67846,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,0,0,0,2,0,0,15,0,0,7,0,2],[72,151,0.4768,0.66518,0.22759,0.4286,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,4,0,0,2,0,0,11,0,0,6,0,4],[76,151,0.5033,0.6362,0.22679,0.67857,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,1,0,0,1,0,0,2,0,0,2,0,0,18,1,0,5,0,0],[80,151,0.5298,0.66964,0.19377,0.67857,0.71429,0.71429,0.0,0.85714,1,0,1,1,0,1,0,0,1,0,0,0,0,0,5,0,0,17,0,0,7,0,0],[84,151,0.5563,0.59374,0.24251,0.5354,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,2,0,0,2,0,0,2,0,0,3,0,0,17,0,0,4,0,0],[88,151,0.5828,0.59819,0.27067,0.42859,0.71429,0.75,0.0,1.0,3,1,0,3,0,1,0,0,2,0,0,3,0,0,4,0,0,11,0,0,7,0,1],[92,151,0.6093,0.51994,0.2566,0.28571,0.57121,0.71429,0.0,1.0,1,1,0,1,0,2,0,1,7,0,0,4,0,0,5,0,0,6,0,0,5,0,1],[96,151,0.6358,0.52453,0.27292,0.28571,0.5712,0.71429,0.0,1.0,3,1,0,3,0,1,0,0,5,0,0,5,0,1,5,0,0,5,0,0,6,0,1],[100,151,0.6623,0.61152,0.27733,0.39286,0.71429,0.85714,0.14,1.0,0,3,0,0,0,4,0,0,4,0,0,4,0,0,1,0,0,8,0,0,8,0,3],[104,151,0.6887,0.52678,0.33586,0.24999,0.71429,0.75,0.0,1.0,5,3,0,5,0,3,0,0,4,0,0,1,0,0,2,0,0,9,0,0,5,0,3],[108,151,0.7152,0.56919,0.24903,0.39286,0.71429,0.71429,0.0,0.85714,1,0,0,1,1,1,0,0,5,0,0,4,0,0,2,0,0,12,0,0,6,0,0],[112,151,0.7417,0.53113,0.27959,0.2857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,3,0,0,3,0,0,4,0,0,10,0,0,3,0,2],[116,151,0.7682,0.48214,0.26426,0.28571,0.4286,0.60714,0.0,1.0,3,2,0,3,0,1,0,0,6,0,0,8,0,0,6,0,0,3,0,0,3,0,2],[120,151,0.7947,0.54464,0.27067,0.28571,0.64286,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,6,0,0,3,0,0,3,0,0,10,0,0,5,0,1],[124,151,0.8212,0.54453,0.24089,0.28571,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,6,0,0,4,0,0,4,0,0,10,0,0,5,0,0],[128,151,0.8477,0.5491,0.29904,0.28571,0.71429,0.74996,0.0,0.85714,4,0,0,4,0,2,0,0,3,0,0,4,0,0,0,0,0,11,0,0,8,0,0],[132,151,0.8742,0.50445,0.26301,0.33918,0.57121,0.71429,0.0,0.85714,3,0,0,3,0,3,0,0,2,0,1,6,0,0,3,0,0,10,1,0,3,0,0],[136,151,0.9007,0.51785,0.30251,0.28571,0.71429,0.71429,0.0,0.85714,5,0,0,5,0,1,0,0,4,0,0,5,0,0,0,0,0,10,0,0,7,0,0],[140,151,0.9272,0.42634,0.3033,0.24999,0.42857,0.71429,0.0,1.0,6,1,0,6,1,1,0,0,6,0,0,6,0,0,2,0,0,5,0,0,4,0,1],[144,151,0.9536,0.39724,0.32881,0.0,0.35729,0.71429,0.0,0.85714,9,0,0,9,0,3,0,0,4,0,0,3,0,0,1,0,0,7,0,0,5,0,0],[148,151,0.9801,0.15625,0.16888,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,10,0,0,8,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[151,151,1.0,0.12938,0.13533,0.0,0.14286,0.1429,0.0,0.4286,13,0,0,13,0,12,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.32813,"x":0.77887,"p":[[0,316,0.0,0.32813,0.25807,0.14286,0.28571,0.57143,0.0,0.85714,6,0,0,6,1,6,0,0,6,0,0,3,0,0,5,0,0,4,0,0,1,0,0],[4,316,0.0127,0.55356,0.29827,0.28571,0.71429,0.75,0.0,1.0,3,2,1,3,0,2,0,0,5,0,0,3,0,0,2,0,0,9,0,0,6,0,2],[8,316,0.0253,0.65177,0.19865,0.57143,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,2,0,0,5,0,0,15,0,0,4,0,2],[12,316,0.038,0.63839,0.24481,0.53571,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,4,0,0,2,0,0,4,0,0,9,0,0,10,0,1],[16,316,0.0506,0.73213,0.15153,0.71429,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,2,1,0,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On a line $l$ there are three different points $A, B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $B Q$ intersects $B Q$ in $L$ and $B R$ in $T$. The line through $B$ perpendicular to $A R$ intersects $A R$ in $K$ and $A Q$ in $S$.\n\n(a) Prove that $P, T, S$ are collinear.\n\n(b) Prove that $P, K, L$ are collinear.\n\n#","t":[{"b":2,"e":0.14286,"k":"flat","v":0.14732,"x":0.33474,"p":[[0,26,0.0,0.16071,0.13243,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,13,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.14732,0.12619,0.0,0.14286,0.1429,0.0,0.42857,9,0,0,9,0,16,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.2008,0.15096,0.14286,0.14286,0.32143,0.0,0.42857,6,0,0,6,0,15,0,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.21429,0.16367,0.14286,0.14286,0.42857,0.0,0.4286,7,0,0,7,0,12,0,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.22768,0.16698,0.14286,0.14286,0.42857,0.0,0.4286,7,0,0,7,0,10,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.22326,0.16348,0.14286,0.14288,0.42857,0.0,0.43,7,0,0,7,0,10,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.33474,0.13663,0.2857,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,5,0,0,5,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.23206,0.16274,0.14286,0.1429,0.42857,0.0,0.4286,6,0,0,6,0,11,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.05804,"x":0.15625,"p":[[0,75,0.0,0.15625,0.13997,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,16,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.08929,0.1171,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,75,0.1067,0.125,0.13716,0.0,0.14286,0.14286,0.0,0.4286,14,0,0,14,0,11,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,75,0.16,0.12053,0.12428,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,13,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,75,0.2133,0.09822,0.10374,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,75,0.2667,0.08464,0.10002,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,75,0.32,0.125,0.12752,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,15,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,75,0.3733,0.08027,0.09401,0.0,0.07,0.14286,0.0,0.4286,16,0,0,16,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,75,0.4267,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,75,0.48,0.08474,0.07867,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,75,0.5333,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,75,0.64,0.09366,0.08453,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,75,0.6933,0.11607,0.13092,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,15,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,75,0.7467,0.08473,0.10008,0.0,0.07,0.14286,0.0,0.4286,16,0,0,16,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,75,0.8,0.15625,0.19352,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,14,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[64,75,0.8533,0.07125,0.08734,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[75,75,1.0,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"67d4a7cbf34af762","q":"Prove that for infinitely many pairs $(a, b)$ of integers the equation\n\n$$\nx^{2012}=a x+b\n$$\n\nhas among its solutions two distinct real numbers whose product is 1 .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.75446,"x":0.8482,"p":[[0,10,0.0,0.8482,0.17107,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,4,0,16],[4,10,0.4,0.75446,0.11971,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,0,5,0,4],[8,10,0.8,0.76785,0.17768,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,17,0,0,3,0,8],[10,10,1.0,0.80803,0.14555,0.71429,0.78564,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,7,0,9]]},{"b":3,"e":1.0,"k":"flat","v":0.85713,"x":0.98214,"p":[[0,27,0.0,0.91071,0.13243,0.82132,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,21],[4,27,0.1481,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[8,27,0.2963,0.85713,0.13832,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,5,0,14],[12,27,0.4444,0.92856,0.13367,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,24],[16,27,0.5926,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[20,27,0.7407,0.92856,0.13367,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,24],[24,27,0.8889,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[27,27,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"daf12e6f4f206d36","q":"For a given $n$-gon with all sides of equal length, all vertices have rational coordinates. Prove that $n$ is even.","t":[{"b":0,"e":1.0,"k":"rising","v":0.54018,"x":0.93737,"p":[[0,25,0.0,0.54018,0.4721,0.0,0.85714,1.0,0.0,1.0,13,13,0,13,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,13],[4,25,0.16,0.93737,0.13355,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,25],[8,25,0.32,0.90625,0.14987,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,4,0,21],[12,25,0.48,0.89286,0.17128,0.85713,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,5,0,20],[16,25,0.64,0.92857,0.12372,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,21],[20,25,0.8,0.90179,0.1729,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,3,0,22],[24,25,0.96,0.88836,0.12752,0.85714,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,11,0,15],[25,25,1.0,0.79461,0.1888,0.67857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,9,0,10]]},{"b":3,"e":0.71429,"k":"rising","v":0.5982,"x":0.93304,"p":[[0,28,0.0,0.5982,0.44669,0.0,0.78564,1.0,0.0,1.0,11,14,0,11,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,14],[4,28,0.1429,0.93295,0.17163,1.0,1.0,1.0,0.14,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,26],[8,28,0.2857,0.93304,0.15561,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,26],[12,28,0.4286,0.92857,0.15567,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,24],[16,28,0.5714,0.92857,0.14725,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,24],[20,28,0.7143,0.91518,0.16698,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,24],[24,28,0.8571,0.90625,0.1411,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,6,0,20],[28,28,1.0,0.77679,0.2141,0.71429,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,14,0,0,4,0,10]]}]},{"i":"56bd5fd07011d32a","q":"Let $n$ be a natural number. In a village, there live $n$ boys and $n$ girls. For the annual ball, $n$ dance pairs need to be formed, each consisting of one boy and one girl. Each girl provides a list containing the name of the boy she would most like to dance with, plus zero or more names of other boys she would also be willing to dance with. It turns out that it is possible to form $n$ dance pairs such that each girl dances with a boy whose name is on her list.\nProve that it is possible to form $n$ dance pairs such that each girl dances with a boy whose name is on her list and at least one girl dances with the boy she would most like to dance with.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,16,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,21,0.0,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,3,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,21,0.1905,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0134,0.04165,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"838be0382b9d8217","q":"An equilateral triangle is divided into $n^{2}$ congruent equilateral triangles. A spider stands at one of the vertices, a fly at another. Alternately each of them moves to a neighbouring vertex. Prove that the spider can always catch the fly.","t":[{"b":2,"e":0.42857,"k":"rising","v":0.28554,"x":0.60715,"p":[[0,24,0.0,0.28554,0.22599,0.14286,0.21435,0.42857,0.0,1.0,5,1,0,5,0,11,0,0,2,0,0,11,0,0,1,0,0,1,0,0,0,0,1],[4,24,0.1667,0.50447,0.25997,0.42857,0.42857,0.60714,0.0,1.0,2,5,0,2,0,0,0,0,4,0,0,17,0,0,1,0,0,3,0,0,0,0,5],[8,24,0.3333,0.52232,0.20705,0.42857,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,17,0,0,2,0,0,6,0,0,0,0,3],[12,24,0.5,0.48661,0.2854,0.39286,0.42857,0.71429,0.0,1.0,3,4,0,3,0,3,0,0,2,0,0,13,0,0,0,0,0,7,0,0,0,0,4],[16,24,0.6667,0.49558,0.21423,0.42857,0.42857,0.4286,0.14286,1.0,0,4,0,0,0,1,0,0,3,0,0,22,0,0,0,0,0,2,0,0,0,0,4],[20,24,0.8333,0.50447,0.16745,0.42857,0.42857,0.50002,0.1429,1.0,0,1,0,0,0,1,0,0,0,0,0,23,0,0,0,0,0,6,0,0,1,0,1],[24,24,1.0,0.60715,0.21128,0.42857,0.4286,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,10,0,0,0,0,5]]},{"b":7,"e":0.71429,"k":"rising","v":0.38393,"x":0.63839,"p":[[0,26,0.0,0.38393,0.34337,0.14286,0.2143,0.5,0.0,1.0,4,5,0,4,0,12,0,0,2,0,0,6,0,0,0,0,0,1,0,0,2,0,5],[4,26,0.1538,0.46429,0.22868,0.42857,0.42857,0.4286,0.0,1.0,2,3,0,2,0,1,0,0,2,0,0,21,0,0,0,0,0,3,0,0,0,0,3],[8,26,0.3077,0.52679,0.22711,0.42857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,4,0,0,17,0,0,0,0,0,6,0,0,0,0,4],[12,26,0.4615,0.58035,0.23402,0.42857,0.4998,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,14,0,0,4,0,0,7,0,0,0,0,5],[16,26,0.6154,0.55356,0.21354,0.42857,0.42857,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,18,0,0,2,0,0,7,0,0,1,0,3],[20,26,0.7692,0.62481,0.18118,0.42857,0.64071,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,9,0,0,6,0,0,12,0,0,1,0,3],[24,26,0.9231,0.63839,0.18893,0.42857,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,15,0,0,0,0,4],[26,26,1.0,0.5982,0.14914,0.42857,0.64286,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,15,0,0,0,0,1]]}]},{"i":"2f058cbbe15a963d","q":"Find all integers  $a$  such that there are infinitely many positive integers  $n$  such that  $n$  divides  $\\phi(n)!+a$ .","t":[{"b":0,"e":0.42857,"k":"falling","v":0.48213,"x":0.83033,"p":[[0,29,0.0,0.83033,0.17659,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,8,0,13],[4,29,0.1379,0.8125,0.18363,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,6,0,12],[8,29,0.2759,0.80802,0.21313,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,4,0,0,4,0,15],[12,29,0.4138,0.80356,0.20438,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,7,0,0,7,0,12],[16,29,0.5517,0.63838,0.23686,0.42857,0.64286,0.75,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,8,0,0,5,0,0,8,0,0,2,0,6],[20,29,0.6897,0.78125,0.21424,0.71429,0.78564,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,0,4,0,12],[24,29,0.8276,0.58927,0.17768,0.42857,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,13,0,0,9,0,0,6,0,0,1,0,3],[28,29,0.9655,0.48213,0.14173,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,25,0,0,2,0,0,2,0,0,1,0,1],[29,29,1.0,0.48658,0.11767,0.42857,0.42857,0.4642,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,24,0,0,5,0,0,1,0,0,2,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.43303,"x":0.83035,"p":[[0,27,0.0,0.81249,0.20652,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,4,0,0,6,0,14],[4,27,0.1481,0.83035,0.19704,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,8,0,0,8,0,13],[8,27,0.2963,0.78124,0.21719,0.71429,0.85707,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,7,0,0,7,0,11],[12,27,0.4444,0.78569,0.18901,0.57143,0.78571,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,7,0,0,5,0,11],[16,27,0.5926,0.65625,0.23381,0.42857,0.64286,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,8,0,0,5,0,0,4,0,0,7,0,5],[20,27,0.7407,0.56704,0.16157,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,12,0,0,10,0,0,7,0,0,0,0,2],[24,27,0.8889,0.43303,0.17307,0.28571,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,15,0,0,4,0,0,0,0,0,0,0,2],[27,27,1.0,0.45089,0.13883,0.39286,0.42857,0.46431,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,16,0,0,3,0,0,5,0,0,0,0,0]]}]},{"i":"6eb69e955db2cb60","q":"Let  $n,k \\in \\mathbb{Z}^{+}$  with  $k \\leq n$  and let  $S$  be a set containing  $n$  distinct real numbers. Let  $T$  be a set of all real numbers of the form  $x_1 + x_2 + \\ldots + x_k$  where  $x_1, x_2, \\ldots, x_k$  are distinct elements of  $S.$  Prove that  $T$  contains at least  $k(n-k)+1$  distinct elements.","t":[{"b":1,"e":0.0,"k":"falling","v":0.09597,"x":0.58482,"p":[[0,41,0.0,0.44863,0.29515,0.14286,0.4998,0.71429,0.0,1.0,5,1,0,5,0,5,0,0,1,0,0,5,0,0,6,1,0,5,0,0,3,0,1],[4,41,0.0976,0.58482,0.42762,0.14286,0.71429,1.0,0.0,1.0,7,14,0,7,0,4,0,0,2,0,0,0,0,0,0,0,0,5,0,0,0,0,14],[8,41,0.1951,0.38393,0.44383,0.0,0.14286,1.0,0.0,1.0,13,10,0,13,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,10],[12,41,0.2927,0.43079,0.39061,0.14286,0.21429,0.78571,0.0,1.0,6,8,0,6,1,9,0,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,8],[16,41,0.3902,0.3906,0.37625,0.10714,0.14288,0.71429,0.0,1.0,8,6,0,8,0,9,0,0,1,0,1,0,0,0,3,0,0,4,0,0,0,0,6],[20,41,0.4878,0.39732,0.40364,0.10714,0.1429,0.89286,0.0,1.0,8,8,0,8,0,9,0,0,4,0,0,0,0,0,0,0,0,2,0,0,1,0,8],[24,41,0.5854,0.20088,0.27164,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,9,0,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,1],[28,41,0.6829,0.25447,0.3169,0.0,0.14286,0.32143,0.0,1.0,12,3,0,12,0,8,0,0,4,0,0,3,0,0,0,0,0,1,0,0,1,0,3],[32,41,0.7805,0.17411,0.27603,0.0,0.0,0.17857,0.0,1.0,19,1,0,19,0,5,0,0,1,0,0,2,0,0,1,0,0,3,0,0,0,0,1],[36,41,0.878,0.09597,0.18348,0.0,0.0,0.08929,0.0,0.71429,23,0,0,23,1,1,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[40,41,0.9756,0.16071,0.24419,0.0,0.0,0.17857,0.0,1.0,17,1,0,17,0,7,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[41,41,1.0,0.15848,0.25984,0.0,0.0,0.17857,0.0,1.0,19,1,0,19,1,4,0,0,2,0,0,2,0,0,1,0,0,2,0,0,0,0,1]]},{"b":7,"e":0.28571,"k":"falling","v":0.09598,"x":0.83929,"p":[[0,53,0.0,0.43749,0.39759,0.0,0.35714,0.75,0.0,1.0,9,7,0,9,0,6,0,0,1,0,0,2,0,0,1,0,0,5,0,0,1,0,7],[4,53,0.0755,0.83929,0.33455,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[8,53,0.1509,0.67857,0.41955,0.14286,1.0,1.0,0.0,1.0,5,19,0,5,0,4,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,19],[12,53,0.2264,0.51553,0.42444,0.14286,0.42856,1.0,0.0,1.0,6,11,0,6,0,8,0,0,2,0,0,0,0,0,1,0,0,2,0,0,1,1,11],[16,53,0.3019,0.47768,0.44444,0.0,0.28571,1.0,0.0,1.0,10,12,0,10,1,3,0,1,2,0,0,0,0,0,1,0,0,2,0,0,0,0,12],[20,53,0.3774,0.57366,0.43539,0.10714,0.71429,1.0,0.0,1.0,8,14,0,8,0,3,0,1,1,0,0,0,0,0,2,0,0,2,0,0,1,0,14],[24,53,0.4528,0.50893,0.41333,0.14286,0.2857,1.0,0.0,1.0,3,11,0,3,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,3,0,11],[28,53,0.5283,0.20982,0.2618,0.0,0.14286,0.1786,0.0,1.0,9,2,0,9,2,13,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,2],[32,53,0.6038,0.45982,0.42066,0.10714,0.2857,1.0,0.0,1.0,8,11,0,8,0,5,0,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,11],[36,53,0.6792,0.27009,0.36014,0.0,0.14286,0.32143,0.0,1.0,13,5,0,13,1,6,0,2,2,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[40,53,0.7547,0.27009,0.35162,0.0,0.14286,0.2857,0.0,1.0,10,5,0,10,1,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[44,53,0.8302,0.16072,0.2187,0.0,0.14286,0.1429,0.0,1.0,11,1,0,11,2,13,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[48,53,0.9057,0.10259,0.09602,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,53,0.9811,0.12939,0.12037,0.0,0.14286,0.1786,0.0,0.42857,12,0,0,12,0,12,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[53,53,1.0,0.09598,0.12577,0.0,0.03571,0.14286,0.0,0.57143,16,0,0,16,1,11,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"d1ebbff85811be5c","q":"There is a  $2n \\times 2n$  array (matrix)  consisting of  $0's$  and  $1's$  and there are exactly  $3n$  zeroes. Show that it is possible to remove all the zeroes by deleting some  $n$  rows and some  $n$  columns.","t":[{"b":0,"e":1.0,"k":"rising","v":0.29911,"x":0.96429,"p":[[0,39,0.0,0.52679,0.46626,0.0,0.57143,1.0,0.0,1.0,11,15,5,11,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,15],[4,39,0.1026,0.54009,0.39736,0.14286,0.35714,1.0,0.0,1.0,4,12,0,4,0,5,0,0,7,0,0,1,0,0,0,0,0,3,0,0,0,0,12],[8,39,0.2051,0.53571,0.41188,0.24999,0.28571,1.0,0.0,1.0,6,13,0,6,0,2,0,0,9,0,0,0,0,0,1,0,0,1,0,0,0,0,13],[12,39,0.3077,0.29911,0.31816,0.10714,0.21431,0.32143,0.0,1.0,8,4,0,8,0,8,0,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,4],[16,39,0.4103,0.42411,0.37878,0.14286,0.28571,0.89286,0.0,1.0,6,8,0,6,0,5,0,0,10,0,0,1,0,0,0,0,0,1,0,0,1,0,8],[20,39,0.5128,0.76786,0.34947,0.39286,1.0,1.0,0.14286,1.0,0,21,0,0,0,5,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,21],[24,39,0.6154,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[28,39,0.7179,0.91518,0.20782,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,27],[32,39,0.8205,0.84375,0.23517,0.67857,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,2,0,0,1,0,21],[36,39,0.9231,0.72765,0.29959,0.53539,0.78564,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,0,0,0,4,0,0,3,0,0,5,0,0,2,0,14],[39,39,1.0,0.6875,0.30813,0.42857,0.71429,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,0,0,0,8,0,0,1,0,0,5,0,0,1,0,13]]},{"b":7,"e":1.0,"k":"falling","v":0.30803,"x":0.83472,"p":[[0,67,0.0,0.61607,0.47034,0.0,1.0,1.0,0.0,1.0,10,19,2,10,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19],[4,67,0.0597,0.5357,0.42708,0.14286,0.35714,1.0,0.0,1.0,6,14,0,6,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,14],[8,67,0.1194,0.83472,0.28841,0.71429,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,22],[12,67,0.1791,0.75446,0.36637,0.39288,1.0,1.0,0.0,1.0,2,21,0,2,0,3,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,21],[16,67,0.2388,0.81696,0.32387,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,23],[20,67,0.2985,0.67856,0.37965,0.28571,1.0,1.0,0.0,1.0,2,17,0,2,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,0,1,0,17],[24,67,0.3582,0.79018,0.35533,0.71429,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,22],[28,67,0.4179,0.51339,0.41166,0.10714,0.5,1.0,0.0,1.0,8,9,0,8,0,3,0,0,4,0,0,1,0,0,1,0,0,2,0,0,4,0,9],[32,67,0.4776,0.71875,0.39847,0.28571,1.0,1.0,0.0,1.0,4,20,0,4,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,1,0,20],[36,67,0.5373,0.6384,0.34623,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,1,0,0,11,0,0,1,0,0,0,0,0,4,0,0,1,0,13],[40,67,0.597,0.60268,0.37752,0.28571,0.71429,1.0,0.0,1.0,4,12,0,4,0,2,0,0,5,0,0,4,0,0,0,0,0,3,0,0,2,0,12],[44,67,0.6567,0.58034,0.38122,0.28571,0.4998,1.0,0.0,1.0,3,13,0,3,0,3,0,0,8,0,0,2,0,0,1,0,0,2,0,0,0,0,13],[48,67,0.7164,0.46429,0.37965,0.14286,0.28571,0.89286,0.0,1.0,7,8,0,7,0,2,0,0,8,0,0,3,0,0,0,0,0,3,0,0,1,0,8],[52,67,0.7761,0.45093,0.38151,0.24999,0.28571,0.89286,0.0,1.0,7,8,0,7,0,1,0,0,12,0,0,1,0,0,0,0,0,1,0,0,2,0,8],[56,67,0.8358,0.53571,0.37965,0.28571,0.35716,1.0,0.0,1.0,5,10,0,5,0,1,0,0,10,0,0,1,0,0,0,0,0,4,0,0,1,0,10],[60,67,0.8955,0.45536,0.40632,0.10714,0.28571,1.0,0.0,1.0,8,9,0,8,0,4,0,0,6,0,0,2,0,0,0,0,0,1,0,0,2,0,9],[64,67,0.9552,0.39732,0.39566,0.0,0.28571,1.0,0.0,1.0,9,9,0,9,0,3,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[67,67,1.0,0.30803,0.27918,0.0,0.28571,0.42858,0.0,1.0,9,1,0,9,0,3,0,0,9,0,0,5,0,0,1,0,0,2,0,0,2,0,1]]}]},{"i":"21ea5e449bd59179","q":"6. (POL) Let $A B C D E F$ be a convex hexagon such that $\\angle B+\\angle D+\\angle F=$ $360^{\\circ}$ and  $$ \\frac{A B}{B C} \\cdot \\frac{C D}{D E} \\cdot \\frac{E F}{F A}=1 $$  Prove that  $$ \\frac{B C}{C A} \\cdot \\frac{A E}{E F} \\cdot \\frac{F D}{D B}=1 $$","t":[{"b":1,"e":0.0,"k":"falling","v":0.0,"x":0.27231,"p":[[0,30,0.0,0.16516,0.19266,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,4,0,0,10,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[4,30,0.1333,0.27231,0.24052,0.0,0.2857,0.42857,0.0,0.85714,9,0,0,9,0,6,0,0,6,0,0,4,0,0,5,0,0,1,0,0,1,0,0],[8,30,0.2667,0.22768,0.27632,0.0,0.14286,0.32143,0.0,1.0,13,1,0,13,0,7,0,0,4,0,0,3,0,0,1,0,0,2,0,0,1,0,1],[12,30,0.4,0.11598,0.19376,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,6,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[16,30,0.5333,0.16963,0.28667,0.0,0.0,0.17857,0.0,0.85714,22,0,0,22,0,2,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,0],[20,30,0.6667,0.11606,0.22139,0.0,0.0,0.14287,0.0,1.0,22,1,0,22,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[24,30,0.8,0.05804,0.12807,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.00446,"x":0.46427,"p":[[0,43,0.0,0.20088,0.23105,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,3,0,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[4,43,0.093,0.37045,0.34973,0.0,0.28571,0.60714,0.0,1.0,10,4,0,10,0,4,0,0,3,0,0,4,0,0,3,0,0,3,0,0,1,0,4],[8,43,0.186,0.46427,0.35714,0.10714,0.57143,0.71429,0.0,1.0,8,5,0,8,0,3,0,0,2,0,0,1,0,0,7,0,0,5,0,0,1,0,5],[12,43,0.2791,0.21875,0.3589,0.0,0.0,0.32143,0.0,1.0,21,4,0,21,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,4],[16,43,0.3721,0.17411,0.27137,0.0,0.0,0.2857,0.0,1.0,19,1,1,19,0,3,0,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,1],[20,43,0.4651,0.25892,0.31629,0.0,0.07143,0.57111,0.0,1.0,16,1,0,16,0,3,0,0,2,0,0,2,0,0,3,0,0,4,0,0,1,0,1],[24,43,0.5581,0.11607,0.20958,0.0,0.0,0.1786,0.0,1.0,21,1,0,21,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[28,43,0.6512,0.17856,0.32141,0.0,0.0,0.14287,0.0,1.0,21,2,0,21,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,2],[32,43,0.7442,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.12054,0.23449,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[40,43,0.9302,0.05357,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ad47fc464d554b43","q":"Let  $N$  be a positive integer. Let  $R$  denote the smallest positive number that is the sum of  $m$  terms  $\\sum^m_{i=1}{\\pm \\sqrt{a_i}}$ , where each  $a_i, i=1,\\cdots, m$  is an integer not larger than  $N$ . Prove that \\[R\\le C\\cdot N^{-m+\\frac{3}{2}}\\] for some positive real number  $C$ .\n\n*Proposed by Navid*\n\n\n*(Clarification: note that the constant is allowed to depend on  $m$  but should be independent of  $N$ , i.e. the equation  $R(m,N)\\le C(m)\\cdot N^{-m+\\frac{3}{2}}$  should hold for all positive integers  $N$ )*","t":[{"b":2,"e":0.2857,"k":"flat","v":0.08927,"x":0.35271,"p":[[0,18,0.0,0.09821,0.18363,0.0,0.0,0.14286,0.0,0.71429,21,0,5,21,0,6,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[4,18,0.2222,0.35271,0.3214,0.0,0.28571,0.46525,0.0,1.0,9,3,0,9,0,3,0,0,6,0,0,6,0,0,2,0,0,1,0,0,2,0,3],[8,18,0.4444,0.26339,0.22899,0.0,0.28571,0.42857,0.0,0.85714,11,0,0,11,0,2,0,0,6,0,0,9,0,0,3,0,0,0,0,0,1,0,0],[12,18,0.6667,0.08927,0.15042,0.0,0.0,0.14286,0.0,0.571,22,0,0,22,0,3,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,18,0.8889,0.20088,0.20156,0.0,0.21428,0.42857,0.0,0.571,15,0,0,15,0,1,0,0,5,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[18,18,1.0,0.19196,0.28259,0.0,0.0,0.32143,0.0,1.0,18,2,0,18,0,3,0,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,2]]},{"b":5,"e":0.42857,"k":"flat","v":0.0625,"x":0.33035,"p":[[0,33,0.0,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.28571,22,0,8,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.33035,0.28219,0.0,0.42857,0.42857,0.0,1.0,9,1,0,9,0,3,0,0,3,0,0,12,0,0,1,0,0,0,0,0,3,0,1],[8,33,0.2424,0.26786,0.23077,0.10714,0.2857,0.42857,0.0,0.85714,8,0,0,8,0,7,0,0,5,0,0,9,0,0,1,0,0,0,0,0,2,0,0],[12,33,0.3636,0.21875,0.24219,0.0,0.14286,0.42857,0.0,0.85714,14,0,0,14,0,3,0,0,5,0,0,8,0,0,0,0,0,0,0,0,2,0,0],[16,33,0.4848,0.19642,0.19802,0.0,0.14286,0.42857,0.0,0.571,14,0,0,14,0,4,0,0,3,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[20,33,0.6061,0.14286,0.21724,0.0,0.0,0.2857,0.0,0.85714,20,0,0,20,0,2,0,0,4,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[24,33,0.7273,0.19196,0.19434,0.0,0.14286,0.42857,0.0,0.4286,15,0,0,15,0,2,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.15179,0.20183,0.0,0.0,0.42857,0.0,0.42857,20,0,0,20,0,1,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.24554,0.22084,0.0,0.28571,0.42857,0.0,0.85714,12,0,0,12,0,2,0,0,4,0,0,13,0,0,0,0,0,0,0,0,1,0,0],[33,33,1.0,0.09822,0.14914,0.0,0.0,0.1786,0.0,0.4286,21,0,0,21,0,3,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a324c9baad565347","q":"Let  $BB'$ ,  $CC'$  be the altitudes of an acute-angled triangle  $ABC$ . Two circles passing through  $A$  and  $C'$   are tangent to  $BC$  at points  $P$  and  $Q$ . Prove that  $A, B', P, Q$  are concyclic.","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.42855,"p":[[0,76,0.0,0.42855,0.39285,0.14286,0.14286,1.0,0.0,1.0,2,9,0,2,0,17,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,9],[4,76,0.0526,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,76,0.1053,0.05348,0.11703,0.0,0.0,0.035,0.0,0.57143,24,0,0,24,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,76,0.1579,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,76,0.2105,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,76,0.2632,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,76,0.3158,0.04893,0.07646,0.0,0.0,0.14071,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,76,0.3684,0.04018,0.09607,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,76,0.4211,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,76,0.4737,0.05357,0.13243,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,76,0.5263,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,76,0.5789,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,76,0.6842,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,76,0.8421,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,76,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.05171,"x":0.41518,"p":[[0,87,0.0,0.41518,0.40146,0.14286,0.14286,0.89286,0.0,1.0,4,8,0,4,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,8],[4,87,0.046,0.06687,0.07965,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,87,0.092,0.05171,0.07638,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,1,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,87,0.1379,0.05339,0.06893,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,87,0.1839,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,87,0.2299,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,87,0.2759,0.13813,0.02482,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,87,0.3218,0.12938,0.04162,0.14286,0.14286,0.14286,0.0,0.143,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,87,0.3678,0.1292,0.04156,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,87,0.4138,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,87,0.4598,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,87,0.5057,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,87,0.5517,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,87,0.5977,0.16509,0.152,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,87,0.6437,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,87,0.6897,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,87,0.7356,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,87,0.7816,0.1875,0.17655,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[72,87,0.8276,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,87,0.8736,0.13654,0.02662,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,1,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,87,0.9195,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,87,0.9655,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[87,87,1.0,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f8e66637f4a3c755","q":"We define a sequence of natural numbers by the initial values  $a_0 = a_1 = a_2 = 1$  and the recursion   $$  a_n = \\bigg \\lfloor \\frac{n}{a_{n-1}a_{n-2}a_{n-3}} \\bigg \\rfloor  $$   \nfor all  $n \\ge 3$ . Find the value of  $a_{2022}$ .","t":[{"b":4,"e":0.85714,"k":"flat","v":0.78124,"x":0.96429,"p":[[0,56,0.0,0.78124,0.18892,0.57143,0.85707,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,9,0,9],[4,56,0.0714,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[8,56,0.1429,0.89509,0.20824,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,1,0,0,0,2,0,0,2,0,0,3,0,23],[12,56,0.2143,0.91071,0.20124,0.85714,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,22],[16,56,0.2857,0.95534,0.09067,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[20,56,0.3571,0.9375,0.15946,0.96429,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[24,56,0.4286,0.95089,0.13175,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[28,56,0.5,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[32,56,0.5714,0.91518,0.15916,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,24],[36,56,0.6429,0.9464,0.1276,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,26],[40,56,0.7143,0.95088,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[44,56,0.7857,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[48,56,0.8571,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[52,56,0.9286,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[56,56,1.0,0.86606,0.14258,0.857,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,12,0,13]]},{"b":6,"e":0.85714,"k":"flat","v":0.81695,"x":0.98214,"p":[[0,85,0.0,0.83036,0.19045,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,0,6,0,14],[4,85,0.0471,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,85,0.0941,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[12,85,0.1412,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[16,85,0.1882,0.94196,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[20,85,0.2353,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[24,85,0.2824,0.94196,0.18162,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[28,85,0.3294,0.91071,0.18472,0.85714,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,20],[32,85,0.3765,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,85,0.4235,0.95535,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[40,85,0.4706,0.95759,0.12482,1.0,1.0,1.0,0.35714,1.0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,3,0,27],[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],[48,85,0.5647,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[52,85,0.6118,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[56,85,0.6588,0.92411,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[60,85,0.7059,0.89284,0.14289,0.85711,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,18],[64,85,0.7529,0.93749,0.10682,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[68,85,0.8,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[72,85,0.8471,0.91964,0.13803,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,21],[76,85,0.8941,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[80,85,0.9412,0.875,0.09942,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,19,0,9],[84,85,0.9882,0.88839,0.10555,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,13],[85,85,1.0,0.81695,0.10853,0.82132,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,22,0,2]]}]},{"i":"d9b10837b5d00efa","q":"Let $ABC$ be an acute triangle. Let $P$ and $Q$ be two points on the segment [BC]. We denote by $\\mathrm{O}_{1}, \\mathrm{O}_{2}, \\mathrm{O}_{3}$ and $\\mathrm{O}_{4}$ the centers of the circumcircles of triangles ABP, ABQ, ACP and ACQ, respectively. Show that the points $\\mathrm{O}_{1}, \\mathrm{O}_{2}, \\mathrm{O}_{3}$ and $\\mathrm{O}_{4}$ are concyclic if and only if $\\widehat{\\mathrm{BAP}}=\\widehat{\\mathrm{QAC}}$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.13384,"x":0.39724,"p":[[0,100,0.0,0.13384,0.13333,0.0,0.14286,0.1429,0.0,0.42857,11,0,3,11,0,16,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,100,0.04,0.30795,0.21468,0.14286,0.21429,0.42857,0.0,1.0,1,1,0,1,0,15,0,0,1,0,0,13,0,0,0,0,0,0,0,0,1,0,1],[8,100,0.08,0.36158,0.32727,0.14286,0.14286,0.46418,0.0,1.0,2,5,0,2,0,16,0,0,2,0,0,4,0,0,2,0,0,0,0,0,1,0,5],[12,100,0.12,0.23214,0.18814,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,22,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[16,100,0.16,0.32589,0.29066,0.14286,0.14288,0.42857,0.0,1.0,2,4,0,2,0,15,0,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,4],[20,100,0.2,0.34812,0.29873,0.14286,0.21428,0.46428,0.0,1.0,2,4,0,2,0,14,0,0,5,0,0,3,0,0,3,0,0,1,0,0,0,0,4],[24,100,0.24,0.32134,0.31547,0.14286,0.14288,0.46429,0.0,1.0,5,4,0,5,0,14,0,0,2,0,0,3,0,0,3,0,0,1,0,0,0,0,4],[28,100,0.28,0.32143,0.27433,0.14286,0.21431,0.42857,0.0,1.0,4,3,0,4,0,12,0,0,2,0,0,9,0,0,2,0,0,0,0,0,0,0,3],[32,100,0.32,0.24964,0.2552,0.14,0.14286,0.42857,0.0,1.0,7,2,0,7,0,13,0,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,2],[36,100,0.36,0.24552,0.19636,0.14286,0.14288,0.28571,0.0,1.0,4,1,0,4,0,13,0,0,8,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[40,100,0.4,0.39724,0.32097,0.14286,0.28586,0.42858,0.0,1.0,2,6,0,2,0,11,0,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,6],[44,100,0.44,0.29465,0.25489,0.14286,0.2857,0.42857,0.0,1.0,5,2,0,5,0,9,0,0,9,0,0,5,0,0,0,0,0,2,0,0,0,0,2],[48,100,0.48,0.28125,0.25123,0.14286,0.14286,0.42857,0.0,1.0,4,2,0,4,0,14,0,0,4,0,0,6,0,0,1,0,0,1,0,0,0,0,2],[52,100,0.52,0.30349,0.23629,0.14286,0.28571,0.42857,0.0,1.0,3,2,0,3,0,12,0,0,4,0,0,10,0,0,1,0,0,0,0,0,0,0,2],[56,100,0.56,0.22303,0.16351,0.14214,0.14286,0.42857,0.0,0.57143,6,0,0,6,0,12,0,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[60,100,0.6,0.24098,0.1766,0.14286,0.14286,0.42857,0.0,0.71429,4,0,0,4,0,15,0,0,3,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[64,100,0.64,0.23205,0.20442,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,17,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,1],[68,100,0.68,0.31255,0.20961,0.14286,0.28571,0.42858,0.0,1.0,3,1,0,3,0,9,0,0,7,0,0,8,0,0,4,0,0,0,0,0,0,0,1],[72,100,0.72,0.20518,0.15549,0.14214,0.14286,0.28571,0.0,0.57143,6,0,0,6,0,14,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[76,100,0.76,0.24554,0.23483,0.14286,0.14286,0.28571,0.0,1.0,3,2,0,3,0,19,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,2],[80,100,0.8,0.26786,0.16269,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,10,0,0,5,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[84,100,0.84,0.21866,0.12875,0.14286,0.14286,0.28571,0.0,0.4286,2,0,0,2,0,18,0,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[88,100,0.88,0.15616,0.12038,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,21,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[92,100,0.92,0.21875,0.13825,0.14286,0.14286,0.32143,0.0,0.4286,3,0,0,3,0,17,0,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[96,100,0.96,0.22313,0.14703,0.14286,0.14286,0.32143,0.0,0.57143,3,0,0,3,0,17,0,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[100,100,1.0,0.19643,0.11152,0.14286,0.14286,0.28571,0.0,0.4286,2,0,0,2,0,20,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.24553,"x":0.88838,"p":[[0,88,0.0,0.24553,0.16455,0.14286,0.1429,0.42857,0.0,0.57143,4,0,1,4,0,13,0,0,5,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[4,88,0.0455,0.30349,0.2691,0.14286,0.14288,0.42857,0.0,1.0,1,3,0,1,0,18,0,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,3],[8,88,0.0909,0.28116,0.25631,0.14286,0.14286,0.42857,0.0,1.0,3,2,0,3,0,16,0,0,4,0,0,5,0,0,1,0,0,0,0,0,1,0,2],[12,88,0.1364,0.32143,0.19885,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,12,0,0,4,0,0,11,0,0,3,0,0,0,0,0,0,0,1],[16,88,0.1818,0.3525,0.33511,0.14286,0.14288,0.42858,0.0,1.0,3,6,0,3,0,14,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,6],[20,88,0.2273,0.47759,0.3247,0.1429,0.42857,0.71429,0.14,1.0,0,7,0,0,0,10,0,0,5,0,0,4,0,0,4,0,0,2,0,0,0,0,7],[24,88,0.2727,0.46429,0.32537,0.14286,0.42857,0.60714,0.14286,1.0,0,7,0,0,0,11,0,0,4,0,0,5,0,0,4,0,0,1,0,0,0,0,7],[28,88,0.3182,0.44638,0.32487,0.14286,0.28571,0.64286,0.0,1.0,2,5,0,2,0,7,0,0,9,0,0,2,0,0,4,0,0,0,0,0,3,0,5],[32,88,0.3636,0.57143,0.35892,0.2857,0.42859,1.0,0.0,1.0,2,12,0,2,0,4,0,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,12],[36,88,0.4091,0.50893,0.35524,0.14289,0.42857,1.0,0.0,1.0,2,9,0,2,0,7,0,0,5,0,0,4,0,0,4,0,0,0,0,0,1,0,9],[40,88,0.4545,0.58933,0.33832,0.28571,0.50071,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,4,0,0,6,0,0,3,0,0,1,0,0,1,0,11],[44,88,0.5,0.5179,0.34208,0.14286,0.4293,1.0,0.14286,1.0,0,9,0,0,0,10,0,0,3,0,0,4,0,0,5,0,0,1,0,0,0,0,9],[48,88,0.5455,0.47311,0.30195,0.2857,0.35714,0.60714,0.14,1.0,0,6,0,0,0,6,0,0,10,0,0,5,0,0,3,0,0,1,0,0,1,0,6],[52,88,0.5909,0.58033,0.30916,0.28571,0.57143,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,4,0,0,4,0,0,9,0,0,0,0,0,1,0,9],[56,88,0.6364,0.78568,0.30724,0.57132,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,1,0,0,1,0,0,3,0,0,1,0,0,4,0,18],[60,88,0.6818,0.69643,0.28959,0.42857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,7,0,0,3,0,0,3,0,0,3,0,12],[64,88,0.7273,0.74094,0.27766,0.53572,0.85714,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,3,0,0,4,0,0,4,0,0,3,0,0,3,0,14],[68,88,0.7727,0.6116,0.35217,0.28571,0.64286,1.0,0.0,1.0,2,10,0,2,0,3,0,0,6,0,0,3,0,0,2,0,0,1,0,0,5,0,10],[72,88,0.8182,0.65174,0.28334,0.42857,0.64286,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,1,0,0,6,0,0,6,0,0,4,0,0,4,0,8],[76,88,0.8636,0.76783,0.2896,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,0,0,0,5,0,0,2,0,0,1,0,0,6,0,15],[80,88,0.9091,0.88838,0.1812,0.82132,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,2,0,22],[84,88,0.9545,0.82587,0.2544,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,4,0,0,4,0,18],[88,88,1.0,0.8616,0.21572,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,3,0,20]]}]},{"i":"097cab4716e4920f","q":"Let $a, b, c$ be positive real numbers, such that $(a b)^{2}+(b c)^{2}+(c a)^{2}=3$. Prove that\n\n$$\n\\left(a^{2}-a+1\\right)\\left(b^{2}-b+1\\right)\\left(c^{2}-c+1\\right) \\geq 1 .\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,99,0.0,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,99,0.0404,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,99,0.0808,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,99,0.1212,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,99,0.1616,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,99,0.202,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,99,0.2424,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,99,0.2828,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,99,0.3232,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,99,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,99,0.404,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,99,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,99,0.4848,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,99,0.5253,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,99,0.5657,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,99,0.6061,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,99,0.6465,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,99,0.6869,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,99,0.7273,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,99,0.7677,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,99,0.8081,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,99,0.8485,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,99,0.8889,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,99,0.9293,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,99,0.9697,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[99,99,1.0,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,67,0.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,67,0.0597,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,67,0.1194,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,67,0.1791,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,67,0.2388,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,67,0.4776,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,67,0.5373,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,67,0.6567,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,67,0.7164,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,67,0.8955,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,67,0.9552,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[67,67,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a705b9ebb5d47605","q":"Call a positive integer $n$ practical if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$.\n\nFor example, the divisors of 6 are $\\mathbf{1 , 2}, \\mathbf{3}$, and $\\mathbf{6}$. Since\n\n$$\n1=\\mathbf{1}, \\quad 2=\\mathbf{2}, \\quad 3=\\mathbf{3}, \\quad 4=\\mathbf{1}+\\mathbf{3}, \\quad 5=\\mathbf{2}+\\mathbf{3}, \\quad 6=\\mathbf{6},\n$$\n\nwe see that 6 is practical.\n\nProve that the product of two practical numbers is also practical.","t":[{"b":2,"e":1.0,"k":"rising","v":0.46875,"x":1.0,"p":[[0,13,0.0,0.46875,0.49902,0.0,0.0,1.0,0.0,1.0,17,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[4,13,0.3077,0.71875,0.44961,0.0,1.0,1.0,0.0,1.0,9,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23],[8,13,0.6154,0.8125,0.39031,1.0,1.0,1.0,0.0,1.0,6,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[12,13,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":1.0,"k":"rising","v":0.65625,"x":1.0,"p":[[0,9,0.0,0.65625,0.47496,0.0,1.0,1.0,0.0,1.0,11,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[4,9,0.4444,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"8e992e6db29e24d9","q":"In  $\\vartriangle ABC$ ,  $AB = AC$  and  $\\angle BAC = 20^o$ . A point  $D$  lies on the side  $AB$  and  $AD = BC$ . Find  $\\angle BCD$ .\n\n(LF. Sharygin, Moscow)","t":[{"b":0,"e":0.57143,"k":"rising","v":0.29463,"x":0.6294,"p":[[0,29,0.0,0.29463,0.32327,0.0,0.07143,0.71429,0.0,0.71429,16,0,0,16,0,2,0,0,0,0,0,1,0,0,4,0,0,9,0,0,0,0,0],[4,29,0.1379,0.37944,0.34368,0.0,0.42859,0.71429,0.0,1.0,12,2,0,12,0,2,0,0,1,0,0,2,0,0,4,0,0,9,0,0,0,0,2],[8,29,0.2759,0.5044,0.29661,0.42859,0.57143,0.71429,0.0,1.0,7,1,0,7,0,0,0,0,0,0,0,4,0,0,9,0,0,8,0,0,3,0,1],[12,29,0.4138,0.57142,0.19885,0.57132,0.57143,0.71429,0.0,1.0,2,1,0,2,0,0,0,0,1,0,0,4,0,0,14,0,0,9,0,0,1,0,1],[16,29,0.5517,0.5848,0.25843,0.5354,0.71429,0.71429,0.0,1.0,4,1,0,4,0,0,0,0,1,0,0,3,0,0,4,0,0,17,0,0,2,0,1],[20,29,0.6897,0.6294,0.14225,0.571,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,13,0,0,0,0,2],[24,29,0.8276,0.61158,0.18977,0.571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,5,0,0,10,0,0,12,0,0,1,0,2],[28,29,0.9655,0.59371,0.13883,0.4286,0.64286,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,7,0,0,7,0,0,16,0,0,0,0,0],[29,29,1.0,0.60712,0.15153,0.571,0.64286,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,14,0,0,2,0,0]]},{"b":6,"e":0.57143,"k":"rising","v":0.25441,"x":0.7589,"p":[[0,29,0.0,0.25441,0.30453,0.0,0.0,0.571,0.0,1.0,17,1,0,17,0,1,0,0,2,0,0,2,0,0,6,0,0,3,0,0,0,0,1],[4,29,0.1379,0.43743,0.32326,0.0,0.571,0.71429,0.0,1.0,9,3,0,9,0,0,0,0,3,0,0,3,0,0,8,0,0,6,0,0,0,0,3],[8,29,0.2759,0.60711,0.2988,0.571,0.71429,0.71429,0.0,1.0,4,4,0,4,0,2,0,0,0,0,0,1,0,0,4,0,0,15,0,0,2,0,4],[12,29,0.4138,0.65624,0.29421,0.5354,0.71429,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,2,0,0,3,0,0,3,0,0,12,0,0,1,0,8],[16,29,0.5517,0.7589,0.20343,0.71429,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,2,0,9],[20,29,0.6897,0.71426,0.26965,0.71429,0.71429,1.0,0.0,1.0,2,9,0,2,0,1,0,0,0,0,0,2,0,0,2,0,0,14,0,0,2,0,9],[24,29,0.8276,0.616,0.16918,0.571,0.57143,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,4,0,0,13,0,0,11,0,0,2,0,1],[28,29,0.9655,0.70532,0.13337,0.67857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,19,0,0,2,0,3],[29,29,1.0,0.73209,0.13722,0.71429,0.71429,0.75,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,17,0,0,4,0,4]]}]},{"i":"8214d40f78140105","q":"Consider a triangle  $ABC$  with  $BC>AC$ . The circle with center  $C$  and radius  $AC$  intersects the segment  $BC$  in  $D$ . Let  $I$  be the incenter of triangle  $ABC$  and  $\\Gamma$  be the circle that passes through  $I$  and is tangent to the line  $CA$  at  $A$ . The line  $AB$  and  $\\Gamma$  intersect at a point  $F$  with  $F \\neq A$ . Prove that  $BF=BD$ .","t":[{"b":1,"e":0.71429,"k":"falling","v":0.69642,"x":0.91071,"p":[[0,27,0.0,0.91071,0.14618,0.71429,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,0,0,23],[4,27,0.1481,0.81695,0.22085,0.71429,0.9285,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,11,0,0,1,0,16],[8,27,0.2963,0.83929,0.15465,0.71429,0.71429,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,0,0,15],[12,27,0.4444,0.75893,0.10374,0.71429,0.71429,0.71429,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,5],[16,27,0.5926,0.73661,0.15198,0.71429,0.71429,0.71429,0.1429,1.0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,5],[20,27,0.7407,0.76786,0.14617,0.71429,0.71429,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,0,0,1,0,7],[24,27,0.8889,0.69642,0.13718,0.71429,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,0,0,1],[27,27,1.0,0.72308,0.04974,0.71429,0.71429,0.71429,0.71,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1]]},{"b":4,"e":0.57143,"k":"falling","v":0.53562,"x":0.83929,"p":[[0,15,0.0,0.81696,0.22934,0.71429,0.85714,1.0,0.0,1.0,1,16,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,13,0,0,0,0,16],[4,15,0.2667,0.83929,0.23351,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,6,0,0,0,0,20],[8,15,0.5333,0.54463,0.08328,0.57143,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,0,0,0,29,0,0,0,0,0,0,0,0],[12,15,0.8,0.56247,0.04971,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0],[15,15,1.0,0.53562,0.09446,0.571,0.57143,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,0,0,0,28,0,0,0,0,0,0,0,0]]}]},{"i":"dab656fa1428c9d0","q":"Let  $ABCD$  be a trapezoid with bases  $AB$  and  $CD$ . Points  $P$  and  $Q$  lie on diagonals  $AC$  and  $BD$ , respectively and  $\\angle APD = \\angle BQC$ . Prove that  $\\angle AQD = \\angle BPC$ .","t":[{"b":1,"e":0.2857,"k":"falling","v":0.13839,"x":0.36161,"p":[[0,47,0.0,0.36161,0.35712,0.0,0.42857,0.42857,0.0,1.0,12,5,0,12,0,1,0,0,2,0,0,10,0,0,0,0,0,1,0,0,1,0,5],[4,47,0.0851,0.33929,0.17035,0.28571,0.42857,0.42857,0.0,0.57143,6,0,0,6,0,0,0,0,3,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[8,47,0.1702,0.30804,0.21461,0.0,0.42857,0.42857,0.0,0.85714,9,0,0,9,0,1,0,0,1,0,0,20,0,0,0,0,0,0,0,0,1,0,0],[12,47,0.2553,0.34822,0.19865,0.42857,0.42857,0.42857,0.0,0.85714,7,0,0,7,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,1,0,0],[16,47,0.3404,0.33035,0.17288,0.14286,0.42857,0.42857,0.0,0.571,5,0,0,5,0,4,0,0,0,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[20,47,0.4255,0.27232,0.24053,0.0,0.42857,0.42857,0.0,1.0,12,1,0,12,0,1,0,0,1,0,0,17,0,0,0,0,0,0,0,0,0,0,1],[24,47,0.5106,0.34822,0.15542,0.28571,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,2,0,0,3,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[28,47,0.5957,0.3125,0.18707,0.10714,0.42857,0.42857,0.0,0.42857,8,0,0,8,0,1,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.28572,0.19562,0.0,0.42857,0.42857,0.0,0.57143,9,0,0,9,0,2,0,0,2,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[36,47,0.766,0.29018,0.22442,0.0,0.42857,0.42857,0.0,0.85714,11,0,0,11,0,0,0,0,1,0,0,19,0,0,0,0,0,0,0,0,1,0,0],[40,47,0.8511,0.25,0.20203,0.0,0.35714,0.42857,0.0,0.57143,11,0,0,11,0,3,0,0,2,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[44,47,0.9362,0.24554,0.1931,0.0,0.35714,0.42857,0.0,0.42857,10,0,0,10,0,5,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.13839,0.18724,0.0,0.0,0.32143,0.0,0.4286,20,0,0,20,0,1,0,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.24553,"x":0.375,"p":[[0,69,0.0,0.24553,0.26541,0.0,0.1429,0.4286,0.0,0.85714,14,0,0,14,0,3,0,0,2,0,0,9,0,0,1,0,0,1,0,0,2,0,0],[4,69,0.058,0.34821,0.15945,0.39285,0.42857,0.42857,0.0,0.571,4,0,0,4,0,3,0,0,1,0,0,23,0,0,1,0,0,0,0,0,0,0,0],[8,69,0.1159,0.34375,0.15916,0.39286,0.42857,0.42857,0.0,0.42857,5,0,0,5,0,1,0,0,2,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.37054,0.14664,0.42857,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,4,0,0,1,0,0,24,0,0,0,0,0,1,0,0,0,0,0],[16,69,0.2319,0.33482,0.27107,0.0,0.42857,0.42857,0.0,1.0,9,2,0,9,0,1,0,0,3,0,0,16,0,0,0,0,0,0,0,0,1,0,2],[20,69,0.2899,0.375,0.23351,0.25,0.42857,0.42857,0.0,1.0,4,2,0,4,0,4,0,0,2,0,0,19,0,0,0,0,0,1,0,0,0,0,2],[24,69,0.3478,0.32589,0.20589,0.21427,0.42857,0.42857,0.0,0.85714,8,0,0,8,0,0,0,0,2,0,0,21,0,0,0,0,0,0,0,0,1,0,0],[28,69,0.4058,0.32143,0.22016,0.10714,0.42857,0.42857,0.0,1.0,8,1,0,8,0,1,0,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,1],[32,69,0.4638,0.25893,0.19377,0.0,0.42857,0.42857,0.0,0.4286,10,0,0,10,0,3,0,0,2,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.26786,0.21354,0.0,0.42857,0.42857,0.0,0.71429,10,0,0,10,0,4,0,0,1,0,0,15,0,0,1,0,0,1,0,0,0,0,0],[40,69,0.5797,0.3259,0.21498,0.1429,0.42857,0.42857,0.0,1.0,7,1,0,7,0,2,0,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,1],[44,69,0.6377,0.32143,0.19885,0.10714,0.42857,0.42857,0.0,0.71429,8,0,0,8,0,1,0,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0],[48,69,0.6957,0.30804,0.17169,0.14289,0.42857,0.42857,0.0,0.4286,6,0,0,6,0,3,0,0,3,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.25451,0.19804,0.0,0.42857,0.42857,0.0,0.43,11,0,0,11,0,2,0,0,2,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.34822,0.15126,0.28571,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,4,0,0,2,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[60,69,0.8696,0.29017,0.19059,0.10714,0.42857,0.42857,0.0,0.571,8,0,0,8,0,3,0,0,2,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[64,69,0.9275,0.27233,0.19019,0.0,0.42857,0.42857,0.0,0.4286,9,0,0,9,0,3,0,0,2,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.29916,0.17988,0.14286,0.42857,0.42857,0.0,0.43,7,0,0,7,0,3,0,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.24554,0.20277,0.0,0.42857,0.42857,0.0,0.4286,12,0,0,12,0,2,0,0,1,0,0,17,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"327cf1d5e03a0b00","q":"Cyclic quadrilateral  $ABCD$  is such that  $\\angle BAD = 2\\angle ADC$  and  $CD = 2BC$ . Let  $H$  be the projection of  $C$  onto  $AD$ . Prove that  $BH \\parallel CD$ .\n\n*Proposed by Fedir Yudin, Anton Trygub*","t":[{"b":2,"e":0.0,"k":"flat","v":0.01339,"x":0.32143,"p":[[0,98,0.0,0.03571,0.11294,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,98,0.0408,0.16071,0.34764,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[8,98,0.0816,0.24553,0.42593,0.0,0.0,0.21425,0.0,1.0,24,7,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7],[12,98,0.1224,0.29018,0.42028,0.0,0.0,0.71429,0.0,1.0,21,6,0,21,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,6],[16,98,0.1633,0.17857,0.37287,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[20,98,0.2041,0.11161,0.28512,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[24,98,0.2449,0.11159,0.24413,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0],[28,98,0.2857,0.13839,0.31438,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[32,98,0.3265,0.19643,0.37923,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,5],[36,98,0.3673,0.13393,0.3008,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,2],[40,98,0.4082,0.125,0.30252,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,2],[44,98,0.449,0.13839,0.32632,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[48,98,0.4898,0.28125,0.4278,0.0,0.0,0.85714,0.0,1.0,22,6,0,22,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,6],[52,98,0.5306,0.23658,0.39219,0.0,0.0,0.571,0.0,1.0,23,5,0,23,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,5],[56,98,0.5714,0.22768,0.38109,0.0,0.0,0.42857,0.0,1.0,22,5,0,22,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,5],[60,98,0.6122,0.31696,0.44569,0.0,0.0,0.89286,0.0,1.0,21,8,0,21,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,8],[64,98,0.6531,0.28125,0.43519,0.0,0.0,0.78571,0.0,1.0,22,8,0,22,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[68,98,0.6939,0.17857,0.36422,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[72,98,0.7347,0.19197,0.36528,0.0,0.0,0.03571,0.0,1.0,24,4,0,24,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,4],[76,98,0.7755,0.24107,0.42021,0.0,0.0,0.17857,0.0,1.0,24,7,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,7],[80,98,0.8163,0.15625,0.33381,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,3],[84,98,0.8571,0.32143,0.43006,0.0,0.0,0.85714,0.0,1.0,20,6,0,20,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,6],[88,98,0.898,0.23661,0.37561,0.0,0.0,0.42857,0.0,1.0,22,4,0,22,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,4],[92,98,0.9388,0.08929,0.24936,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[96,98,0.9796,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[98,98,1.0,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04018,"x":0.38393,"p":[[0,165,0.0,0.04911,0.15815,0.0,0.0,0.0,0.0,0.71429,29,0,1,29,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,165,0.0242,0.30357,0.42671,0.0,0.0,0.71429,0.0,1.0,21,6,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,6],[8,165,0.0485,0.36161,0.45314,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,9],[12,165,0.0727,0.24554,0.39807,0.0,0.0,0.71429,0.0,1.0,23,4,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,4],[16,165,0.097,0.19643,0.38919,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[20,165,0.1212,0.26339,0.42425,0.0,0.0,0.75,0.0,1.0,23,6,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,6],[24,165,0.1455,0.32366,0.45104,0.0,0.0,0.89286,0.0,1.0,21,8,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,8],[28,165,0.1697,0.20089,0.36397,0.0,0.0,0.10714,0.0,1.0,24,4,0,24,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,4],[32,165,0.1939,0.3125,0.44383,0.0,0.0,0.89286,0.0,1.0,21,8,0,21,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,8],[36,165,0.2182,0.23659,0.39383,0.0,0.0,0.46418,0.0,1.0,23,5,0,23,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,5],[40,165,0.2424,0.13839,0.33405,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[44,165,0.2667,0.18304,0.33548,0.0,0.0,0.10714,0.0,1.0,24,2,0,24,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,2],[48,165,0.2909,0.30357,0.42521,0.0,0.0,0.75,0.0,1.0,20,7,0,20,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,7],[52,165,0.3152,0.30357,0.44428,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[56,165,0.3394,0.38393,0.46762,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,9],[60,165,0.3636,0.22768,0.38608,0.0,0.0,0.2857,0.0,1.0,22,5,0,22,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,5],[64,165,0.3879,0.2232,0.38121,0.0,0.0,0.35704,0.0,1.0,23,5,0,23,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,5],[68,165,0.4121,0.33929,0.46941,0.0,0.0,1.0,0.0,1.0,21,10,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,10],[72,165,0.4364,0.25892,0.39517,0.0,0.0,0.60714,0.0,1.0,22,4,0,22,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,4],[76,165,0.4606,0.32143,0.45457,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,9],[80,165,0.4848,0.37054,0.45717,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,9],[84,165,0.5091,0.35714,0.46429,0.0,0.0,1.0,0.0,1.0,20,9,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,9],[88,165,0.5333,0.28571,0.43595,0.0,0.0,0.78571,0.0,1.0,22,8,0,22,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,8],[92,165,0.5576,0.23213,0.39244,0.0,0.0,0.35704,0.0,1.0,23,5,0,23,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,5],[96,165,0.5818,0.2232,0.39436,0.0,0.0,0.14275,0.0,1.0,24,5,0,24,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,5],[100,165,0.6061,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[104,165,0.6303,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[108,165,0.6545,0.25893,0.39357,0.0,0.0,0.5,0.0,1.0,21,5,0,21,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,5],[112,165,0.6788,0.26786,0.41611,0.0,0.0,0.71429,0.0,1.0,22,6,0,22,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,6],[116,165,0.703,0.33482,0.44264,0.0,0.0,0.78571,0.0,1.0,20,8,0,20,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,8],[120,165,0.7273,0.20982,0.38957,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,5],[124,165,0.7515,0.18304,0.34669,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,3],[128,165,0.7758,0.2723,0.40303,0.0,0.0,0.60682,0.0,1.0,21,5,0,21,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,5],[132,165,0.8,0.04018,0.16458,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[136,165,0.8242,0.33482,0.4512,0.0,0.0,1.0,0.0,1.0,20,9,0,20,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,9],[140,165,0.8485,0.21875,0.39445,0.0,0.0,0.10714,0.0,1.0,24,6,0,24,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,6],[144,165,0.8727,0.18304,0.38172,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[148,165,0.897,0.16517,0.35554,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,4],[152,165,0.9212,0.24554,0.41069,0.0,0.0,0.39286,0.0,1.0,23,6,0,23,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,6],[156,165,0.9455,0.19196,0.3739,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,4],[160,165,0.9697,0.19196,0.36528,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,2],[164,165,0.9939,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[165,165,1.0,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"97c61e3082f371ca","q":"In a triangle  $ABC$  the midpoints of  $BC, CA$  and  $AB$  are  $D, E$  and  $F$ , respectively. Prove that the circumcircles of triangles  $AEF, BFD$  and  $CDE$  intersect all in one point.","t":[{"b":4,"e":1.0,"k":"rising","v":0.00893,"x":0.76326,"p":[[0,50,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.14731,0.23276,0.0,0.0,0.2857,0.0,0.85714,20,0,0,20,0,3,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0],[8,50,0.16,0.09375,0.21312,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,50,0.24,0.09821,0.21852,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[16,50,0.32,0.08929,0.19805,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[20,50,0.4,0.125,0.28959,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,2],[24,50,0.48,0.17411,0.36023,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[28,50,0.56,0.29463,0.37616,0.0,0.0,0.60682,0.0,1.0,18,4,0,18,0,0,0,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,4],[32,50,0.64,0.23661,0.36001,0.0,0.0,0.35714,0.0,1.0,20,4,0,20,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,0,0,0,4],[36,50,0.72,0.23661,0.36702,0.0,0.0,0.60714,0.0,1.0,21,2,0,21,0,2,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,2],[40,50,0.8,0.42844,0.41486,0.0,0.35714,0.85714,0.0,1.0,12,6,0,12,0,3,0,0,1,0,0,3,0,0,0,0,0,2,0,0,5,0,6],[44,50,0.88,0.58033,0.38949,0.25001,0.57143,1.0,0.0,1.0,7,10,0,7,0,1,0,0,2,0,0,2,0,0,5,0,0,1,0,0,4,0,10],[48,50,0.96,0.70088,0.39019,0.49968,0.85714,1.0,0.0,1.0,6,14,0,6,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,7,0,14],[50,50,1.0,0.76326,0.31279,0.571,0.92857,1.0,0.0,1.0,1,16,0,1,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,16]]},{"b":6,"e":0.42857,"k":"rising","v":0.00446,"x":0.39283,"p":[[0,67,0.0,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,67,0.0597,0.06695,0.15557,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,67,0.1194,0.05356,0.14612,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,67,0.1791,0.04464,0.14032,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,67,0.2388,0.09821,0.2659,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1],[20,67,0.2985,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,67,0.3582,0.06241,0.18532,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,67,0.4179,0.20088,0.3622,0.0,0.0,0.14286,0.0,1.0,21,5,0,21,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[32,67,0.4776,0.05804,0.14664,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,67,0.5373,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,67,0.597,0.04018,0.15663,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,67,0.6567,0.125,0.27835,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,1],[48,67,0.7164,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,67,0.7761,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,67,0.8358,0.14955,0.28924,0.0,0.0,0.16071,0.0,1.0,23,1,0,23,0,1,0,1,2,0,0,1,0,0,0,0,0,1,0,0,2,0,1],[60,67,0.8955,0.08482,0.20782,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[64,67,0.9552,0.03124,0.12228,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[67,67,1.0,0.39283,0.30721,0.14286,0.35714,0.71429,0.0,1.0,6,2,0,6,0,6,0,0,4,0,0,4,0,0,3,0,0,6,0,0,1,0,2]]}]},{"i":"4430da8665d6f249","q":"A frog jumps around on the grid points in the plane, from one grid point to another. The frog starts at the point  $(0, 0)$ . Then it makes, successively, a jump of one step horizontally, a jump of  $2$  steps vertically, a jump of  $3$  steps horizontally, a jump of  $4$  steps vertically, et cetera. Determine all  $n > 0$  such that the frog can be back in  $(0, 0)$  after  $n$  jumps.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.63837,"x":0.99554,"p":[[0,44,0.0,0.63837,0.15967,0.57143,0.57143,0.57143,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,1,0,0,0,0,5],[4,44,0.0909,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,44,0.1818,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,44,0.2727,0.97768,0.08828,1.0,1.0,1.0,0.5714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[16,44,0.3636,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[20,44,0.4545,0.87946,0.18935,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,17],[24,44,0.5455,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[28,44,0.6364,0.84375,0.12556,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,13,0,9],[32,44,0.7273,0.91071,0.13243,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,8,0,19],[36,44,0.8182,0.875,0.12242,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,8,0,14],[40,44,0.9091,0.87054,0.11495,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,11,0,12],[44,44,1.0,0.83482,0.11904,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,20,0,5]]},{"b":1,"e":0.85714,"k":"rising","v":0.6875,"x":1.0,"p":[[0,53,0.0,0.6875,0.19704,0.57143,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,0,0,9],[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],[8,53,0.1509,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,53,0.2264,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,53,0.3019,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[20,53,0.3774,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,53,0.4528,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[28,53,0.5283,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[32,53,0.6038,0.93304,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,23],[36,53,0.6792,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[40,53,0.7547,0.92857,0.17128,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,26],[44,53,0.8302,0.9375,0.11812,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,23],[48,53,0.9057,0.95982,0.11425,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[52,53,0.9811,0.84375,0.17261,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,7,0,14],[53,53,1.0,0.9241,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21]]}]},{"i":"563cbee67f75c650","q":"An acute triangle  $ ABC$  with  $ AB \\neq AC$  is given. Let  $ V$  and  $ D$  be the feet of the altitude and angle bisector from  $ A$ , and let  $ E$  and  $ F$  be the intersection points of the circumcircle of  $ \\triangle AVD$  with sides  $ AC$  and  $ AB$ , respectively. Prove that  $ AD$ ,  $ BE$  and  $ CF$  have a common point.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.03116,"x":0.17857,"p":[[0,97,0.0,0.17857,0.14286,0.0,0.28571,0.28571,0.0,0.42857,12,0,9,12,0,1,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,97,0.0412,0.12491,0.13715,0.0,0.14286,0.17857,0.0,0.57143,14,0,0,14,0,10,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,97,0.0825,0.12945,0.17624,0.0,0.0,0.28571,0.0,0.571,19,0,0,19,0,3,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,97,0.1237,0.11607,0.16146,0.0,0.0,0.1429,0.0,0.71429,17,0,0,17,0,8,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,97,0.1649,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,97,0.2062,0.15178,0.14258,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,9,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,97,0.2474,0.08036,0.14698,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,97,0.2887,0.10268,0.13474,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,7,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,97,0.3299,0.11161,0.1461,0.0,0.0,0.2857,0.0,0.57143,18,0,0,18,0,5,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,97,0.3711,0.125,0.1948,0.0,0.0,0.17857,0.0,0.57143,20,0,0,20,0,4,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[40,97,0.4124,0.11161,0.1665,0.0,0.0,0.17857,0.0,0.71429,19,0,0,19,0,5,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[44,97,0.4536,0.09821,0.1448,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,5,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,97,0.4948,0.06687,0.1335,0.0,0.0,0.035,0.0,0.57143,24,0,0,24,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[52,97,0.5361,0.09375,0.13175,0.0,0.0,0.17857,0.0,0.42857,20,0,0,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,97,0.5773,0.09375,0.13651,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,97,0.6186,0.08482,0.11214,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,97,0.6598,0.08473,0.12295,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,97,0.701,0.06251,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,97,0.7423,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,97,0.7835,0.09821,0.13092,0.0,0.0,0.17857,0.0,0.42857,19,0,0,19,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,97,0.8247,0.05803,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,97,0.866,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,97,0.9072,0.10713,0.17493,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,4,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[92,97,0.9485,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,97,0.9897,0.09822,0.14914,0.0,0.0,0.14287,0.0,0.57143,20,0,0,20,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[97,97,1.0,0.03116,0.09257,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.01786,"x":0.20089,"p":[[0,80,0.0,0.20089,0.13296,0.10714,0.2857,0.28571,0.0,0.42857,8,0,4,8,0,5,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,80,0.05,0.18295,0.13942,0.105,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,10,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,80,0.1,0.09822,0.11538,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,80,0.15,0.09375,0.11071,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,80,0.2,0.08482,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,80,0.25,0.10259,0.13472,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,9,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,80,0.3,0.11152,0.12232,0.0,0.14143,0.14286,0.0,0.4286,15,0,0,15,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,80,0.35,0.125,0.15047,0.0,0.07143,0.2857,0.0,0.57143,16,0,0,16,0,7,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,80,0.4,0.09375,0.11633,0.0,0.0,0.1429,0.0,0.28571,18,0,0,18,0,7,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,80,0.45,0.06696,0.10092,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,80,0.5,0.15625,0.1488,0.0,0.14286,0.28571,0.0,0.4286,13,0,0,13,0,6,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,80,0.55,0.14732,0.145,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,10,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[48,80,0.6,0.10259,0.14388,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,80,0.65,0.11161,0.13236,0.0,0.07143,0.14287,0.0,0.42857,16,0,0,16,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,80,0.7,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,80,0.75,0.07589,0.10705,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,80,0.8,0.06697,0.11837,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,80,0.85,0.06695,0.12864,0.0,0.0,0.14286,0.0,0.571,23,0,0,23,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,80,0.9,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,80,0.95,0.03562,0.0713,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,80,1.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8ab8758215681f22","q":"Let $p, q, r$ be positive real numbers and $n \\in \\mathbb{N}$. Show that if $p q r=1$, then\n\n$$\n\\frac{1}{p^{n}+q^{n}+1}+\\frac{1}{q^{n}+r^{n}+1}+\\frac{1}{r^{n}+p^{n}+1} \\leq 1\n$$","t":[{"b":0,"e":0.28571,"k":"flat","v":0.14286,"x":0.25,"p":[[0,74,0.0,0.22313,0.26232,0.0,0.14286,0.28571,0.0,1.0,9,2,0,9,0,11,0,0,8,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[4,74,0.0541,0.16964,0.24338,0.0,0.14286,0.17857,0.0,1.0,13,2,0,13,0,11,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[8,74,0.1081,0.15179,0.18536,0.0,0.14286,0.1786,0.0,1.0,11,1,0,11,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,74,0.1622,0.24554,0.2237,0.14286,0.28571,0.28571,0.0,1.0,5,2,0,5,0,10,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[16,74,0.2162,0.25,0.25754,0.14286,0.14286,0.28571,0.0,1.0,4,3,0,4,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[20,74,0.2703,0.21875,0.23141,0.14286,0.14286,0.2857,0.0,1.0,7,2,0,7,0,12,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[24,74,0.3243,0.14732,0.09771,0.14286,0.14286,0.1786,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,74,0.3784,0.20536,0.23673,0.0,0.14286,0.28571,0.0,1.0,10,2,0,10,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,74,0.4324,0.16964,0.10374,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,17,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,74,0.4865,0.16072,0.17768,0.10714,0.14286,0.14287,0.0,1.0,8,1,0,8,0,17,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,74,0.5405,0.16071,0.12242,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,74,0.5946,0.17411,0.17762,0.14286,0.14286,0.2857,0.0,1.0,7,1,0,7,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,74,0.6486,0.14732,0.16935,0.14286,0.14286,0.14286,0.0,1.0,7,1,0,7,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,74,0.7027,0.14732,0.09771,0.14286,0.14286,0.17857,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,74,0.7568,0.14732,0.10248,0.10714,0.14286,0.23214,0.0,0.28571,8,0,0,8,0,14,0,2,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,74,0.8108,0.2008,0.09359,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,16,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,74,0.8649,0.22312,0.23131,0.14214,0.14286,0.28571,0.0,1.0,7,2,0,7,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[68,74,0.9189,0.16518,0.1017,0.14286,0.14286,0.28571,0.0,0.28571,6,0,0,6,0,15,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,74,0.973,0.14286,0.20203,0.0,0.14286,0.1786,0.0,1.0,15,1,0,15,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[74,74,1.0,0.1517,0.10063,0.14214,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.11161,"x":0.20312,"p":[[0,76,0.0,0.14062,0.10786,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,1,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,76,0.0526,0.18304,0.18293,0.14286,0.14286,0.28571,0.0,1.0,7,1,0,7,0,15,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,76,0.1053,0.18071,0.18558,0.105,0.14286,0.2857,0.0,1.0,8,1,0,8,0,13,0,1,8,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,76,0.1579,0.20312,0.26068,0.05357,0.14286,0.14286,0.0,1.0,8,2,0,8,1,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[16,76,0.2105,0.16072,0.18123,0.0,0.14286,0.1786,0.0,1.0,9,1,0,9,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,76,0.2632,0.1875,0.23538,0.0,0.14286,0.1786,0.0,1.0,9,2,0,9,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[24,76,0.3158,0.11161,0.16263,0.0,0.14286,0.14286,0.0,0.85714,15,0,0,15,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[28,76,0.3684,0.16518,0.18595,0.0,0.14286,0.2857,0.0,1.0,10,1,0,10,0,12,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,76,0.4211,0.14286,0.12372,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,11,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,76,0.4737,0.13393,0.12339,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,11,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,76,0.5263,0.1875,0.23538,0.0,0.14286,0.2857,0.0,1.0,10,2,0,10,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,76,0.5789,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.16071,0.1171,0.14286,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,16,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,76,0.6842,0.14947,0.11623,0.105,0.14286,0.16075,0.0,0.42857,8,0,0,8,0,16,0,1,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.12946,0.10326,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.15625,0.19019,0.0,0.14286,0.1786,0.0,1.0,11,1,0,11,0,13,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[64,76,0.8421,0.1875,0.16917,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[68,76,0.8947,0.19187,0.18425,0.14286,0.14286,0.2857,0.0,1.0,6,1,0,6,0,16,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[72,76,0.9474,0.14286,0.10101,0.10714,0.14286,0.17857,0.0,0.28571,8,0,0,8,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,76,1.0,0.17857,0.07986,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,20,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5218e315804ecc3f","q":"Show that no rectangle of the form $1 \\times k$ or $2 \\times n$, where $4 \\nmid n$, is $(1,2)$-tileable.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.16062,"x":0.8705,"p":[[0,39,0.0,0.16062,0.23077,0.0,0.14286,0.14286,0.0,1.0,13,1,3,13,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[4,39,0.1026,0.50444,0.38129,0.14286,0.42836,0.89286,0.0,1.0,1,8,0,1,0,14,0,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,8],[8,39,0.2051,0.56248,0.3387,0.14286,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,10,0,0,1,0,0,2,0,0,5,0,0,4,0,0,2,0,8],[12,39,0.3077,0.74111,0.29755,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,3,0,0,9,0,11],[16,39,0.4103,0.73213,0.28959,0.67857,0.85714,1.0,0.0,1.0,1,9,0,1,0,3,0,0,0,0,0,3,0,0,1,0,0,5,0,0,10,0,9],[20,39,0.5128,0.6875,0.30605,0.53572,0.71429,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,0,0,0,2,0,0,2,0,0,7,0,0,6,0,9],[24,39,0.6154,0.79463,0.23675,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,5,0,0,3,0,0,9,0,12],[28,39,0.7179,0.79017,0.23415,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,1,0,0,4,0,0,6,0,0,7,0,12],[32,39,0.8205,0.80802,0.23586,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,9,0,0,3,0,15],[36,39,0.9231,0.8705,0.14451,0.85711,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,13,0,13],[39,39,1.0,0.86603,0.13339,0.857,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,13,0,12]]},{"b":1,"e":0.71429,"k":"volatile","v":0.21429,"x":0.9375,"p":[[0,11,0.0,0.21429,0.29667,0.0,0.14286,0.14286,0.0,1.0,11,3,2,11,0,14,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[4,11,0.3636,0.75446,0.36984,0.35714,1.0,1.0,0.14286,1.0,0,21,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,21],[8,11,0.7273,0.91518,0.22548,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,26],[11,11,1.0,0.9375,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24]]}]},{"i":"a9d45c9bbc7cea90","q":"Let $k$ be a natural number. Prove that for positive real numbers $x, y, z$ whose sum is equal to 1, the inequality\n\n$$\n\\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}} \\geqslant \\frac{1}{7}\n$$\n\nholds. When does equality hold?\n\nTime for work 270 minutes.\n\nEach task is worth 7 points.\n\n## 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altitudes  $AA_1, BB_1, CC_1$  of an acute triangle  $ABC$  concur at  $H$ . The perpendicular lines from  $H$  to  $B_1C_1, A_1C_1$  meet rays  $CA, CB$  at  $P, Q$  respectively. Prove that the line from  $C$  perpendicular to  $A_1B_1$  passes through the midpoint of  $PQ$ 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 $\\mathbb{Q^+}$  denote the set of positive rational numbers. Determine all functions  $f: \\mathbb{Q^+} \\to \\mathbb{Q^+}$  that satisfy the conditions\n\n\\[ f \\left( \\frac{x}{x+1}\\right) = \\frac{f(x)}{x+1}  \\qquad \\text{and} \\qquad f \\left(\\frac{1}{x}\\right)=\\frac{f(x)}{x^3}\\]\n\nfor all  $x \\in \\mathbb{Q^+}.$","t":[{"b":1,"e":1.0,"k":"flat","v":0.85266,"x":1.0,"p":[[0,87,0.0,0.85266,0.22726,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,4,0,0,2,0,20],[4,87,0.046,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,87,0.092,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[12,87,0.1379,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,87,0.1839,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,87,0.2299,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,87,0.2759,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,87,0.3218,0.97326,0.10351,1.0,1.0,1.0,0.43,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[32,87,0.3678,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,87,0.4138,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,87,0.4598,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[44,87,0.5057,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,87,0.5517,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,87,0.5977,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,87,0.6437,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[60,87,0.6897,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,87,0.7356,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,87,0.7816,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,87,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,87,0.8736,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,87,0.9195,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,87,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[87,87,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":2,"e":1.0,"k":"flat","v":0.77232,"x":0.99554,"p":[[0,184,0.0,0.77232,0.31106,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,4,0,0,1,0,0,2,0,0,2,0,0,3,0,0,2,0,18],[4,184,0.0217,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,184,0.0435,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[12,184,0.0652,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[16,184,0.087,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,184,0.1087,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,184,0.1304,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,184,0.1522,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[32,184,0.1739,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,184,0.1957,0.99116,0.03424,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,184,0.2174,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,184,0.2391,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[48,184,0.2609,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,184,0.2826,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[56,184,0.3043,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[60,184,0.3261,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[64,184,0.3478,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[68,184,0.3696,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,184,0.3913,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[76,184,0.413,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[80,184,0.4348,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[84,184,0.4565,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[88,184,0.4783,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[92,184,0.5,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[96,184,0.5217,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[100,184,0.5435,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[104,184,0.5652,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[108,184,0.587,0.97321,0.10374,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[112,184,0.6087,0.94196,0.11769,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,24],[116,184,0.6304,0.9375,0.11259,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,21],[120,184,0.6522,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[124,184,0.6739,0.96429,0.12877,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[128,184,0.6957,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[132,184,0.7174,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[136,184,0.7391,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[140,184,0.7609,0.92411,0.14278,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,9,0,21],[144,184,0.7826,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[148,184,0.8043,0.93303,0.14279,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,23],[152,184,0.8261,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[156,184,0.8478,0.94642,0.09279,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[160,184,0.8696,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[164,184,0.8913,0.9241,0.07974,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[168,184,0.913,0.88392,0.10971,0.85714,0.85714,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,20,0,10],[172,184,0.9348,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[176,184,0.9565,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[180,184,0.9783,0.9241,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[184,184,1.0,0.90625,0.12682,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,14,0,16]]}]},{"i":"f18a954bea6f3691","q":"(a) Find all strictly monotone functions  $f:\\mathbb{R}\\rightarrow\\mathbb{R}$  such that\n\\[f(x+f(y))=f(x)+y\\quad\\text{for all real}\\ x,y. \\]\n(b) If  $n>1$  is an integer, prove that there is no strictly monotone function  $f:\\mathbb{R}\\rightarrow\\mathbb{R}$  such that\n\\[  f(x+f(y))=f(x)+y^n\\quad \\text{for all real}\\ x, y.\\]","t":[{"b":6,"e":0.42857,"k":"falling","v":0.45541,"x":0.87945,"p":[[0,19,0.0,0.86161,0.19393,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,1,0,20],[4,19,0.2105,0.77218,0.23924,0.57143,0.85707,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,2,0,0,2,0,15],[8,19,0.4211,0.77676,0.23944,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,8,0,0,4,0,0,3,0,14],[12,19,0.6316,0.87945,0.18937,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,3,0,21],[16,19,0.8421,0.45541,0.07522,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0],[19,19,1.0,0.46875,0.11425,0.42857,0.42857,0.42858,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,26,0,0,1,0,0,3,0,0,1,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,13,0.0,0.82143,0.22303,0.67857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,5,0,0,1,0,18],[4,13,0.3077,0.94196,0.13767,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,26],[8,13,0.6154,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,13,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[13,13,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]}]},{"i":"66a93dddaa944931","q":"Let  $l$  some line, that is not parallel to the coordinate axes. Find minimal  $d$  that always exists point  $A$  with integer coordinates, and distance from  $A$  to  $l$  is   $\\leq d$","t":[{"b":2,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,38,0.0,0.93304,0.18893,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[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],[12,38,0.3158,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,38,0.4211,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[24,38,0.6316,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,38,0.7368,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":7,"e":0.85714,"k":"flat","v":0.94643,"x":0.99554,"p":[[0,61,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,61,0.0656,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,61,0.1311,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[12,61,0.1967,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,61,0.2623,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,61,0.3279,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,61,0.3934,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,61,0.459,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[32,61,0.5246,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,61,0.5902,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[40,61,0.6557,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,61,0.7213,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[48,61,0.7869,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[52,61,0.8525,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[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],[60,61,0.9836,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[61,61,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"41284cb55f852a8c","q":"Let  $N$  be the greatest four-digit integer with the property that whenever one of its digits is changed to  $1$ , the resulting number is divisible by  $7$ . Let  $Q$  and  $R$  be the quotient and remainder, respectively, when  $N$  is divided by  $1000$ . Find  $Q+R$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.89285,"x":0.94643,"p":[[0,35,0.0,0.92857,0.07143,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[4,35,0.1143,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[8,35,0.2286,0.91518,0.07873,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[12,35,0.3429,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[16,35,0.4571,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[20,35,0.5714,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[24,35,0.6857,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[28,35,0.8,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[32,35,0.9143,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[35,35,1.0,0.90625,0.06785,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11]]},{"b":2,"e":0.85714,"k":"flat","v":0.88392,"x":0.92857,"p":[[0,82,0.0,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[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],[8,82,0.0976,0.89732,0.08917,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,12],[12,82,0.1463,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[16,82,0.1951,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[20,82,0.2439,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[24,82,0.2927,0.89732,0.08171,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,11],[28,82,0.3415,0.9107,0.08565,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[32,82,0.3902,0.88392,0.05576,0.85714,0.85714,0.85714,0.857,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[36,82,0.439,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[40,82,0.4878,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[44,82,0.5366,0.91071,0.06917,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[48,82,0.5854,0.90179,0.06622,0.85714,0.85714,1.0,0.85714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[52,82,0.6341,0.91071,0.06916,0.85714,0.85714,1.0,0.85714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[56,82,0.6829,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[60,82,0.7317,0.91964,0.07087,0.85714,0.85714,1.0,0.857,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[64,82,0.7805,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[68,82,0.8293,0.90625,0.06785,0.85714,0.85714,1.0,0.85714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[72,82,0.878,0.90178,0.06622,0.85714,0.85714,1.0,0.857,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,10],[76,82,0.9268,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[80,82,0.9756,0.88838,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7],[82,82,1.0,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8]]}]},{"i":"f9285b55308bfab5","q":"The sequence  $\\{a_{n}\\}_{n \\ge 1}$  is defined by  $a_{1}=1$  and \\[a_{n+1}= \\frac{a_{n}}{2}+\\frac{1}{4a_{n}}\\; (n \\in \\mathbb{N}).\\] Prove that  $\\sqrt{\\frac{2}{2a_{n}^{2}-1}}$  is a positive integer for  $n>1$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.24998,"x":0.43302,"p":[[0,52,0.0,0.36606,0.39437,0.0,0.14286,0.67857,0.0,1.0,9,8,0,9,0,9,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,8],[4,52,0.0769,0.34375,0.30275,0.14286,0.2143,0.4286,0.0,1.0,2,4,0,2,0,14,0,0,6,0,0,3,0,0,2,0,0,0,0,0,1,0,4],[8,52,0.1538,0.33482,0.36528,0.10714,0.14286,0.46429,0.0,1.0,8,6,0,8,0,10,0,0,5,0,0,1,0,0,1,0,0,0,0,0,1,0,6],[12,52,0.2308,0.33032,0.31221,0.14286,0.14286,0.57111,0.0,1.0,7,1,0,7,0,11,0,0,2,0,0,1,0,0,4,0,0,3,0,0,3,0,1],[16,52,0.3077,0.30357,0.3067,0.14286,0.14286,0.42858,0.0,1.0,7,2,0,7,0,11,0,0,4,0,0,3,0,0,2,0,0,0,0,0,3,0,2],[20,52,0.3846,0.25893,0.24073,0.10714,0.14288,0.42857,0.0,0.85714,8,0,0,8,0,9,0,0,5,0,0,6,0,0,1,0,0,1,0,0,2,0,0],[24,52,0.4615,0.34373,0.30692,0.14286,0.2857,0.57111,0.0,1.0,6,2,0,6,0,8,0,0,7,0,0,2,0,0,3,0,0,1,0,0,3,0,2],[28,52,0.5385,0.28125,0.27078,0.14286,0.14286,0.46429,0.0,0.85714,7,0,0,7,0,12,0,0,3,0,0,2,0,0,4,0,0,1,0,0,3,0,0],[32,52,0.6154,0.32588,0.2654,0.14286,0.2143,0.57111,0.0,0.85714,3,0,0,3,0,13,0,0,6,0,0,1,0,0,3,0,0,3,0,0,3,0,0],[36,52,0.6923,0.30355,0.29176,0.14286,0.14286,0.57111,0.0,0.85714,7,0,0,7,0,11,0,0,4,0,0,1,0,0,3,0,0,2,0,0,4,0,0],[40,52,0.7692,0.26338,0.28816,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,13,0,0,4,0,0,0,0,0,1,0,0,2,0,0,4,0,0],[44,52,0.8462,0.24998,0.26723,0.10714,0.14286,0.32143,0.0,0.85714,8,0,0,8,0,14,0,0,2,0,0,1,0,0,2,0,0,3,0,0,2,0,0],[48,52,0.9231,0.31238,0.27768,0.14286,0.14286,0.57111,0.0,0.85714,6,0,0,6,0,11,0,0,4,0,0,1,0,0,5,0,0,2,0,0,3,0,0],[52,52,1.0,0.43302,0.33019,0.14286,0.42859,0.71429,0.0,1.0,4,3,0,4,0,9,0,0,2,0,0,4,0,0,2,0,0,5,0,0,3,0,3]]},{"b":7,"e":0.14286,"k":"flat","v":0.09822,"x":0.23639,"p":[[0,5,0.0,0.23639,0.19761,0.14214,0.14286,0.32143,0.0,0.71429,6,0,0,6,0,13,0,0,5,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[4,5,0.8,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.10714,0.11845,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"610c9e127ebdffdf","q":"Let  $M>1$  be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function  $f: \\mathbb{N} \\rightarrow \\mathbb{N}$  satisfying \n\n    \n- $f(M) \\ne M$  [/*]\n- $f(k)<2k$  for all  $k \\in \\mathbb{N}$ [/*]\n- $f^{f(n)}(n)=n$  for all  $n \\in \\mathbb{N}$ . For each  $\\ell>0$  we define  $f^{\\ell}(n)=f\\left(f^{\\ell-1}(n)\\right)$  and  $f^0(n)=n$ [/*]\n\n    Tom wins otherwise. Prove that for infinitely many  $M$ , Tom wins, and for infinitely many  $M$ , Jerry wins. \n\n*Proposed by Anant Mudgal*","t":[{"b":0,"e":1.0,"k":"flat","v":0.83035,"x":0.94196,"p":[[0,45,0.0,0.86161,0.16164,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,5,0,16],[4,45,0.0889,0.83035,0.16536,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,10,0,0,7,0,12],[8,45,0.1778,0.84821,0.13803,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,5,0,13],[12,45,0.2667,0.83928,0.17035,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,6,0,14],[16,45,0.3556,0.86592,0.15962,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,4,0,17],[20,45,0.4444,0.89732,0.15251,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,4,0,20],[24,45,0.5333,0.875,0.14617,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,7,0,16],[28,45,0.6222,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[32,45,0.7111,0.875,0.14174,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,16],[36,45,0.8,0.9107,0.14177,0.82143,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,2,0,22],[40,45,0.8889,0.89285,0.12877,0.82132,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,7,0,17],[44,45,0.9778,0.91964,0.13803,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,21],[45,45,1.0,0.875,0.15465,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,6,0,17]]},{"b":4,"e":1.0,"k":"flat","v":0.80356,"x":0.87931,"p":[[0,22,0.0,0.87931,0.13435,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,6,0,16],[4,22,0.1818,0.8571,0.16758,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,8,0,15],[8,22,0.3636,0.80356,0.17769,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,9,0,10],[12,22,0.5455,0.87499,0.13717,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,5,0,16],[16,22,0.7273,0.83035,0.1729,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,7,0,13],[20,22,0.9091,0.83928,0.15872,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,12,0,11],[22,22,1.0,0.83467,0.16028,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,6,0,13]]}]},{"i":"5266705dc1f9b5c4","q":"Let $a, b$ be two co-prime positive integers. A number is called good if it can be written in the form $a x+b y$ for non-negative integers $x, y$. Define the function $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ as $f(n)=n-n_{a}-n_{b}$, where $s_{t}$ represents the remainder of $s$ upon division by $t$. Show that an integer $n$ is good if and only if the infinite sequence $n, f(n), f(f(n)), \\ldots$ contains only non-negative integers.","t":[{"b":1,"e":1.0,"k":"flat","v":0.58925,"x":0.74539,"p":[[0,29,0.0,0.68287,0.25186,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,6,0,0,4,0,0,5,0,0,5,0,8],[4,29,0.1379,0.70534,0.2788,0.42857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,6,0,0,5,0,0,1,0,0,3,0,0,7,0,10],[8,29,0.2759,0.74539,0.26424,0.63964,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,4,0,0,3,0,0,0,0,0,4,0,0,11,0,9],[12,29,0.4138,0.71424,0.22305,0.5354,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,7,0,0,5,0,8],[16,29,0.5517,0.58925,0.29179,0.28571,0.571,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,9,0,0,5,0,0,3,0,0,3,0,0,5,0,6],[20,29,0.6897,0.64729,0.24481,0.42857,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,9,0,0,6,0,0,3,0,0,4,0,7],[24,29,0.8276,0.66963,0.25614,0.4286,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,3,0,0,7,0,0,3,0,0,4,0,0,8,0,6],[28,29,0.9655,0.73657,0.22901,0.571,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,8,0,0,2,0,11],[29,29,1.0,0.68747,0.27303,0.42857,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,1,0,0,9,0,0,4,0,0,3,0,0,4,0,10]]},{"b":3,"e":0.85714,"k":"flat","v":0.66958,"x":0.93302,"p":[[0,70,0.0,0.80356,0.22233,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,2,0,0,7,0,14],[4,70,0.0571,0.71873,0.2435,0.5354,0.857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,2,0,0,9,0,8],[8,70,0.1143,0.71423,0.27896,0.42857,0.857,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,6,0,0,4,0,0,2,0,0,3,0,0,6,0,11],[12,70,0.1714,0.71871,0.25378,0.4286,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,4,0,0,2,0,12],[16,70,0.2286,0.70979,0.2868,0.42857,0.857,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,6,0,0,5,0,0,2,0,0,2,0,0,5,0,12],[20,70,0.2857,0.66958,0.25616,0.5354,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,5,0,0,2,0,0,6,0,0,5,0,0,7,0,6],[24,70,0.3429,0.82588,0.21048,0.85711,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,15,0,11],[28,70,0.4,0.74995,0.24746,0.571,0.85707,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,3,0,0,7,0,11],[32,70,0.4571,0.86605,0.11812,0.857,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,16,0,10],[36,70,0.5143,0.74553,0.23347,0.57143,0.85714,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,2,0,0,13,0,7],[40,70,0.5714,0.93302,0.1129,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[44,70,0.6286,0.86607,0.14698,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,16,0,11],[48,70,0.6857,0.79907,0.14668,0.71429,0.85707,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,0,13,0,6],[52,70,0.7429,0.87053,0.12037,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,17,0,10],[56,70,0.8,0.87053,0.12555,0.85714,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,15,0,11],[60,70,0.8571,0.86161,0.20356,0.85714,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,12,0,15],[64,70,0.9143,0.87944,0.12934,0.85714,0.85714,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,16,0,12],[68,70,0.9714,0.91516,0.11218,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,10,0,18],[70,70,1.0,0.8348,0.14775,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,14,0,9]]}]},{"i":"6a2f820064f6f451","q":"USA   Suppose that $s_{1}, s_{2}, s_{3}, \\ldots$ is a strictly increasing sequence of positive integers such that the subsequences  $$ s_{s_{1}}, s_{s_{2}}, s_{s_{3}}, \\ldots \\quad \\text { and } \\quad s_{s_{1}+1}, s_{s_{2}+1}, s_{s_{3}+1}, \\ldots $$  are both arithmetic progressions. Prove that $s_{1}, s_{2}, s_{3}, \\ldots$ is itself an arithmetic progression.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.34371,"x":0.46428,"p":[[0,146,0.0,0.34371,0.1707,0.24999,0.42857,0.42857,0.0,0.57143,3,0,2,3,0,5,0,0,5,0,0,14,0,0,5,0,0,0,0,0,0,0,0],[4,146,0.0274,0.42409,0.13591,0.39286,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,3,0,0,5,0,0,15,0,0,8,0,0,1,0,0,0,0,0],[8,146,0.0548,0.43302,0.16553,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,11,0,0,8,0,0,10,0,0,1,0,0,1,0,0],[12,146,0.0822,0.46428,0.09448,0.42857,0.42859,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,16,0,0,12,0,0,0,0,0,0,0,0],[16,146,0.1096,0.37053,0.10012,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,11,0,0,17,0,0,2,0,0,0,0,0,0,0,0],[20,146,0.137,0.44207,0.11474,0.42857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,6,0,0,14,0,0,11,0,0,0,0,0,0,0,0],[24,146,0.1644,0.42409,0.13113,0.28571,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,2,0,0,7,0,0,14,0,0,8,0,0,1,0,0,0,0,0],[28,146,0.1918,0.42855,0.11842,0.42857,0.42857,0.571,0.14286,0.57143,0,0,0,0,0,2,0,0,5,0,0,16,0,0,9,0,0,0,0,0,0,0,0],[32,146,0.2192,0.39286,0.14286,0.28571,0.42857,0.42858,0.0,0.57143,2,0,0,2,0,0,0,0,9,0,0,14,0,0,7,0,0,0,0,0,0,0,0],[36,146,0.2466,0.42856,0.08746,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,20,0,0,6,0,0,0,0,0,0,0,0],[40,146,0.274,0.43746,0.10057,0.42857,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,16,0,0,9,0,0,0,0,0,0,0,0],[44,146,0.3014,0.42409,0.13591,0.28571,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,9,0,0,11,0,0,11,0,0,0,0,0,0,0,0],[48,146,0.3288,0.39731,0.13234,0.28571,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,4,0,0,6,0,0,15,0,0,7,0,0,0,0,0,0,0,0],[52,146,0.3562,0.39735,0.12744,0.28571,0.42857,0.42895,0.14286,0.57143,0,0,0,0,0,3,0,0,8,0,0,14,0,0,7,0,0,0,0,0,0,0,0],[56,146,0.3836,0.40626,0.11355,0.28571,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,2,0,0,7,0,0,17,0,0,6,0,0,0,0,0,0,0,0],[60,146,0.411,0.40625,0.08828,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,19,0,0,4,0,0,0,0,0,0,0,0],[64,146,0.4384,0.41964,0.10677,0.28571,0.42857,0.46431,0.28571,0.57143,0,0,0,0,0,0,0,0,10,0,0,14,0,0,8,0,0,0,0,0,0,0,0],[68,146,0.4658,0.39732,0.12745,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,0,0,0,11,0,0,13,0,0,7,0,0,0,0,0,0,0,0],[72,146,0.4932,0.42411,0.0977,0.42857,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,1,0,0,5,0,0,20,0,0,6,0,0,0,0,0,0,0,0],[76,146,0.5205,0.45982,0.09933,0.42857,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,15,0,0,12,0,0,0,0,0,0,0,0],[80,146,0.5479,0.39732,0.12745,0.39286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,5,0,0,19,0,0,5,0,0,0,0,0,0,0,0],[84,146,0.5753,0.41963,0.12338,0.28571,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,2,0,0,7,0,0,14,0,0,9,0,0,0,0,0,0,0,0],[88,146,0.6027,0.40625,0.12428,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,0,0,0,9,0,0,15,0,0,7,0,0,0,0,0,0,0,0],[92,146,0.6301,0.44197,0.09006,0.42857,0.42857,0.46431,0.28571,0.57143,0,0,0,0,0,0,0,0,5,0,0,19,0,0,8,0,0,0,0,0,0,0,0],[96,146,0.6575,0.43741,0.10702,0.42857,0.42857,0.57143,0.14,0.57143,0,0,0,0,0,1,0,0,5,0,0,17,0,0,9,0,0,0,0,0,0,0,0],[100,146,0.6849,0.41071,0.09941,0.28571,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,16,0,0,6,0,0,0,0,0,0,0,0],[104,146,0.7123,0.42409,0.10401,0.28571,0.42857,0.4642,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,15,0,0,8,0,0,0,0,0,0,0,0],[108,146,0.7397,0.40625,0.12428,0.28571,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,12,0,0,10,0,0,9,0,0,0,0,0,0,0,0],[112,146,0.7671,0.38836,0.12488,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,12,0,0,11,0,0,7,0,0,0,0,0,0,0,0],[116,146,0.7945,0.43304,0.0977,0.42857,0.42857,0.4286,0.1429,0.57143,0,0,0,0,0,1,0,0,4,0,0,20,0,0,7,0,0,0,0,0,0,0,0],[120,146,0.8219,0.40621,0.12423,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,0,0,0,9,0,0,15,0,0,7,0,0,0,0,0,0,0,0],[124,146,0.8493,0.41518,0.11495,0.39286,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,2,0,0,6,0,0,17,0,0,7,0,0,0,0,0,0,0,0],[128,146,0.8767,0.39289,0.14285,0.28571,0.42857,0.46525,0.0,0.57143,1,0,0,1,0,2,0,0,9,0,0,12,0,0,8,0,0,0,0,0,0,0,0],[132,146,0.9041,0.43299,0.14928,0.28593,0.4286,0.57143,0.14286,0.57143,0,0,0,0,0,4,0,0,5,0,0,9,0,0,14,0,0,0,0,0,0,0,0],[136,146,0.9315,0.42411,0.10403,0.28571,0.42857,0.46431,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,15,0,0,8,0,0,0,0,0,0,0,0],[140,146,0.9589,0.43307,0.09092,0.42857,0.42857,0.42893,0.28571,0.57143,0,0,0,0,0,0,0,0,6,0,0,19,0,0,7,0,0,0,0,0,0,0,0],[144,146,0.9863,0.45086,0.09518,0.42857,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,17,0,0,10,0,0,0,0,0,0,0,0],[146,146,1.0,0.43753,0.1181,0.39286,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,7,0,0,13,0,0,11,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.27679,"x":0.44191,"p":[[0,70,0.0,0.32141,0.13361,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,14,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[4,70,0.0571,0.39287,0.12369,0.28571,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,3,0,0,8,0,0,15,0,0,6,0,0,0,0,0,0,0,0],[8,70,0.1143,0.41071,0.12753,0.42857,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,5,0,0,20,0,0,4,0,0,1,0,0,0,0,0],[12,70,0.1714,0.4107,0.17033,0.28571,0.42857,0.4642,0.0,1.0,1,1,0,1,0,1,0,0,10,0,0,12,0,0,7,0,0,0,0,0,0,0,1],[16,70,0.2286,0.35714,0.17128,0.28571,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,4,0,0,5,0,0,14,0,0,6,0,0,0,0,0,0,0,0],[20,70,0.2857,0.41515,0.12552,0.28571,0.42857,0.571,0.14286,0.57143,0,0,0,0,0,2,0,0,8,0,0,13,0,0,9,0,0,0,0,0,0,0,0],[24,70,0.3429,0.39728,0.12739,0.28571,0.42857,0.4642,0.14286,0.57143,0,0,0,0,0,2,0,0,11,0,0,11,0,0,8,0,0,0,0,0,0,0,0],[28,70,0.4,0.41515,0.14442,0.28571,0.42857,0.57111,0.0,0.57143,1,0,0,1,0,2,0,0,6,0,0,13,0,0,10,0,0,0,0,0,0,0,0],[32,70,0.4571,0.3749,0.15883,0.2857,0.42857,0.4642,0.0,0.57143,1,0,0,1,0,5,0,0,7,0,0,11,0,0,8,0,0,0,0,0,0,0,0],[36,70,0.5143,0.44191,0.1255,0.42857,0.42857,0.57111,0.14286,0.57143,0,0,0,0,0,2,0,0,5,0,0,13,0,0,12,0,0,0,0,0,0,0,0],[40,70,0.5714,0.36598,0.15958,0.28571,0.42857,0.4286,0.0,0.57143,2,0,0,2,0,5,0,0,3,0,0,17,0,0,5,0,0,0,0,0,0,0,0],[44,70,0.6286,0.35719,0.12879,0.28571,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,5,0,0,10,0,0,13,0,0,4,0,0,0,0,0,0,0,0],[48,70,0.6857,0.32589,0.11971,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,10,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[52,70,0.7429,0.38393,0.14914,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,3,0,0,15,0,0,7,0,0,0,0,0,0,0,0],[56,70,0.8,0.39727,0.13702,0.28571,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,5,0,0,4,0,0,16,0,0,7,0,0,0,0,0,0,0,0],[60,70,0.8571,0.30804,0.17169,0.14286,0.28571,0.4286,0.0,0.57143,2,0,0,2,0,9,0,0,9,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[64,70,0.9143,0.38837,0.13471,0.28571,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,4,0,0,8,0,0,13,0,0,7,0,0,0,0,0,0,0,0],[68,70,0.9714,0.36606,0.14256,0.28571,0.35714,0.42858,0.14286,0.57143,0,0,0,0,0,5,0,0,11,0,0,9,0,0,7,0,0,0,0,0,0,0,0],[70,70,1.0,0.27679,0.12846,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,12,0,0,8,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"e337e09904917e65","q":"Let  $P$  be a point inside  $\\triangle ABC$ . Let the perpendicular bisectors of  $PA,PB,PC$  be  $\\ell_1,\\ell_2,\\ell_3$ . Let  $D =\\ell_1 \\cap \\ell_2$  ,  $E=\\ell_2 \\cap \\ell_3$ ,  $F=\\ell_3 \\cap \\ell_1$ . If  $A,B,C,D,E,F$  lie on a circle, prove that  $C, P,D$  are 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(BLR) In an acute-angled triangle $A B C$, let $A D, B E$ be altitudes and $A P, B Q$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcenter of the triangle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are 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are  $300$  participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of  $n$  for which it is possible to satisfy the following conditions at the same time: each contestant plays at most  $n$  games of chess, and for each  $m$  with  $1 \\le m \\le n$ , there is a contestant playing exactly  $m$  games of chess.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.4241,"x":0.82588,"p":[[0,92,0.0,0.65177,0.29653,0.57132,0.71429,0.85714,0.0,1.0,4,2,3,4,0,0,0,0,2,0,0,1,0,0,2,0,0,9,0,0,12,0,2],[4,92,0.0435,0.54016,0.36549,0.2857,0.57121,0.85714,0.0,1.0,7,5,0,7,0,0,0,0,4,0,0,4,0,0,2,0,0,2,0,0,8,0,5],[8,92,0.087,0.60267,0.39243,0.28571,0.71429,1.0,0.0,1.0,6,12,0,6,0,1,0,0,3,0,0,4,0,0,1,0,0,2,0,0,3,0,12],[12,92,0.1304,0.65187,0.31332,0.42857,0.71429,0.895,0.0,1.0,3,8,0,3,0,0,0,0,1,0,0,9,0,0,1,0,0,3,0,0,7,0,8],[16,92,0.1739,0.75445,0.28846,0.57132,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,0,8,0,12],[20,92,0.2174,0.70535,0.31529,0.42857,0.85714,1.0,0.0,1.0,3,9,0,3,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,10,0,9],[24,92,0.2609,0.64731,0.3214,0.42857,0.71429,1.0,0.0,1.0,3,9,0,3,0,0,0,0,2,0,0,8,0,0,2,0,0,2,0,0,6,0,9],[28,92,0.3043,0.56696,0.38213,0.2857,0.64286,1.0,0.0,1.0,7,9,0,7,0,0,0,0,3,0,0,5,0,0,1,0,0,3,0,0,4,0,9],[32,92,0.3478,0.53572,0.35892,0.28571,0.50001,0.85714,0.0,1.0,6,7,0,6,0,1,0,0,3,0,0,6,0,0,3,0,0,2,0,0,4,0,7],[36,92,0.3913,0.62491,0.37596,0.39286,0.71429,1.0,0.0,1.0,5,12,0,5,0,1,0,0,2,0,0,6,0,0,0,0,0,3,0,0,3,0,12],[40,92,0.4348,0.50893,0.38289,0.10714,0.42859,0.85714,0.0,1.0,8,7,0,8,0,1,0,0,3,0,0,5,0,0,1,0,0,3,0,0,4,0,7],[44,92,0.4783,0.55804,0.33949,0.28571,0.42859,0.85714,0.0,1.0,4,7,0,4,0,0,0,0,6,0,0,8,0,0,0,0,0,2,0,0,5,0,7],[48,92,0.5217,0.64286,0.3481,0.42857,0.71429,1.0,0.0,1.0,4,10,0,4,0,1,0,0,2,0,0,4,0,0,3,0,0,3,0,0,5,0,10],[52,92,0.5652,0.48661,0.37434,0.14286,0.42857,0.89286,0.0,1.0,6,8,0,6,0,3,0,0,6,0,0,3,0,0,3,0,0,1,0,0,2,0,8],[56,92,0.6087,0.47767,0.38234,0.0,0.42859,0.85714,0.0,1.0,9,5,0,9,0,1,0,0,3,0,0,5,0,0,2,0,0,0,0,0,7,0,5],[60,92,0.6522,0.4241,0.33213,0.10714,0.42857,0.60714,0.0,1.0,8,4,0,8,0,1,0,0,4,0,0,8,0,0,3,0,0,2,0,0,2,0,4],[64,92,0.6957,0.69196,0.30327,0.53572,0.857,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,1,0,0,4,0,0,4,0,0,3,0,0,9,0,8],[68,92,0.7391,0.74551,0.20121,0.71429,0.85714,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,19,0,1],[72,92,0.7826,0.79017,0.23415,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,14,0,9],[76,92,0.8261,0.77679,0.20183,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,7,0,8],[80,92,0.8696,0.71427,0.26964,0.57132,0.78571,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,1,0,0,4,0,0,2,0,0,7,0,0,9,0,7],[84,92,0.913,0.77678,0.21409,0.71429,0.85714,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,4,0,0,16,0,6],[88,92,0.9565,0.82588,0.14167,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,15,0,7],[92,92,1.0,0.79017,0.14281,0.71429,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,9,0,0,20,0,1]]},{"b":4,"e":0.57143,"k":"falling","v":0.35268,"x":0.92411,"p":[[0,118,0.0,0.66518,0.30849,0.57143,0.71429,0.85714,0.0,1.0,4,7,1,4,0,0,0,0,1,0,0,1,0,0,6,0,0,7,0,0,6,0,7],[4,118,0.0339,0.92411,0.18552,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,26],[8,118,0.0678,0.89732,0.23483,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,26],[12,118,0.1017,0.74554,0.30458,0.4286,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,2,0,0,6,0,0,1,0,0,0,0,0,7,0,14],[16,118,0.1356,0.85714,0.21724,0.82143,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,17],[20,118,0.1695,0.79465,0.31122,0.67857,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,0,3,0,19],[24,118,0.2034,0.79018,0.29011,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,1,0,0,4,0,0,2,0,0,1,0,0,5,0,17],[28,118,0.2373,0.75893,0.28445,0.53571,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,1,0,0,5,0,0,1,0,0,3,0,0,7,0,13],[32,118,0.2712,0.69197,0.35912,0.42857,0.85714,1.0,0.0,1.0,4,14,0,4,0,0,0,0,3,0,0,3,0,0,3,0,0,0,0,0,5,0,14],[36,118,0.3051,0.75,0.29666,0.42859,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,0,0,0,7,0,0,1,0,0,3,0,0,5,0,14],[40,118,0.339,0.6875,0.34337,0.28571,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,6,0,0,2,0,0,1,0,0,2,0,0,5,0,13],[44,118,0.3729,0.65179,0.35881,0.42857,0.85714,1.0,0.0,1.0,5,11,0,5,0,0,0,0,0,0,0,8,0,0,1,0,0,1,0,0,6,0,11],[48,118,0.4068,0.54464,0.35613,0.28571,0.42857,1.0,0.0,1.0,5,9,0,5,0,1,0,0,3,0,0,9,0,0,2,0,0,1,0,0,2,0,9],[52,118,0.4407,0.6741,0.34669,0.42857,0.78571,1.0,0.0,1.0,4,11,0,4,0,1,0,0,1,0,0,4,0,0,1,0,0,5,0,0,5,0,11],[56,118,0.4746,0.67857,0.32927,0.42857,0.85714,1.0,0.0,1.0,3,11,0,3,0,0,0,0,1,0,0,9,0,0,1,0,0,0,0,0,7,0,11],[60,118,0.5085,0.64286,0.33881,0.42857,0.71429,1.0,0.0,1.0,3,10,0,3,0,1,0,0,3,0,0,6,0,0,1,0,0,3,0,0,5,0,10],[64,118,0.5424,0.63393,0.34798,0.42857,0.64286,1.0,0.0,1.0,4,11,0,4,0,1,0,0,1,0,0,6,0,0,4,0,0,2,0,0,3,0,11],[68,118,0.5763,0.5625,0.38122,0.2857,0.42859,1.0,0.0,1.0,5,10,0,5,0,1,0,0,7,0,0,4,0,0,0,0,0,1,0,0,4,0,10],[72,118,0.6102,0.64277,0.29682,0.42857,0.57143,1.0,0.0,1.0,1,9,0,1,0,2,0,0,1,0,0,9,0,0,4,0,0,2,0,0,4,0,9],[76,118,0.6441,0.54465,0.30813,0.42857,0.42859,0.85714,0.0,1.0,3,7,0,3,0,0,0,0,4,0,0,13,0,0,1,0,0,2,0,0,2,0,7],[80,118,0.678,0.73214,0.30462,0.42857,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,1,0,0,7,0,0,1,0,0,2,0,0,6,0,13],[84,118,0.7119,0.62054,0.29582,0.39286,0.50001,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,7,0,0,8,0,0,2,0,0,0,0,0,6,0,8],[88,118,0.7458,0.62947,0.29636,0.42857,0.50001,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,10,0,0,1,0,0,2,0,0,4,0,9],[92,118,0.7797,0.59375,0.29038,0.42857,0.42857,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,2,0,0,15,0,0,1,0,0,1,0,0,3,0,8],[96,118,0.8136,0.74107,0.25111,0.42857,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,10,0,0,1,0,0,0,0,0,10,0,10],[100,118,0.8475,0.61607,0.28446,0.42857,0.4286,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,4,0,0,12,0,0,1,0,0,0,0,0,8,0,6],[104,118,0.8814,0.53571,0.28122,0.39286,0.42857,0.75,0.0,1.0,2,5,0,2,0,0,0,0,6,0,0,11,0,0,3,0,0,2,0,0,3,0,5],[108,118,0.9153,0.47767,0.25407,0.39286,0.42857,0.60714,0.0,1.0,3,2,0,3,0,0,0,0,5,0,0,14,0,0,2,0,0,3,0,0,3,0,2],[112,118,0.9492,0.35268,0.21124,0.2857,0.42857,0.42858,0.0,1.0,5,1,0,5,0,0,0,0,10,0,0,14,0,0,0,0,0,2,0,0,0,0,1],[116,118,0.9831,0.38839,0.11426,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,10,0,0,17,0,0,4,0,0,0,0,0,0,0,0],[118,118,1.0,0.375,0.13243,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,1,0,0,7,0,0,19,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"cb7dcfc42b0fa9f5","q":"Let  $(a_n)^\\infty_{n=1}$  be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms  $a_n,a_{n+1},a_{n+2},a_{n+3}$  belongs to the same sequence. Prove that the sequence  $\\frac{a_{n+1}}{a_n}$  converges and find all possible values of its limit.","t":[{"b":3,"e":0.85714,"k":"flat","v":0.65624,"x":0.82589,"p":[[0,94,0.0,0.65624,0.27166,0.53571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,1,0,0,3,0,0,7,0,0,11,0,3],[4,94,0.0426,0.82589,0.1665,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,0,10,0,10],[8,94,0.0851,0.79016,0.22301,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,0,8,0,11],[12,94,0.1277,0.77231,0.17808,0.71429,0.78569,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,10,0,0,9,0,7],[16,94,0.1702,0.76339,0.19759,0.67857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,9,0,0,7,0,8],[20,94,0.2128,0.79462,0.22852,0.67857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,5,0,0,6,0,13],[24,94,0.2553,0.70536,0.22851,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,3,0,0,1,0,0,5,0,0,10,0,0,6,0,6],[28,94,0.2979,0.82143,0.15567,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,7,0,11],[32,94,0.3404,0.79463,0.21111,0.71429,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,8,0,0,4,0,13],[36,94,0.383,0.79464,0.19212,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,8,0,0,10,0,9],[40,94,0.4255,0.82588,0.23074,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,4,0,0,7,0,15],[44,94,0.4681,0.77678,0.18189,0.71429,0.78564,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,9,0,0,8,0,8],[48,94,0.5106,0.80789,0.17361,0.71429,0.85707,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,10,0,0,8,0,10],[52,94,0.5532,0.77679,0.21706,0.67857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,5,0,0,10,0,9],[56,94,0.5957,0.71873,0.22725,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,7,0,0,10,0,5],[60,94,0.6383,0.74999,0.20826,0.57142,0.78571,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,6,0,0,8,0,8],[64,94,0.6809,0.79018,0.19557,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,10,0,0,10,0,8],[68,94,0.7234,0.74553,0.25438,0.71429,0.78564,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,0,0,0,1,0,0,2,0,0,10,0,0,8,0,8],[72,94,0.766,0.75,0.19562,0.57143,0.78571,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,7,0,0,10,0,6],[76,94,0.8085,0.82142,0.18558,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,5,0,0,12,0,10],[80,94,0.8511,0.76338,0.20396,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,10,0,0,7,0,8],[84,94,0.8936,0.75893,0.24338,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,7,0,0,9,0,9],[88,94,0.9362,0.73661,0.20858,0.71429,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,11,0,0,11,0,4],[92,94,0.9787,0.78571,0.18558,0.71429,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,8,0,0,15,0,5],[94,94,1.0,0.75444,0.12996,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,0,12,0,2]]},{"b":7,"e":1.0,"k":"rising","v":0.65178,"x":0.88839,"p":[[0,49,0.0,0.65178,0.24728,0.53569,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,2,0,0,6,0,0,4,0,0,12,0,2],[4,49,0.0816,0.78571,0.21129,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,6,0,0,9,0,10],[8,49,0.1633,0.81695,0.22085,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,7,0,0,6,0,14],[12,49,0.2449,0.88839,0.12745,0.82132,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,16],[16,49,0.3265,0.86159,0.18725,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,17],[20,49,0.4082,0.85713,0.16366,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,9,0,14],[24,49,0.4898,0.86607,0.13803,0.82143,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,11,0,13],[28,49,0.5714,0.84375,0.19678,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,5,0,0,7,0,15],[32,49,0.6531,0.84375,0.13533,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,12,0,10],[36,49,0.7347,0.77232,0.20781,0.57143,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,6,0,0,7,0,10],[40,49,0.8163,0.83482,0.13882,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,12,0,9],[44,49,0.898,0.79018,0.13825,0.71429,0.85714,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,12,0,5],[48,49,0.9796,0.87054,0.09689,0.85714,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,20,0,8],[49,49,1.0,0.83481,0.11907,0.71429,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,15,0,7]]}]},{"i":"cc06caa4c6413bcd","q":"Let $ABC$ be a triangle and $\\Gamma$ its circumcircle. Let $A_{1}$ be the midpoint of the arc $\\widehat{BC}$ not containing $A$; $B_{1}$ the midpoint of the arc $\\overparen{CA}$ not containing $B$; $C_{1}$ the midpoint of the arc $\\overparen{AB}$ not containing $C$. Finally, let $A_{2}$ be the point such that $A B_{1} A_{2} C_{1}$ is a parallelogram; $B_{2}$ the point such that $B C_{1} B_{2} A_{1}$ is a parallelogram; $C_{2}$ the point such that $C A_{1} C_{2} B_{1}$ is a parallelogram.\nProve that the circumcircles of $ABC$ and $A_{2} B_{2} C_{2}$ are concentric.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.51786,"x":0.85714,"p":[[0,136,0.0,0.75446,0.33548,0.42857,1.0,1.0,0.0,1.0,1,20,1,1,0,2,0,0,3,0,0,3,0,0,3,0,0,0,0,0,0,0,20],[4,136,0.0294,0.80802,0.24384,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,1,0,0,2,0,18],[8,136,0.0588,0.85714,0.22588,0.57143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,0,0,0,1,0,22],[12,136,0.0882,0.77677,0.24985,0.57132,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,2,0,0,2,0,16],[16,136,0.1176,0.79464,0.2922,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,2,0,0,1,0,0,4,0,0,3,0,0,1,0,19],[20,136,0.1471,0.8482,0.21708,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,2,0,20],[24,136,0.1765,0.74107,0.29329,0.42857,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,3,0,0,5,0,0,2,0,0,4,0,0,2,0,15],[28,136,0.2059,0.80804,0.249,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,2,0,0,6,0,0,1,0,0,4,0,17],[32,136,0.2353,0.66964,0.27301,0.4286,0.64286,1.0,0.0,1.0,1,10,0,1,0,0,0,0,3,0,0,5,0,0,7,0,0,5,0,0,1,0,10],[36,136,0.2647,0.64285,0.27664,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,5,0,0,6,0,0,5,0,0,4,0,0,2,0,9],[40,136,0.2941,0.69193,0.26753,0.53539,0.64286,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,4,0,0,8,0,0,3,0,0,2,0,11],[44,136,0.3235,0.63383,0.29882,0.39286,0.64271,0.89286,0.14,1.0,0,8,0,0,0,4,0,0,4,0,0,2,0,0,6,0,0,4,0,0,4,0,8],[48,136,0.3529,0.60267,0.34392,0.28571,0.57143,1.0,0.0,1.0,2,10,0,2,0,3,0,0,5,0,0,4,0,0,3,0,0,2,0,0,3,0,10],[52,136,0.3824,0.63837,0.29448,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,5,0,0,5,0,0,6,0,0,2,0,0,2,0,10],[56,136,0.4118,0.54017,0.29824,0.39286,0.4998,0.85714,0.0,1.0,2,5,1,2,0,3,0,0,3,0,0,8,0,0,6,0,0,1,0,0,4,0,5],[60,136,0.4412,0.62052,0.23585,0.5354,0.57143,0.75,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,3,0,0,10,0,0,6,0,0,4,0,4],[64,136,0.4706,0.53125,0.27254,0.28571,0.57143,0.57143,0.0,1.0,1,6,0,1,0,2,0,0,6,0,0,6,0,0,10,0,0,1,0,0,0,0,6],[68,136,0.5,0.51786,0.23619,0.42859,0.57121,0.57143,0.0,1.0,1,2,0,1,0,4,0,0,1,0,0,7,0,0,12,0,0,3,0,0,2,0,2],[72,136,0.5294,0.56245,0.26949,0.42857,0.571,0.75,0.0,1.0,1,5,0,1,0,2,0,0,3,0,0,9,0,0,7,0,0,2,0,0,3,0,5],[76,136,0.5588,0.62051,0.23854,0.4286,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,9,0,0,10,0,0,1,0,0,2,0,7],[80,136,0.5882,0.54015,0.26901,0.28571,0.571,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,7,0,0,5,0,0,8,0,0,2,0,0,2,0,5],[84,136,0.6176,0.56696,0.27545,0.28571,0.57143,0.75,0.0,1.0,1,6,0,1,0,0,0,0,8,0,0,6,0,0,6,0,0,3,0,0,2,0,6],[88,136,0.6471,0.53567,0.27664,0.28571,0.57121,0.57143,0.0,1.0,2,5,0,2,0,1,0,0,6,0,0,4,0,0,12,0,0,0,0,0,2,0,5],[92,136,0.6765,0.6428,0.2113,0.4286,0.57143,0.74996,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,7,0,0,9,0,0,6,0,0,3,0,5],[96,136,0.7059,0.5357,0.26486,0.28571,0.57121,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,7,0,0,5,0,0,8,0,0,3,0,0,1,0,5],[100,136,0.7353,0.54464,0.29329,0.28571,0.42857,0.71429,0.0,1.0,1,7,0,1,0,2,0,0,6,0,0,10,0,0,1,0,0,5,0,0,0,0,7],[104,136,0.7647,0.65619,0.23655,0.57075,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,3,0,0,13,0,0,1,0,0,4,0,7],[108,136,0.7941,0.61606,0.26351,0.39286,0.57143,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,7,0,0,1,0,0,11,0,0,3,0,0,2,0,7],[112,136,0.8235,0.55801,0.28873,0.28571,0.57143,0.75,0.0,1.0,1,6,0,1,0,3,0,0,5,0,0,4,0,0,9,0,0,2,0,0,2,0,6],[116,136,0.8529,0.69195,0.27919,0.53539,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,5,0,0,2,0,0,7,0,0,3,0,0,3,0,11],[120,136,0.8824,0.61603,0.27067,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,5,0,0,3,0,0,10,0,0,2,0,0,3,0,7],[124,136,0.9118,0.66518,0.27342,0.53569,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,3,0,0,3,0,0,7,0,0,4,0,0,6,0,7],[128,136,0.9412,0.61605,0.25615,0.42857,0.57143,0.78571,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,9,0,0,7,0,0,4,0,0,0,0,8],[132,136,0.9706,0.54457,0.29983,0.2857,0.5005,0.857,0.0,1.0,1,5,0,1,0,3,0,0,7,0,0,5,0,0,5,0,0,1,0,0,5,0,5],[136,136,1.0,0.65183,0.21105,0.42965,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,7,0,0,8,0,0,5,0,0,6,0,4]]},{"b":4,"e":0.2857,"k":"falling","v":0.49101,"x":0.85714,"p":[[0,105,0.0,0.78125,0.29877,0.42857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,4,0,0,4,0,0,1,0,0,1,0,0,2,0,19],[4,105,0.0381,0.79909,0.24708,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,1,0,0,3,0,17],[8,105,0.0762,0.73214,0.24157,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,3,0,0,4,0,11],[12,105,0.1143,0.80357,0.22517,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,2,0,0,3,0,16],[16,105,0.1524,0.85268,0.21572,0.78571,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,0,0,0,5,0,19],[20,105,0.1905,0.85714,0.21129,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,1,0,0,3,0,20],[24,105,0.2286,0.8304,0.23258,0.67857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,2,0,0,4,0,18],[28,105,0.2667,0.8482,0.22852,0.57143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,0,0,0,2,0,21],[32,105,0.3048,0.7232,0.23403,0.57143,0.57143,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,0,0,0,2,0,12],[36,105,0.3429,0.76338,0.26393,0.57132,1.0,1.0,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 $n,k$  be positive integers such that  $n\\ge k$ .  $n$  lamps are placed on a circle, which are all off. In any step we can change the state of  $k$  consecutive lamps. In the following three cases, how many states of lamps are there in all  $2^n$  possible states that can be obtained from the initial state by a certain series of operations?\ni) $k$  is a prime number greater than  $2$ ;\nii)  $k$  is odd;\niii)  $k$  is even.","t":[{"b":4,"e":0.28571,"k":"falling","v":0.58926,"x":0.79464,"p":[[0,54,0.0,0.79464,0.24984,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,6,0,0,1,0,0,5,0,0,3,0,16],[4,54,0.0741,0.73661,0.30537,0.42857,0.85714,1.0,0.0,1.0,1,16,1,1,0,1,0,0,2,0,0,6,0,0,0,0,0,6,0,0,0,0,16],[8,54,0.1481,0.74106,0.27766,0.53539,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,2,0,0,0,0,16],[12,54,0.2222,0.70535,0.29867,0.42857,0.78564,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,5,0,0,6,0,0,0,0,0,4,0,0,3,0,13],[16,54,0.2963,0.64732,0.25501,0.42857,0.57144,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,14,0,0,0,0,0,5,0,0,3,0,8],[20,54,0.3704,0.69196,0.27689,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,3,0,0,7,0,0,3,0,0,4,0,0,3,0,11],[24,54,0.4444,0.67856,0.26964,0.42857,0.71429,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,1,0,0,11,0,0,2,0,0,5,0,0,1,0,11],[28,54,0.5185,0.70089,0.3059,0.42857,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,7,0,0,6,0,0,0,0,0,4,0,0,0,0,15],[32,54,0.5926,0.70536,0.27418,0.42857,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,3,0,0,9,0,0,2,0,0,4,0,0,1,0,13],[36,54,0.6667,0.65177,0.2765,0.42857,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,6,0,0,7,0,0,4,0,0,2,0,0,4,0,9],[40,54,0.7407,0.60267,0.25688,0.42857,0.57143,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,7,0,0,8,0,0,3,0,0,4,0,0,5,0,5],[44,54,0.8148,0.69642,0.28292,0.42857,0.64286,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,8,0,0,4,0,0,1,0,0,2,0,13],[48,54,0.8889,0.62054,0.27804,0.39286,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,8,0,0,7,0,0,2,0,0,3,0,0,5,0,7],[52,54,0.963,0.68303,0.25439,0.42859,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,4,0,0,3,0,0,6,0,0,8,0,6],[54,54,1.0,0.58926,0.24157,0.28571,0.64286,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,5,0,0,7,0,0,8,0,1]]},{"b":7,"e":0.85714,"k":"falling","v":0.62499,"x":0.88393,"p":[[0,59,0.0,0.81249,0.26109,0.71429,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,5,0,0,1,0,0,6,0,0,0,0,19],[4,59,0.0678,0.6875,0.27533,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,5,0,0,7,0,0,1,0,0,6,0,0,2,0,11],[8,59,0.1356,0.71427,0.28572,0.42857,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,5,0,0,6,0,0,2,0,0,4,0,0,1,0,14],[12,59,0.2034,0.70536,0.26229,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,10,0,0,1,0,0,6,0,0,1,0,12],[16,59,0.2712,0.77678,0.28107,0.42857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,4,0,0,5,0,0,1,0,0,3,0,0,1,0,18],[20,59,0.339,0.78124,0.22013,0.67857,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,9,0,0,1,0,14],[24,59,0.4068,0.85714,0.23419,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,3,0,0,1,0,22],[28,59,0.4746,0.67411,0.27947,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,10,0,0,1,0,0,5,0,0,0,0,12],[32,59,0.5424,0.77679,0.25238,0.4286,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,6,0,0,1,0,16],[36,59,0.6102,0.75,0.25254,0.42857,0.71429,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,7,0,0,1,0,14],[40,59,0.678,0.78125,0.27195,0.42857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,10,0,0,0,0,0,1,0,0,2,0,18],[44,59,0.7458,0.88393,0.21261,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,2,0,23],[48,59,0.8136,0.79018,0.23954,0.67857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,7,0,0,1,0,16],[52,59,0.8814,0.75892,0.26351,0.53571,0.85707,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,4,0,0,1,0,0,6,0,0,3,0,14],[56,59,0.9492,0.8125,0.26351,0.53571,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,2,0,0,1,0,20],[59,59,1.0,0.62499,0.29828,0.28571,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,11,0,0,2,0,0,4,0,0,4,0,0,1,0,10]]}]},{"i":"0fc444ce3131284e","q":"The functions  $f_0, f_1, f_2, ...$  are defined on the reals by  $f_0(x) = 8$  for all  $x$ ,  $f_{n+1}(x) = \\sqrt{x^2 + 6f_n(x)}$ . For all  $n$  solve the equation  $f_n(x) = 2x$ .","t":[{"b":3,"e":0.4286,"k":"flat","v":0.45982,"x":0.58928,"p":[[0,13,0.0,0.45982,0.25439,0.39286,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,6,0,0,2,0,0,18,0,0,0,0,0,1,0,0,1,0,4],[4,13,0.3077,0.51785,0.31288,0.28571,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,5,0,0,7,0,0,10,0,0,0,0,0,0,0,0,3,0,7],[8,13,0.6154,0.55803,0.30797,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,6,0,0,12,0,0,0,0,0,1,0,0,1,0,9],[12,13,0.9231,0.58928,0.31083,0.39286,0.42857,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,5,0,0,11,0,0,0,0,0,1,0,0,3,0,9],[13,13,1.0,0.52232,0.18073,0.42857,0.42857,0.46431,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,24,0,0,1,0,0,4,0,0,0,0,3]]},{"b":7,"e":1.0,"k":"rising","v":0.45089,"x":0.99107,"p":[[0,21,0.0,0.51339,0.27632,0.28571,0.42857,0.50002,0.14286,1.0,0,7,0,0,0,2,0,0,7,0,0,15,0,0,0,0,0,1,0,0,0,0,7],[4,21,0.1905,0.45089,0.22335,0.28571,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,2,0,0,7,0,0,19,0,0,0,0,0,0,0,0,0,0,4],[8,21,0.381,0.51339,0.29851,0.39286,0.42857,0.57143,0.14286,1.0,0,8,0,0,0,5,0,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,8],[12,21,0.5714,0.83482,0.25532,0.42859,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,22],[16,21,0.7619,0.86605,0.21412,0.82132,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,3,0,21],[20,21,0.9524,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[21,21,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"4d08b5cba3489847","q":"Find all pairs of positive integers  $(p; q) $ such that both the equations  $x^2- px + q = 0 $  and  $ x^2 -qx + p = 0 $  have integral solutions.","t":[{"b":6,"e":0.57143,"k":"flat","v":0.63838,"x":0.74551,"p":[[0,15,0.0,0.74551,0.23348,0.57132,0.78564,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,4,0,0,5,0,11],[4,15,0.2667,0.70533,0.26712,0.571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,3,0,0,3,0,11],[8,15,0.5333,0.63838,0.18205,0.4286,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,7,0,0,4,0,3],[12,15,0.8,0.65627,0.17802,0.571,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,12,0,0,3,0,3],[15,15,1.0,0.65177,0.16343,0.57143,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,10,0,0,2,0,3]]},{"b":7,"e":0.71429,"k":"flat","v":0.66961,"x":0.90623,"p":[[0,19,0.0,0.7009,0.27282,0.53571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,4,0,0,3,0,0,6,0,0,4,0,0,3,0,11],[4,19,0.2105,0.66961,0.24338,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,5,0,0,6,0,0,7,0,0,2,0,8],[8,19,0.4211,0.74552,0.20744,0.57143,0.71429,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,7,0,0,8,0,0,6,0,8],[12,19,0.6316,0.79003,0.23147,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,6,0,0,2,0,15],[16,19,0.8421,0.90623,0.16605,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,6,0,21],[19,19,1.0,0.84821,0.18189,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,7,0,15]]}]},{"i":"20cb517e388bb1ca","q":"7. (POL 1b) Let $I=(0,1]$ be the unit interval of the real line. For a given number $a \\in(0,1)$ we define a map $T: I \\rightarrow I$ by the formula  $$ T(x, y)= \\begin{cases}x+(1-a) & \\text { if } 0<x \\leq a \\\\ x-a & \\text { if } a<x \\leq 1\\end{cases} $$  Show that for every interval $J \\subset I$ there exists an integer $n>0$ such that $T^{n}(J) \\cap J \\neq \\emptyset$.","t":[{"b":1,"e":0.42857,"k":"flat","v":0.53569,"x":0.75445,"p":[[0,31,0.0,0.71873,0.27776,0.42859,0.857,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,8,0,0,2,0,0,2,0,0,5,0,12],[4,31,0.129,0.75445,0.23753,0.42857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,10,0,0,1,0,0,1,0,0,10,0,10],[8,31,0.2581,0.53569,0.2369,0.28571,0.42859,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,9,0,0,10,0,0,3,0,0,2,0,0,6,0,2],[12,31,0.3871,0.54907,0.23448,0.42857,0.4286,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,13,0,0,4,0,0,2,0,0,3,0,4],[16,31,0.5161,0.54461,0.2214,0.42857,0.4286,0.60714,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,12,0,0,7,0,0,2,0,0,3,0,3],[20,31,0.6452,0.5982,0.24856,0.42857,0.49979,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,5,0,0,11,0,0,4,0,0,2,0,0,5,0,5],[24,31,0.7742,0.58928,0.28959,0.28571,0.4286,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,10,0,0,8,0,0,1,0,0,1,0,0,5,0,7],[28,31,0.9032,0.62052,0.25657,0.42857,0.57121,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,9,0,0,2,0,0,2,0,0,9,0,4],[31,31,1.0,0.625,0.26905,0.42857,0.5,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,6,0,0,10,0,0,1,0,0,2,0,0,7,0,6]]},{"b":4,"e":1.0,"k":"flat","v":0.6116,"x":0.74106,"p":[[0,10,0.0,0.6116,0.26058,0.42857,0.57121,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,3,0,0,11,0,0,5,0,0,2,0,0,3,0,7],[4,10,0.4,0.74106,0.27066,0.42857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,1,0,0,11,0,10],[8,10,0.8,0.69639,0.27838,0.42857,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,4,0,0,8,0,0,3,0,0,2,0,0,3,0,12],[10,10,1.0,0.6429,0.2624,0.42857,0.57214,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,12,0,0,0,0,0,2,0,0,8,0,6]]}]},{"i":"23354b0d8b84677b","q":"Find all the integers  $n$  for which  $\\frac{8n-25}{n+5}$  is cube of a rational number.","t":[{"b":3,"e":0.57143,"k":"flat","v":0.65179,"x":0.7366,"p":[[0,61,0.0,0.66069,0.12244,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,10,0,0,4,0,1],[4,61,0.0656,0.71427,0.18558,0.57143,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,2,0,0,5,0,7],[8,61,0.1311,0.66071,0.15047,0.57143,0.57143,0.71429,0.5714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,4,0,0,2,0,4],[12,61,0.1967,0.65625,0.11769,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,8,0,0,4,0,1],[16,61,0.2623,0.67409,0.15252,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,8,0,0,2,0,4],[20,61,0.3279,0.65625,0.14664,0.57143,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,5,0,0,3,0,3],[24,61,0.3934,0.65179,0.12846,0.57143,0.57143,0.71429,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,6,0,0,5,0,1],[28,61,0.459,0.65625,0.13767,0.57143,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,9,0,0,1,0,3],[32,61,0.5246,0.67857,0.16751,0.57143,0.57143,0.85704,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,4,0,0,5,0,4],[36,61,0.5902,0.66295,0.16193,0.57143,0.57143,0.71429,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,3,0,0,2,1,4],[40,61,0.6557,0.65625,0.15092,0.57143,0.57143,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,3,0,0,2,0,4],[44,61,0.7213,0.66964,0.1357,0.57143,0.57143,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,6,0,0,5,0,2],[48,61,0.7869,0.66071,0.13243,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,9,0,0,3,0,2],[52,61,0.8525,0.7366,0.16409,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,7,0,0,8,0,5],[56,61,0.918,0.71429,0.18211,0.57143,0.64286,0.89286,0.4286,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,7,0,0,1,0,8],[60,61,0.9836,0.67857,0.14725,0.57143,0.57143,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,8,0,0,2,0,4],[61,61,1.0,0.69639,0.14618,0.57143,0.64286,0.857,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,7,0,0,6,0,3]]},{"b":5,"e":0.71429,"k":"flat","v":0.67857,"x":0.77677,"p":[[0,82,0.0,0.71875,0.18029,0.57143,0.57143,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,5,0,0,2,0,8],[4,82,0.0488,0.7723,0.17808,0.57143,0.78564,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,7,0,9],[8,82,0.0976,0.71875,0.14054,0.57143,0.71429,0.75,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,13,0,0,4,0,4],[12,82,0.1463,0.77677,0.1819,0.57143,0.71429,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,7,0,0,3,0,11],[16,82,0.1951,0.73214,0.15047,0.57143,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,11,0,0,5,0,5],[20,82,0.2439,0.72321,0.15947,0.57143,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,10,0,0,3,0,6],[24,82,0.2927,0.75446,0.14826,0.67857,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,13,0,0,5,0,6],[28,82,0.3415,0.69194,0.15199,0.57143,0.57143,0.74996,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,7,0,0,4,0,4],[32,82,0.3902,0.71429,0.15567,0.57143,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,9,0,0,4,0,5],[36,82,0.439,0.69643,0.1915,0.57143,0.71429,0.85714,0.0,1.0,1,4,1,1,0,0,0,0,0,0,0,0,0,0,12,0,0,10,0,0,5,0,4],[40,82,0.4878,0.75446,0.15663,0.57143,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,12,0,0,4,0,7],[44,82,0.5366,0.70982,0.15355,0.57143,0.71429,0.75,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,10,0,0,3,0,5],[48,82,0.5854,0.75,0.16366,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,12,0,0,4,0,7],[52,82,0.6341,0.71427,0.14727,0.57143,0.71429,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,10,0,0,5,0,4],[56,82,0.6829,0.73214,0.17405,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,11,0,0,3,0,7],[60,82,0.7317,0.72321,0.13803,0.57143,0.71429,0.75,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,14,0,0,4,0,4],[64,82,0.7805,0.67857,0.12877,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,13,0,0,3,0,2],[68,82,0.8293,0.73657,0.17173,0.57143,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,9,0,0,2,0,8],[72,82,0.878,0.69195,0.13883,0.57143,0.71429,0.71429,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,10,0,0,4,0,3],[76,82,0.9268,0.76786,0.16269,0.57143,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,10,0,0,5,0,8],[80,82,0.9756,0.71427,0.12878,0.57143,0.71429,0.71429,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,15,0,0,4,0,3],[82,82,1.0,0.74554,0.12745,0.71429,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,17,0,0,5,0,4]]}]},{"i":"72a592c2a401d479","q":"Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point $B$ in the unit square. Then Alice chooses a sequence of points $P_{0}, P_{1}, \\ldots, P_{N}$ in the plane. After choosing $P_{k}$ (but before choosing $P_{k+1}$ ) for $k \\geqslant 1$, Bob tells \"warmer\" if $P_{k}$ is closer to $B$ than $P_{k-1}$, otherwise he says \"colder\". After Alice has chosen $P_{N}$ and heard Bob's answer, Alice chooses a final point $A$. Alice wins if the distance $A B$ is at most $\\frac{1}{2020}$, otherwise Bob wins. Show that if $N=18$, Alice cannot guarantee a win.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.77232,"x":0.79911,"p":[[0,3,0.0,0.77232,0.16698,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,22,0,0,1,0,8],[3,3,1.0,0.79911,0.14664,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,0,1,0,10]]},{"b":2,"e":0.71429,"k":"flat","v":0.72768,"x":0.78571,"p":[[0,5,0.0,0.78571,0.13832,0.71429,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,21,0,0,2,0,8],[4,5,0.8,0.72768,0.09006,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0,2],[5,5,1.0,0.77219,0.11221,0.71429,0.71429,0.71429,0.71,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,1,0,6]]}]},{"i":"aa4e6021de5a0175","q":"11. (VIE 1) Prove that there exist infinitely many positive integers $n$ such that the decimal representation of $5^{n}$ contains a block of 1976 consecutive zeros.","t":[{"b":0,"e":1.0,"k":"rising","v":0.70088,"x":1.0,"p":[[0,18,0.0,0.70088,0.33189,0.4286,0.71429,1.0,0.0,1.0,1,15,0,1,0,4,0,0,1,0,0,3,0,0,3,0,0,5,0,0,0,0,15],[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],[18,18,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":0.42857,"k":"flat","v":0.34379,"x":0.72322,"p":[[0,27,0.0,0.67409,0.343,0.42857,0.71429,1.0,0.0,1.0,3,13,0,3,0,2,0,0,1,0,0,4,0,0,2,0,0,6,0,0,1,0,13],[4,27,0.1481,0.57586,0.27312,0.39286,0.57143,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,5,0,0,3,0,0,6,0,0,9,0,0,2,0,4],[8,27,0.2963,0.62039,0.1771,0.4286,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,15,0,0,1,0,2],[12,27,0.4444,0.47773,0.25656,0.39286,0.50071,0.71429,0.0,1.0,3,1,0,3,0,4,0,0,1,0,0,8,0,0,6,0,0,8,0,0,1,0,1],[16,27,0.5926,0.34379,0.25967,0.14286,0.28571,0.60714,0.0,0.71429,7,0,0,7,0,3,0,0,9,0,0,4,0,0,1,0,0,8,0,0,0,0,0],[20,27,0.7407,0.59822,0.14032,0.42857,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,18,0,0,0,0,0],[24,27,0.8889,0.66515,0.20706,0.5354,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,14,0,0,3,0,4],[27,27,1.0,0.72322,0.17473,0.71429,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,16,0,0,6,0,4]]}]},{"i":"fa155942c75cf363","q":"Let  $A$  be the set of all binary sequences of length  $n$  and denote  $o =(0, 0, \\ldots , 0) \\in A$ . Define the addition on  $A$  as  $(a_1, \\ldots , a_n)+(b_1, \\ldots , b_n) =(c_1, \\ldots , c_n)$ , where  $c_i = 0$  when  $a_i = b_i$  and  $c_i = 1$  otherwise. Suppose that  $f\\colon A \\to A$  is a function such that  $f(0) = 0$ , and for each  $a, b \\in A$ , the sequences  $f(a)$  and  $f(b)$  differ in exactly as many places as  $a$  and  $b$  do. Prove that if  $a$  ,  $b$ ,  $c \\in A$  satisfy  $a+ b + c = 0$ , then  $f(a)+ f(b) + f(c) = 0$ .","t":[{"b":1,"e":0.85714,"k":"rising","v":0.71873,"x":0.94196,"p":[[0,13,0.0,0.71873,0.3164,0.49968,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,5,0,0,0,0,0,4,0,0,2,0,0,4,0,14],[4,13,0.3077,0.875,0.21943,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,7,0,20],[8,13,0.6154,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[12,13,0.9231,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[13,13,1.0,0.92857,0.13363,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,0,20]]},{"b":3,"e":0.71429,"k":"rising","v":0.74097,"x":0.97768,"p":[[0,24,0.0,0.74097,0.29564,0.57132,0.85714,1.0,0.14,1.0,0,14,0,0,0,2,0,0,4,0,0,1,0,0,5,0,0,1,0,0,5,0,14],[4,24,0.1667,0.91518,0.19516,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,23],[8,24,0.3333,0.94196,0.15916,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,25],[12,24,0.5,0.93303,0.17852,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,26],[16,24,0.6667,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[20,24,0.8333,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[24,24,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]}]},{"i":"7dafd1b2d3612e7c","q":"Let $n \\geq 2$ be an integer. Initially, the number 1 is written $n$ times on a board. Every minute, Vishal picks two numbers written on the board, say $a$ and $b$, erases them, and writes either $a+b$ or $\\min \\left\\{a^{2}, b^{2}\\right\\}$. After $n-1$ minutes there is one number left on the board. Let the largest possible value for this final number be $f(n)$. Prove that\n\n$$\n2^{n / 3}<f(n) \\leq 3^{n / 3}\n$$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.15625,"x":0.40624,"p":[[0,57,0.0,0.36159,0.17121,0.2857,0.35714,0.4642,0.0,0.71429,1,0,0,1,0,6,0,0,9,0,0,8,0,0,7,0,0,1,0,0,0,0,0],[4,57,0.0702,0.38839,0.25563,0.14289,0.42857,0.4286,0.0,1.0,4,1,0,4,0,5,0,0,4,0,0,12,0,0,0,0,0,5,0,0,1,0,1],[8,57,0.1404,0.29911,0.22689,0.14286,0.2857,0.4286,0.0,0.71429,6,0,0,6,0,8,0,0,5,0,0,6,0,0,4,0,0,3,0,0,0,0,0],[12,57,0.2105,0.3482,0.27878,0.14286,0.35714,0.42858,0.0,1.0,5,2,0,5,0,8,0,0,3,0,0,10,0,0,2,0,0,0,0,0,2,0,2],[16,57,0.2807,0.36604,0.25735,0.14286,0.35714,0.571,0.0,1.0,3,1,0,3,0,9,0,0,4,0,0,7,0,0,5,0,0,1,0,0,2,0,1],[20,57,0.3509,0.32581,0.26788,0.14214,0.42857,0.4286,0.0,1.0,7,1,0,7,0,7,0,0,1,0,0,11,0,0,3,0,0,0,0,0,2,0,1],[24,57,0.4211,0.31246,0.2407,0.14286,0.2857,0.4286,0.0,1.0,4,1,0,4,0,11,0,0,4,0,0,6,0,0,4,0,0,2,0,0,0,0,1],[28,57,0.4912,0.40624,0.26026,0.14286,0.42857,0.57111,0.0,1.0,3,2,0,3,0,6,0,0,3,0,0,11,0,0,4,0,0,2,0,0,1,0,2],[32,57,0.5614,0.32119,0.26732,0.14286,0.21428,0.571,0.0,1.0,5,1,0,5,0,11,0,0,3,0,0,4,0,0,5,0,0,2,0,0,1,0,1],[36,57,0.6316,0.3125,0.19045,0.14286,0.2857,0.42857,0.0,0.71429,2,0,0,2,0,11,0,0,5,0,0,9,0,0,3,0,0,2,0,0,0,0,0],[40,57,0.7018,0.30792,0.28599,0.14286,0.14286,0.4286,0.0,1.0,5,2,0,5,0,13,0,0,2,0,0,6,0,0,2,0,0,0,0,0,2,0,2],[44,57,0.7719,0.2991,0.23785,0.14286,0.2857,0.4286,0.0,0.85714,6,0,0,6,0,8,0,0,6,0,0,6,0,0,2,0,0,3,0,0,1,0,0],[48,57,0.8421,0.20515,0.18537,0.14,0.14286,0.2857,0.0,0.71429,7,0,0,7,0,15,0,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[52,57,0.9123,0.15625,0.16115,0.0,0.14286,0.14286,0.0,0.71429,9,0,0,9,0,18,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[56,57,0.9825,0.20517,0.18194,0.14286,0.14286,0.2857,0.0,0.857,5,0,0,5,0,18,0,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[57,57,1.0,0.17848,0.16753,0.14286,0.14286,0.14286,0.0,0.85714,6,0,0,6,0,19,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.16063,"x":0.49097,"p":[[0,42,0.0,0.32132,0.17504,0.14286,0.28571,0.42857,0.14,0.71429,0,0,0,0,0,10,0,0,12,0,0,5,0,0,2,0,0,3,0,0,0,0,0],[4,42,0.0952,0.29464,0.2257,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,10,0,0,7,0,0,7,0,0,1,0,0,2,0,0,0,0,1],[8,42,0.1905,0.30357,0.28065,0.14286,0.14286,0.42857,0.0,1.0,6,2,0,6,0,11,0,0,4,0,0,4,0,0,2,0,0,3,0,0,0,0,2],[12,42,0.2857,0.49097,0.30301,0.24999,0.42857,0.71429,0.0,1.0,1,5,0,1,0,7,0,0,3,0,0,9,0,0,1,0,0,5,0,0,1,0,5],[16,42,0.381,0.28569,0.24997,0.14286,0.14295,0.42857,0.0,1.0,6,1,0,6,0,11,0,0,4,0,0,4,0,0,4,0,0,2,0,0,0,0,1],[20,42,0.4762,0.42848,0.29888,0.14286,0.42857,0.71429,0.0,1.0,2,2,0,2,0,10,0,0,3,0,0,5,0,0,1,0,0,7,0,0,2,0,2],[24,42,0.5714,0.34375,0.2547,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,10,0,0,5,0,0,8,0,0,0,0,0,4,0,0,1,0,1],[28,42,0.6667,0.34818,0.25735,0.14286,0.28571,0.4642,0.0,0.85714,4,0,0,4,0,9,0,0,4,0,0,7,0,0,3,0,0,2,0,0,3,0,0],[32,42,0.7619,0.39284,0.26243,0.14286,0.42857,0.4642,0.0,1.0,1,2,0,1,0,10,0,0,4,0,0,9,0,0,3,0,0,1,0,0,2,0,2],[36,42,0.8571,0.30801,0.24769,0.14286,0.14288,0.46418,0.0,1.0,3,1,0,3,0,14,0,0,4,0,0,3,0,0,4,0,0,3,0,0,0,0,1],[40,42,0.9524,0.16063,0.14618,0.0,0.14286,0.1429,0.0,0.4286,9,0,0,9,0,16,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.20526,0.22286,0.0,0.14286,0.32142,0.0,1.0,9,1,0,9,0,14,0,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,1]]}]},{"i":"9b7e98845304192c","q":"10. (FIN 1) Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1 / q$ and a polynomial $P$ with integral coefficients such that  $$ \\left|P(x)-\\frac{p}{q}\\right|<\\frac{1}{q^{2}} $$  for all $x \\in I$.","t":[{"b":2,"e":1.0,"k":"rising","v":0.17409,"x":0.85714,"p":[[0,45,0.0,0.17409,0.3383,0.0,0.0,0.14286,0.0,1.0,23,3,23,23,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,3],[4,45,0.0889,0.72322,0.2141,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,9,0,0,8,0,6],[8,45,0.1778,0.7232,0.23129,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,9,0,0,7,0,7],[12,45,0.2667,0.7098,0.19393,0.67857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,14,0,0,6,0,4],[16,45,0.3556,0.69179,0.22899,0.57132,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,2,0,0,6,0,0,11,0,0,4,0,6],[20,45,0.4444,0.71427,0.25506,0.57132,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,4,0,0,2,0,0,3,0,0,7,0,0,7,0,8],[24,45,0.5333,0.79464,0.18877,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,8,0,0,10,0,9],[28,45,0.6222,0.76338,0.19435,0.71429,0.85707,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,9,0,0,11,0,6],[32,45,0.7111,0.85714,0.15972,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,7,0,14],[36,45,0.8,0.79909,0.19187,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,8,0,0,12,0,8],[40,45,0.8889,0.80804,0.23037,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,7,0,0,9,0,12],[44,45,0.9778,0.83481,0.15198,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,12,0,10],[45,45,1.0,0.82588,0.11701,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,17,0,5]]},{"b":5,"e":0.57143,"k":"rising","v":0.25446,"x":0.75446,"p":[[0,79,0.0,0.25446,0.41301,0.0,0.0,0.32143,0.0,1.0,21,6,19,21,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,6],[4,79,0.0506,0.73211,0.18125,0.67857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,13,0,0,6,0,5],[8,79,0.1013,0.70534,0.22571,0.5354,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,5,0,0,1,0,0,12,0,0,4,0,7],[12,79,0.1519,0.68747,0.17657,0.57132,0.71429,0.75,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,14,0,0,5,0,3],[16,79,0.2025,0.67408,0.2059,0.53539,0.71429,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,12,0,0,3,0,5],[20,79,0.2532,0.59372,0.26513,0.42857,0.57143,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,2,0,0,6,0,0,6,0,0,8,0,0,3,0,4],[24,79,0.3038,0.66516,0.19759,0.57132,0.71429,0.857,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,11,0,0,7,0,2],[28,79,0.3544,0.63393,0.23673,0.42857,0.71429,0.75,0.14286,1.0,0,4,0,0,0,2,0,0,2,0,0,6,0,0,4,0,0,10,0,0,4,0,4],[32,79,0.4051,0.6071,0.25504,0.42857,0.57143,0.74996,0.14286,1.0,0,4,0,0,0,3,0,0,4,0,0,2,0,0,8,0,0,7,0,0,4,0,4],[36,79,0.4557,0.60712,0.26001,0.42857,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,5,0,0,0,0,0,5,0,0,5,0,0,10,0,0,3,0,4],[40,79,0.5063,0.52676,0.25862,0.39286,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,6,0,0,1,0,0,5,0,0,5,0,0,9,0,0,5,0,0],[44,79,0.557,0.64286,0.25,0.42859,0.71429,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,4,0,0,3,0,0,3,0,0,12,0,0,3,0,5],[48,79,0.6076,0.65177,0.24984,0.42859,0.71429,0.85704,0.14286,1.0,0,4,0,0,0,3,0,0,2,0,0,4,0,0,1,0,0,13,0,0,5,0,4],[52,79,0.6582,0.49106,0.26711,0.28571,0.4286,0.71429,0.14286,1.0,0,3,0,0,0,7,0,0,4,0,0,6,0,0,5,0,0,6,0,0,1,0,3],[56,79,0.7089,0.66068,0.22799,0.5354,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,5,0,0,5,0,0,8,0,0,8,0,3],[60,79,0.7595,0.54463,0.27065,0.28571,0.64286,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,3,0,0,6,0,0,1,0,0,10,0,0,4,0,2],[64,79,0.8101,0.70532,0.21707,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,0,0,0,5,0,0,12,0,0,6,0,5],[68,79,0.8608,0.70089,0.21237,0.57143,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,11,0,0,8,0,4],[72,79,0.9114,0.5848,0.23786,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,2,0,0,5,0,0,8,0,0,8,0,0,4,0,2],[76,79,0.962,0.72321,0.21997,0.67857,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,11,0,0,7,0,6],[79,79,1.0,0.75446,0.18977,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,2,0,9]]}]},{"i":"8b93a78fc2d08d25","q":"Let  $ABCD$  be a trapezoid such that  $AD\\parallel BC$  and  $|AB|=|BC|$ . Let  $E$  and  $F$  be the midpoints of  $[BC]$  and  $[AD]$ , respectively. If the internal angle bisector of  $\\triangle ABC$  passes through  $F$ , find  $|BD|/|EF|$ .","t":[{"b":1,"e":0.1429,"k":"flat","v":0.16071,"x":0.28107,"p":[[0,24,0.0,0.28107,0.11586,0.14286,0.28571,0.28571,0.14,0.57143,0,0,0,0,0,9,0,0,17,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[4,24,0.1667,0.16071,0.10564,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,25,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,24,0.3333,0.2008,0.12811,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,21,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,24,0.5,0.27665,0.22825,0.14286,0.14286,0.2857,0.14286,0.85714,0,0,0,0,0,21,0,0,4,0,0,2,0,0,0,0,0,3,0,0,2,0,0],[16,24,0.6667,0.16964,0.10967,0.14286,0.14286,0.14287,0.0,0.571,2,0,0,2,0,26,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,24,0.8333,0.17847,0.09447,0.14286,0.14286,0.14286,0.14,0.571,0,0,0,0,0,27,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,24,1.0,0.17857,0.15152,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.14286,"k":"flat","v":0.14277,"x":0.27677,"p":[[0,58,0.0,0.27677,0.12843,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,9,0,0,15,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[4,58,0.069,0.19644,0.11152,0.14286,0.14286,0.14292,0.14286,0.57143,0,0,0,0,0,25,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,58,0.1379,0.20079,0.12299,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,0,22,0,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,58,0.2069,0.22768,0.14664,0.14286,0.1429,0.28571,0.14286,0.85714,0,0,0,0,0,20,0,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[16,58,0.2759,0.17411,0.08552,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,25,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.20089,0.14223,0.14286,0.14286,0.2857,0.0,0.71429,2,0,0,2,0,21,0,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[24,58,0.4138,0.20536,0.12339,0.14286,0.14286,0.1786,0.14286,0.57143,0,0,0,0,0,24,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[28,58,0.4828,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.17856,0.10096,0.14286,0.14286,0.14287,0.0,0.571,1,0,0,1,0,25,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,58,0.6207,0.19632,0.09942,0.14286,0.14286,0.2857,0.14,0.571,0,0,0,0,0,23,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,58,0.6897,0.2009,0.15093,0.14286,0.14286,0.1429,0.0,0.71429,2,0,0,2,0,23,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[44,58,0.7586,0.1517,0.04973,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.19195,0.14552,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,22,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[52,58,0.8966,0.16063,0.07786,0.14286,0.14286,0.14287,0.0,0.42857,2,0,0,2,0,25,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.14732,0.09094,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,27,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"59a4df1aab0d3ab8","q":"Let  $\\triangle ABC$  be an acute-angled triangle. Let  $P$  be the midpoint of  $BC$ , and  $K$  the foot of the altitude from  $A$  to side  $BC$ . Let  $D$  be a point on segment  $AP$  such that  $\\angle BDC = 90^\\circ$ . Let  $E$  be the second point of intersection of line  $BC$  with the circumcircle of  $\\triangle ADK$ . Let  $F$  be the second point of intersection of line  $AE$  with the circumcircle of  $\\triangle ABC$ . Prove that  $\\angle AFD = 90^\\circ$ 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 $n$  be a positive square free integer,  $S$  is a subset of  $[n]:=\\{1,2,\\ldots ,n\\}$  such that  $|S|\\ge n/2.$  Prove that there exists three elements  $a,b,c\\in S$  (can be same), satisfy  $ab\\equiv c\\pmod n.$ *Created by Zhenhua Qu*","t":[{"b":3,"e":0.28571,"k":"rising","v":0.12473,"x":0.40178,"p":[[0,63,0.0,0.12473,0.12749,0.0,0.14,0.2857,0.0,0.42857,14,0,1,14,0,9,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,63,0.0635,0.33479,0.22755,0.14286,0.28571,0.4642,0.0,0.85714,4,0,0,4,0,8,0,0,5,0,0,7,0,0,5,0,0,2,0,0,1,0,0],[8,63,0.127,0.33479,0.1735,0.2857,0.2857,0.4286,0.0,0.71429,2,0,0,2,0,3,0,0,17,0,0,5,0,0,2,0,0,3,0,0,0,0,0],[12,63,0.1905,0.3705,0.2365,0.2857,0.2857,0.4642,0.0,1.0,3,1,0,3,0,3,0,0,14,0,0,4,0,0,3,0,0,3,0,0,1,0,1],[16,63,0.254,0.32571,0.16476,0.2857,0.28571,0.32143,0.14,0.857,0,0,0,0,0,6,0,0,18,0,0,5,0,0,0,0,0,2,0,0,1,0,0],[20,63,0.3175,0.33481,0.21236,0.25,0.28571,0.4286,0.0,0.85714,3,0,0,3,1,4,0,0,11,0,1,7,0,0,2,0,0,1,0,0,2,0,0],[24,63,0.381,0.40178,0.24336,0.28571,0.28571,0.57143,0.0,1.0,1,1,0,1,0,4,0,0,15,0,0,3,0,0,3,0,0,2,0,0,3,0,1],[28,63,0.4444,0.32145,0.17496,0.2857,0.28571,0.28571,0.0,0.857,2,0,0,2,0,3,0,0,20,0,0,2,0,0,3,0,0,1,0,0,1,0,0],[32,63,0.5079,0.3125,0.18013,0.25,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,6,0,0,15,0,0,5,0,0,2,0,0,1,0,0,1,0,0],[36,63,0.5714,0.35475,0.18343,0.2857,0.28571,0.44645,0.0,0.85714,1,0,0,1,0,5,0,0,14,0,0,4,0,1,5,0,0,1,0,0,1,0,0],[40,63,0.6349,0.33032,0.22986,0.24999,0.28571,0.4286,0.0,1.0,4,1,0,4,0,4,0,0,15,0,0,2,0,0,3,0,0,3,0,0,0,0,1],[44,63,0.6984,0.32133,0.19568,0.2857,0.28571,0.4286,0.0,0.71429,4,0,0,4,0,3,0,0,15,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[48,63,0.7619,0.33479,0.20702,0.2857,0.28571,0.32143,0.0,1.0,3,1,0,3,0,2,0,0,19,0,0,2,0,0,3,0,0,2,0,0,0,0,1],[52,63,0.8254,0.3482,0.13801,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,21,0,0,4,0,0,3,0,0,2,0,0,0,0,0],[56,63,0.8889,0.34364,0.20165,0.2857,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,3,0,0,18,0,0,4,0,0,1,0,0,2,0,0,2,0,0],[60,63,0.9524,0.34587,0.19497,0.24999,0.2857,0.4286,0.14,0.85714,0,0,0,0,0,8,0,0,14,0,1,2,0,0,3,0,0,3,0,0,1,0,0],[63,63,1.0,0.35711,0.15969,0.2857,0.28571,0.4286,0.0,0.71429,2,0,0,2,0,1,0,0,16,0,0,6,0,0,6,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.4286,"k":"flat","v":0.18304,"x":0.48873,"p":[[0,64,0.0,0.18304,0.13474,0.10714,0.1429,0.28571,0.0,0.4286,8,0,0,8,0,10,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,64,0.0625,0.48873,0.28865,0.2857,0.39279,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,12,0,1,4,0,0,2,0,0,4,0,0,1,0,5],[8,64,0.125,0.37719,0.24884,0.2857,0.28571,0.571,0.0,1.0,4,2,0,4,0,1,0,0,15,0,0,3,0,0,4,1,0,2,0,0,0,0,2],[12,64,0.1875,0.30354,0.25438,0.0,0.28571,0.42858,0.0,1.0,9,1,0,9,0,1,0,0,11,0,0,5,0,0,2,0,0,3,0,0,0,0,1],[16,64,0.25,0.27437,0.21299,0.14214,0.2857,0.42857,0.0,0.71429,6,0,0,6,0,9,0,0,6,0,1,4,0,0,4,0,0,2,0,0,0,0,0],[20,64,0.3125,0.30133,0.17651,0.14296,0.28571,0.37489,0.0,0.85714,2,0,0,2,0,7,0,0,14,0,1,5,0,0,1,0,0,1,0,0,1,0,0],[24,64,0.375,0.30804,0.25781,0.14286,0.2857,0.28571,0.0,1.0,4,2,0,4,0,8,0,0,13,0,0,2,0,0,1,0,0,1,0,0,1,0,2],[28,64,0.4375,0.33469,0.20083,0.24999,0.28571,0.4286,0.0,0.71429,4,0,0,4,0,4,0,0,10,0,0,8,0,0,3,0,0,3,0,0,0,0,0],[32,64,0.5,0.2856,0.20519,0.14286,0.2857,0.28571,0.0,1.0,4,1,0,4,0,6,0,0,16,0,0,2,0,0,2,0,0,1,0,0,0,0,1],[36,64,0.5625,0.26336,0.16788,0.14286,0.2857,0.32143,0.0,0.57143,6,0,0,6,0,4,0,0,14,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[40,64,0.625,0.31464,0.17101,0.26785,0.28571,0.4286,0.0,0.571,3,0,0,3,0,4,0,1,14,0,0,3,0,0,7,0,0,0,0,0,0,0,0],[44,64,0.6875,0.29018,0.1838,0.2857,0.28571,0.32144,0.0,0.71429,6,0,0,6,0,1,0,0,17,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[48,64,0.75,0.27231,0.16885,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,9,0,0,13,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[52,64,0.8125,0.24772,0.13603,0.14289,0.2857,0.28571,0.0,0.71429,3,0,0,3,1,6,0,0,19,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[56,64,0.875,0.28338,0.14568,0.14286,0.2857,0.42857,0.0,0.643,2,0,0,2,0,8,0,0,13,0,0,7,0,0,1,1,0,0,0,0,0,0,0],[60,64,0.9375,0.3013,0.17466,0.26785,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,3,0,1,15,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[64,64,1.0,0.23437,0.12401,0.1429,0.2857,0.28571,0.0,0.4286,5,0,0,5,0,5,0,0,18,0,1,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"60b6d4668c9d871b","q":"The sequences $\\left(a_{n}\\right)_{n \\geqslant 0}$ and $\\left(b_{n}\\right)_{n \\geqslant 0}$ are defined by\n\n$$\n\\left\\{\\begin{array} { l l } \n{ a _ { n } = 0 } & { \\text { if } n = 0 ; } \\\\\n{ a _ { n } = 2 a _ { \\lfloor n / 2 \\rfloor } + n } & { \\text { if } n \\geqslant 1 ; }\n\\end{array} \\quad \\text { and } \\quad \\left\\{\\begin{array}{ll}\nb_{n}=0 & \\text { if } n=0 ; \\\\\nb_{n}=3 b_{\\lfloor n / 3\\rfloor}+n & \\text { if } n \\geqslant 1 .\n\\end{array}\\right.\\right.\n$$\n\nProve that the sequence with general term $2^{a_{n}}-3^{b_{n}}$ changes sign infinitely often.","t":[{"b":0,"e":1.0,"k":"flat","v":0.61159,"x":0.96875,"p":[[0,53,0.0,0.6696,0.22992,0.42859,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,6,0,0,5,0,0,9,0,0,2,0,7],[4,53,0.0755,0.96875,0.08553,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[8,53,0.1509,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[12,53,0.2264,0.8839,0.1766,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,5,0,20],[16,53,0.3019,0.87945,0.16796,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,19],[20,53,0.3774,0.89272,0.20215,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,21],[24,53,0.4528,0.94643,0.11152,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[28,53,0.5283,0.83485,0.2175,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,4,0,17],[32,53,0.6038,0.86155,0.18038,0.67857,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,5,0,18],[36,53,0.6792,0.84819,0.205,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,4,0,18],[40,53,0.7547,0.79461,0.24471,0.57143,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,2,0,0,7,0,14],[44,53,0.8302,0.7455,0.25938,0.57142,0.78571,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,1,0,0,8,0,0,4,0,0,4,0,12],[48,53,0.9057,0.82587,0.18119,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,9,0,12],[52,53,0.9811,0.70087,0.28203,0.53539,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,7,0,0,1,0,0,7,0,0,2,0,0,3,0,12],[53,53,1.0,0.61159,0.27719,0.28571,0.64286,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,10,0,0,4,0,0,2,0,0,5,0,0,5,0,6]]},{"b":6,"e":0.71429,"k":"flat","v":0.68745,"x":0.92855,"p":[[0,32,0.0,0.72318,0.25741,0.571,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,4,0,0,7,0,0,5,0,0,3,0,11],[4,32,0.125,0.88839,0.21646,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,1,0,24],[8,32,0.25,0.87496,0.17775,0.82132,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,5,0,19],[12,32,0.375,0.92855,0.14729,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,24],[16,32,0.5,0.81694,0.23756,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,5,0,0,3,0,17],[20,32,0.625,0.8214,0.24487,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,3,0,0,5,0,17],[24,32,0.75,0.77679,0.26471,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,2,0,0,5,0,15],[28,32,0.875,0.82365,0.25443,0.67856,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,2,0,1,2,0,0,3,0,0,1,0,20],[32,32,1.0,0.68745,0.23541,0.571,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,2,0,0,9,0,0,6,0,0,3,0,8]]}]},{"i":"390ddaad6968f664","q":"Given an acute triangle  $ABC$  with  $AB>AC$  that has circumcenter  $O$ . Line  $BO$  and  $CO$  meet the bisector of  $\\angle BAC$  at  $P$  and  $Q$ , respectively. Moreover, line  $BQ$  and  $CP$  meet at  $R$ . Show that  $AR$  is perpendicular to  $BC$ .\n\n*Proposer: Soewono and Fajar Yuliawan*","t":[{"b":1,"e":1.0,"k":"rising","v":0.65179,"x":1.0,"p":[[0,94,0.0,0.65179,0.4164,0.14286,0.85714,1.0,0.0,1.0,5,15,1,5,0,5,0,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,15],[4,94,0.0426,0.94643,0.17405,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[8,94,0.0851,0.9375,0.2257,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[12,94,0.1277,0.88839,0.29824,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[16,94,0.1702,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,94,0.2128,0.92857,0.20203,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,26],[24,94,0.2553,0.92857,0.21129,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[28,94,0.2979,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,94,0.3404,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,94,0.383,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[40,94,0.4255,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[44,94,0.4681,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,28],[48,94,0.5106,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[52,94,0.5532,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[56,94,0.5957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,94,0.6383,0.98212,0.07792,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[64,94,0.6809,0.92411,0.22864,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,28],[68,94,0.7234,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[72,94,0.766,0.95089,0.14555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[76,94,0.8085,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,94,0.8511,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[84,94,0.8936,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[88,94,0.9362,0.95982,0.1439,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[92,94,0.9787,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[94,94,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"rising","v":0.59375,"x":0.95981,"p":[[0,179,0.0,0.59375,0.43317,0.10714,0.78571,1.0,0.0,1.0,8,14,3,8,0,2,0,0,3,0,0,0,0,0,0,0,0,3,0,0,2,0,14],[4,179,0.0223,0.93304,0.24218,1.0,1.0,1.0,0.0,1.0,2,29,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[8,179,0.0447,0.95089,0.17717,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[12,179,0.067,0.83929,0.33834,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[16,179,0.0894,0.90179,0.25364,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,26],[20,179,0.1117,0.85714,0.32341,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,26],[24,179,0.1341,0.85267,0.30615,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,22],[28,179,0.1564,0.80357,0.37754,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,25],[32,179,0.1788,0.84375,0.34508,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,26],[36,179,0.2011,0.84821,0.31326,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[40,179,0.2235,0.87945,0.26514,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,25],[44,179,0.2458,0.78124,0.32534,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,3,0,0,1,0,0,0,0,0,3,0,0,4,0,18],[48,179,0.2682,0.90178,0.23266,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,25],[52,179,0.2905,0.76338,0.36876,0.5354,1.0,1.0,0.0,1.0,4,21,0,4,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,21],[56,179,0.3128,0.8125,0.31831,0.67857,1.0,1.0,0.0,1.0,1,22,0,1,0,3,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,22],[60,179,0.3352,0.9375,0.16342,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,26],[64,179,0.3575,0.87051,0.27286,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,25],[68,179,0.3799,0.74552,0.3707,0.53539,1.0,1.0,0.0,1.0,3,19,0,3,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,19],[72,179,0.4022,0.91518,0.22548,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,27],[76,179,0.4246,0.86607,0.26949,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,4,0,23],[80,179,0.4469,0.90179,0.27994,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[84,179,0.4693,0.83034,0.29762,0.82143,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,3,0,21],[88,179,0.4916,0.86161,0.29771,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,25],[92,179,0.514,0.79464,0.36759,0.82143,1.0,1.0,0.0,1.0,4,23,0,4,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,23],[96,179,0.5363,0.85714,0.27199,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,0,1,0,23],[100,179,0.5587,0.80357,0.33834,0.82132,1.0,1.0,0.0,1.0,3,21,0,3,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,21],[104,179,0.581,0.73214,0.36727,0.39286,1.0,1.0,0.0,1.0,2,19,0,2,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,0,2,0,19],[108,179,0.6034,0.86158,0.29122,0.96425,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,24],[112,179,0.6257,0.80356,0.31895,0.57143,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,22],[116,179,0.648,0.80354,0.29615,0.57132,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,0,4,0,19],[120,179,0.6704,0.81249,0.29112,0.82143,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,7,0,17],[124,179,0.6927,0.86161,0.31234,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,25],[128,179,0.7151,0.86161,0.29121,0.96429,1.0,1.0,0.0,1.0,2,24,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,24],[132,179,0.7374,0.86605,0.29221,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,25],[136,179,0.7598,0.83033,0.30608,0.78561,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,0,1,0,23],[140,179,0.7821,0.85713,0.31136,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,26],[144,179,0.8045,0.88837,0.23349,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,24],[148,179,0.8268,0.87946,0.28596,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,25],[152,179,0.8492,0.75445,0.38835,0.49968,1.0,1.0,0.0,1.0,5,21,0,5,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,21],[156,179,0.8715,0.91071,0.20748,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,26],[160,179,0.8939,0.92409,0.21424,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[164,179,0.9162,0.77678,0.35344,0.67857,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,2,0,20],[168,179,0.9385,0.89731,0.22086,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,25],[172,179,0.9609,0.90622,0.19764,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,24],[176,179,0.9832,0.89283,0.21133,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,4,0,22],[179,179,1.0,0.95981,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25]]}]},{"i":"22e72a5dabf862e8","q":"Find all  $n>1$  and integers  $a_1,a_2,\\dots,a_n$  satisfying the following three conditions:\n(i)  $2<a_1\\le a_2\\le \\cdots\\le a_n$ (ii)  $a_1,a_2,\\dots,a_n$  are divisors of  $15^{25}+1$ .\n(iii)  $2-\\frac{2}{15^{25}+1}=\\left(1-\\frac{2}{a_1}\\right)+\\left(1-\\frac{2}{a_2}\\right)+\\cdots+\\left(1-\\frac{2}{a_n}\\right)$","t":[{"b":4,"e":1.0,"k":"flat","v":0.94195,"x":0.99554,"p":[[0,110,0.0,0.96428,0.0945,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[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],[12,110,0.1091,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,110,0.1455,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,110,0.1818,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,110,0.2182,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,110,0.2545,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,110,0.2909,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,110,0.3273,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,110,0.3636,0.97767,0.08074,1.0,1.0,1.0,0.5714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[44,110,0.4,0.94195,0.133,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[48,110,0.4364,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[52,110,0.4727,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[56,110,0.5091,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[60,110,0.5455,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[64,110,0.5818,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[68,110,0.6182,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,110,0.6545,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,25],[76,110,0.6909,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[80,110,0.7273,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[84,110,0.7636,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[88,110,0.8,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[92,110,0.8364,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[96,110,0.8727,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[100,110,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[104,110,0.9455,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[108,110,0.9818,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[110,110,1.0,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27]]},{"b":7,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,88,0.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[4,88,0.0455,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,88,0.0909,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,88,0.1364,0.95089,0.18073,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[16,88,0.1818,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,88,0.2273,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,88,0.2727,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[28,88,0.3182,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[32,88,0.3636,0.9598,0.10858,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[36,88,0.4091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,88,0.4545,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,88,0.5,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,88,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,88,0.5909,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[56,88,0.6364,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,88,0.6818,0.94643,0.1915,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[64,88,0.7273,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[68,88,0.7727,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[72,88,0.8182,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[76,88,0.8636,0.95981,0.09611,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[80,88,0.9091,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[84,88,0.9545,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[88,88,1.0,0.95982,0.13474,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28]]}]},{"i":"2337c61eccbe6d88","q":"Let $a, b, c$ be positive real numbers. Prove that\n\n$$\n\\frac{2 a}{a^{2}+b c}+\\frac{2 b}{b^{2}+c a}+\\frac{2 c}{c^{2}+a b} \\leq \\frac{a}{b c}+\\frac{b}{c a}+\\frac{c}{a b} .\n$$","t":[{"b":2,"e":1.0,"k":"flat","v":0.46875,"x":0.63393,"p":[[0,19,0.0,0.54911,0.4448,0.0,0.42857,1.0,0.0,1.0,10,15,0,10,0,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,15],[4,19,0.2105,0.63393,0.43439,0.21429,1.0,1.0,0.0,1.0,8,18,0,8,0,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,18],[8,19,0.4211,0.46875,0.43775,0.0,0.42857,1.0,0.0,1.0,12,12,0,12,0,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,12],[12,19,0.6316,0.55804,0.45926,0.0,0.7143,1.0,0.0,1.0,11,16,0,11,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,16],[16,19,0.8421,0.53571,0.46015,0.0,0.5,1.0,0.0,1.0,12,15,0,12,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,15],[19,19,1.0,0.62054,0.36001,0.42857,0.42857,1.0,0.0,1.0,4,14,0,4,0,0,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,14]]},{"b":6,"e":1.0,"k":"rising","v":0.48661,"x":1.0,"p":[[0,64,0.0,0.50446,0.45032,0.0,0.42857,1.0,0.0,1.0,12,13,0,12,0,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,13],[4,64,0.0625,0.48661,0.45995,0.0,0.35714,1.0,0.0,1.0,13,13,0,13,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,13],[8,64,0.125,0.68304,0.3908,0.28571,1.0,1.0,0.0,1.0,4,18,0,4,0,1,0,0,4,0,0,4,0,0,0,0,0,0,0,0,1,0,18],[12,64,0.1875,0.54911,0.44047,0.0,0.42857,1.0,0.0,1.0,9,15,0,9,0,0,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,15],[16,64,0.25,0.51339,0.44872,0.0,0.42857,1.0,0.0,1.0,11,14,0,11,0,0,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,14],[20,64,0.3125,0.55804,0.45926,0.0,0.7143,1.0,0.0,1.0,11,16,0,11,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,16],[24,64,0.375,0.53125,0.47814,0.0,0.71429,1.0,0.0,1.0,13,16,0,13,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,16],[28,64,0.4375,0.91518,0.26693,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[32,64,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,64,0.5625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,64,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,64,0.6875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,64,0.75,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,64,0.8125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,64,0.875,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,64,0.9375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,64,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"44b1b2911cb16e95","q":"Isaac repeatedly flips a fair coin. Whenever a particular face appears for the  $2n+1$ th time, for any nonnegative integer  $n$ , he earns a point. The expected number of flips it takes for Isaac to get  $10$  points is  $\\tfrac ab$  for coprime positive integers  $a$  and  $b$ . Find  $a + b$ .\n\n*Proposed by Isaac Chen*","t":[{"b":2,"e":0.14286,"k":"flat","v":0.04911,"x":0.12054,"p":[[0,82,0.0,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,13,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,82,0.0488,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,82,0.0976,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,82,0.1463,0.10714,0.10714,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,20,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,82,0.1951,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,82,0.2439,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,82,0.2927,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,82,0.3415,0.08473,0.07867,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,82,0.3902,0.04911,0.06786,0.0,0.0,0.14286,0.0,0.143,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,82,0.439,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,82,0.4878,0.10268,0.08171,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,82,0.5366,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,82,0.5854,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,82,0.6341,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,82,0.6829,0.12054,0.0724,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,82,0.7317,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,82,0.7805,0.08473,0.08639,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,82,0.8293,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,82,0.878,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,82,0.9268,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,82,0.9756,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[82,82,1.0,0.07589,0.07973,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.05804,"x":0.09821,"p":[[0,46,0.0,0.08482,0.11214,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,46,0.087,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.08929,0.09943,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"68605041bdd12ffd","q":"Mr. Murgatroyd decides to throw his class a pizza party, but he's going to make them hunt for it first. He chooses eleven locations in the school, which we'll call  $1, 2, \\ldots, 11$ . His plan is to tell students to start at location  $1$ , and at each location  $n$  from  $1$  to  $10$ , they will find a  message directing them to go to location  $n+1$ ; at location  $11$ , there's pizza!\n\nMr. Murgatroyd sends his teaching assistant to post the ten messages in locations  $1$  to  $10$ . Unfortunately, the assistant jumbles up the message cards at random before posting them. If the students begin at location  $1$  as planned and follow the directions at each location, show that they will still get to the pizza.","t":[{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"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],[4,25,0.16,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,25,0.32,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,25,0.64,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,25,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":4,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"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],[4,26,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"8b5e233f8798151f","q":"Let  $ABCD$  be an isosceles trapezoid with  $AD\\parallel BC$  and  $BC>AD$  such that the distance between the incenters of  $\\triangle ABC$  and  $\\triangle DBC$  is  $16$ .  If the perimeters of  $ABCD$  and  $ABC$  are  $120$  and  $114$  respectively, then the area of  $ABCD$  can be written as  $m\\sqrt n,$  where  $m$  and  $n$  are positive integers with  $n$  not divisible by the square of any prime.  Find  $100m+n$ .\n\n*Proposed by David Altizio and Evan Chen*","t":[{"b":0,"e":0.71429,"k":"flat","v":0.67409,"x":0.73661,"p":[[0,30,0.0,0.72768,0.16888,0.71429,0.71429,0.75,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,5,0,3],[4,30,0.1333,0.70982,0.16164,0.71429,0.71429,0.71429,0.0,1.0,1,2,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,24,0,0,3,0,2],[8,30,0.2667,0.67409,0.17216,0.71429,0.71429,0.71429,0.0,0.85714,1,0,1,1,0,0,0,0,1,0,0,2,0,0,2,0,0,21,0,0,5,0,0],[12,30,0.4,0.6875,0.13092,0.71429,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,1,0,0],[16,30,0.5333,0.70982,0.08364,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,27,0,0,3,0,0],[20,30,0.6667,0.70982,0.10999,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,24,0,0,5,0,0],[24,30,0.8,0.69643,0.09279,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,26,0,0,2,0,0],[28,30,0.9333,0.71875,0.05629,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0],[30,30,1.0,0.73661,0.05187,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0]]},{"b":3,"e":0.71429,"k":"flat","v":0.71874,"x":0.75446,"p":[[0,70,0.0,0.75446,0.14827,0.71429,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,16,0,0,2,0,7],[4,70,0.0571,0.73661,0.16016,0.71429,0.71429,0.71429,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,4,0,3],[8,70,0.1143,0.75446,0.22934,0.71429,0.71429,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,3,0,8],[12,70,0.1714,0.71875,0.06667,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,0,3,0,0],[16,70,0.2286,0.73214,0.08564,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,0,4,0,1],[20,70,0.2857,0.72321,0.04971,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1],[24,70,0.3429,0.74107,0.06622,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,4,0,1],[28,70,0.4,0.72768,0.04164,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0],[32,70,0.4571,0.73661,0.06298,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,3,0,1],[36,70,0.5143,0.73661,0.05187,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0],[40,70,0.5714,0.72321,0.03458,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0],[44,70,0.6286,0.73214,0.04725,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,4,0,0],[48,70,0.6857,0.74107,0.05576,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0],[52,70,0.7429,0.73214,0.04725,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,4,0,0],[56,70,0.8,0.72768,0.05486,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,4,0,0],[60,70,0.8571,0.73661,0.05187,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0],[64,70,0.9143,0.71874,0.05633,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0],[68,70,0.9714,0.72767,0.04164,0.71429,0.71429,0.71429,0.714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0],[70,70,1.0,0.72321,0.03458,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0]]}]},{"i":"897f4e7c060579f1","q":"Suppose  $f:\\mathbb{R}\\to\\mathbb{R}$  is a differentiable function for which the inequality  $f'(x) \\leq f'(x+\\frac{1}{n})$  holds for every  $x\\in\\mathbb{R}$  and every  $n\\in\\mathbb{N}$ .Prove that f is continiously differentiable","t":[{"b":4,"e":0.71429,"k":"flat","v":0.53125,"x":0.64284,"p":[[0,19,0.0,0.54907,0.15612,0.42857,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,12,0,0,7,0,0,10,0,0,1,0,0],[4,19,0.2105,0.64284,0.14726,0.57132,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,19,0,0,1,0,1],[8,19,0.4211,0.58033,0.13802,0.42857,0.57143,0.71429,0.42857,0.857,0,0,0,0,0,0,0,0,0,0,0,13,0,0,5,0,0,13,0,0,1,0,0],[12,19,0.6316,0.56471,0.14326,0.42857,0.57121,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,1,12,0,0,4,0,0,14,0,0,0,0,0],[16,19,0.8421,0.61157,0.11972,0.571,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,14,0,0,1,0,0],[19,19,1.0,0.53125,0.15663,0.42857,0.57141,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,12,0,0,8,0,0,8,0,0,1,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.54013,"x":0.91293,"p":[[0,26,0.0,0.54013,0.22512,0.42857,0.57121,0.71429,0.0,1.0,2,1,1,2,0,0,0,0,5,0,0,5,0,0,7,0,0,11,0,0,1,0,1],[4,26,0.1538,0.61607,0.16823,0.48215,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,6,0,2,5,0,0,15,0,0,0,0,2],[8,26,0.3077,0.66963,0.13571,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,21,0,0,0,0,2],[12,26,0.4615,0.65402,0.13959,0.67857,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,5,0,1,1,0,0,23,0,0,0,0,1],[16,26,0.6154,0.58926,0.18123,0.42857,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,6,0,0,5,0,0,16,0,0,0,0,1],[20,26,0.7692,0.88839,0.1665,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,3,0,20],[24,26,0.9231,0.85268,0.16554,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,3,0,16],[26,26,1.0,0.91293,0.15022,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,1,22]]}]},{"i":"6f44e2b11316ca18","q":"Let  $M$  and  $I$  be the centroid and the incenter of a scalene triangle  $ABC$ , and let  $r$  be its inradius. Prove that  $MI = r/3$  if and only if  $MI$  is perpendicular to one of the sides of the triangle.\n\n(A.Karlyuchenko)","t":[{"b":2,"e":0.14286,"k":"flat","v":0.09375,"x":0.34822,"p":[[0,75,0.0,0.24549,0.14402,0.14286,0.2857,0.42857,0.0,0.43,4,0,1,4,0,10,0,0,9,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.20526,0.15543,0.14286,0.14286,0.2857,0.0,0.857,2,0,0,2,0,21,0,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[8,75,0.1067,0.2991,0.22687,0.14286,0.2857,0.42857,0.0,0.85714,3,0,0,3,0,12,0,0,5,0,0,9,0,0,0,0,0,0,0,0,3,0,0],[12,75,0.16,0.2232,0.18186,0.14286,0.14286,0.2857,0.0,0.85714,3,0,0,3,0,18,0,0,6,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[16,75,0.2133,0.18749,0.14476,0.14286,0.14286,0.2857,0.0,0.571,6,0,0,6,0,16,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,75,0.2667,0.22322,0.23128,0.14286,0.14286,0.17868,0.0,0.85714,5,0,0,5,0,19,0,0,2,0,0,3,0,0,0,0,0,0,0,0,3,0,0],[24,75,0.32,0.20982,0.19555,0.14286,0.14286,0.32143,0.0,0.857,7,0,0,7,0,15,0,0,2,0,0,6,0,0,1,0,0,0,0,0,1,0,0],[28,75,0.3733,0.20974,0.17494,0.14286,0.14286,0.28571,0.0,0.85714,5,0,0,5,0,16,0,0,5,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[32,75,0.4267,0.19196,0.15405,0.14286,0.14286,0.1786,0.0,0.857,3,0,0,3,0,21,0,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[36,75,0.48,0.28554,0.22879,0.14286,0.14286,0.42857,0.0,0.85714,2,0,0,2,0,17,0,0,1,0,0,9,0,0,0,0,0,0,0,0,3,0,0],[40,75,0.5333,0.19197,0.13651,0.14286,0.14286,0.28571,0.0,0.4286,5,0,0,5,0,17,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.19643,0.16656,0.10714,0.14286,0.32143,0.0,0.57143,8,0,0,8,0,13,0,0,3,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[48,75,0.64,0.34822,0.25238,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,10,0,0,7,0,0,8,0,0,0,0,0,0,0,0,5,0,0],[52,75,0.6933,0.2857,0.23418,0.14286,0.14286,0.42857,0.0,0.85714,3,0,0,3,0,14,0,0,6,0,0,4,0,0,2,0,0,0,0,0,3,0,0],[56,75,0.7467,0.21427,0.21426,0.14286,0.14286,0.2857,0.0,0.85714,6,0,0,6,0,17,0,0,3,0,0,3,0,0,1,0,0,0,0,0,2,0,0],[60,75,0.8,0.09375,0.11633,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,75,0.8533,0.24098,0.26594,0.0,0.14286,0.32143,0.0,0.85714,10,0,0,10,0,10,0,0,4,0,0,3,0,0,1,0,0,1,0,0,3,0,0],[68,75,0.9067,0.20964,0.14727,0.14286,0.14286,0.32143,0.0,0.4286,5,0,0,5,0,15,0,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.25875,0.14494,0.14286,0.14286,0.42857,0.0,0.57143,1,0,0,1,0,16,0,0,4,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[75,75,1.0,0.25,0.12372,0.14286,0.14288,0.42857,0.14286,0.4286,0,0,0,0,0,17,0,0,6,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.19641,"x":0.33036,"p":[[0,72,0.0,0.20982,0.14719,0.14286,0.14288,0.28571,0.0,0.4286,6,0,1,6,0,12,0,0,7,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.20527,0.16731,0.14286,0.14286,0.28571,0.0,0.85714,4,0,0,4,0,18,0,0,5,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[8,72,0.1111,0.24982,0.20524,0.14286,0.14286,0.28571,0.0,0.85714,3,0,0,3,0,16,0,0,6,0,0,4,0,0,1,0,0,0,0,0,2,0,0],[12,72,0.1667,0.24554,0.14827,0.14286,0.14286,0.42857,0.0,0.4286,3,0,0,3,0,14,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[16,72,0.2222,0.22768,0.18509,0.14286,0.14286,0.32143,0.0,0.85714,6,0,0,6,0,12,0,0,6,0,0,7,0,0,0,0,0,0,0,0,1,0,0],[20,72,0.2778,0.20982,0.14718,0.14286,0.14286,0.2857,0.0,0.71429,4,0,0,4,0,15,0,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[24,72,0.3333,0.26339,0.20857,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,10,0,0,8,0,0,7,0,0,0,0,0,0,0,0,2,0,0],[28,72,0.3889,0.33036,0.24074,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,9,0,0,7,0,0,9,0,0,0,0,0,0,0,0,4,0,0],[32,72,0.4444,0.27669,0.2081,0.14286,0.2857,0.42857,0.0,0.85714,4,0,0,4,0,11,0,0,6,0,0,9,0,0,0,0,0,0,0,0,2,0,0],[36,72,0.5,0.25447,0.17029,0.14286,0.1429,0.32143,0.0,0.85714,1,0,0,1,0,17,0,0,6,0,0,6,0,0,1,0,0,0,0,0,1,0,0],[40,72,0.5556,0.19641,0.12749,0.14286,0.14286,0.28571,0.0,0.571,3,0,0,3,0,19,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[44,72,0.6111,0.21428,0.18209,0.14286,0.14286,0.2857,0.0,0.857,4,0,0,4,0,17,0,0,7,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[48,72,0.6667,0.25447,0.21646,0.14286,0.14286,0.42857,0.0,0.85714,5,0,0,5,0,14,0,0,2,0,0,9,0,0,0,0,0,0,0,0,2,0,0],[52,72,0.7222,0.26786,0.23623,0.14286,0.14286,0.42857,0.0,0.85714,5,0,0,5,0,13,0,0,4,0,0,7,0,0,0,0,0,0,0,0,3,0,0],[56,72,0.7778,0.23652,0.17721,0.14286,0.14288,0.28571,0.0,0.85714,3,0,0,3,0,16,0,0,6,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[60,72,0.8333,0.21875,0.18552,0.14286,0.14286,0.28571,0.0,0.85714,5,0,0,5,0,16,0,0,4,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[64,72,0.8889,0.20982,0.13355,0.14286,0.14288,0.28571,0.0,0.42857,5,0,0,5,0,12,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.26339,0.19269,0.14286,0.21428,0.28571,0.0,0.85714,2,0,0,2,0,14,0,0,9,0,0,5,0,0,0,0,0,0,0,0,2,0,0],[72,72,1.0,0.26786,0.16269,0.14286,0.14286,0.42857,0.14286,0.85714,0,0,0,0,0,17,0,0,5,0,0,9,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"7c925ee8ff165902","q":"A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?","t":[{"b":1,"e":0.28571,"k":"falling","v":0.22312,"x":1.0,"p":[[0,51,0.0,0.83036,0.35792,1.0,1.0,1.0,0.0,1.0,4,26,1,4,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[4,51,0.0784,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,51,0.1569,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,51,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,51,0.3137,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,51,0.3922,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,51,0.4706,0.90625,0.29148,1.0,1.0,1.0,0.0,1.0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[28,51,0.549,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[32,51,0.6275,0.93304,0.24218,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[36,51,0.7059,0.9375,0.20183,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[40,51,0.7843,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[44,51,0.8627,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[48,51,0.9412,0.42848,0.45181,0.0,0.2143,1.0,0.0,1.0,12,12,0,12,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[51,51,1.0,0.22312,0.31532,0.0,0.14286,0.28571,0.0,1.0,14,4,0,14,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,4]]},{"b":4,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,31,0.0,0.91071,0.27837,1.0,1.0,1.0,0.0,1.0,2,29,2,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[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],[8,31,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,31,0.3871,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[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],[24,31,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"be50bdd02e7cb6bf","q":"Let  $f : \\mathbb{N} \\longrightarrow \\mathbb{N}$  be a function such that\n(a)  $ f(m) < f(n)$  whenever  $m < n$ .\n(b)  $f(2n) = f(n) + n$  for all  $n \\in \\mathbb{N}$ .\n(c)  $n$  is prime whenever  $f(n)$  is prime.\nFind   $$ \\sum_{n=1}^{2022} f(n). $$","t":[{"b":4,"e":0.57143,"k":"falling","v":0.61159,"x":0.95536,"p":[[0,110,0.0,0.9375,0.17835,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[4,110,0.0364,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[8,110,0.0727,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[12,110,0.1091,0.93304,0.12869,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[16,110,0.1455,0.9375,0.17474,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[20,110,0.1818,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[24,110,0.2182,0.91518,0.17445,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,24],[28,110,0.2545,0.95089,0.12682,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[32,110,0.2909,0.88839,0.21349,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,0,1,0,24],[36,110,0.3273,0.91964,0.18189,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,4,0,24],[40,110,0.3636,0.88839,0.20119,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,3,0,22],[44,110,0.4,0.91071,0.12753,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,11,0,18],[48,110,0.4364,0.83482,0.19597,0.67857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,0,7,0,15],[52,110,0.4727,0.89731,0.18295,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,4,0,22],[56,110,0.5091,0.85268,0.21275,0.82143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,1,0,0,6,0,18],[60,110,0.5455,0.80357,0.22798,0.57143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,9,0,0,2,0,0,3,0,16],[64,110,0.5818,0.85714,0.20825,0.78571,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,0,0,0,5,0,19],[68,110,0.6182,0.91518,0.16698,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,3,0,24],[72,110,0.6545,0.85267,0.20042,0.57143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,0,1,0,20],[76,110,0.6909,0.89286,0.17128,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,21],[80,110,0.7273,0.89732,0.16457,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,5,0,21],[84,110,0.7636,0.87946,0.16409,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,7,0,18],[88,110,0.8,0.88839,0.1665,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,5,0,20],[92,110,0.8364,0.875,0.16656,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,10,0,16],[96,110,0.8727,0.86607,0.19212,0.82143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,6,0,18],[100,110,0.9091,0.71875,0.25376,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,5,0,0,1,0,0,7,0,0,4,0,0,5,0,10],[104,110,0.9455,0.78125,0.22012,0.57143,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,6,0,0,3,0,0,10,0,10],[108,110,0.9818,0.61159,0.1996,0.53539,0.57143,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,3,0,0,11,0,0,5,0,0,7,0,1],[110,110,1.0,0.63391,0.26472,0.39286,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,8,0,0,2,0,0,9,0,0,0,0,0,7,0,6]]},{"b":5,"e":0.85714,"k":"flat","v":0.92857,"x":1.0,"p":[[0,74,0.0,0.92857,0.13832,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24],[4,74,0.0541,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,74,0.1081,0.95982,0.10248,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[12,74,0.1622,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,74,0.2162,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,74,0.2703,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,74,0.3243,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,74,0.3784,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,74,0.4324,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[36,74,0.4865,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[40,74,0.5405,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[44,74,0.5946,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,74,0.6486,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[52,74,0.7027,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[56,74,0.7568,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[60,74,0.8108,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[64,74,0.8649,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[68,74,0.9189,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,27],[72,74,0.973,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[74,74,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"e59a3867e0385ef7","q":"The vertices of a regular  $2005$ -gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color. \n(a) Prove that there exists a \ufb01nite sequence of allowed recolorings after which all the vertices are of the same color. \n(b) Is that color uniquely determined by the initial coloring?","t":[{"b":6,"e":0.71429,"k":"rising","v":0.62054,"x":1.0,"p":[[0,33,0.0,0.62054,0.09182,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,8,0,0,0,0,1],[4,33,0.1212,0.93304,0.18552,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[8,33,0.2424,0.90624,0.13651,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,18],[12,33,0.3636,0.91071,0.18814,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,21],[16,33,0.4848,0.91071,0.13717,0.85711,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,20],[20,33,0.6061,0.94196,0.1551,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,24],[24,33,0.7273,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[28,33,0.8485,0.9375,0.17105,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[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],[33,33,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"rising","v":0.61158,"x":0.98214,"p":[[0,51,0.0,0.61158,0.06425,0.57143,0.57143,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,9,0,0,0,0,0],[4,51,0.0784,0.91057,0.17783,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,22],[8,51,0.1569,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[12,51,0.2353,0.9241,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19],[16,51,0.3137,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[20,51,0.3922,0.94195,0.12303,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[24,51,0.4706,0.91071,0.18472,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,23],[28,51,0.549,0.93748,0.11263,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[32,51,0.6275,0.94194,0.11778,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,24],[36,51,0.7059,0.91963,0.18537,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,22],[40,51,0.7843,0.94196,0.13296,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,8,0,23],[44,51,0.8627,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[48,51,0.9412,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[51,51,1.0,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]}]},{"i":"83a15b30584949dd","q":"Let  $n$  be a positive integer and  $A=\\{ 1,2,\\ldots ,n\\}$ . A subset of  $A$  is said to be connected if it consists of one element or several consecutive elements. Determine the maximum  $k$  for which there exist  $k$  distinct subsets of  $A$  such that the intersection of any two of them is connected.","t":[{"b":4,"e":0.85714,"k":"rising","v":0.54909,"x":0.95089,"p":[[0,29,0.0,0.54909,0.05186,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0],[4,29,0.1379,0.95089,0.13175,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[8,29,0.2759,0.91071,0.14617,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,22],[12,29,0.4138,0.90179,0.14032,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,20],[16,29,0.5517,0.88393,0.14032,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,19],[20,29,0.6897,0.87946,0.14334,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,2,0,18],[24,29,0.8276,0.83481,0.14334,0.71429,0.857,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,5,0,12],[28,29,0.9655,0.8125,0.12595,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,4,0,9],[29,29,1.0,0.87946,0.13882,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,1,0,18]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.84375,"p":[[0,33,0.0,0.53571,0.06186,0.53571,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0],[4,33,0.1212,0.84375,0.21829,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,4,0,17],[8,33,0.2424,0.82589,0.23619,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,2,0,17],[12,33,0.3636,0.79909,0.2283,0.71429,0.78571,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,2,0,14],[16,33,0.4848,0.82589,0.1665,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,11,0,0,4,0,13],[20,33,0.6061,0.58482,0.39344,0.14286,0.71429,1.0,0.0,1.0,7,10,0,7,0,2,0,0,2,0,0,0,0,0,2,0,0,7,0,0,2,0,10],[24,33,0.7273,0.33482,0.43684,0.0,0.0,0.85714,0.0,1.0,19,7,0,19,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,7],[28,33,0.8485,0.10267,0.24801,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[32,33,0.9697,0.11161,0.28288,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[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]]}]},{"i":"2a428a994b9d28b8","q":"Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\\operatorname{deg} f>\\operatorname{deg} g$. Suppose that for infinitely many primes $p$ the polynomial $p f+g$ has a rational root. Prove that $f$ has a rational root.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.30804,"x":0.63393,"p":[[0,26,0.0,0.30804,0.24251,0.14286,0.28571,0.42857,0.0,0.85714,5,0,2,5,0,8,0,0,8,0,0,6,0,0,0,0,0,3,0,0,2,0,0],[4,26,0.1538,0.51785,0.31083,0.28571,0.42857,0.85704,0.0,1.0,2,4,0,2,0,4,0,0,6,0,0,5,0,0,3,0,0,3,0,0,5,0,4],[8,26,0.3077,0.53116,0.30989,0.28571,0.42857,0.85714,0.14,1.0,0,4,0,0,0,6,0,0,8,0,0,3,0,0,1,0,0,4,0,0,6,0,4],[12,26,0.4615,0.63393,0.2878,0.39286,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,6,0,0,4,0,0,4,0,0,3,0,0,6,0,7],[16,26,0.6154,0.47768,0.29148,0.28571,0.42857,0.75,0.14286,1.0,0,4,0,0,0,6,0,0,8,0,0,8,0,0,1,0,0,1,0,0,4,0,4],[20,26,0.7692,0.41517,0.22405,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,21,0,0,5,0,0,0,0,0,2,0,0,2,0,2],[24,26,0.9231,0.41965,0.23128,0.28571,0.28586,0.42858,0.14286,0.85714,0,0,0,0,0,4,0,0,13,0,0,8,0,0,1,0,0,0,0,0,6,0,0],[26,26,1.0,0.3125,0.06622,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.02232,"x":0.54909,"p":[[0,40,0.0,0.35705,0.26494,0.14286,0.28571,0.46429,0.0,1.0,3,2,1,3,0,7,0,0,11,0,0,3,0,0,3,0,0,2,0,0,1,0,2],[4,40,0.1,0.54909,0.29903,0.28571,0.64286,0.74996,0.0,1.0,2,3,0,2,0,3,0,0,6,0,0,3,0,0,2,0,0,8,0,0,5,0,3],[8,40,0.2,0.35714,0.28121,0.14286,0.28571,0.57143,0.0,0.85714,7,0,0,7,0,4,0,0,7,0,0,4,0,0,3,0,0,4,0,0,3,0,0],[12,40,0.3,0.22768,0.2942,0.0,0.14286,0.28571,0.0,1.0,13,2,0,13,0,8,0,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,2],[16,40,0.4,0.19197,0.22192,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,11,0,0,4,0,0,4,0,0,0,0,0,0,0,0,2,0,0],[20,40,0.5,0.25447,0.30248,0.0,0.14286,0.42857,0.0,1.0,12,2,0,12,0,9,0,0,1,0,0,3,0,0,2,0,0,3,0,0,0,0,2],[24,40,0.6,0.15179,0.19212,0.0,0.07143,0.2857,0.0,0.71429,16,0,0,16,0,6,0,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[28,40,0.7,0.12947,0.14446,0.0,0.14286,0.2857,0.0,0.4286,15,0,0,15,0,8,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.18749,0.2185,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,10,0,0,4,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[36,40,0.9,0.10268,0.11425,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,12,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"70154d8a82911c30","q":"We are given a square  $ ABCD$ . Let  $ P$  be a point not equal to a corner of the square or to its center  $ M$ . For any such  $ P$ , we let  $ E$  denote the common point of the lines  $ PD$  and  $ AC$ , if such a point exists. Furthermore, we let  $ F$  denote the common point of the lines  $ PC$  and  $ BD$ , if such a point exists. All such points  $ P$ , for which  $ E$  and  $ F$  exist are called acceptable points. Determine the set of all acceptable points, for which the line  $ EF$  is parallel to  $ AD$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.82589,"x":0.87945,"p":[[0,19,0.0,0.83035,0.15335,0.71429,0.85707,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,5,0,12],[4,19,0.2105,0.84375,0.16506,0.71429,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,4,0,14],[8,19,0.4211,0.83481,0.13882,0.71429,0.85707,1.0,0.4286,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,9,0,10],[12,19,0.6316,0.82589,0.1504,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,3,0,12],[16,19,0.8421,0.87053,0.12556,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,7,0,14],[19,19,1.0,0.87945,0.12931,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,5,0,16]]},{"b":6,"e":0.71429,"k":"flat","v":0.7991,"x":0.88839,"p":[[0,77,0.0,0.8482,0.12846,0.71429,0.857,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,6,0,12],[4,77,0.0519,0.88839,0.12745,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,5,0,17],[8,77,0.1039,0.86169,0.12619,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,7,0,13],[12,77,0.1558,0.7991,0.20158,0.71429,0.78564,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,6,0,10],[16,77,0.2078,0.81249,0.14913,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,2,0,11],[20,77,0.2597,0.79911,0.14664,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,0,1,0,10],[24,77,0.3117,0.82142,0.14725,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,4,0,11],[28,77,0.3636,0.80357,0.15465,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,15,0,0,6,0,9],[32,77,0.4156,0.82588,0.12234,0.71429,0.78564,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,7,0,9],[36,77,0.4675,0.86147,0.13607,0.71429,0.85714,1.0,0.71,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,3,0,15],[40,77,0.5195,0.83035,0.12595,0.71429,0.78564,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,6,0,10],[44,77,0.5714,0.83928,0.13243,0.71429,0.78564,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,4,0,12],[48,77,0.6234,0.84821,0.13333,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,4,0,13],[52,77,0.6753,0.80804,0.14555,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,3,0,10],[56,77,0.7273,0.83008,0.1262,0.71429,0.78564,1.0,0.71,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,6,0,10],[60,77,0.7792,0.8616,0.13115,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,5,0,14],[64,77,0.8312,0.83928,0.12242,0.71429,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,8,0,10],[68,77,0.8831,0.88392,0.12595,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,6,0,16],[72,77,0.9351,0.85267,0.13116,0.71429,0.85714,1.0,0.714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,5,0,13],[76,77,0.987,0.81696,0.1439,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,5,0,10],[77,77,1.0,0.84821,0.12846,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,6,0,12]]}]},{"i":"03f0ef657512b715","q":"For all positive integers  $n$ , denote by  $\\sigma(n)$  the sum of the positive divisors of  $n$  and  $\\nu_p(n)$  the largest power of  $p$  which divides  $n$ .  Compute the largest positive integer  $k$  such that  $5^k$  divides \\[\\sum_{d|N}\\nu_3(d!)(-1)^{\\sigma(d)},\\] where  $N=6^{1999}$ .\n\n*Proposed by David Altizio*","t":[{"b":0,"e":0.85714,"k":"flat","v":0.82143,"x":0.90179,"p":[[0,54,0.0,0.90179,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,12],[4,54,0.0741,0.82143,0.14725,0.85714,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,28,0,1],[8,54,0.1481,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[12,54,0.2222,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[16,54,0.2963,0.86161,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,3],[20,54,0.3704,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[24,54,0.4444,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[28,54,0.5185,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[32,54,0.5926,0.85714,0.05051,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,2],[36,54,0.6667,0.86161,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,3],[40,54,0.7407,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[44,54,0.8148,0.875,0.04725,0.85714,0.85714,0.85714,0.85714,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,4],[48,54,0.8889,0.85714,0.05051,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,2],[52,54,0.963,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[54,54,1.0,0.88839,0.05906,0.85714,0.85714,0.85714,0.85714,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7]]},{"b":7,"e":1.0,"k":"flat","v":0.87054,"x":1.0,"p":[[0,6,0.0,0.87054,0.1488,0.85714,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,9],[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":"d3cd02136b4c641c","q":"Let \\( N \\) be a natural number and \\( x_{1}, x_{2}, \\ldots, x_{n} \\) be other natural numbers less than \\( N \\) such that the least common multiple of any two of these \\( n \\) numbers is greater than \\( N \\).\nProve that the sum of the reciprocals of these \\( n \\) numbers is always less than 2; that is,\n\n$$\n\\frac{1}{x_{1}}+\\frac{1}{x_{2}}+\\cdots+\\frac{1}{x_{n}}<2\n$$","t":[{"b":3,"e":0.0,"k":"falling","v":0.06696,"x":0.98661,"p":[[0,60,0.0,0.47321,0.45237,0.0,0.42857,1.0,0.0,1.0,14,11,0,14,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,1,0,11],[4,60,0.0667,0.91518,0.24707,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[8,60,0.1333,0.89732,0.29284,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,28],[12,60,0.2,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,60,0.2667,0.93304,0.24218,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[20,60,0.3333,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[24,60,0.4,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[28,60,0.4667,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[32,60,0.5333,0.84375,0.36309,1.0,1.0,1.0,0.0,1.0,5,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[36,60,0.6,0.86607,0.33108,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[40,60,0.6667,0.71429,0.43595,0.21429,1.0,1.0,0.0,1.0,8,22,0,8,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,22],[44,60,0.7333,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[48,60,0.8,0.68304,0.4542,0.0,1.0,1.0,0.0,1.0,9,21,0,9,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,21],[52,60,0.8667,0.56697,0.49162,0.0,1.0,1.0,0.0,1.0,13,18,0,13,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[56,60,0.9333,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[60,60,1.0,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]},{"b":6,"e":0.0,"k":"falling","v":0.00446,"x":0.94643,"p":[[0,36,0.0,0.59821,0.44527,0.0,0.85714,1.0,0.0,1.0,9,15,0,9,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,15],[4,36,0.1111,0.94643,0.18123,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[8,36,0.2222,0.86161,0.30196,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,24],[12,36,0.3333,0.74107,0.3984,0.60714,1.0,1.0,0.0,1.0,6,20,0,6,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,20],[16,36,0.4444,0.50893,0.4883,0.0,0.64287,1.0,0.0,1.0,15,15,0,15,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,15],[20,36,0.5556,0.35714,0.46839,0.0,0.0,1.0,0.0,1.0,19,11,0,19,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[24,36,0.6667,0.13393,0.32915,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[28,36,0.7778,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,36,1.0,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"a7db2a281a2a252e","q":"For every positive integer  $n$ , denote by  $D_n$  the number of permutations  $(x_1, \\dots, x_n)$  of  $(1,2,\\dots, n)$  such that  $x_j\\neq j$  for every  $1\\le j\\le n$ . For  $1\\le k\\le \\frac{n}{2}$ , denote by  $\\Delta (n,k)$  the number of permutations  $(x_1,\\dots, x_n)$  of  $(1,2,\\dots, n)$  such that  $x_i=k+i$  for every  $1\\le i\\le k$  and  $x_j\\neq j$  for every  $1\\le j\\le n$ . Prove that  $$ \\Delta (n,k)=\\sum_{i=0}^{k=1} \\binom{k-1}{i} \\frac{D_{(n+1)-(k+i)}}{n-(k+i)} $$  (Proposed by Combinatorics; Ferdowsi University of Mashhad, Iran; Mirzavaziri)","t":[{"b":4,"e":0.571,"k":"flat","v":0.62051,"x":0.7946,"p":[[0,53,0.0,0.62942,0.25219,0.42857,0.71414,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,0,0,0,5,0,0,6,0,0,8,0,0,5,0,4],[4,53,0.0755,0.7946,0.19217,0.71429,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,11,0,0,3,0,12],[8,53,0.1509,0.70524,0.2449,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,1,0,0,6,0,0,12,0,0,2,0,8],[12,53,0.2264,0.62051,0.29148,0.5354,0.71429,0.85714,0.0,1.0,1,6,0,1,0,5,0,0,0,0,0,2,0,0,7,0,0,8,0,0,3,0,6],[16,53,0.3019,0.68295,0.20438,0.571,0.71429,0.85704,0.14286,1.0,0,4,0,0,0,2,0,0,0,0,0,1,0,0,10,0,0,10,0,0,5,0,4],[20,53,0.3774,0.74993,0.19887,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,1,0,0,8,0,0,7,0,0,8,0,7],[24,53,0.4528,0.71873,0.22725,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,0,0,0,0,0,0,6,0,0,8,0,0,11,0,4],[28,53,0.5283,0.73659,0.23177,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,0,0,0,1,0,0,4,0,0,13,0,0,4,0,8],[32,53,0.6038,0.68294,0.1948,0.571,0.71429,0.85704,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,0,0,0,11,0,0,10,0,0,6,0,3],[36,53,0.6792,0.683,0.18468,0.57143,0.71429,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,2,0,0,7,0,0,16,0,0,3,0,3],[40,53,0.7547,0.76338,0.14989,0.71429,0.71429,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,14,0,0,4,0,7],[44,53,0.8302,0.73658,0.15615,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,11,0,0,6,0,5],[48,53,0.9057,0.69201,0.15624,0.57143,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,11,0,0,3,0,4],[52,53,0.9811,0.73211,0.16271,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,13,0,0,5,0,5],[53,53,1.0,0.71424,0.14729,0.57143,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,16,0,0,3,0,4]]},{"b":6,"e":0.14,"k":"falling","v":0.22295,"x":0.73212,"p":[[0,38,0.0,0.50871,0.23659,0.42857,0.571,0.60607,0.0,1.0,3,1,0,3,0,2,0,0,1,0,0,5,0,0,13,0,0,6,0,0,1,0,1],[4,38,0.1053,0.72762,0.18685,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,13,0,0,4,0,6],[8,38,0.2105,0.73212,0.29398,0.57143,0.78571,1.0,0.0,1.0,2,11,0,2,0,2,0,0,0,0,0,0,0,0,5,0,0,7,0,0,5,0,11],[12,38,0.3158,0.6831,0.2416,0.67857,0.71429,0.75,0.0,1.0,1,5,0,1,0,2,0,0,1,0,0,0,0,0,4,0,0,16,0,0,3,0,5],[16,38,0.4211,0.70981,0.27313,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,1,0,0,0,0,0,4,0,0,10,0,0,4,0,9],[20,38,0.5263,0.59372,0.2769,0.42859,0.64286,0.71429,0.0,1.0,1,4,0,1,0,5,0,0,0,0,0,3,0,0,7,0,0,9,0,0,3,0,4],[24,38,0.6316,0.67405,0.25566,0.571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,2,0,0,1,0,0,5,0,0,12,0,0,2,0,7],[28,38,0.7368,0.59374,0.31157,0.42857,0.71429,0.75,0.0,1.0,3,5,0,3,0,4,0,0,0,0,0,2,0,0,5,0,0,10,0,0,3,0,5],[32,38,0.8421,0.65169,0.29452,0.53571,0.71429,0.85714,0.14,1.0,0,7,0,0,0,6,0,0,1,0,0,1,0,0,2,0,0,12,0,0,3,0,7],[36,38,0.9474,0.30354,0.21646,0.14286,0.14286,0.46418,0.14286,0.857,0,0,0,0,0,18,0,0,4,0,0,2,0,0,5,0,0,2,0,0,1,0,0],[38,38,1.0,0.22295,0.14713,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,20,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"c57212c298a8041a","q":"$m,n$  are given integer numbers such that  $m+n$  is an odd number. Edges of a complete bipartie graph  $K_{m,n}$  are labeled by  ${-1,1}$  such that the sum of all labels is  $0$ . Prove that there exists a spanning tree such that the sum of the labels of its edges is equal to  $0$ .\n\nProposed by Nima Dolatabadi","t":[{"b":1,"e":0.71429,"k":"flat","v":0.50893,"x":0.83481,"p":[[0,66,0.0,0.50893,0.3387,0.14289,0.42857,0.89286,0.0,1.0,1,8,0,1,0,8,0,0,3,0,0,8,0,0,1,0,0,2,0,0,1,0,8],[4,66,0.0606,0.74999,0.27895,0.57132,0.71429,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,5,0,0,3,0,0,7,0,0,0,0,15],[8,66,0.1212,0.75447,0.22083,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,4,0,0,2,0,0,13,0,0,1,0,11],[12,66,0.1818,0.73659,0.24252,0.53539,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,11,0,0,0,0,12],[16,66,0.2424,0.83481,0.20239,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,11,0,0,0,0,17],[20,66,0.303,0.73661,0.26513,0.42857,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,8,0,0,1,0,0,8,0,0,1,0,13],[24,66,0.3636,0.73659,0.17897,0.71429,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,16,0,0,2,0,7],[28,66,0.4242,0.80804,0.24382,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,9,0,0,2,0,16],[32,66,0.4848,0.70536,0.28107,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,5,0,0,1,0,0,9,0,0,2,0,11],[36,66,0.5455,0.76337,0.20082,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,13,0,0,1,0,11],[40,66,0.6061,0.76786,0.22232,0.71429,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,13,0,0,1,0,12],[44,66,0.6667,0.76786,0.1948,0.71429,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,15,0,0,2,0,10],[48,66,0.7273,0.76784,0.228,0.67857,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,10,0,0,1,0,13],[52,66,0.7879,0.75893,0.18363,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,15,0,0,0,0,10],[56,66,0.8485,0.64732,0.23686,0.53571,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,3,0,0,1,0,0,4,0,0,2,0,0,17,0,0,0,0,5],[60,66,0.9091,0.60713,0.17496,0.4286,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,11,0,0,1,0,0,17,0,0,1,0,1],[64,66,0.9697,0.64728,0.13358,0.571,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,21,0,0,0,0,1],[66,66,1.0,0.64284,0.12878,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,19,0,0,0,0,1]]},{"b":3,"e":1.0,"k":"rising","v":0.48213,"x":0.91517,"p":[[0,45,0.0,0.48213,0.34022,0.14286,0.42857,0.89286,0.0,1.0,1,8,0,1,0,9,0,0,3,0,0,9,0,0,1,0,0,0,0,0,1,0,8],[4,45,0.0889,0.77232,0.2809,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,3,0,0,1,0,0,6,0,0,2,0,16],[8,45,0.1778,0.67857,0.26245,0.42859,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,3,0,0,5,0,0,3,0,0,9,0,0,3,0,8],[12,45,0.2667,0.53569,0.26963,0.28571,0.42859,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,6,0,0,8,0,0,3,0,0,6,0,0,2,0,4],[16,45,0.3556,0.56247,0.26471,0.42857,0.571,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,2,0,0,10,0,0,4,0,0,7,0,0,2,0,4],[20,45,0.4444,0.58929,0.31288,0.42857,0.50001,1.0,0.0,1.0,2,9,0,2,0,2,0,0,2,0,0,10,0,0,2,0,0,5,0,0,0,0,9],[24,45,0.5333,0.63393,0.2549,0.53571,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,4,0,0,2,0,0,2,0,0,2,0,0,17,0,0,0,0,5],[28,45,0.6222,0.69196,0.24513,0.53572,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,3,0,0,4,0,0,2,0,0,12,0,0,2,0,8],[32,45,0.7111,0.76785,0.24936,0.53572,0.85707,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,6,0,0,3,0,14],[36,45,0.8,0.8125,0.18707,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,10,0,0,4,0,13],[40,45,0.8889,0.76775,0.25214,0.57143,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,1,0,0,6,0,0,6,0,0,5,0,12],[44,45,0.9778,0.91517,0.123,0.82132,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,21],[45,45,1.0,0.87946,0.13415,0.82143,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,9,0,15]]}]},{"i":"61b8abc59a137cc8","q":"A pentagon  $ABCDE$  is inscribed in a circle  $O$ , and satis\fes  $AB = BC , AE =  DE$ . The circle that is tangent to  $DE$  at  $E$  and passing  $A$  hits  $EC$  at  $F$  and  $BF$  at  $G (\\ne F)$ . Let  $DG\\cap O = H (\\ne D)$ . Prove that the tangent to  $O$  at  $E$  is perpendicular to  $HA$ .","t":[{"b":3,"e":0.0,"k":"falling","v":0.04464,"x":0.24554,"p":[[0,75,0.0,0.22759,0.36047,0.0,0.0,0.2857,0.0,1.0,19,4,0,19,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,4],[4,75,0.0533,0.24554,0.29284,0.0,0.21435,0.28571,0.0,1.0,14,1,0,14,0,2,0,0,10,0,0,0,0,0,1,0,0,2,0,0,2,0,1],[8,75,0.1067,0.1875,0.25111,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,2,0,0,8,0,0,0,0,0,3,0,0,0,0,0,2,0,0],[12,75,0.16,0.1607,0.21051,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,2,0,0,10,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[16,75,0.2133,0.14732,0.18723,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,75,0.2667,0.10268,0.1525,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,1,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,75,0.32,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,75,0.3733,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,75,0.4267,0.06251,0.1126,0.0,0.0,0.03571,0.0,0.286,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,75,0.48,0.09375,0.12682,0.0,0.0,0.2857,0.0,0.28571,20,0,0,20,0,3,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,75,0.5333,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.05357,0.14617,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,75,0.64,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,75,0.6933,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,75,0.7467,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,75,0.8,0.08036,0.12339,0.0,0.0,0.17857,0.0,0.28571,22,0,0,22,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,75,0.8533,0.08482,0.12807,0.0,0.0,0.2857,0.0,0.28571,22,0,0,22,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.12053,0.13882,0.0,0.0,0.28571,0.0,0.28571,18,0,0,18,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.11161,0.13235,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,3,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[75,75,1.0,0.07143,0.11845,0.0,0.0,0.14287,0.0,0.28571,23,0,0,23,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.04911,"x":0.21872,"p":[[0,85,0.0,0.07588,0.20817,0.0,0.0,0.0,0.0,1.0,27,1,3,27,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,85,0.0471,0.21872,0.22295,0.0,0.21428,0.28571,0.0,0.85714,12,0,0,12,0,4,0,0,10,0,0,1,0,0,4,0,0,0,0,0,1,0,0],[8,85,0.0941,0.1875,0.20957,0.0,0.14288,0.28571,0.0,0.857,14,0,0,14,0,3,0,0,11,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[12,85,0.1412,0.21427,0.22014,0.0,0.14288,0.28571,0.0,0.85714,11,0,0,11,0,6,0,0,10,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[16,85,0.1882,0.12946,0.15302,0.0,0.0,0.2857,0.0,0.57143,17,0,0,17,0,3,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,85,0.2353,0.17409,0.20431,0.0,0.14286,0.2857,0.0,0.85714,14,0,0,14,0,5,0,0,10,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[24,85,0.2824,0.17411,0.19144,0.0,0.14286,0.28571,0.0,0.85714,13,0,0,13,0,5,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[28,85,0.3294,0.10259,0.15248,0.0,0.0,0.17857,0.0,0.5714,20,0,0,20,0,4,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,85,0.3765,0.11152,0.1776,0.0,0.0,0.17857,0.0,0.85714,19,0,0,19,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[36,85,0.4235,0.07589,0.11836,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,85,0.4706,0.14284,0.22585,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,3,0,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[44,85,0.5176,0.14723,0.19061,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,2,0,0,11,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[48,85,0.5647,0.09821,0.1448,0.0,0.0,0.2857,0.0,0.4286,21,0,0,21,0,2,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,85,0.6118,0.15178,0.234,0.0,0.0,0.28571,0.0,0.85714,19,0,0,19,0,2,0,0,8,0,0,0,0,0,1,0,0,0,0,0,2,0,0],[56,85,0.6588,0.10713,0.21126,0.0,0.0,0.1429,0.0,1.0,22,1,0,22,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,85,0.7059,0.08929,0.14617,0.0,0.0,0.1786,0.0,0.57143,22,0,0,22,0,2,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,85,0.7529,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,85,0.8,0.10714,0.18556,0.0,0.0,0.2857,0.0,0.857,22,0,0,22,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[72,85,0.8471,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,85,0.8941,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,85,0.9412,0.08929,0.12242,0.0,0.0,0.1786,0.0,0.28571,20,0,0,20,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,85,0.9882,0.06687,0.11279,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[85,85,1.0,0.09821,0.13092,0.0,0.0,0.2857,0.0,0.28571,20,0,0,20,0,2,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"57fabf053897a735","q":"Let  $\\mathbb{N} =\\{1, 2, 3, ...\\}$  be the set of positive integers. Let  $f : \\mathbb{N} \\rightarrow \\mathbb{N}$  be a function that gives a positive integer value, to every positive integer. Suppose that  $f$  satisfies the following conditions: $f(1)=1$  $f(a+b+ab)=a+b+f(ab)$ Find the value of  $f(2015)$ Proposed by Jose Antonio Gomez Ortega","t":[{"b":2,"e":0.14286,"k":"falling","v":0.27679,"x":0.81696,"p":[[0,69,0.0,0.54464,0.35614,0.28571,0.42857,1.0,0.0,1.0,4,9,0,4,0,1,0,0,8,0,0,4,0,0,2,0,0,2,0,0,2,0,9],[4,69,0.058,0.69643,0.24936,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,8,0,0,8,0,0,0,0,0,6,0,9],[8,69,0.1159,0.81696,0.20277,0.67857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,9,0,13],[12,69,0.1739,0.70089,0.23787,0.57143,0.64286,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,4,0,0,11,0,0,3,0,0,5,0,8],[16,69,0.2319,0.74552,0.2135,0.57132,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,4,0,0,9,0,8],[20,69,0.2899,0.71427,0.23146,0.53539,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,4,0,0,5,0,9],[24,69,0.3478,0.62054,0.2101,0.42859,0.57143,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,9,0,0,11,0,0,3,0,0,4,0,4],[28,69,0.4058,0.74998,0.22869,0.57143,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,5,0,0,5,0,0,2,0,0,11,0,8],[32,69,0.4638,0.70535,0.24727,0.42857,0.85707,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,7,0,0,5,0,0,1,0,0,10,0,7],[36,69,0.5217,0.62933,0.3006,0.42857,0.64071,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,1,0,0,6,0,0,4,0,0,4,0,0,4,0,8],[40,69,0.5797,0.67856,0.21725,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,4,0,0,9,0,0,6,0,0,6,0,5],[44,69,0.6377,0.66517,0.2687,0.57143,0.71429,0.85714,0.0,1.0,3,4,0,3,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,0,8,0,4],[48,69,0.6957,0.63393,0.25489,0.42859,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,4,0,0,10,0,0,1,0,0,7,0,5],[52,69,0.7536,0.75,0.25254,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,1,0,0,6,0,12],[56,69,0.8116,0.63839,0.2575,0.42857,0.57143,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,7,0,0,8,0,0,2,0,0,7,0,5],[60,69,0.8696,0.59375,0.22619,0.42857,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,11,0,0,9,0,0,1,0,0,6,0,3],[64,69,0.9275,0.50893,0.3213,0.14286,0.42857,0.75,0.0,1.0,1,5,0,1,0,9,0,0,1,0,0,6,0,0,3,0,0,4,0,0,3,0,5],[68,69,0.9855,0.35268,0.2789,0.14286,0.21428,0.42857,0.0,1.0,1,2,0,1,0,15,0,0,2,0,0,8,0,0,1,0,0,0,0,0,3,0,2],[69,69,1.0,0.27679,0.24984,0.14286,0.14286,0.28571,0.0,1.0,1,2,0,1,0,19,0,0,6,0,0,1,0,0,2,0,0,0,0,0,1,0,2]]},{"b":7,"e":0.14286,"k":"flat","v":0.47768,"x":0.82143,"p":[[0,188,0.0,0.50891,0.26949,0.28571,0.42857,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,8,0,0,9,0,0,4,0,0,3,0,0,2,0,4],[4,188,0.0213,0.69642,0.21355,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,6,0,0,6,0,6],[8,188,0.0426,0.76786,0.25692,0.57143,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,4,0,0,3,0,0,4,0,0,7,0,12],[12,188,0.0638,0.66963,0.25363,0.42857,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,11,0,0,3,0,0,3,0,0,8,0,6],[16,188,0.0851,0.69642,0.22233,0.5354,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,8,0,0,9,0,0,2,0,0,5,0,8],[20,188,0.1064,0.65179,0.24468,0.42859,0.64286,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,6,0,0,7,0,0,5,0,0,6,0,5],[24,188,0.1277,0.71428,0.18898,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,3,0,0,9,0,5],[28,188,0.1489,0.74107,0.21558,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,3,0,0,5,0,10],[32,188,0.1702,0.72321,0.24468,0.42857,0.78571,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,3,0,0,6,0,10],[36,188,0.1915,0.82143,0.18558,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,7,0,13],[40,188,0.2128,0.6875,0.23538,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,7,0,0,5,0,0,3,0,0,10,0,5],[44,188,0.234,0.6875,0.24074,0.53572,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,6,0,0,10,0,0,1,0,0,4,0,9],[48,188,0.2553,0.80355,0.2075,0.57143,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,2,0,0,9,0,12],[52,188,0.2766,0.66517,0.2276,0.42859,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,8,0,0,8,0,0,4,0,0,5,0,6],[56,188,0.2979,0.66518,0.249,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,12,0,0,4,0,0,0,0,0,9,0,6],[60,188,0.3191,0.5982,0.2822,0.42857,0.57143,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,0,0,0,9,0,0,8,0,0,0,0,0,5,0,6],[64,188,0.3404,0.73214,0.22517,0.42857,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,0,0,0,12,0,7],[68,188,0.3617,0.62053,0.25155,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,9,0,0,8,0,0,2,0,0,4,0,6],[72,188,0.383,0.71875,0.23278,0.57143,0.57143,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,1,0,0,3,0,11],[76,188,0.4043,0.61606,0.27302,0.42857,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,2,0,0,3,0,0,10,0,0,3,0,0,4,0,6],[80,188,0.4255,0.74998,0.22017,0.57143,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,1,0,0,11,0,8],[84,188,0.4468,0.69196,0.25532,0.57143,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,1,0,0,3,0,0,5,0,0,3,0,0,13,0,4],[88,188,0.4681,0.63392,0.2878,0.42857,0.64286,0.85714,0.0,1.0,1,6,0,1,0,2,0,0,3,0,0,5,0,0,5,0,0,3,0,0,7,0,6],[92,188,0.4894,0.73661,0.23987,0.57142,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,2,0,0,7,0,10],[96,188,0.5106,0.65625,0.17445,0.57143,0.57143,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,14,0,0,2,0,0,10,0,1],[100,188,0.5319,0.74554,0.24153,0.57143,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,2,0,0,8,0,10],[104,188,0.5532,0.69196,0.2372,0.53571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,6,0,0,6,0,0,5,0,0,6,0,7],[108,188,0.5745,0.64732,0.2448,0.42857,0.57143,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,1,0,0,7,0,0,7,0,0,2,0,0,9,0,4],[112,188,0.5957,0.6741,0.26542,0.53571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,1,0,0,4,0,0,7,0,0,2,0,0,9,0,6],[116,188,0.617,0.69196,0.26752,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,1,0,0,4,0,0,8,0,0,2,0,0,7,0,8],[120,188,0.6383,0.66964,0.20958,0.57143,0.57143,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,5,0,0,10,0,0,3,0,0,8,0,4],[124,188,0.6596,0.67466,0.23256,0.53571,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,6,0,0,9,0,0,1,0,0,9,0,5],[128,188,0.6809,0.70087,0.23517,0.57143,0.71429,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,1,0,0,2,0,0,11,0,0,4,0,0,6,0,7],[132,188,0.7021,0.72321,0.2922,0.57143,0.85714,1.0,0.0,1.0,2,9,0,2,0,1,0,0,0,0,0,4,0,0,5,0,0,0,0,0,11,0,9],[136,188,0.7234,0.70087,0.27049,0.53539,0.85707,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,3,0,0,4,0,0,6,0,0,1,0,0,9,0,8],[140,188,0.7447,0.70088,0.24054,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,5,0,0,5,0,8],[144,188,0.766,0.72768,0.1868,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,6,0,0,11,0,4],[148,188,0.7872,0.63836,0.2474,0.42859,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,5,0,0,11,0,0,2,0,0,3,0,7],[152,188,0.8085,0.71874,0.23004,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,2,0,0,11,0,0,2,0,0,7,0,8],[156,188,0.8298,0.66964,0.23538,0.42857,0.57143,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,8,0,0,7,0,0,3,0,0,5,0,7],[160,188,0.8511,0.6875,0.22428,0.57143,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,4,0,0,13,0,0,3,0,0,3,0,8],[164,188,0.8723,0.72772,0.24572,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,5,0,0,5,0,0,5,0,0,5,0,10],[168,188,0.8936,0.70087,0.23245,0.53539,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,6,0,0,7,0,0,2,0,0,8,0,7],[172,188,0.9149,0.65178,0.23673,0.42857,0.64286,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,8,0,0,6,0,0,3,0,0,10,0,3],[176,188,0.9362,0.69196,0.23449,0.42857,0.64286,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,9,0,0,6,0,0,1,0,0,8,0,7],[180,188,0.9574,0.64285,0.28121,0.53572,0.57143,0.85714,0.0,1.0,1,5,0,1,0,3,0,0,1,0,0,3,0,0,9,0,0,1,0,0,9,0,5],[184,188,0.9787,0.71426,0.22589,0.57132,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,5,0,0,9,0,6],[188,188,1.0,0.47768,0.29366,0.28571,0.42857,0.64286,0.0,1.0,2,3,0,2,0,3,0,0,10,0,0,3,0,0,6,0,0,0,0,0,5,0,3]]}]},{"i":"a07b23648832235a","q":"Determine the number of ten-digit positive integers with the following properties: $\\bullet$  Each of the digits  $0, 1, 2, . . . , 8$  and  $9$  is contained exactly once. $\\bullet$  Each digit, except  $9$ , has a neighbouring digit that is larger than it.\n(Note. For example, in the number  $1230$ , the digits  $1$  and  $3$  are the neighbouring digits of  $2$   while  $2$  and  $0$  are the neighbouring digits of  $3$ . The digits  $1$  and  $0$  have only one neighbouring digit.)\n\n*(Karl Czakler)*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.6741,"x":0.9554,"p":[[0,60,0.0,0.77677,0.2313,0.57143,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,6,0,0,6,0,0,2,0,14],[4,60,0.0667,0.89732,0.17941,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,2,0,23],[8,60,0.1333,0.86606,0.21112,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,5,0,20],[12,60,0.2,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[16,60,0.2667,0.90625,0.17717,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,6,0,22],[20,60,0.3333,0.90179,0.18707,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,6,0,22],[24,60,0.4,0.9554,0.10951,1.0,1.0,1.0,0.43,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[28,60,0.4667,0.81696,0.20896,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,2,0,0,11,0,12],[32,60,0.5333,0.92857,0.16366,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,24],[36,60,0.6,0.89286,0.18898,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,4,0,22],[40,60,0.6667,0.86607,0.21998,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,0,0,0,6,0,20],[44,60,0.7333,0.87054,0.20316,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,9,0,18],[48,60,0.8,0.89286,0.16751,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,10,0,18],[52,60,0.8667,0.85714,0.22588,0.85714,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,8,0,18],[56,60,0.9333,0.76786,0.26666,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,2,0,0,3,0,0,1,0,0,8,0,13],[60,60,1.0,0.6741,0.3159,0.28571,0.78571,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,10,0,0,3,0,0,2,0,0,1,0,0,3,0,13]]},{"b":5,"e":1.0,"k":"flat","v":0.79464,"x":0.95536,"p":[[0,98,0.0,0.79464,0.26471,0.57143,0.85714,1.0,0.0,1.0,1,15,1,1,0,0,0,0,1,0,0,4,0,0,3,0,0,1,0,0,7,0,15],[4,98,0.0408,0.90625,0.18073,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,2,0,24],[8,98,0.0816,0.91964,0.15947,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,23],[12,98,0.1224,0.90179,0.14914,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,6,0,20],[16,98,0.1633,0.89732,0.1996,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,6,0,22],[20,98,0.2041,0.90625,0.249,1.0,1.0,1.0,0.0,1.0,2,25,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,25],[24,98,0.2449,0.90625,0.21609,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,24],[28,98,0.2857,0.87052,0.22408,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,20],[32,98,0.3265,0.89732,0.21793,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,4,0,23],[36,98,0.3673,0.8125,0.26351,0.82143,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,10,0,14],[40,98,0.4082,0.86607,0.20183,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,6,0,19],[44,98,0.449,0.79911,0.32313,0.78571,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,5,0,19],[48,98,0.4898,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[52,98,0.5306,0.86606,0.22288,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,0,4,0,21],[56,98,0.5714,0.90179,0.16536,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,6,0,21],[60,98,0.6122,0.79911,0.26453,0.57143,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,6,0,0,2,0,0,1,0,0,6,0,16],[64,98,0.6531,0.89286,0.20825,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,2,0,24],[68,98,0.6939,0.83482,0.23987,0.78561,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,0,7,0,17],[72,98,0.7347,0.85714,0.21724,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,0,0,0,6,0,19],[76,98,0.7755,0.90626,0.16978,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,5,0,22],[80,98,0.8163,0.90625,0.16982,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,3,0,23],[84,98,0.8571,0.91071,0.16656,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,6,0,22],[88,98,0.898,0.90625,0.21011,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,24],[92,98,0.9388,0.875,0.24157,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,4,0,22],[96,98,0.9796,0.91964,0.14698,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,8,0,21],[98,98,1.0,0.9375,0.1234,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24]]}]},{"i":"cb6c2ebc52123b22","q":"Let  $a,b,c$  be real numbers such that  $ab\\not= 0$  and  $c>0$ . Let  $(a_{n})_{n\\geq 1}$  be the sequence of real numbers defined by:  $a_{1}=a, a_{2}=b$  and\n\\[a_{n+1}=\\frac{a_{n}^{2}+c}{a_{n-1}}\\]\nfor all  $n\\geq 2$ .\nShow that all the terms of the sequence are integer numbers if and only if the numbers  $a,b$  and  $\\frac{a^{2}+b^{2}+c}{ab}$  are integers.","t":[{"b":0,"e":0.42857,"k":"falling","v":0.42857,"x":0.87932,"p":[[0,27,0.0,0.87932,0.17551,0.85714,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,6,0,19],[4,27,0.1481,0.73213,0.21944,0.57132,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,4,0,0,6,0,9],[8,27,0.2963,0.75892,0.22143,0.57143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,3,0,0,6,0,11],[12,27,0.4444,0.80356,0.21652,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,3,0,0,4,0,15],[16,27,0.5926,0.82589,0.22794,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,4,0,18],[20,27,0.7407,0.78122,0.22014,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,3,0,0,5,0,13],[24,27,0.8889,0.5,0.17857,0.42857,0.42857,0.4286,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,27,0,0,1,0,0,0,0,0,1,0,3],[27,27,1.0,0.42857,1e-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":5,"e":0.42857,"k":"falling","v":0.4375,"x":0.90176,"p":[[0,56,0.0,0.87497,0.19484,0.82143,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,4,0,20],[4,56,0.0714,0.83035,0.21852,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,5,0,17],[8,56,0.1429,0.90176,0.14483,0.857,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,20],[12,56,0.2143,0.83927,0.20749,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,5,0,17],[16,56,0.2857,0.83927,0.21055,0.57143,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,2,0,19],[20,56,0.3571,0.81246,0.20963,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,2,0,0,7,0,14],[24,56,0.4286,0.87054,0.19019,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,7,0,18],[28,56,0.5,0.89732,0.16457,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,21],[32,56,0.5714,0.73661,0.27689,0.42859,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,8,0,0,3,0,0,3,0,0,1,0,15],[36,56,0.6429,0.69645,0.2618,0.42857,0.57143,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,2,0,0,0,0,13],[40,56,0.7143,0.70979,0.26843,0.42857,0.85714,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,9,0,0,5,0,0,0,0,0,7,0,10],[44,56,0.7857,0.66965,0.25363,0.42857,0.57143,1.0,0.286,1.0,0,10,0,0,0,0,0,0,1,0,0,12,0,0,5,0,0,2,0,0,2,0,10],[48,56,0.8571,0.6875,0.2683,0.42857,0.64286,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,14,0,0,1,0,0,1,0,0,4,0,11],[52,56,0.9286,0.51786,0.21052,0.42857,0.42857,0.46431,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,21,0,0,2,0,0,1,0,0,1,0,4],[56,56,1.0,0.4375,0.04971,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,28,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"c1f1155a0d0eb640","q":"Let  $ A_0 \\equal{} (a_1,\\dots,a_n)$  be a finite sequence of real numbers. For each  $ k\\geq 0$ , from the sequence  $ A_k \\equal{} (x_1,\\dots,x_k)$  we construct a new sequence  $ A_{k \\plus{} 1}$  in the following way.\r\n1. We choose a partition  $ \\{1,\\dots,n\\} \\equal{} I\\cup J$ , where  $ I$  and  $ J$  are two disjoint sets, such that the expression\r\n\\[ \\left|\\sum_{i\\in I}x_i \\minus{} \\sum_{j\\in J}x_j\\right|\r\n\\]\r\nattains the smallest value. (We allow  $ I$  or  $ J$  to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily.\r\n2. We set  $ A_{k \\plus{} 1} \\equal{} (y_1,\\dots,y_n)$  where  $ y_i \\equal{} x_i \\plus{} 1$  if  $ i\\in I$ , and  $ y_i \\equal{} x_i \\minus{} 1$  if  $ i\\in J$ .\r\nProve that for some  $ k$ , the sequence  $ A_k$  contains an element  $ x$  such that  $ |x|\\geq\\frac n2$ .\r\n\r\n*Author: Omid Hatami, Iran*","t":[{"b":1,"e":0.42857,"k":"flat","v":0.5089,"x":0.70087,"p":[[0,37,0.0,0.5089,0.13332,0.42857,0.57121,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,9,0,0,13,0,0,5,0,0,0,0,0],[4,37,0.1081,0.64283,0.23691,0.571,0.71429,0.75,0.0,1.0,1,3,0,1,0,2,0,0,0,0,0,4,0,0,6,0,0,11,0,0,5,0,3],[8,37,0.2162,0.62048,0.23314,0.571,0.57143,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,5,0,0,11,0,0,8,0,0,2,0,4],[12,37,0.3243,0.62273,0.20844,0.57132,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,1,1,0,0,2,0,0,13,0,0,9,0,0,2,0,3],[16,37,0.4324,0.70087,0.17629,0.57143,0.71429,0.75,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,10,0,0,2,0,6],[20,37,0.5405,0.64283,0.18899,0.57132,0.71429,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,2,0,0,9,0,0,9,0,0,7,0,1],[24,37,0.6486,0.54463,0.22428,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,3,0,0,1,0,0,7,0,0,8,0,0,9,0,0,2,0,1],[28,37,0.7568,0.64732,0.18893,0.57143,0.71429,0.75,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,2,0,0,10,0,0,9,0,0,7,0,1],[32,37,0.8649,0.65624,0.17807,0.57143,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,3,0,0,7,0,0,15,0,0,3,0,2],[36,37,0.973,0.61158,0.17216,0.571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,5,0,0,11,0,0,10,0,0,3,0,1],[37,37,1.0,0.6428,0.12375,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,12,0,0,14,0,0,3,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.4955,"x":0.66514,"p":[[0,42,0.0,0.4955,0.15964,0.42857,0.571,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,11,0,0,12,0,0,5,0,0,0,0,0],[4,42,0.0952,0.62943,0.17807,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,8,0,0,8,0,0,9,0,0,4,0,2],[8,42,0.1905,0.61605,0.27535,0.5354,0.64286,0.85714,0.0,1.0,3,4,0,3,0,0,0,0,2,0,0,3,0,0,8,0,0,7,0,0,5,0,4],[12,42,0.2857,0.62052,0.23313,0.57132,0.71429,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,0,0,0,4,0,0,6,0,0,14,0,0,3,0,2],[16,42,0.381,0.54015,0.23618,0.42859,0.57143,0.71429,0.0,1.0,3,1,0,3,0,1,0,0,1,0,0,4,0,0,13,0,0,7,0,0,2,0,1],[20,42,0.4762,0.58034,0.19541,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,6,0,0,9,0,0,8,0,0,2,0,2],[24,42,0.5714,0.58926,0.25191,0.42857,0.57143,0.75,0.14286,1.0,0,4,0,0,0,3,0,0,3,0,0,5,0,0,9,0,0,4,0,0,4,0,4],[28,42,0.6667,0.62054,0.18073,0.42857,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,5,0,0,9,0,0,4,0,2],[32,42,0.7619,0.66514,0.16216,0.57132,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,11,0,0,3,0,3],[36,42,0.8571,0.58033,0.15542,0.42859,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,6,0,0,12,0,0,8,0,0,3,0,0],[40,42,0.9524,0.6339,0.16729,0.57132,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,5,0,0,12,0,0,10,0,0,1,0,3],[42,42,1.0,0.65177,0.19213,0.57132,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,4,0,0,6,0,0,13,0,0,3,0,3]]}]},{"i":"2d76f0deed2d3b7f","q":"Define the sequence of real numbers  $\\{x_n\\}_{n \\geq 1}$ , where  $x_1$  is any real number and \\[x_n = 1 - x_1x_2\\ldots x_{n-1} \\text{ for all } n > 1.\\] Show that  $x_{2011} >  \\frac{2011}{2012}$ .","t":[{"b":3,"e":0.42857,"k":"flat","v":0.51786,"x":0.85714,"p":[[0,31,0.0,0.51786,0.23891,0.28571,0.42857,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,9,0,0,13,0,0,1,0,0,2,0,0,4,0,3],[4,31,0.129,0.82143,0.20825,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,0,7,0,14],[8,31,0.2581,0.85714,0.18898,0.82143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,8,0,16],[12,31,0.3871,0.80804,0.18766,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,8,0,11],[16,31,0.5161,0.81695,0.17942,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,0,11,0,10],[20,31,0.6452,0.75893,0.19377,0.57143,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,5,0,0,10,0,7],[24,31,0.7742,0.60713,0.17128,0.42857,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,13,0,0,5,0,0,7,0,0,7,0,0],[28,31,0.9032,0.63392,0.18537,0.42857,0.71429,0.75,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,10,0,0,4,0,0,9,0,0,7,0,1],[31,31,1.0,0.63393,0.15947,0.42857,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,14,0,0,5,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.4732,"x":0.86607,"p":[[0,38,0.0,0.4732,0.16535,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,5,0,0,20,0,0,3,0,0,1,0,0,2,0,1],[4,38,0.1053,0.81696,0.22084,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,0,6,0,15],[8,38,0.2105,0.80357,0.17405,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,0,10,0,9],[12,38,0.3158,0.82589,0.19475,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,7,0,14],[16,38,0.4211,0.79911,0.18509,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,16,0,7],[20,38,0.5263,0.86607,0.16728,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,11,0,14],[24,38,0.6316,0.62947,0.18161,0.42857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,11,0,0,4,0,2],[28,38,0.7368,0.68304,0.13709,0.67857,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,19,0,0,4,0,1],[32,38,0.8421,0.70981,0.14502,0.71429,0.71429,0.75,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,17,0,0,6,0,2],[36,38,0.9474,0.66964,0.17655,0.42859,0.71429,0.75,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,15,0,0,7,0,1],[38,38,1.0,0.7232,0.1126,0.71429,0.71429,0.75,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,20,0,0,8,0,0]]}]},{"i":"2d00f793a78a44d6","q":"Let  $ G$  be a graph with  $ 2n$  vertexes and  $ 2n(n\\minus{}1)$  edges.If we color some edge to red,then vertexes,which are connected by this edge,must be colored to red too. But not necessary that all edges from the red vertex are red.\nProve that it is possible to color some vertexes and edges in  $ G$ ,such that all red vertexes has exactly  $ n$  red edges.","t":[{"b":4,"e":0.14286,"k":"falling","v":0.12947,"x":0.30357,"p":[[0,30,0.0,0.30357,0.27374,0.14286,0.14286,0.28571,0.14286,1.0,0,2,0,0,0,21,0,0,4,0,0,0,0,0,1,0,0,3,0,0,1,0,2],[4,30,0.1333,0.15625,0.08268,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,24,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.13384,"x":0.28124,"p":[[0,33,0.0,0.28124,0.23277,0.14286,0.14286,0.32143,0.14286,1.0,0,1,0,0,0,21,0,0,3,0,0,2,0,0,3,0,0,1,0,0,1,0,1],[4,33,0.1212,0.15625,0.16115,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,33,0.2424,0.13393,0.08703,0.14286,0.14286,0.14286,0.0,0.4286,6,0,0,6,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.17858,0.16366,0.14286,0.14286,0.14287,0.0,1.0,2,1,0,2,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,33,0.4848,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.17411,0.09268,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,28,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,33,0.7273,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.14268,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4b3fedbf061118f9","q":"Let $\\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each  line $\\ell \\in \\mathcal{L}$ a point $f(\\ell)$ on $\\ell$. Suppose that for any point $X$, and for any three lines $\\ell_{1}, \\ell_{2}, \\ell_{3}$ passing through $X$, the points $f\\left(\\ell_{1}\\right), f\\left(\\ell_{2}\\right), f\\left(\\ell_{3}\\right)$ and $X$ lie on a circle.  Prove that there is a unique point $P$ such that $f(\\ell)=P$ for any line $\\ell$ passing through $P$. (Australia) Common remarks. The condition on $f$ is equivalent to the following: There is some function $g$ that assigns to each point $X$ a circle $g(X)$ passing through $X$ such that for any line $\\ell$ passing through $X$, the point $f(\\ell)$ lies on $g(X)$. (The function $g$ may not be uniquely defined for all points, if some points $X$ have at most one value of $f(\\ell)$ other than $X$; for such points, an arbitrary choice is made.)  If there were two points $P$ and $Q$ with the given property, $f(P Q)$ would have to be both $P$ and $Q$, so there is at most one such point, and it will suffice to show that such a point exists.","t":[{"b":0,"e":1.0,"k":"rising","v":0.37054,"x":0.96875,"p":[[0,49,0.0,0.37054,0.23107,0.14286,0.42857,0.42858,0.0,1.0,1,1,1,1,0,9,0,0,5,0,0,12,0,0,1,0,0,1,0,0,2,0,1],[4,49,0.0816,0.95536,0.14032,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[8,49,0.1633,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[12,49,0.2449,0.93748,0.09408,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[16,49,0.3265,0.90165,0.21272,0.96425,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,24],[20,49,0.4082,0.87052,0.24055,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,22],[24,49,0.4898,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,28],[28,49,0.5714,0.91072,0.18813,0.85714,1.0,1.0,0.1429,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,23],[32,49,0.6531,0.76783,0.29828,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,14],[36,49,0.7347,0.76338,0.30011,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,1,0,0,1,0,0,2,0,0,6,0,0,2,0,16],[40,49,0.8163,0.82134,0.23172,0.71429,0.92857,1.0,0.14,1.0,0,16,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,11,0,0,2,0,16],[44,49,0.898,0.7856,0.26986,0.67857,0.85714,1.0,0.14,1.0,0,14,0,0,0,3,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,14],[48,49,0.9796,0.66963,0.2822,0.53572,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,4,0,0,2,0,0,2,0,0,4,0,0,7,0,0,6,0,7],[49,49,1.0,0.7321,0.25694,0.571,0.78564,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,5,0,0,6,0,10]]},{"b":6,"e":1.0,"k":"rising","v":0.41964,"x":0.98661,"p":[[0,48,0.0,0.41964,0.27649,0.14286,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,9,0,0,9,0,0,5,0,0,0,0,0,4,0,0,3,0,2],[4,48,0.0833,0.90616,0.1846,0.85714,1.0,1.0,0.14,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,23],[8,48,0.1667,0.96427,0.10107,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[12,48,0.25,0.92857,0.15153,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[16,48,0.3333,0.89731,0.17215,0.82132,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,2,0,22],[20,48,0.4167,0.85267,0.20355,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,9,0,0,2,0,18],[24,48,0.5,0.92191,0.13759,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,1,22],[28,48,0.5833,0.86607,0.17104,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,10,0,0,2,0,18],[32,48,0.6667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,48,0.75,0.96427,0.09455,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[40,48,0.8333,0.98213,0.04728,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,48,0.9167,0.94642,0.16269,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[48,48,1.0,0.8973,0.212,0.85711,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,6,0,22]]}]},{"i":"7124b70e4f9e5d30","q":"Suppose  $ A $  is a subset of   $ n $ -elements taken from  $ 1,2,3,4,...,2009 $  such that the difference of any two numbers in  $ A $  is not a prime number. Find the largest value of   $ n $  and the set   $ A $  with this number of elements.","t":[{"b":2,"e":0.85714,"k":"rising","v":0.29464,"x":0.72767,"p":[[0,61,0.0,0.29464,0.13333,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,61,0.0656,0.72767,0.25843,0.57142,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,6,0,0,1,0,0,4,0,0,2,0,0,11,0,8],[8,61,0.1311,0.55803,0.27283,0.28571,0.57143,0.85714,0.0,1.0,1,1,0,1,0,3,0,0,6,0,0,2,0,0,7,0,0,3,0,0,9,0,1],[12,61,0.1967,0.56695,0.26603,0.28571,0.57143,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,12,0,0,0,0,0,5,0,0,4,0,0,8,0,2],[16,61,0.2623,0.53571,0.25754,0.28571,0.57143,0.75,0.0,1.0,1,1,0,1,0,1,0,0,9,0,0,4,0,0,5,0,0,4,0,0,7,0,1],[20,61,0.3279,0.52231,0.23853,0.28571,0.57143,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,7,0,0,3,0,0,8,0,0,7,0,0,3,0,1],[24,61,0.3934,0.60714,0.22303,0.42859,0.57143,0.74996,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,3,0,0,9,0,0,6,0,0,6,0,2],[28,61,0.459,0.56696,0.25123,0.28571,0.57143,0.75,0.14286,1.0,0,3,0,0,0,2,0,0,7,0,0,3,0,0,9,0,0,3,0,0,5,0,3],[32,61,0.5246,0.58479,0.2183,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,3,0,0,9,0,0,6,0,0,6,0,1],[36,61,0.5902,0.49106,0.18188,0.39286,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,2,0,0,19,0,0,2,0,0,1,0,0],[40,61,0.6557,0.62497,0.2594,0.42859,0.57143,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,4,0,0,4,0,0,12,0,0,0,0,0,5,0,6],[44,61,0.7213,0.53569,0.24484,0.28571,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,11,0,0,2,0,0,8,0,0,3,0,0,5,0,2],[48,61,0.7869,0.58929,0.20748,0.42857,0.57143,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,5,0,0,9,0,0,3,0,0,9,0,0],[52,61,0.8525,0.59374,0.18935,0.57132,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,5,0,0,2,0,0,16,0,0,2,0,0,6,0,1],[56,61,0.918,0.60712,0.17497,0.57132,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,3,0,0,16,0,0,5,0,0,3,0,2],[60,61,0.9836,0.60265,0.19799,0.57132,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,1,0,0,15,0,0,4,0,0,6,0,1],[61,61,1.0,0.51785,0.20438,0.39286,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,5,0,0,13,0,0,2,0,0,3,0,1]]},{"b":7,"e":0.0,"k":"falling","v":0.00446,"x":0.59375,"p":[[0,77,0.0,0.28125,0.06667,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,4,0,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.54906,0.29038,0.28571,0.57121,0.75,0.0,1.0,2,5,1,2,0,2,0,0,5,0,0,4,0,0,9,0,0,2,0,0,3,0,5],[8,77,0.1039,0.59375,0.21162,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,7,0,0,9,0,0,5,0,0,5,0,2],[12,77,0.1558,0.56238,0.23418,0.42857,0.57143,0.71429,0.14,1.0,0,3,0,0,0,2,0,0,5,0,0,5,0,0,10,0,0,4,0,0,3,0,3],[16,77,0.2078,0.53122,0.21497,0.28571,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,8,0,0,5,0,0,9,0,0,4,0,0,4,0,1],[20,77,0.2597,0.47318,0.26105,0.28571,0.42859,0.71429,0.0,0.85714,4,0,0,4,0,0,0,0,7,0,0,6,0,0,6,0,0,4,0,0,5,0,0],[24,77,0.3117,0.48213,0.34022,0.24999,0.42857,0.85714,0.0,1.0,4,5,0,4,0,4,0,0,7,0,0,3,0,0,3,0,0,2,0,0,4,0,5],[28,77,0.3636,0.51339,0.34043,0.28571,0.5,0.85704,0.0,1.0,4,5,0,4,0,3,0,0,7,0,0,2,0,0,2,0,0,5,0,0,4,0,5],[32,77,0.4156,0.55357,0.28065,0.39286,0.57143,0.75,0.0,1.0,3,2,0,3,0,1,0,0,4,0,0,4,0,0,7,0,0,5,0,0,6,0,2],[36,77,0.4675,0.40616,0.28155,0.24999,0.35714,0.57143,0.0,1.0,4,2,0,4,0,4,0,0,8,0,0,5,0,0,6,0,0,0,0,0,3,0,2],[40,77,0.5195,0.4196,0.30078,0.24999,0.28571,0.60714,0.0,1.0,5,2,0,5,0,3,0,0,9,0,0,2,0,0,5,0,0,3,0,0,3,0,2],[44,77,0.5714,0.45089,0.26513,0.2857,0.50001,0.57143,0.0,0.85714,3,0,0,3,0,4,0,0,6,0,0,3,0,0,9,0,0,2,0,0,5,0,0],[48,77,0.6234,0.52677,0.30813,0.28571,0.57143,0.85714,0.0,1.0,3,3,0,3,0,4,0,0,3,0,0,3,0,0,8,0,0,2,0,0,6,0,3],[52,77,0.6753,0.4375,0.25489,0.2857,0.42859,0.57143,0.0,0.85714,3,0,0,3,0,4,0,0,6,0,0,4,0,0,9,0,0,2,0,0,4,0,0],[56,77,0.7273,0.55356,0.19804,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,6,0,0,10,0,0,6,0,0,3,0,1],[60,77,0.7792,0.45535,0.26591,0.28571,0.42857,0.60714,0.0,1.0,1,1,0,1,0,6,0,0,6,0,0,7,0,0,4,0,0,2,0,0,5,0,1],[64,77,0.8312,0.48213,0.27374,0.28571,0.4286,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,4,0,0,7,0,0,6,0,0,3,0,0,5,0,1],[68,77,0.8831,0.41071,0.30877,0.24999,0.28571,0.60714,0.0,1.0,7,1,0,7,0,1,0,0,9,0,0,2,0,0,5,0,0,2,0,0,5,0,1],[72,77,0.9351,0.34372,0.26208,0.10714,0.42857,0.4642,0.0,1.0,8,1,0,8,0,2,0,0,5,0,0,9,0,0,5,0,0,1,0,0,1,0,1],[76,77,0.987,0.04018,0.15251,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[77,77,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6f98530d67754f5f","q":"Let $n$ be an integer and $0<\\mathfrak{u}_{1}<\\mathfrak{u}_{2}<\\ldots<\\boldsymbol{u}_{\\boldsymbol{n}}$ be real numbers such that\n\n$$\n\\mathfrak{u}_{1}+\\mathfrak{u}_{2}+\\ldots+\\mathfrak{u}_{n}=\\frac{1}{\\mathfrak{u}_{1}^{2}}+\\frac{1}{\\mathfrak{u}_{2}^{2}}+\\ldots+\\frac{1}{\\mathfrak{u}_{n}^{2}}\n$$\n\nShow that for any integer $k$ less than or equal to $n$, there exist $k$ real numbers among $\\mathfrak{u}_{1}, \\mathfrak{u}_{2}, \\ldots, \\mathfrak{u}_{n}$ whose sum is greater than or equal to $k$.","t":[{"b":0,"e":1.0,"k":"rising","v":0.76339,"x":0.98661,"p":[[0,13,0.0,0.76339,0.3001,0.67857,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,0,0,0,3,0,0,1,0,0,7,0,0,0,0,17],[4,13,0.3077,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[8,13,0.6154,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,13,0.9231,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[13,13,1.0,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28]]},{"b":4,"e":0.2857,"k":"falling","v":0.58928,"x":0.75893,"p":[[0,31,0.0,0.75893,0.2976,0.71429,0.78571,1.0,0.0,1.0,1,15,1,1,0,3,0,0,0,0,0,2,0,0,0,0,0,10,0,0,1,0,15],[4,31,0.129,0.70089,0.26088,0.67857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,14,0,0,1,0,9],[8,31,0.2581,0.63392,0.16728,0.42857,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,19,0,0,0,0,2],[12,31,0.3871,0.66518,0.1365,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,25,0,0,0,0,1],[16,31,0.5161,0.64731,0.1988,0.42857,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,14,0,0,0,0,5],[20,31,0.6452,0.69643,0.15872,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,22,0,0,0,0,4],[24,31,0.7742,0.66964,0.1729,0.53572,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,18,0,0,0,0,4],[28,31,0.9032,0.66964,0.16146,0.64286,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,21,0,0,0,0,3],[31,31,1.0,0.58928,0.19804,0.42857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,11,0,0,0,0,0,16,0,0,0,0,2]]}]},{"i":"720bb1eb0a44776a","q":"Let  $p$  be an odd prime. Show that for every integer  $c$ , there exists an integer  $a$  such that  $$ a^{\\frac{p+1}{2}} + (a+c)^{\\frac{p+1}{2}} \\equiv c\\pmod p. $$","t":[{"b":1,"e":0.71429,"k":"flat","v":0.90623,"x":0.97321,"p":[[0,21,0.0,0.9375,0.15542,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,27],[4,21,0.1905,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,28],[8,21,0.381,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[12,21,0.5714,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[16,21,0.7619,0.90623,0.14558,0.85711,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,21],[20,21,0.9524,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[21,21,1.0,0.94642,0.08565,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22]]},{"b":5,"e":1.0,"k":"flat","v":0.86607,"x":0.97321,"p":[[0,45,0.0,0.91072,0.17767,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,2,0,24],[4,45,0.0889,0.92857,0.17496,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,27],[8,45,0.1778,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[12,45,0.2667,0.95536,0.1448,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,29],[16,45,0.3556,0.90625,0.19434,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,25],[20,45,0.4444,0.86607,0.21998,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,3,0,0,3,0,21],[24,45,0.5333,0.92857,0.14286,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,25],[28,45,0.6222,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[32,45,0.7111,0.9107,0.16271,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,0,0,24],[36,45,0.8,0.94642,0.11714,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[40,45,0.8889,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,27],[44,45,0.9778,0.95981,0.10858,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[45,45,1.0,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29]]}]},{"i":"c3499578fd668d41","q":"The quartic polynomial  $p$  has integer coefficients and 4 distinct positive integer roots.  If  $p(4) = -256$  and  $p(5) = -135$ , what are its roots?","t":[{"b":0,"e":1.0,"k":"flat","v":0.91964,"x":0.99554,"p":[[0,12,0.0,0.91964,0.17835,0.85714,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,20],[4,12,0.3333,0.9375,0.15126,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[8,12,0.6667,0.9375,0.15947,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,25],[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]]},{"b":7,"e":1.0,"k":"flat","v":0.87946,"x":0.89286,"p":[[0,7,0.0,0.88839,0.24415,0.85714,1.0,1.0,0.0,1.0,2,21,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,21],[4,7,0.5714,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[7,7,1.0,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8]]}]},{"i":"edfadb3916b8aafc","q":"Let $p \\geqslant 5$ be a prime number. Show that there exists an integer $n$ such that for all $x \\in\\{n-1, n, n+1\\}, p^{2} \\nmid x^{p-1}-1$ and $p \\nmid x$.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.6339,"x":0.73213,"p":[[0,16,0.0,0.73213,0.20749,0.57143,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,6,0,0,2,0,10],[4,16,0.25,0.6339,0.16343,0.57143,0.64286,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,4,0,0,11,0,0,13,0,0,1,0,2],[8,16,0.5,0.64283,0.13363,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,11,0,0,1,0,2],[12,16,0.75,0.64729,0.13826,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,15,0,0,2,0,1],[16,16,1.0,0.66069,0.13245,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,15,0,0,1,0,2]]},{"b":3,"e":1.0,"k":"flat","v":0.54463,"x":0.71427,"p":[[0,30,0.0,0.69191,0.21165,0.571,0.71429,0.85704,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,4,0,0,9,0,0,9,0,0,2,0,7],[4,30,0.1333,0.71427,0.13833,0.57143,0.71429,0.71429,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,17,0,0,2,0,4],[8,30,0.2667,0.6875,0.19045,0.57143,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,13,0,0,1,0,6],[12,30,0.4,0.62496,0.15466,0.57132,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,4,0,0,15,0,0,8,0,0,2,0,2],[16,30,0.5333,0.60712,0.13833,0.57132,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,16,0,0,8,0,0,0,0,2],[20,30,0.6667,0.59372,0.11904,0.53539,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,8,0,0,12,0,0,11,0,0,1,0,0],[24,30,0.8,0.54463,0.0974,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,16,0,0,5,0,0,0,0,0],[28,30,0.9333,0.6428,0.12879,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,10,0,0,1,0,2],[30,30,1.0,0.62496,0.15466,0.57132,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,17,0,0,6,0,0,1,0,3]]}]},{"i":"1e46909433f0f140","q":"Written on a blackboard are  $n$  nonnegative integers whose greatest common divisor is  $1$ . A *move* consists of erasing two numbers  $x$  and  $y$ , where  $x\\ge y$ , on the blackboard and replacing them with the numbers  $x-y$  and  $2y$ . Determine for which original  $n$ -tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where  $n-1$  of the numbers of the blackboard are zeroes.","t":[{"b":1,"e":1.0,"k":"rising","v":0.55357,"x":1.0,"p":[[0,34,0.0,0.55357,0.33264,0.28571,0.42857,0.85714,0.0,1.0,1,7,1,1,0,6,0,0,4,0,0,6,0,0,1,0,0,3,0,0,4,0,7],[4,34,0.1176,0.71425,0.29016,0.42857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,4,0,0,5,0,0,3,0,0,4,0,0,1,0,14],[8,34,0.2353,0.77231,0.21088,0.57143,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,7,0,0,3,0,12],[12,34,0.3529,0.74107,0.27534,0.53569,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,4,0,0,2,0,0,3,0,0,8,0,11],[16,34,0.4706,0.73197,0.28962,0.4286,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,4,0,0,4,0,0,3,0,0,4,0,0,1,0,15],[20,34,0.5882,0.86158,0.19723,0.71429,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,3,0,19],[24,34,0.7059,0.79461,0.21413,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,6,0,13],[28,34,0.8235,0.71427,0.2342,0.42857,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,8,0,0,2,0,10],[32,34,0.9412,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[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":2,"e":1.0,"k":"rising","v":0.5089,"x":0.96429,"p":[[0,52,0.0,0.52223,0.36362,0.14286,0.42857,1.0,0.0,1.0,3,9,3,3,0,6,0,0,3,0,0,7,0,0,1,0,0,1,0,0,2,0,9],[4,52,0.0769,0.91517,0.15513,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,23],[8,52,0.1538,0.80356,0.25693,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,1,0,0,4,0,0,6,0,0,3,0,16],[12,52,0.2308,0.70981,0.30407,0.42857,0.71429,1.0,0.0,1.0,1,14,0,1,0,2,0,0,0,0,0,7,0,0,3,0,0,4,0,0,1,0,14],[16,52,0.3077,0.75892,0.33205,0.4286,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,19],[20,52,0.3846,0.71427,0.29451,0.4286,0.78571,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,1,0,0,5,0,0,4,0,0,3,0,0,3,0,13],[24,52,0.4615,0.69639,0.30672,0.42857,0.78571,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,4,0,0,2,0,0,4,0,0,3,0,0,4,0,12],[28,52,0.5385,0.5089,0.27879,0.2857,0.4998,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,6,0,0,5,0,0,7,0,0,3,0,0,2,0,4],[32,52,0.6154,0.82143,0.19561,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,0,6,0,14],[36,52,0.6923,0.83929,0.20748,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,5,0,17],[40,52,0.7692,0.8839,0.12085,0.85714,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,14,0,13],[44,52,0.8462,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[48,52,0.9231,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[52,52,1.0,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26]]}]},{"i":"2f1ba32856d3bfba","q":"12. (GRE 2) In a triangle $A B C$, choose any points $K \\in B C, L \\in A C$, $M \\in A B, N \\in L M, R \\in M K$, and $F \\in K L$. If $E_{1}, E_{2}, E_{3}, E_{4}, E_{5}$, $E_{6}$, and $E$ denote the areas of the triangles $A M R, C K R, B K F, A L F$, $B N M, C L N$, and $A B C$ respectively, show that  $$ E \\geq 8 \\sqrt[6]{E_{1} E_{2} E_{3} E_{4} E_{5} E_{6}} $$  Remark. Points $K, L, M, N, R, F$ lie on segments $B C, A C, A B, L M$, $M K, K L$ respectively.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.56247,"x":0.60267,"p":[[0,72,0.0,0.60267,0.06902,0.57143,0.57143,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,8,0,0,0,0,0],[4,72,0.0556,0.58926,0.06916,0.57143,0.57143,0.57143,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,0,6,0,0,0,0,0],[8,72,0.1111,0.56247,0.13803,0.57143,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,1,0,0,3,0,0,20,0,0,7,0,0,0,0,0],[12,72,0.1667,0.58032,0.07088,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,27,0,0,4,0,0,0,0,0],[16,72,0.2222,0.58927,0.05923,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,5,0,0,0,0,0],[20,72,0.2778,0.56694,0.10999,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,1,0,0,22,0,0,6,0,0,0,0,0],[24,72,0.3333,0.56691,0.09094,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,25,0,0,4,0,0,0,0,0],[28,72,0.3889,0.59372,0.1078,0.57143,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,1,0,0,21,0,0,9,0,0,0,0,0],[32,72,0.4444,0.57587,0.08364,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,2,0,0,24,0,0,5,0,0,0,0,0],[36,72,0.5,0.57588,0.05629,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0,0,0,0],[40,72,0.5556,0.60267,0.05906,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0],[44,72,0.6111,0.59373,0.05186,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0],[48,72,0.6667,0.57585,0.05629,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,3,0,0,0,0,0],[52,72,0.7222,0.5669,0.10998,0.57132,0.57143,0.57143,0.1429,0.71429,0,0,0,0,0,1,0,0,1,0,0,1,0,0,24,0,0,5,0,0,0,0,0],[56,72,0.7778,0.58926,0.04726,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0],[60,72,0.8333,0.58033,0.07936,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,25,0,0,5,0,0,0,0,0],[64,72,0.8889,0.58033,0.03459,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0],[68,72,0.9444,0.57585,0.07563,0.57143,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,26,0,0,4,0,0,0,0,0],[72,72,1.0,0.57141,0.03571,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"flat","v":0.47766,"x":0.58925,"p":[[0,71,0.0,0.58925,0.07785,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,25,0,0,6,0,0,0,0,0],[4,71,0.0563,0.5625,0.10677,0.57143,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0],[8,71,0.1127,0.57136,0.09449,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,24,0,0,5,0,0,0,0,0],[12,71,0.169,0.55803,0.09689,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,3,0,0,25,0,0,3,0,0,0,0,0],[16,71,0.2254,0.55802,0.15303,0.57143,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,0,0,0,0,0,0,0,0,0,25,0,0,5,0,0,0,0,0],[20,71,0.2817,0.5357,0.19232,0.57143,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,0,0,0,1,0,0,1,0,0,20,0,0,7,0,0,0,0,0],[24,71,0.338,0.54463,0.1357,0.57143,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,2,0,0,23,0,0,4,0,0,0,0,0],[28,71,0.3944,0.51342,0.13764,0.53575,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,3,0,0,22,0,0,2,0,0,0,0,0],[32,71,0.4507,0.54904,0.09521,0.571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,1,0,0,26,0,0,2,0,0,0,0,0],[36,71,0.507,0.54015,0.18808,0.57142,0.57143,0.57143,0.0,0.71429,3,0,0,3,0,0,0,0,0,0,0,2,0,0,20,0,0,7,0,0,0,0,0],[40,71,0.5634,0.47766,0.15814,0.42859,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,5,0,0,6,0,0,17,0,0,2,0,0,0,0,0],[44,71,0.6197,0.52679,0.16917,0.57143,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,0,0,0,2,0,0,3,0,0,20,0,0,5,0,0,0,0,0],[48,71,0.6761,0.48659,0.17806,0.39288,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,0,0,0,6,0,0,2,0,0,19,0,0,3,0,0,0,0,0],[52,71,0.7324,0.54909,0.13882,0.57143,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,2,0,0,22,0,0,5,0,0,0,0,0],[56,71,0.7887,0.56246,0.15947,0.5713,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,1,0,0,2,0,0,18,0,0,9,0,0,0,0,0],[60,71,0.8451,0.54907,0.09522,0.57142,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,1,0,0,26,0,0,2,0,0,0,0,0],[64,71,0.9014,0.55802,0.10326,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,4,0,0,21,0,0,5,0,0,0,0,0],[68,71,0.9577,0.49107,0.21997,0.42857,0.57143,0.60714,0.0,0.71429,3,0,0,3,0,2,0,0,2,0,0,4,0,0,13,0,0,8,0,0,0,0,0],[71,71,1.0,0.50441,0.14716,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,4,0,0,6,0,0,18,0,0,3,0,0,0,0,0]]}]},{"i":"061f325e939774fb","q":"Let  $f_n\\ (n=1,\\ 2,\\ \\cdots)$  be a linear transformation expressed by a matrix  $\\left(\n\\begin{array}{cc}\n1-n & 1  \n-n(n+1) & n+2\n\\end{array}\n\\right)$  on the  $xy$  plane. Answer the following questions:\n\n(1) Prove that there exists 2 lines passing through the origin  $O(0,\\ 0)$  such that  all points of the lines are mapped to the same lines, then find the equation of the lines.\n\n(2) Find the area  $S_n$  of the figure enclosed by the lines obtained in (1) and the curve  $y=x^2$ .\n\n(3) Find  $\\sum_{n=1}^{\\infty} \\frac{1}{S_n-\\frac 16}.$ *2011 Tokyo Institute of Technlogy entrance exam, Problem 1*","t":[{"b":1,"e":1.0,"k":"flat","v":0.92857,"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],[4,35,0.1143,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,35,0.2286,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[12,35,0.3429,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[16,35,0.4571,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[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":4,"e":0.57143,"k":"falling","v":0.51339,"x":1.0,"p":[[0,147,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,147,0.0272,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,147,0.0544,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,147,0.1088,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,147,0.1361,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,147,0.1633,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[28,147,0.1905,0.90625,0.19434,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,25],[32,147,0.2177,0.91518,0.20782,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,27],[36,147,0.2449,0.94197,0.15915,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,28],[40,147,0.2721,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[44,147,0.2993,0.92857,0.17128,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,27],[48,147,0.3265,0.94197,0.15915,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,28],[52,147,0.3537,0.96875,0.12234,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[56,147,0.381,0.93304,0.18205,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,28],[60,147,0.4082,0.91518,0.17807,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,26],[64,147,0.4354,0.96205,0.11006,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,1,0,1,0,28],[68,147,0.4626,0.93304,0.15966,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,27],[72,147,0.4898,0.83929,0.24419,0.57143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,0,0,0,0,0,22],[76,147,0.517,0.86606,0.21999,0.57143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,0,0,0,23],[80,147,0.5442,0.88393,0.24856,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,26],[84,147,0.5714,0.7857,0.23421,0.57143,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,0,0,0,0,0,17],[88,147,0.5986,0.68303,0.27603,0.42857,0.57143,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,6,0,0,9,0,0,0,0,0,0,0,13],[92,147,0.6259,0.75,0.28121,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,4,0,0,8,0,0,0,0,0,0,0,17],[96,147,0.6531,0.74554,0.27603,0.57143,0.78571,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,1,0,0,12,0,0,0,0,0,0,0,16],[100,147,0.6803,0.75893,0.25364,0.57143,0.78571,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,3,0,0,12,0,0,0,0,0,0,0,16],[104,147,0.7075,0.70536,0.26229,0.57143,0.57143,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,2,0,0,14,0,0,0,0,0,0,0,13],[108,147,0.7347,0.70089,0.23787,0.57143,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,3,0,0,14,0,0,1,0,0,1,0,11],[112,147,0.7619,0.74554,0.26663,0.57143,0.78571,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,3,0,0,10,0,0,0,0,0,0,0,16],[116,147,0.7891,0.73661,0.25781,0.57143,0.57143,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,0,0,0,0,0,15],[120,147,0.8163,0.65626,0.26691,0.42859,0.57143,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,5,0,0,4,0,0,12,0,0,0,0,0,0,0,11],[124,147,0.8435,0.68304,0.26663,0.57143,0.57143,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,2,0,0,14,0,0,0,0,0,0,0,12],[128,147,0.8707,0.66518,0.29148,0.42857,0.57143,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,6,0,0,6,0,0,7,0,0,0,0,0,0,0,13],[132,147,0.898,0.70089,0.28428,0.42859,0.57143,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,3,0,0,5,0,0,8,0,0,1,0,0,0,0,14],[136,147,0.9252,0.78125,0.2575,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,0,0,0,0,0,18],[140,147,0.9524,0.65625,0.2621,0.42857,0.57143,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,3,0,0,8,0,0,10,0,0,0,0,0,0,0,11],[144,147,0.9796,0.70088,0.24579,0.57143,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,4,0,0,13,0,0,1,0,0,0,0,12],[147,147,1.0,0.51339,0.17807,0.42857,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,6,0,0,4,0,0,19,0,0,0,0,0,0,0,2]]}]},{"i":"10856dcfe963837d","q":"If  $f:\\mathbb{R} \\rightarrow \\mathbb{R}$  satisfies  $f(x^2 +f(y))=y+xf(x)$  for all  $x,y \\in \\mathbb{R}$ , find  $f(x)$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.71874,"x":0.96428,"p":[[0,20,0.0,0.89732,0.17941,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,20],[4,20,0.2,0.89285,0.18558,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,5,0,21],[8,20,0.4,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[12,20,0.6,0.71874,0.31438,0.42859,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,1,0,0,4,0,0,4,0,0,2,0,0,2,0,15],[16,20,0.8,0.86605,0.23675,0.85711,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,0,3,0,22],[20,20,1.0,0.83034,0.26351,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,3,0,0,2,0,20]]},{"b":2,"e":1.0,"k":"flat","v":0.83926,"x":0.99554,"p":[[0,31,0.0,0.9241,0.20513,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[4,31,0.129,0.83926,0.25193,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,1,0,0,0,0,0,3,0,0,3,0,0,4,0,19],[8,31,0.2581,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27],[12,31,0.3871,0.88839,0.22794,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,23],[16,31,0.5161,0.92857,0.19884,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[20,31,0.6452,0.91963,0.20499,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,27],[24,31,0.7742,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,31,0.9032,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[31,31,1.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30]]}]},{"i":"4a7ce5f93feda311","q":"Find all functions  $f : R \\to  R$  which satisfy for all  $x, y \\in R$  the relation  $f(f(f(x) + y) + y) = x + y + f(y)$","t":[{"b":4,"e":0.71429,"k":"flat","v":0.3794,"x":0.49107,"p":[[0,17,0.0,0.41071,0.21651,0.2857,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,12,0,0,6,0,0,2,0,0,6,0,0,0,0,1],[4,17,0.2353,0.49107,0.23128,0.28571,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,2,0,0,8,0,0,11,0,0,3,0,0,3,0,0,3,0,2],[8,17,0.4706,0.41963,0.23673,0.28571,0.42857,0.57111,0.0,1.0,2,1,0,2,0,3,0,0,9,0,0,9,0,0,3,0,0,3,0,0,2,0,1],[12,17,0.7059,0.45088,0.22898,0.28571,0.42857,0.4642,0.14286,1.0,0,2,0,0,0,4,0,0,7,0,0,13,0,0,2,0,0,2,0,0,2,0,2],[16,17,0.9412,0.46875,0.22654,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,9,0,0,9,0,0,2,0,0,7,0,0,0,0,2],[17,17,1.0,0.3794,0.20714,0.24999,0.28571,0.5711,0.14,0.857,0,0,0,0,0,8,0,0,10,0,0,5,0,0,4,0,0,4,0,0,1,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.32141,"x":0.44196,"p":[[0,75,0.0,0.36606,0.17103,0.2857,0.28571,0.4286,0.0,0.71429,1,0,0,1,0,4,0,0,12,0,0,9,0,0,3,0,0,3,0,0,0,0,0],[4,75,0.0533,0.39283,0.22301,0.24999,0.28571,0.571,0.14286,0.85714,0,0,0,0,0,8,0,0,9,0,0,6,0,0,4,0,0,2,0,0,3,0,0],[8,75,0.1067,0.37501,0.20124,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,9,0,0,5,0,0,12,0,0,3,0,0,2,0,0,0,0,1],[12,75,0.16,0.37945,0.14984,0.28571,0.42857,0.42857,0.14286,0.857,0,0,0,0,0,4,0,0,10,0,0,13,0,0,4,0,0,0,0,0,1,0,0],[16,75,0.2133,0.40177,0.18362,0.28571,0.42857,0.4642,0.14286,0.85714,0,0,0,0,0,4,0,0,11,0,0,9,0,0,5,0,0,1,0,0,2,0,0],[20,75,0.2667,0.40618,0.17162,0.28571,0.42857,0.571,0.14286,1.0,0,1,0,0,0,3,0,0,11,0,0,9,0,0,8,0,0,0,0,0,0,0,1],[24,75,0.32,0.44196,0.16506,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,9,0,0,9,0,0,9,0,0,2,0,0,1,0,0],[28,75,0.3733,0.32141,0.15565,0.24999,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,12,0,0,8,0,0,3,0,0,1,0,0,0,0,0],[32,75,0.4267,0.38393,0.16146,0.28571,0.28571,0.46431,0.14286,0.71429,0,0,0,0,0,3,0,0,15,0,0,6,0,0,5,0,0,3,0,0,0,0,0],[36,75,0.48,0.3661,0.15947,0.2857,0.35714,0.42858,0.14286,0.71429,0,0,0,0,0,6,0,0,10,0,0,10,0,0,4,0,0,2,0,0,0,0,0],[40,75,0.5333,0.34372,0.11209,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,16,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[44,75,0.5867,0.37945,0.1268,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,10,0,0,15,0,0,3,0,0,1,0,0,0,0,0],[48,75,0.64,0.33479,0.14549,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,6,0,0,15,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[52,75,0.6933,0.40624,0.13414,0.28571,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,1,0,0,11,0,0,14,0,0,5,0,0,0,0,0,1,0,0],[56,75,0.7467,0.42862,0.17128,0.28571,0.42857,0.42895,0.0,1.0,1,1,0,1,0,0,0,0,9,0,0,15,0,0,4,0,0,2,0,0,0,0,1],[60,75,0.8,0.33918,0.15056,0.14286,0.35714,0.42857,0.14,0.57143,0,0,0,0,0,9,0,0,7,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[64,75,0.8533,0.38392,0.17653,0.28571,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,13,0,0,12,0,0,2,0,0,0,0,0,1,0,1],[68,75,0.9067,0.36606,0.16726,0.2857,0.35714,0.42857,0.14286,0.85714,0,0,0,0,0,6,0,0,10,0,0,11,0,0,3,0,0,1,0,0,1,0,0],[72,75,0.96,0.41071,0.15465,0.28571,0.42857,0.46431,0.0,0.71429,1,0,0,1,0,2,0,0,7,0,0,14,0,0,6,0,0,2,0,0,0,0,0],[75,75,1.0,0.3659,0.19213,0.25,0.28571,0.46395,0.14,1.0,0,1,0,0,0,8,0,0,9,0,0,7,0,0,7,0,0,0,0,0,0,0,1]]}]},{"i":"e552139942f6e7d1","q":"At the beginning of school year in one of the first grade classes: $i)$  every student had exatly  $20$  acquaintances $ii)$  every two students knowing each other had exactly  $13$  mutual acquaintances $iii)$  every two students not knowing each other had exactly  $12$  mutual acquaintances\nFind number of students in this class","t":[{"b":4,"e":0.71429,"k":"falling","v":0.52678,"x":0.84375,"p":[[0,17,0.0,0.73661,0.27919,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,0,0,0,0,0,0,6,0,0,8,0,0,1,0,13],[4,17,0.2353,0.7142,0.26745,0.57143,0.71429,1.0,0.14,1.0,0,11,0,0,0,3,0,0,1,0,0,2,0,0,3,0,0,12,0,0,0,0,11],[8,17,0.4706,0.84375,0.20628,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,11,0,0,0,0,18],[12,17,0.7059,0.80801,0.21905,0.71429,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,0,0,0,16],[16,17,0.9412,0.54901,0.26765,0.35715,0.57143,0.71429,0.14,1.0,0,3,0,0,0,8,0,0,0,0,0,2,0,0,7,0,0,12,0,0,0,0,3],[17,17,1.0,0.52678,0.2126,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,1,0,0,6,0,0,9,0,0,10,0,0,0,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.69195,"x":0.79908,"p":[[0,15,0.0,0.7232,0.26472,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,1,0,0,4,0,0,11,0,0,1,0,11],[4,15,0.2667,0.74553,0.21939,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,1,0,10],[8,15,0.5333,0.79908,0.23384,0.71429,0.78571,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,10,0,0,1,0,15],[12,15,0.8,0.76784,0.17036,0.67857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,3,0,9],[15,15,1.0,0.69195,0.13884,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,22,0,0,1,0,2]]}]},{"i":"3a7fafaa81cd9d45","q":"Solve in  $ \\mathbb{C} $  the following equation:  $ |z|+|z-5i|=|z-2i|+|z-3i|. $","t":[{"b":0,"e":0.71429,"k":"flat","v":0.67409,"x":0.86161,"p":[[0,88,0.0,0.72765,0.20629,0.571,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,13,0,0,1,0,9],[4,88,0.0455,0.8125,0.18363,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,15,0,0,0,0,14],[8,88,0.0909,0.8125,0.14914,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,2,0,11],[12,88,0.1364,0.79464,0.16342,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,19,0,0,0,0,11],[16,88,0.1818,0.73213,0.15466,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,21,0,0,0,0,6],[20,88,0.2273,0.86161,0.14934,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,17],[24,88,0.2727,0.82143,0.15567,0.71429,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,13],[28,88,0.3182,0.77679,0.11811,0.71429,0.71429,0.71429,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,7],[32,88,0.3636,0.77679,0.17105,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,19,0,0,0,0,10],[36,88,0.4091,0.75,0.13832,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,24,0,0,0,0,6],[40,88,0.4545,0.6875,0.10972,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,27,0,0,0,0,1],[44,88,0.5,0.72768,0.11495,0.71429,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,3],[48,88,0.5455,0.70089,0.13533,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,24,0,0,0,0,3],[52,88,0.5909,0.73214,0.09942,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,3],[56,88,0.6364,0.72767,0.11494,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,26,0,0,1,0,3],[60,88,0.6818,0.75893,0.12595,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,25,0,0,0,0,6],[64,88,0.7273,0.67857,0.11845,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,26,0,0,0,0,1],[68,88,0.7727,0.68737,0.08324,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,29,0,0,0,0,0],[72,88,0.8182,0.72321,0.13333,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,25,0,0,0,0,4],[76,88,0.8636,0.69643,0.14174,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,24,0,0,0,0,3],[80,88,0.9091,0.67409,0.13941,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,23,0,0,0,0,2],[84,88,0.9545,0.70981,0.07565,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,29,0,0,0,0,1],[88,88,1.0,0.67856,0.11295,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,25,0,0,0,0,1]]},{"b":4,"e":1.0,"k":"rising","v":0.73658,"x":1.0,"p":[[0,30,0.0,0.73658,0.22336,0.57132,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,12,0,0,0,0,11],[4,30,0.1333,0.88393,0.14032,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,19],[8,30,0.2667,0.83482,0.17169,0.71429,0.78564,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,14,0,0,1,0,15],[12,30,0.4,0.80802,0.15817,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,18,0,0,0,0,12],[16,30,0.5333,0.83929,0.17405,0.71429,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,14,0,0,0,0,16],[20,30,0.6667,0.87054,0.13997,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,17],[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],[28,30,0.9333,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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":"a50951a4082f7651","q":"Suppose  $P$  is a cubic polynomial satisfying  $P(0) = 3$  and \\[(x^3 - 2x + 1 - P(x))(2x^3 - 5x^2 + 4 - P(x))\\leq 0\\] for all  $x\\in\\mathbb R$ .  Determine all possible values of  $P(-1)$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.99107,"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],[8,33,0.2424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,33,0.6061,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":0.98661,"x":1.0,"p":[[0,28,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,28,0.2857,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,28,0.4286,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,28,0.5714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,28,0.7143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"4eb124ee82536dd8","q":"Given a tetrahedron  $PABC$ , draw the height  $PH$  from vertex  $P$  to  $ABC$ . From point  $H$ , draw perpendiculars  $HA\u2019,HB\u2019,HC\u2019$  to the lines  $PA,PB,PC$ . Suppose the planes  $ABC$  and  $A\u2019B\u2019C\u2019$  intersects at line  $\\ell$ . Let  $O$  be the circumcenter of triangle  $ABC$ . Prove that  $OH\\perp \\ell$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.79013,"x":0.97321,"p":[[0,63,0.0,0.89731,0.21202,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,0,1,0,24],[4,63,0.0635,0.95536,0.16143,1.0,1.0,1.0,0.143,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[8,63,0.127,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,63,0.1905,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[16,63,0.254,0.83928,0.27837,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,3,0,20],[20,63,0.3175,0.90625,0.13175,0.82132,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[24,63,0.381,0.89284,0.20517,0.85711,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,6,0,21],[28,63,0.4444,0.86605,0.25987,0.85711,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,22],[32,63,0.5079,0.88392,0.23538,0.96425,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,0,1,0,24],[36,63,0.5714,0.90625,0.13651,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,21],[40,63,0.6349,0.90179,0.13092,0.82143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,19],[44,63,0.6984,0.85714,0.21429,0.71429,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,6,0,0,5,0,18],[48,63,0.7619,0.91517,0.20159,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,23],[52,63,0.8254,0.79013,0.24744,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,7,0,0,3,0,15],[56,63,0.8889,0.87496,0.20754,0.85711,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,20],[60,63,0.9524,0.88839,0.20743,0.85714,1.0,1.0,0.1429,1.0,0,21,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,0,6,0,21],[63,63,1.0,0.80353,0.2643,0.67846,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,2,0,0,6,0,16]]},{"b":5,"e":1.0,"k":"flat","v":0.87498,"x":0.99107,"p":[[0,66,0.0,0.91071,0.17405,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,21],[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],[8,66,0.1212,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,66,0.1818,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[16,66,0.2424,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,66,0.303,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[24,66,0.3636,0.9107,0.15468,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,22],[28,66,0.4242,0.92411,0.19228,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,26],[32,66,0.4848,0.90623,0.1772,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,22],[36,66,0.5455,0.94642,0.11156,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[40,66,0.6061,0.87498,0.21945,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,21],[44,66,0.6667,0.94196,0.14665,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[48,66,0.7273,0.94195,0.12811,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,24],[52,66,0.7879,0.91516,0.12808,0.85711,1.0,1.0,0.4286,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,19],[56,66,0.8485,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,66,0.9091,0.92857,0.13832,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,24],[64,66,0.9697,0.91517,0.11215,0.857,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,7,0,19],[66,66,1.0,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25]]}]},{"i":"6b3f7d2f523b6c93","q":"How many  $7$ -digit positive integers are there such that the number remains same when its digits are reversed and is multiple of  $11$ ? $ \\textbf{(A)}\\ 900\n\\qquad\\textbf{(B)}\\ 854 \n\\qquad\\textbf{(C)}\\ 818\n\\qquad\\textbf{(D)}\\ 726\n\\qquad\\textbf{(E)}\\ \\text{None}\n$","t":[{"b":1,"e":1.0,"k":"flat","v":0.71429,"x":0.91517,"p":[[0,17,0.0,0.91517,0.17807,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,23],[4,17,0.2353,0.91071,0.12243,0.82132,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,20],[8,17,0.4706,0.89286,0.2369,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,23],[12,17,0.7059,0.71429,0.31135,0.4286,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,3,0,0,4,0,0,2,0,0,3,0,0,3,0,14],[16,17,0.9412,0.91071,0.14174,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,2,0,22],[17,17,1.0,0.90625,0.13175,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20]]},{"b":6,"e":1.0,"k":"flat","v":0.85267,"x":0.95982,"p":[[0,101,0.0,0.89496,0.13379,0.82143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,6,0,0,6,0,18],[4,101,0.0396,0.90178,0.18363,0.85711,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,22],[8,101,0.0792,0.9241,0.13825,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,23],[12,101,0.1188,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[16,101,0.1584,0.87945,0.13885,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,4,0,17],[20,101,0.198,0.89732,0.1394,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,20],[24,101,0.2376,0.90179,0.13092,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,2,0,20],[28,101,0.2772,0.89286,0.13832,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,19],[32,101,0.3168,0.85267,0.14503,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,2,0,15],[36,101,0.3564,0.87499,0.14177,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,16],[40,101,0.396,0.92411,0.13356,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,23],[44,101,0.4356,0.86161,0.15355,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,1,0,17],[48,101,0.4752,0.89284,0.13833,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,19],[52,101,0.5149,0.89061,0.13245,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,5,1,17],[56,101,0.5545,0.875,0.14617,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,4,0,17],[60,101,0.5941,0.92855,0.11849,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[64,101,0.6337,0.86161,0.14054,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,4,0,15],[68,101,0.6733,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[72,101,0.7129,0.91964,0.11812,0.85711,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,21],[76,101,0.7525,0.90625,0.12682,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,3,0,20],[80,101,0.7921,0.94196,0.10631,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[84,101,0.8317,0.90179,0.1357,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,3,0,20],[88,101,0.8713,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[92,101,0.9109,0.875,0.14174,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,3,0,17],[96,101,0.9505,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[100,101,0.9901,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[101,101,1.0,0.89732,0.12993,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,19]]}]},{"i":"01e036af2bbab8e9","q":"Determine all integers  $n$  for which there exist an integer  $k\\geq 2$  and positive integers  $x_1,x_2,\\hdots,x_k$  so that  $$ x_1x_2+x_2x_3+\\hdots+x_{k-1}x_k=n\\text{ and } x_1+x_2+\\hdots+x_k=2019. $$","t":[{"b":2,"e":0.42857,"k":"flat","v":0.35268,"x":0.41963,"p":[[0,32,0.0,0.38839,0.12492,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,11,0,0,14,0,0,4,0,0,1,0,0,0,0,0],[4,32,0.125,0.38839,0.15251,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,16,0,0,10,0,0,2,0,0,2,0,0,1,0,0],[8,32,0.25,0.39732,0.16263,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,14,0,0,13,0,0,1,0,0,2,0,0,0,0,1],[12,32,0.375,0.41963,0.17834,0.28571,0.35714,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,16,0,0,9,0,0,2,0,0,4,0,0,0,0,1],[16,32,0.5,0.37945,0.13649,0.2857,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,14,0,0,11,0,0,3,0,0,2,0,0,0,0,0],[20,32,0.625,0.35714,0.08748,0.28571,0.35714,0.42857,0.1429,0.57143,0,0,0,0,0,1,0,0,15,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[24,32,0.75,0.35268,0.09439,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,17,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[28,32,0.875,0.35713,0.09446,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,16,0,0,13,0,0,2,0,0,0,0,0,0,0,0],[32,32,1.0,0.35268,0.10092,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,15,0,0,13,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.34375,"x":0.45982,"p":[[0,15,0.0,0.41963,0.2285,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,3,0,0,14,0,0,8,0,0,2,0,0,1,0,0,2,0,2],[4,15,0.2667,0.45982,0.21939,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,11,0,0,13,0,0,0,0,0,3,0,0,2,0,2],[8,15,0.5333,0.34375,0.09354,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,16,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[12,15,0.8,0.34375,0.07016,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.35267,0.09439,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,19,0,0,12,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"c6dbe4d1aceaf419","q":"Let $a$ and $b$ be positive real numbers. Define two sequences of real numbers $\\left\\{a_{n}\\right\\}$ and $\\left\\{b_{n}\\right\\}$ for all positive integers $n$ by $(a+b i)^{n}=a_{n}+b_{n} i$. Prove that\n\n$$\n\\frac{\\left|a_{n+1}\\right|+\\left|b_{n+1}\\right|}{\\left|a_{n}\\right|+\\left|b_{n}\\right|} \\geq \\frac{a^{2}+b^{2}}{a+b}\n$$\n\nfor all positive integers $n$.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.42857,"x":0.51787,"p":[[0,55,0.0,0.43304,0.11564,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,1],[4,55,0.0727,0.50447,0.18893,0.42857,0.42857,0.4286,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,27,0,0,1,0,0,0,0,0,0,0,4],[8,55,0.1455,0.44643,0.10564,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,29,0,0,1,0,0,0,0,0,0,0,1],[12,55,0.2182,0.50447,0.18205,0.42857,0.42857,0.42857,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,1,0,0,1,0,3],[16,55,0.2909,0.49554,0.19228,0.42857,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,4],[20,55,0.3636,0.47321,0.1729,0.42857,0.42857,0.42857,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,27,0,0,0,0,0,0,0,0,0,0,3],[24,55,0.4364,0.45982,0.14167,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,2],[28,55,0.5091,0.49554,0.19228,0.42857,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,4],[32,55,0.5818,0.46429,0.14286,0.42857,0.42857,0.42857,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,2],[36,55,0.6545,0.43304,0.17672,0.42857,0.42857,0.42857,0.0,1.0,1,2,0,1,0,2,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,2],[40,55,0.7273,0.46428,0.16365,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,26,0,0,0,0,0,0,0,0,1,0,2],[44,55,0.8,0.42857,1e-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],[48,55,0.8727,0.46429,0.13832,0.42857,0.42857,0.42857,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,2],[52,55,0.9455,0.51787,0.23622,0.42857,0.42857,0.42857,0.28571,1.0,0,6,0,0,0,0,0,0,4,0,0,22,0,0,0,0,0,0,0,0,0,0,6],[55,55,1.0,0.48214,0.1915,0.42857,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,24,0,0,0,0,0,2,0,0,0,0,3]]},{"b":2,"e":0.42857,"k":"flat","v":0.43303,"x":0.57143,"p":[[0,66,0.0,0.43303,0.10999,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,1],[4,66,0.0606,0.54018,0.23619,0.42857,0.42857,0.5,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,21,0,0,0,0,0,2,0,0,0,0,6],[8,66,0.1212,0.47768,0.16982,0.42857,0.42857,0.42857,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,3],[12,66,0.1818,0.57143,0.24744,0.42857,0.42857,0.57143,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,8],[16,66,0.2424,0.45089,0.14773,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,27,0,0,0,0,0,0,0,0,0,0,2],[20,66,0.303,0.54018,0.23619,0.42857,0.42857,0.42858,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,23,0,0,0,0,0,0,0,0,1,0,6],[24,66,0.3636,0.5,0.18898,0.42857,0.42857,0.42858,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,4],[28,66,0.4242,0.5,0.18898,0.42857,0.42857,0.42857,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,4],[32,66,0.4848,0.45536,0.1448,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,28,0,0,0,0,0,0,0,0,0,0,2],[36,66,0.5455,0.53134,0.2265,0.42857,0.42857,0.42895,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,6],[40,66,0.6061,0.54911,0.23987,0.42857,0.42857,0.4286,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,7],[44,66,0.6667,0.51339,0.21086,0.42857,0.42857,0.42857,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,5],[48,66,0.7273,0.46875,0.16065,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,27,0,0,0,0,0,0,0,0,1,0,2],[52,66,0.7879,0.51786,0.23891,0.42857,0.42857,0.42858,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,23,0,0,0,0,0,0,0,0,0,0,6],[56,66,0.8485,0.47768,0.20395,0.42857,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,23,0,0,0,0,0,0,0,0,0,0,4],[60,66,0.9091,0.56696,0.24086,0.42857,0.42857,0.5357,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,1,0,7],[64,66,0.9697,0.5,0.18898,0.42857,0.42857,0.42857,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,4],[66,66,1.0,0.44643,0.09942,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"866d03123a365993","q":"Let  $n$  be a positive integer, and let  $x_1, x_2, \\dots, x_n$  be distinct positive integers with  $x_1 = 1$ . Construct an  $n \\times 3$  table where the entries of the  $k$ -th row are  $x_k, 2x_k, 3x_k$  for  $k = 1, 2, \\dots, n$ . Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.\n\n[list=a]\n[*] Prove that at least three entries remain at the end of the procedure.\n[*] Prove that there are infinitely many possible choices for  $n$  and  $x_1, x_2, \\dots, x_n$  such that only three entries remain.\n[/list]","t":[{"b":0,"e":0.71429,"k":"flat","v":0.33928,"x":0.61605,"p":[[0,80,0.0,0.33928,0.26904,0.2857,0.28571,0.4643,0.0,1.0,7,2,7,7,0,0,0,0,16,0,0,1,0,0,3,0,0,3,0,0,0,0,2],[4,80,0.05,0.47301,0.28014,0.14286,0.4998,0.71429,0.14,1.0,0,3,0,0,0,9,0,0,5,0,0,2,0,0,5,0,0,8,0,0,0,0,3],[8,80,0.1,0.52676,0.20024,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,6,0,0,7,0,0,11,0,0,1,0,0],[12,80,0.15,0.51786,0.23623,0.28571,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,4,0,0,4,0,0,4,0,0,5,0,0,12,0,0,2,0,0],[16,80,0.2,0.5801,0.27887,0.42859,0.57143,0.71429,0.14,1.0,0,6,0,0,0,6,0,0,0,0,0,6,0,0,6,0,0,8,0,0,0,0,6],[20,80,0.25,0.54014,0.25934,0.28571,0.571,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,4,0,0,5,0,0,4,0,0,10,0,0,1,0,3],[24,80,0.3,0.56248,0.19212,0.39286,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,3,0,0,6,0,0,14,0,0,0,0,1],[28,80,0.35,0.57142,0.26726,0.28571,0.71429,0.71429,0.14286,1.0,0,3,0,0,0,5,0,0,4,0,0,3,0,0,2,0,0,13,0,0,2,0,3],[32,80,0.4,0.57128,0.26015,0.4286,0.57143,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,4,0,0,3,0,0,8,0,0,10,0,0,0,0,4],[36,80,0.45,0.59373,0.28818,0.39286,0.57143,0.75,0.14286,1.0,0,7,0,0,0,4,0,0,4,0,0,5,0,0,4,0,0,7,0,0,1,0,7],[40,80,0.5,0.54908,0.24513,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,3,0,0,3,0,0,4,0,0,9,0,0,9,0,0,0,0,3],[44,80,0.55,0.61605,0.25861,0.39285,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,6,0,0,2,0,0,4,0,0,11,0,0,2,0,5],[48,80,0.6,0.60697,0.22579,0.53575,0.57143,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,2,0,0,4,0,0,9,0,0,10,0,0,2,0,3],[52,80,0.65,0.56242,0.26229,0.42857,0.57121,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,3,0,0,3,0,0,9,0,0,8,0,0,2,0,3],[56,80,0.7,0.50433,0.301,0.14286,0.571,0.71429,0.0,1.0,2,3,0,2,0,7,0,0,1,0,0,5,0,0,5,0,0,6,0,0,3,0,3],[60,80,0.75,0.54898,0.27002,0.39285,0.64286,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,1,0,0,3,0,0,5,0,0,12,0,0,2,0,2],[64,80,0.8,0.47767,0.29797,0.14289,0.42859,0.71429,0.0,1.0,1,3,0,1,0,9,0,0,3,0,0,4,0,0,2,0,0,9,0,0,1,0,3],[68,80,0.85,0.55357,0.2714,0.42857,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,6,0,0,1,0,0,6,0,0,7,0,0,7,0,0,0,0,5],[72,80,0.9,0.46874,0.26057,0.2857,0.42859,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,5,0,0,6,0,0,3,0,0,9,0,0,0,0,2],[76,80,0.95,0.45531,0.24596,0.2857,0.4286,0.71429,0.0,1.0,1,1,0,1,0,6,0,0,6,0,0,4,0,0,5,0,0,9,0,0,0,0,1],[80,80,1.0,0.42851,0.19878,0.28571,0.42857,0.57111,0.14286,0.857,0,0,0,0,0,7,0,0,4,0,0,8,0,0,9,0,0,3,0,0,1,0,0]]},{"b":1,"e":1.0,"k":"rising","v":0.21872,"x":0.76337,"p":[[0,96,0.0,0.21872,0.22295,0.0,0.2857,0.28571,0.0,1.0,12,1,11,12,0,2,0,0,13,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[4,96,0.0417,0.60264,0.27137,0.42859,0.57143,0.71429,0.0,1.0,1,6,0,1,0,3,0,0,2,0,0,3,0,0,8,0,0,9,0,0,0,0,6],[8,96,0.0833,0.66509,0.23873,0.57143,0.71429,0.71429,0.14,1.0,0,6,0,0,0,2,0,0,3,0,0,2,0,0,2,0,0,17,0,0,0,0,6],[12,96,0.125,0.6472,0.23706,0.4286,0.71429,0.75,0.14,1.0,0,5,0,0,0,1,0,0,4,0,0,4,0,0,4,0,0,11,0,0,3,0,5],[16,96,0.1667,0.62042,0.22211,0.5354,0.71429,0.71429,0.14,1.0,0,3,0,0,0,3,0,0,1,0,0,4,0,0,5,0,0,15,0,0,1,0,3],[20,96,0.2083,0.60262,0.27833,0.42857,0.571,0.78571,0.14286,1.0,0,8,0,0,0,3,0,0,3,0,0,7,0,0,6,0,0,5,0,0,0,0,8],[24,96,0.25,0.66069,0.27607,0.4286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,0,0,0,5,0,0,4,0,0,10,0,0,0,0,9],[28,96,0.2917,0.66067,0.21355,0.4286,0.71429,0.85704,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,7,0,0,6,0,0,8,0,0,4,0,5],[32,96,0.3333,0.6026,0.26423,0.5354,0.64286,0.71429,0.0,1.0,2,5,0,2,0,1,0,0,3,0,0,2,0,0,8,0,0,11,0,0,0,0,5],[36,96,0.375,0.65155,0.25252,0.5354,0.71429,0.74996,0.14,1.0,0,5,0,0,0,4,0,0,0,0,0,4,0,0,3,0,0,13,0,0,3,0,5],[40,96,0.4167,0.63391,0.2549,0.42857,0.71429,0.74996,0.14286,1.0,0,6,0,0,0,2,0,0,4,0,0,3,0,0,6,0,0,9,0,0,2,0,6],[44,96,0.4583,0.67407,0.29285,0.571,0.71429,1.0,0.0,1.0,2,9,0,2,0,2,0,0,0,0,0,3,0,0,5,0,0,9,0,0,2,0,9],[48,96,0.5,0.56676,0.32064,0.25001,0.57143,0.85714,0.0,1.0,1,7,0,1,0,7,0,0,1,0,0,3,0,0,7,0,0,4,0,0,2,0,7],[52,96,0.5417,0.66512,0.22193,0.571,0.64286,0.74996,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,8,0,0,1,0,7],[56,96,0.5833,0.66068,0.23624,0.57132,0.71429,0.75,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,3,0,0,7,0,0,11,0,0,3,0,5],[60,96,0.625,0.66924,0.24354,0.571,0.71,0.85714,0.14,1.0,0,7,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,7,0,0,3,0,7],[64,96,0.6667,0.66506,0.29812,0.5354,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,2,0,0,2,0,0,6,0,0,4,0,0,6,0,8],[68,96,0.7083,0.65612,0.2671,0.5354,0.71429,0.78571,0.0,1.0,1,8,0,1,0,1,0,0,3,0,0,3,0,0,5,0,0,11,0,0,0,0,8],[72,96,0.75,0.76337,0.24644,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,1,0,0,4,0,0,11,0,0,2,0,12],[76,96,0.7917,0.57131,0.29246,0.39285,0.57143,0.71429,0.0,1.0,2,5,0,2,0,3,0,0,3,0,0,3,0,0,7,0,0,7,0,0,2,0,5],[80,96,0.8333,0.69636,0.24681,0.571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,3,0,0,9,0,0,5,0,0,6,0,7],[84,96,0.875,0.62486,0.28306,0.42857,0.57143,0.85704,0.0,1.0,1,7,0,1,0,2,0,0,3,0,0,4,0,0,7,0,0,5,0,0,3,0,7],[88,96,0.9167,0.61599,0.26352,0.42859,0.57143,0.75,0.0,1.0,1,6,0,1,0,1,0,0,4,0,0,3,0,0,9,0,0,6,0,0,2,0,6],[92,96,0.9583,0.59358,0.32943,0.42859,0.57121,1.0,0.0,1.0,3,9,0,3,0,3,0,0,0,0,0,7,0,0,5,0,0,4,0,0,1,0,9],[96,96,1.0,0.53997,0.28979,0.42857,0.4998,0.71429,0.0,1.0,2,5,0,2,0,3,0,0,2,0,0,9,0,0,5,0,0,4,0,0,2,0,5]]}]},{"i":"1970a04b1064a9ea","q":"Let  $f,g:\\mathbb{R}\\to\\mathbb{R}$  be functions with  $g(x)=2f(x)+f(x^2),$  for all  $x \\in \\mathbb{R}.$ a) Prove that, if  $f$  is bounded in a neighbourhood of the origin and  $g$  is continuous in the origin, then  $f$  is continuous in the origin.\nb) Provide an example of function  $f$ , discontinuous in the origin, for which the function  $g$  is continuous in the origin.","t":[{"b":4,"e":1.0,"k":"flat","v":0.88393,"x":0.99107,"p":[[0,74,0.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[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],[8,74,0.1081,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,74,0.1622,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[16,74,0.2162,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[20,74,0.2703,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,74,0.3243,0.94195,0.12811,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,25],[28,74,0.3784,0.95536,0.12595,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[32,74,0.4324,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,74,0.4865,0.94642,0.1372,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,27],[40,74,0.5405,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[44,74,0.5946,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[48,74,0.6486,0.95535,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[52,74,0.7027,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[56,74,0.7568,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[60,74,0.8108,0.91517,0.15093,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,5,0,22],[64,74,0.8649,0.93303,0.14279,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,5,0,24],[68,74,0.9189,0.9241,0.11837,0.85714,1.0,1.0,0.5714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[72,74,0.973,0.94196,0.11214,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,23],[74,74,1.0,0.88393,0.17655,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,5,0,20]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.99554,"p":[[0,144,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,144,0.0278,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,144,0.0556,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,144,0.0833,0.95536,0.12595,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[16,144,0.1111,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[20,144,0.1389,0.93304,0.14279,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,25],[24,144,0.1667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,144,0.1944,0.95089,0.1411,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,28],[32,144,0.2222,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,144,0.25,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[40,144,0.2778,0.95536,0.14032,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,29],[44,144,0.3056,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[48,144,0.3333,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[52,144,0.3611,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[56,144,0.3889,0.91951,0.2051,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[60,144,0.4167,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[64,144,0.4444,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[68,144,0.4722,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[72,144,0.5,0.91964,0.15947,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,24],[76,144,0.5278,0.93302,0.12873,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,24],[80,144,0.5556,0.92857,0.16751,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,26],[84,144,0.5833,0.9554,0.12059,1.0,1.0,1.0,0.43,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[88,144,0.6111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,144,0.6389,0.10714,0.29014,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[96,144,0.6667,0.12053,0.31965,0.0,0.0,0.0,0.0,1.0,28,3,1,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[100,144,0.6944,0.10714,0.28794,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,2],[104,144,0.7222,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[108,144,0.75,0.07129,0.22831,0.0,0.0,0.0,0.0,1.0,29,1,1,29,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[112,144,0.7778,0.08482,0.26693,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[116,144,0.8056,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,144,0.8333,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[124,144,0.8611,0.05357,0.21053,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[128,144,0.8889,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[132,144,0.9167,0.06696,0.2172,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[136,144,0.9444,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[140,144,0.9722,0.11161,0.29824,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[144,144,1.0,0.79464,0.3008,0.71429,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,1,0,18]]}]},{"i":"23f3641076a72bc1","q":"Let  $b = \\tfrac 12 (-1 + 3\\sqrt{5})$ .  Determine the number of rational numbers which can be written in the form \\[ a_{2014}b^{2014} + a_{2013}b^{2013} + \\dots + a_1b + a_0 \\] where  $a_0, a_1, \\dots, a_{2014}$  are nonnegative integers less than  $b$ .\n\n*Proposed by Michael Kural and Evan Chen*","t":[{"b":1,"e":1.0,"k":"flat","v":0.85268,"x":1.0,"p":[[0,114,0.0,0.95981,0.08923,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[4,114,0.0351,0.92857,0.16367,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,25],[8,114,0.0702,0.875,0.24679,0.96425,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,24],[12,114,0.1053,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,114,0.1404,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[20,114,0.1754,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[24,114,0.2105,0.90625,0.19759,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,0,3,0,24],[28,114,0.2456,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[32,114,0.2807,0.94195,0.13771,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,27],[36,114,0.3158,0.91071,0.23077,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,27],[40,114,0.3509,0.89732,0.19638,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,1,0,24],[44,114,0.386,0.87054,0.22968,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,0,2,0,22],[48,114,0.4211,0.85268,0.26603,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,2,0,23],[52,114,0.4561,0.8616,0.20973,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,0,0,0,8,0,18],[56,114,0.4912,0.88393,0.18708,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,2,0,22],[60,114,0.5263,0.92857,0.13832,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[64,114,0.5614,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,114,0.5965,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[72,114,0.6316,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[76,114,0.6667,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,114,0.7018,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[84,114,0.7368,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[88,114,0.7719,0.94196,0.07874,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[92,114,0.807,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[96,114,0.8421,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,114,0.8772,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,114,0.9123,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[108,114,0.9474,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[112,114,0.9825,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[114,114,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]},{"b":7,"e":1.0,"k":"flat","v":0.73661,"x":1.0,"p":[[0,103,0.0,0.92857,0.15152,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24],[4,103,0.0388,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,103,0.0777,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,103,0.1165,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,103,0.1553,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[20,103,0.1942,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[24,103,0.233,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[28,103,0.2718,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,103,0.3107,0.95089,0.13175,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[36,103,0.3495,0.90179,0.21558,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,0,1,0,25],[40,103,0.3883,0.73661,0.31766,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,4,0,0,5,0,0,1,0,0,1,0,0,2,0,17],[44,103,0.4272,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[48,103,0.466,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[52,103,0.5049,0.90625,0.19434,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,2,0,24],[56,103,0.5437,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[60,103,0.5825,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[64,103,0.6214,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,103,0.6602,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,103,0.699,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[76,103,0.7379,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[80,103,0.7767,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,103,0.8155,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[88,103,0.8544,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,103,0.8932,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,103,0.932,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[100,103,0.9709,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[103,103,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"bc32fc764325f64c","q":"Sequences $\\left(a_{n}\\right)_{n=0}^{\\infty}$ and $\\left(b_{n}\\right)_{n=0}^{\\infty}$ are defined by the recurrence relations\n\n$$\na_{0}=0, \\quad a_{1}=1, \\quad a_{n+1}=\\frac{2018}{n} a_{n}+a_{n-1} \\quad \\text { for } n \\geqslant 1\n$$\n\nand\n\n$$\nb_{0}=0, \\quad b_{1}=1, \\quad b_{n+1}=\\frac{2020}{n} b_{n}+b_{n-1} \\quad \\text { for } n \\geqslant 1\n$$\n\nProve:\n\n$$\n\\frac{a_{1010}}{1010}=\\frac{b_{1009}}{1009}\n$$\n\n(Dushan Dukic)\n\nTime for work 270 minutes.\n\nSolutions to the problems should be explained in detail.\n\nEach problem is worth 7 points.\n\n## SOLUTIONS","t":[{"b":2,"e":0.2857,"k":"flat","v":0.5133,"x":0.91963,"p":[[0,119,0.0,0.5133,0.35876,0.14286,0.64286,0.85704,0.0,1.0,2,6,1,2,0,10,0,0,3,0,0,0,0,0,1,0,0,7,0,0,3,0,6],[4,119,0.0336,0.91963,0.13807,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,7,0,21],[8,119,0.0672,0.91071,0.17768,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,22],[12,119,0.1008,0.89732,0.21199,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,21],[16,119,0.1345,0.87945,0.1825,0.82132,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,4,0,20],[20,119,0.1681,0.86159,0.2328,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,5,0,20],[24,119,0.2017,0.70982,0.35443,0.42857,0.85714,1.0,0.0,1.0,2,14,0,2,0,4,0,0,1,0,0,3,0,0,0,0,0,2,0,0,6,0,14],[28,119,0.2353,0.86607,0.22286,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,7,0,19],[32,119,0.2689,0.76785,0.32291,0.64286,0.92857,1.0,0.14286,1.0,0,16,0,0,0,5,0,0,1,0,0,2,0,0,0,0,0,1,0,0,7,0,16],[36,119,0.3025,0.77677,0.30711,0.67846,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,3,0,0,1,0,0,1,0,0,4,0,0,2,0,18],[40,119,0.3361,0.6116,0.34485,0.14289,0.71429,1.0,0.14286,1.0,0,9,0,0,0,9,0,0,0,0,0,4,0,0,2,0,0,3,0,0,5,0,9],[44,119,0.3697,0.68304,0.3655,0.25001,0.85714,1.0,0.0,1.0,1,14,0,1,0,7,0,0,1,0,0,1,0,0,1,0,0,3,0,0,4,0,14],[48,119,0.4034,0.77678,0.28557,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,1,0,0,0,0,0,1,0,0,5,0,0,8,0,13],[52,119,0.437,0.74106,0.32229,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,5,0,0,7,0,13],[56,119,0.4706,0.79911,0.28762,0.82143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,9,0,15],[60,119,0.5042,0.64732,0.3912,0.14286,0.85714,1.0,0.0,1.0,3,14,0,3,0,6,0,0,1,0,0,2,0,0,0,0,0,3,0,0,3,0,14],[64,119,0.5378,0.67857,0.38299,0.14289,0.85714,1.0,0.0,1.0,2,14,0,2,0,7,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,14],[68,119,0.5714,0.79911,0.2854,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,3,0,0,5,0,17],[72,119,0.605,0.77677,0.24469,0.71429,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,5,0,0,9,0,11],[76,119,0.6387,0.72768,0.27747,0.42857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,1,0,0,6,0,0,3,0,0,3,0,0,5,0,12],[80,119,0.6723,0.73212,0.30041,0.53539,0.85714,1.0,0.14286,1.0,0,13,0,0,0,4,0,0,0,0,0,4,0,0,4,0,0,1,0,0,6,0,13],[84,119,0.7059,0.7232,0.2922,0.67846,0.85714,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,2,0,0,1,0,0,1,0,0,7,0,0,7,0,10],[88,119,0.7395,0.73212,0.27375,0.57132,0.85707,1.0,0.0,1.0,1,9,0,1,0,1,0,0,2,0,0,3,0,0,2,0,0,5,0,0,9,0,9],[92,119,0.7731,0.65623,0.30064,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,1,0,0,8,0,0,1,0,0,4,0,0,5,0,9],[96,119,0.8067,0.63393,0.29653,0.42857,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,6,0,0,1,0,0,4,0,0,1,0,0,6,0,0,10,0,4],[100,119,0.8403,0.66072,0.31083,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,2,0,0,4,0,0,1,0,0,6,0,0,5,0,9],[104,119,0.8739,0.60714,0.32537,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,5,0,0,4,0,0,6,0,0,2,0,0,3,0,0,2,0,10],[108,119,0.9076,0.69643,0.28291,0.53572,0.78564,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,2,0,0,3,0,0,2,0,0,6,0,0,9,0,7],[112,119,0.9412,0.57588,0.32238,0.14289,0.71429,0.85714,0.0,1.0,1,3,0,1,0,8,0,0,1,0,0,2,0,0,2,0,0,6,0,0,9,0,3],[116,119,0.9748,0.59819,0.3203,0.28571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,6,0,0,2,0,0,4,0,0,1,0,0,4,0,0,10,0,4],[119,119,1.0,0.57141,0.31743,0.14286,0.71429,0.85714,0.0,1.0,1,2,0,1,0,8,0,0,2,0,0,0,0,0,2,0,0,8,0,0,9,0,2]]},{"b":7,"e":0.14286,"k":"rising","v":0.43747,"x":0.98214,"p":[[0,168,0.0,0.43747,0.3029,0.14286,0.42857,0.71429,0.0,1.0,1,2,0,1,0,11,0,0,3,0,0,5,0,0,3,0,0,2,0,0,5,0,2],[4,168,0.0238,0.80803,0.27107,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,3,0,0,7,0,16],[8,168,0.0476,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[12,168,0.0714,0.9375,0.17105,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,27],[16,168,0.0952,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[20,168,0.119,0.93304,0.15561,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,26],[24,168,0.1429,0.86161,0.22156,0.82143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,0,5,0,19],[28,168,0.1667,0.91964,0.11811,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,20],[32,168,0.1905,0.9107,0.18126,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,23],[36,168,0.2143,0.81696,0.2506,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,2,0,0,7,0,16],[40,168,0.2381,0.87946,0.23449,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,23],[44,168,0.2619,0.875,0.21053,0.82132,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,0,4,0,20],[48,168,0.2857,0.81695,0.25061,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,4,0,0,4,0,17],[52,168,0.3095,0.86607,0.21706,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,3,0,21],[56,168,0.3333,0.81249,0.23809,0.67846,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,2,0,0,6,0,16],[60,168,0.3571,0.93749,0.14261,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,26],[64,168,0.381,0.85713,0.25507,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,22],[68,168,0.4048,0.86161,0.20973,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,2,0,0,7,0,18],[72,168,0.4286,0.8125,0.31428,0.75001,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,0,2,0,22],[76,168,0.4524,0.82142,0.24223,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,2,0,0,7,0,16],[80,168,0.4762,0.875,0.19804,0.82132,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,3,0,21],[84,168,0.5,0.90179,0.16917,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,5,0,21],[88,168,0.5238,0.90625,0.14987,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,9,0,19],[92,168,0.5476,0.87054,0.22689,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,0,6,0,20],[96,168,0.5714,0.87946,0.22619,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,4,0,22],[100,168,0.5952,0.85714,0.26245,0.82143,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,22],[104,168,0.619,0.87947,0.2055,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,4,0,21],[108,168,0.6429,0.86161,0.20973,0.82143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,0,6,0,18],[112,168,0.6667,0.83035,0.23538,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,5,0,0,3,0,18],[116,168,0.6905,0.80802,0.26634,0.57143,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,1,0,0,4,0,18],[120,168,0.7143,0.77232,0.309,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,3,0,0,1,0,0,0,0,0,3,0,0,7,0,15],[124,168,0.7381,0.78124,0.27196,0.67857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,0,0,0,3,0,0,2,0,0,2,0,0,9,0,13],[128,168,0.7619,0.79463,0.29001,0.82132,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,0,0,0,2,0,0,1,0,0,1,0,0,9,0,15],[132,168,0.7857,0.85714,0.22304,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,0,7,0,18],[136,168,0.8095,0.82143,0.24485,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,6,0,0,7,0,15],[140,168,0.8333,0.78124,0.30302,0.57132,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,0,0,0,3,0,0,2,0,0,1,0,0,5,0,17],[144,168,0.8571,0.73214,0.28515,0.42857,0.85707,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,4,0,0,5,0,0,0,0,0,5,0,0,4,0,13],[148,168,0.881,0.79463,0.31732,0.67846,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,0,0,0,3,0,0,1,0,0,2,0,0,2,0,20],[152,168,0.9048,0.76339,0.35465,0.64286,1.0,1.0,0.0,1.0,3,19,0,3,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,0,3,0,19],[156,168,0.9286,0.74998,0.31135,0.5354,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,1,0,0,3,0,0,3,0,0,0,0,0,6,0,15],[160,168,0.9524,0.72322,0.31731,0.42857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,1,0,0,6,0,0,0,0,0,2,0,0,5,0,14],[164,168,0.9762,0.64729,0.33309,0.39286,0.78564,1.0,0.0,1.0,1,10,0,1,0,4,0,0,3,0,0,4,0,0,3,0,0,1,0,0,6,0,10],[168,168,1.0,0.6071,0.29881,0.42859,0.71429,0.85714,0.0,1.0,2,5,0,2,0,3,0,0,2,0,0,3,0,0,5,0,0,7,0,0,5,0,5]]}]},{"i":"002a91cb5621c434","q":"Let  $ABC$   ne a right triangle with  $\\angle ACB=90^o$ . Let  $E, F$  be respecitvely the midpoints of the  $BC, AC$  and  $CD$  be it's altitude. Next, let  $P$  be the intersection of the internal angle bisector from  $A$  and the line  $EF$ . Prove that  $P$  is the center of the circle inscribed in the triangle  $CDE$  .","t":[{"b":1,"e":0.0,"k":"falling","v":0.01786,"x":0.42856,"p":[[0,47,0.0,0.38837,0.30141,0.10714,0.42857,0.57143,0.0,1.0,8,3,0,8,0,2,0,0,4,0,0,5,0,0,9,0,0,1,0,0,0,0,3],[4,47,0.0851,0.37947,0.32851,0.0,0.42857,0.57143,0.0,1.0,11,2,0,11,0,1,0,0,1,0,0,7,0,0,6,0,0,1,0,0,3,0,2],[8,47,0.1702,0.41061,0.239,0.28571,0.42859,0.57143,0.0,1.0,6,1,0,6,0,1,0,0,2,0,0,8,0,0,14,0,0,0,0,0,0,0,1],[12,47,0.2553,0.37946,0.25407,0.28571,0.42857,0.57143,0.0,1.0,7,1,0,7,0,0,0,0,7,0,0,6,0,0,10,0,0,0,0,0,1,0,1],[16,47,0.3404,0.42856,0.26963,0.28571,0.42857,0.57143,0.0,1.0,5,3,0,5,0,1,0,0,4,0,0,12,0,0,5,0,0,2,0,0,0,0,3],[20,47,0.4255,0.27231,0.30796,0.0,0.28571,0.42857,0.0,1.0,14,3,0,14,0,1,0,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,3],[24,47,0.5106,0.0625,0.11258,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.08487,0.19847,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,47,0.766,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,47,0.9362,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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":0.0,"k":"volatile","v":0.07589,"x":0.83036,"p":[[0,19,0.0,0.67857,0.32143,0.39286,0.78564,1.0,0.0,1.0,2,11,0,2,0,0,0,0,6,0,0,2,0,0,3,0,0,3,0,0,5,0,11],[4,19,0.2105,0.54018,0.31081,0.28571,0.57143,0.85714,0.0,1.0,3,5,0,3,0,1,0,0,7,0,0,4,0,0,5,0,0,3,0,0,4,0,5],[8,19,0.4211,0.43304,0.32239,0.24999,0.42857,0.57143,0.0,1.0,6,5,0,6,0,2,0,0,7,0,0,4,0,0,6,0,0,2,0,0,0,0,5],[12,19,0.6316,0.07589,0.15966,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,19,0.8421,0.83036,0.29974,0.82143,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,0,3,0,21],[19,19,1.0,0.81249,0.30607,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,1,0,21]]}]},{"i":"5cc3a6ce71120892","q":"Let  $n > 1$  be a positive integer. Prove that every term of the sequence  $$   n - 1, n^n - 1, n^{n^2} - 1, n^{n^3} - 1, \\dots $$  has a prime divisor that does not divide any of the previous terms.","t":[{"b":5,"e":1.0,"k":"flat","v":0.94196,"x":0.95535,"p":[[0,8,0.0,0.94196,0.10012,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[4,8,0.5,0.95535,0.06623,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[8,8,1.0,0.94641,0.06918,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20]]},{"b":7,"e":1.0,"k":"flat","v":0.95088,"x":0.99107,"p":[[0,5,0.0,0.95088,0.0846,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[4,5,0.8,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[5,5,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"b2257c2513cfad64","q":"Let $ABC$ be a triangle and $\\omega$ a circle passing through $A$ and $B$ and intersecting the sides $[BC]$ and $[AC]$ at $D$ and $E$ respectively. The points $K$ and $L$ are the centers of the incircles of $DAC$ and $BEC$ respectively. Let $N$ be the intersection of the lines $(EL)$ and $(DK)$.\n\nProve that the triangle $KNL$ is isosceles.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.11607,"x":0.55802,"p":[[0,43,0.0,0.55802,0.26812,0.28571,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,11,0,0,1,0,0,8,0,0,5,0,0,0,0,6],[4,43,0.093,0.44196,0.25843,0.28571,0.28571,0.57143,0.14286,1.0,0,4,0,0,0,4,0,0,13,0,0,5,0,0,4,0,0,2,0,0,0,0,4],[8,43,0.186,0.41964,0.27185,0.2857,0.28571,0.57143,0.0,1.0,1,4,0,1,0,5,0,0,12,0,0,5,0,0,3,0,0,2,0,0,0,0,4],[12,43,0.2791,0.43303,0.29339,0.2857,0.28571,0.57143,0.0,1.0,2,5,0,2,0,4,0,0,12,0,0,3,0,0,5,0,0,1,0,0,0,0,5],[16,43,0.3721,0.31247,0.18704,0.2857,0.28571,0.28571,0.0,1.0,2,1,0,2,0,5,0,0,18,0,0,2,0,0,4,0,0,0,0,0,0,0,1],[20,43,0.4651,0.31248,0.21851,0.2857,0.28571,0.28571,0.0,1.0,4,2,0,4,0,1,0,0,22,0,0,1,0,0,2,0,0,0,0,0,0,0,2],[24,43,0.5581,0.36607,0.23941,0.24999,0.28571,0.46431,0.14286,1.0,0,2,0,0,0,8,0,0,15,0,0,1,0,0,4,0,0,1,0,0,1,0,2],[28,43,0.6512,0.30355,0.2223,0.14286,0.2857,0.28571,0.0,1.0,2,2,0,2,0,8,0,0,17,0,0,0,0,0,3,0,0,0,0,0,0,0,2],[32,43,0.7442,0.16071,0.12242,0.14286,0.14286,0.17857,0.0,0.42857,7,0,0,7,0,17,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.13393,0.13333,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,16,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.11607,0.08328,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.14732,0.07563,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.15616,"x":0.62499,"p":[[0,36,0.0,0.62499,0.29179,0.39286,0.57143,1.0,0.0,1.0,1,10,1,1,0,0,0,0,7,0,0,2,0,0,10,0,0,2,0,0,0,0,10],[4,36,0.1111,0.60266,0.30249,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,8,0,0,3,0,0,7,0,0,2,0,0,0,0,10],[8,36,0.2222,0.54017,0.27371,0.28571,0.42859,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,14,0,0,3,0,0,3,0,0,6,0,0,0,0,6],[12,36,0.3333,0.50889,0.28106,0.28571,0.49979,0.57143,0.0,1.0,1,6,0,1,0,1,0,0,12,0,0,2,0,0,9,0,0,1,0,0,0,0,6],[16,36,0.4444,0.42409,0.21274,0.2857,0.28571,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,16,0,0,3,0,0,7,0,0,2,0,0,0,0,2],[20,36,0.5556,0.39285,0.23146,0.2857,0.28571,0.35714,0.14286,1.0,0,3,0,0,0,2,0,0,22,0,0,0,0,0,4,0,0,1,0,0,0,0,3],[24,36,0.6667,0.4508,0.28606,0.2857,0.35714,0.71429,0.0,1.0,1,4,0,1,0,6,0,0,9,0,0,4,0,0,3,0,0,5,0,0,0,0,4],[28,36,0.7778,0.26331,0.25032,0.14214,0.2143,0.32143,0.0,1.0,7,2,0,7,0,9,0,0,8,0,0,4,0,0,2,0,0,0,0,0,0,0,2],[32,36,0.8889,0.22322,0.15542,0.14286,0.14286,0.42857,0.0,0.57143,4,0,0,4,0,16,0,0,3,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[36,36,1.0,0.15616,0.20628,0.0,0.14286,0.1786,0.0,1.0,13,1,0,13,0,11,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,1]]}]},{"i":"46040eec217e003e","q":"5. (COL 3) Let $A B C D$ be a convex quadrilateral such that $A C=$ $B D$. Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_{1}, O_{2}, O_{3}, O_{4}$ be the centers of the triangles constructed on $A B, B C, C D, D A$ respectively. Show that $O_{1} O_{3}$ is perpendicular to $O_{2} O_{4}$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.01339,"x":0.12722,"p":[[0,46,0.0,0.12722,0.24981,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,3,0,1,2,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[4,46,0.087,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.03571,0.10714,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.04911,0.09852,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.05804,0.13296,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,46,0.5217,0.04455,0.10366,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,46,0.7826,0.02679,0.08329,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.04009,0.09597,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0625,0.13803,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[46,46,1.0,0.04464,0.10972,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00893,"x":0.1116,"p":[[0,95,0.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,95,0.0421,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,95,0.0842,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,95,0.1263,0.04018,0.15251,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,95,0.1684,0.05357,0.15872,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[20,95,0.2105,0.04911,0.13651,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,95,0.2526,0.05803,0.14664,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,95,0.2947,0.06241,0.12334,0.0,0.0,0.035,0.0,0.4286,24,0,0,24,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,95,0.3368,0.04463,0.11532,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,95,0.3789,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,95,0.4211,0.04018,0.11425,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,95,0.4632,0.06249,0.14694,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[48,95,0.5053,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,95,0.5474,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,95,0.5895,0.06697,0.1472,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,4,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[60,95,0.6316,0.1116,0.18463,0.0,0.0,0.17857,0.0,0.57143,22,0,0,22,0,2,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[64,95,0.6737,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,95,0.7158,0.09821,0.18013,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,4,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[72,95,0.7579,0.06697,0.13825,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,95,0.8,0.03124,0.10548,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,95,0.8421,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,95,0.8842,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,95,0.9263,0.04018,0.12993,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[92,95,0.9684,0.0536,0.14181,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[95,95,1.0,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"13d16943470c6ac0","q":"$ABC$  be a triangle. Its incircle touches the sides  $CB, AC, AB$  respectively at  $N_{A},N_{B},N_{C}$ . The orthic triangle of  $ABC$  is  $H_{A}H_{B}H_{C}$  with  $H_{A}, H_{B}, H_{C}$  are respectively on  $BC, AC, AB$ . The incenter of  $AH_{C}H_{B}$  is  $I_{A}$ ;  $I_{B}$  and  $I_{C}$  were defined similarly.\r\nProve that the hexagon  $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$  has all sides equal.","t":[{"b":0,"e":0.0,"k":"falling","v":0.03563,"x":0.37054,"p":[[0,50,0.0,0.30803,0.12931,0.2857,0.28571,0.42857,0.0,0.4286,3,0,1,3,0,2,0,0,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.34375,0.21683,0.28571,0.28571,0.42857,0.0,1.0,4,2,0,4,0,1,0,0,13,0,0,12,0,0,0,0,0,0,0,0,0,0,2],[8,50,0.16,0.37054,0.23381,0.28571,0.28571,0.42857,0.0,1.0,3,3,0,3,0,0,0,0,16,0,0,10,0,0,0,0,0,0,0,0,0,0,3],[12,50,0.24,0.34375,0.21387,0.28571,0.35714,0.42857,0.0,1.0,5,1,0,5,0,1,0,0,10,0,0,13,0,0,0,0,0,2,0,0,0,0,1],[16,50,0.32,0.35714,0.18898,0.28571,0.28571,0.42857,0.0,1.0,2,1,0,2,0,2,0,0,14,0,0,11,0,0,0,0,0,2,0,0,0,0,1],[20,50,0.4,0.30349,0.12254,0.2857,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,4,0,0,14,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.31254,0.16149,0.2857,0.42857,0.42857,0.0,0.43,6,0,0,6,0,0,0,0,8,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.32146,0.15569,0.28571,0.42857,0.42857,0.0,0.571,5,0,0,5,0,0,0,0,10,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[32,50,0.64,0.29465,0.15126,0.1429,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,5,0,0,8,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.3125,0.20341,0.24999,0.35714,0.42857,0.0,1.0,6,1,0,6,0,2,0,0,8,0,0,15,0,0,0,0,0,0,0,0,0,0,1],[40,50,0.8,0.25006,0.18216,0.0,0.28571,0.42857,0.0,0.43,10,0,0,10,0,1,0,0,8,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.26768,0.15886,0.14214,0.28571,0.42857,0.0,0.4286,6,0,0,6,0,4,0,0,10,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.0625,0.11812,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.03563,0.09439,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.22321,"x":0.35714,"p":[[0,76,0.0,0.30357,0.1171,0.28571,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,3,0,0,16,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[4,76,0.0526,0.32586,0.15246,0.28571,0.35714,0.42857,0.0,0.571,4,0,0,4,0,1,0,0,11,0,0,14,0,0,2,0,0,0,0,0,0,0,0],[8,76,0.1053,0.29018,0.14934,0.2857,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,2,0,0,12,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[12,76,0.1579,0.26786,0.14617,0.14286,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,4,0,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[16,76,0.2105,0.32148,0.20519,0.14286,0.42857,0.42857,0.0,0.85714,6,0,0,6,0,3,0,0,5,0,0,16,0,0,0,0,0,1,0,0,1,0,0],[20,76,0.2632,0.22321,0.15542,0.0,0.28571,0.28571,0.0,0.4286,9,0,0,9,0,2,0,0,15,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[24,76,0.3158,0.24554,0.14826,0.14289,0.28571,0.28571,0.0,0.4286,7,0,0,7,0,2,0,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[28,76,0.3684,0.32143,0.18898,0.2857,0.28571,0.42857,0.0,1.0,5,1,0,5,0,0,0,0,13,0,0,13,0,0,0,0,0,0,0,0,0,0,1],[32,76,0.4211,0.30357,0.13716,0.2857,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,1,0,0,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[36,76,0.4737,0.34822,0.10677,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,2,0,0,11,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[40,76,0.5263,0.33036,0.13092,0.2857,0.42857,0.42857,0.0,0.4286,3,0,0,3,0,1,0,0,11,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[44,76,0.5789,0.29469,0.12345,0.28571,0.28571,0.42857,0.0,0.43,3,0,0,3,0,2,0,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.29464,0.15126,0.2857,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,2,0,0,11,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[52,76,0.6842,0.29018,0.14054,0.2857,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,0,0,0,16,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.32143,0.12877,0.2857,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,1,0,0,13,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.33482,0.09852,0.28571,0.28571,0.42857,0.0,0.42857,1,0,0,1,0,1,0,0,16,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[64,76,0.8421,0.35268,0.10092,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,1,0,0,12,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.35714,0.07143,0.28571,0.35714,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.35714,0.10715,0.28571,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,14,0,0,15,0,0,0,0,0,1,0,0,0,0,0],[76,76,1.0,0.34826,0.11815,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,15,0,0,14,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"e61d14ec9b9c2258","q":"Let  $I$  be the incenter of an acute triangle  $ABC$ . Circle  $o$  passes through  $I$  and is tangent to  $BC$  at  $C$ . Ray  $BI$  meets  $o$  again at  $D\\ne I$ . Ray  $BA$  meets the circumcircle of  $ADI$  again at  $E\\ne A$ , which lies outside the segment  $AB$ . Prove that the intersection of lines  $DE$  and  $AI$  lies on  $o$ 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$a, b, \\alpha, \\beta$ be real numbers such that $0 \\leq a, b \\leq 1$, and $0 \\leq \\alpha, \\beta \\leq \\frac{\\pi}{2}$. Show that if\n\n$$\na b \\cos (\\alpha-\\beta) \\leq \\sqrt{\\left(1-a^{2}\\right)\\left(1-b^{2}\\right)}\n$$\n\nthen\n\n$$\na \\cos \\alpha+b \\sin \\beta \\leq 1+a b \\sin (\\beta-\\alpha)\n$$","t":[{"b":1,"e":1.0,"k":"rising","v":0.44196,"x":1.0,"p":[[0,104,0.0,0.44196,0.36832,0.14286,0.42857,0.75,0.0,1.0,7,7,0,7,0,5,0,0,1,0,0,9,0,0,0,0,0,2,0,0,1,0,7],[4,104,0.0385,0.625,0.34209,0.42857,0.71429,1.0,0.0,1.0,4,10,0,4,0,2,0,0,0,0,0,6,0,0,0,0,0,10,0,0,0,0,10],[8,104,0.0769,0.625,0.32093,0.42857,0.71429,1.0,0.0,1.0,3,9,0,3,0,2,0,0,1,0,0,5,0,0,2,0,0,10,0,0,0,0,9],[12,104,0.1154,0.70536,0.29867,0.53571,0.71429,1.0,0.0,1.0,2,12,0,2,0,1,0,0,1,0,0,4,0,0,1,0,0,11,0,0,0,0,12],[16,104,0.1538,0.59821,0.33586,0.28571,0.71429,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,4,0,0,3,0,0,1,0,0,10,0,0,1,0,8],[20,104,0.1923,0.71875,0.26841,0.71429,0.71429,1.0,0.0,1.0,2,10,0,2,0,0,0,0,1,0,0,3,0,0,1,0,0,14,0,0,1,0,10],[24,104,0.2308,0.55357,0.3549,0.28571,0.64286,0.89286,0.0,1.0,5,8,0,5,0,2,0,0,3,0,0,5,0,0,1,0,0,7,0,0,1,0,8],[28,104,0.2692,0.63839,0.3589,0.42857,0.71429,1.0,0.0,1.0,5,10,0,5,0,1,0,0,1,0,0,4,0,0,1,0,0,6,0,0,4,0,10],[32,104,0.3077,0.62946,0.31106,0.42857,0.71429,1.0,0.0,1.0,2,9,0,2,0,1,0,0,4,0,0,6,0,0,0,0,0,9,0,0,1,0,9],[36,104,0.3462,0.71429,0.28571,0.57143,0.71429,1.0,0.0,1.0,2,12,0,2,0,1,0,0,0,0,0,2,0,0,6,0,0,9,0,0,0,0,12],[40,104,0.3846,0.70982,0.31029,0.53571,0.71429,1.0,0.0,1.0,2,14,0,2,0,1,0,0,1,0,0,4,0,0,4,0,0,6,0,0,0,0,14],[44,104,0.4231,0.60268,0.29175,0.42857,0.71429,0.71429,0.0,1.0,3,4,0,3,0,3,0,0,0,0,0,3,0,0,1,0,0,17,0,0,1,0,4],[48,104,0.4615,0.62946,0.34416,0.42857,0.71429,1.0,0.0,1.0,2,11,0,2,0,5,0,0,0,0,0,5,0,0,2,0,0,6,0,0,1,0,11],[52,104,0.5,0.68741,0.3124,0.42857,0.71429,1.0,0.0,1.0,2,12,0,2,0,1,0,0,2,0,0,5,0,0,1,0,0,8,0,0,1,0,12],[56,104,0.5385,0.66071,0.31894,0.64286,0.71429,0.89286,0.0,1.0,3,8,0,3,0,2,0,0,2,0,0,1,0,0,0,0,0,13,0,0,3,0,8],[60,104,0.5769,0.58929,0.37244,0.28571,0.71429,1.0,0.0,1.0,6,11,0,6,0,0,0,0,3,0,0,5,0,0,1,0,0,6,0,0,0,0,11],[64,104,0.6154,0.68304,0.34206,0.42857,0.71429,1.0,0.0,1.0,3,13,0,3,0,2,0,0,1,0,0,4,0,0,0,0,0,8,0,0,1,0,13],[68,104,0.6538,0.59373,0.34462,0.28571,0.71429,0.89275,0.0,1.0,4,8,0,4,0,3,0,0,2,0,0,2,0,0,2,0,0,10,0,0,1,0,8],[72,104,0.6923,0.82141,0.28123,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,2,0,19],[76,104,0.7308,0.78571,0.36943,0.71429,1.0,1.0,0.0,1.0,5,22,0,5,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,22],[80,104,0.7692,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[84,104,0.8077,0.69196,0.33141,0.42857,0.71429,1.0,0.0,1.0,2,13,0,2,0,3,0,0,0,0,0,5,0,0,1,0,0,6,0,0,2,0,13],[88,104,0.8462,0.90625,0.25155,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,26],[92,104,0.8846,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,104,0.9231,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[100,104,0.9615,0.95981,0.1143,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[104,104,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]},{"b":7,"e":1.0,"k":"rising","v":0.25,"x":0.97321,"p":[[0,144,0.0,0.44643,0.38091,0.0,0.42857,0.71429,0.0,1.0,10,7,0,10,0,2,0,0,0,0,0,7,0,0,2,0,0,4,0,0,0,0,7],[4,144,0.0278,0.67411,0.27718,0.57143,0.71429,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,1,0,0,3,0,0,3,0,0,13,0,0,1,0,8],[8,144,0.0556,0.71875,0.29984,0.53571,0.71429,1.0,0.0,1.0,1,14,0,1,0,1,0,0,3,0,0,3,0,0,4,0,0,5,0,0,1,0,14],[12,144,0.0833,0.44196,0.3542,0.10714,0.5,0.71429,0.0,1.0,8,4,0,8,0,4,0,0,3,0,0,1,0,0,2,0,0,10,0,0,0,0,4],[16,144,0.1111,0.64286,0.32537,0.42857,0.71429,1.0,0.0,1.0,2,10,0,2,0,4,0,0,0,0,0,5,0,0,0,0,0,11,0,0,0,0,10],[20,144,0.1389,0.53571,0.38465,0.14286,0.64286,1.0,0.0,1.0,7,9,0,7,0,3,0,0,1,0,0,3,0,0,2,0,0,7,0,0,0,0,9],[24,144,0.1667,0.52679,0.36498,0.24999,0.64286,0.75,0.0,1.0,7,7,0,7,0,1,0,0,3,0,0,4,0,0,1,0,0,8,0,0,1,0,7],[28,144,0.1944,0.46429,0.34442,0.14286,0.42857,0.71429,0.0,1.0,6,6,0,6,0,4,0,0,1,0,0,9,0,0,1,0,0,5,0,0,0,0,6],[32,144,0.2222,0.46875,0.41224,0.14286,0.28571,1.0,0.0,1.0,6,10,0,6,0,9,0,0,2,0,0,2,0,0,0,0,0,2,0,0,1,0,10],[36,144,0.25,0.50893,0.36759,0.10714,0.64286,0.71429,0.0,1.0,8,6,0,8,0,1,0,0,3,0,0,2,0,0,2,0,0,9,0,0,1,0,6],[40,144,0.2778,0.36161,0.3499,0.0,0.21428,0.71429,0.0,1.0,9,4,0,9,0,7,0,0,2,0,0,4,0,0,0,0,0,6,0,0,0,0,4],[44,144,0.3056,0.3482,0.32524,0.0,0.35714,0.57143,0.0,1.0,12,3,0,12,0,0,0,0,4,0,0,6,0,0,4,0,0,3,0,0,0,0,3],[48,144,0.3333,0.37945,0.36702,0.0,0.28571,0.71429,0.0,1.0,10,5,0,10,0,5,0,0,2,0,0,4,0,0,2,0,0,3,0,0,1,0,5],[52,144,0.3611,0.41072,0.34395,0.0,0.42857,0.71429,0.0,1.0,9,3,0,9,0,3,0,0,2,0,0,6,0,0,1,0,0,6,0,0,2,0,3],[56,144,0.3889,0.32143,0.31135,0.0,0.28571,0.42858,0.0,1.0,9,3,0,9,0,6,0,0,4,0,0,6,0,0,1,0,0,3,0,0,0,0,3],[60,144,0.4167,0.37946,0.33045,0.0,0.42857,0.71429,0.0,1.0,9,3,0,9,0,4,0,0,2,0,0,7,0,0,0,0,0,7,0,0,0,0,3],[64,144,0.4444,0.33036,0.32427,0.0,0.21429,0.5,0.0,1.0,9,3,0,9,0,7,0,0,3,0,0,5,0,0,0,0,0,5,0,0,0,0,3],[68,144,0.4722,0.35268,0.36067,0.0,0.21431,0.5,0.0,1.0,10,5,0,10,0,6,0,0,2,0,0,6,0,0,0,0,0,2,0,0,1,0,5],[72,144,0.5,0.49554,0.33309,0.14286,0.4286,0.71429,0.0,1.0,4,5,0,4,0,5,0,0,3,0,0,5,0,0,1,0,0,8,0,0,1,0,5],[76,144,0.5278,0.46429,0.32927,0.14286,0.42857,0.71429,0.0,1.0,5,5,0,5,0,4,0,0,4,0,0,5,0,0,3,0,0,6,0,0,0,0,5],[80,144,0.5556,0.37945,0.35284,0.0,0.28571,0.71429,0.0,1.0,10,3,0,10,0,5,0,0,2,0,0,2,0,0,2,0,0,7,0,0,1,0,3],[84,144,0.5833,0.40179,0.37532,0.0,0.35714,0.71429,0.0,1.0,10,5,0,10,0,5,0,0,1,0,0,3,0,0,2,0,0,5,0,0,1,0,5],[88,144,0.6111,0.34812,0.34435,0.0,0.2857,0.46429,0.0,1.0,9,5,0,9,0,5,0,0,6,0,0,4,0,0,1,0,0,2,0,0,0,0,5],[92,144,0.6389,0.34375,0.37091,0.10714,0.14286,0.71429,0.0,1.0,8,6,0,8,0,11,0,0,3,0,0,1,0,0,0,0,0,3,0,0,0,0,6],[96,144,0.6667,0.34375,0.40384,0.0,0.14286,0.75,0.0,1.0,13,6,0,13,0,7,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,6],[100,144,0.6944,0.32143,0.33312,0.0,0.14286,0.57143,0.0,1.0,10,3,0,10,0,7,0,0,3,0,0,3,0,0,2,0,0,3,0,0,1,0,3],[104,144,0.7222,0.25,0.32341,0.0,0.14286,0.42857,0.0,1.0,13,4,0,13,0,8,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,4],[108,144,0.75,0.27677,0.28779,0.0,0.21429,0.42857,0.0,1.0,12,2,0,12,0,4,0,0,3,0,0,7,0,0,3,0,0,1,0,0,0,0,2],[112,144,0.7778,0.38839,0.38504,0.0,0.28571,0.71429,0.0,1.0,12,6,0,12,0,2,0,0,3,0,0,3,0,0,3,0,0,2,0,0,1,0,6],[116,144,0.8056,0.66071,0.3458,0.39286,0.71429,1.0,0.0,1.0,4,11,0,4,0,0,0,0,4,0,0,2,0,0,1,0,0,7,0,0,3,0,11],[120,144,0.8333,0.4375,0.37105,0.10714,0.42857,0.75,0.0,1.0,8,7,0,8,0,3,0,0,3,0,0,8,0,0,0,0,0,2,0,0,1,0,7],[124,144,0.8611,0.47322,0.40475,0.14286,0.35714,1.0,0.0,1.0,7,11,1,7,0,4,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,11],[128,144,0.8889,0.59821,0.43071,0.14286,0.85714,1.0,0.0,1.0,7,16,0,7,0,2,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,16],[132,144,0.9167,0.89732,0.21199,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,25],[136,144,0.9444,0.95536,0.1448,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,29],[140,144,0.9722,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[144,144,1.0,0.96875,0.09269,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28]]}]},{"i":"10afa3d0c8ed4287","q":"Turbo the snail sits on a point on a circle with circumference  $1$ . Given an infinite sequence of positive real numbers  $c_1, c_2, c_3, \\dots$ , Turbo successively crawls distances  $c_1, c_2, c_3, \\dots$  around the circle, each time choosing to crawl either clockwise or counterclockwise.\nDetermine the largest constant  $C > 0$  with the following property: for every sequence of positive real numbers  $c_1, c_2, c_3, \\dots$  with  $c_i < C$  for all  $i$ , Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.","t":[{"b":2,"e":0.42857,"k":"rising","v":0.07589,"x":0.67857,"p":[[0,42,0.0,0.07589,0.2448,0.0,0.0,0.0,0.0,1.0,28,2,14,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,42,0.0952,0.60713,0.28347,0.42857,0.57143,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,2,0,0,10,0,0,5,0,0,3,0,0,3,0,7],[8,42,0.1905,0.53124,0.22933,0.42857,0.42857,0.57143,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,20,0,0,3,0,0,1,0,0,2,0,4],[12,42,0.2857,0.58481,0.31615,0.42857,0.4998,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,1,0,0,11,0,0,4,0,0,1,0,0,3,0,8],[16,42,0.381,0.64285,0.26244,0.42857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,13,0,0,4,0,0,3,0,0,3,0,8],[20,42,0.4762,0.67857,0.26,0.42857,0.57143,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,2,0,0,2,0,11],[24,42,0.5714,0.53572,0.26964,0.42857,0.42857,0.71429,0.0,1.0,2,5,0,2,0,1,0,0,1,0,0,16,0,0,3,0,0,2,0,0,2,0,5],[28,42,0.6667,0.4866,0.15915,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,26,0,0,1,0,0,0,0,0,3,0,1],[32,42,0.7619,0.51786,0.19804,0.42857,0.42857,0.42858,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,26,0,0,1,0,0,0,0,0,1,0,4],[36,42,0.8571,0.52232,0.19759,0.42857,0.42857,0.4286,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,24,0,0,0,0,0,2,0,0,2,0,3],[40,42,0.9524,0.43304,0.12619,0.42857,0.42857,0.42857,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,1],[42,42,1.0,0.44196,0.07457,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,29,0,0,0,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"rising","v":0.0625,"x":0.75445,"p":[[0,92,0.0,0.0625,0.16342,0.0,0.0,0.0,0.0,0.71429,27,0,20,27,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,92,0.0435,0.71427,0.30094,0.42857,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,0,0,0,8,0,0,4,0,0,1,0,0,4,0,13],[8,92,0.087,0.69643,0.28959,0.42857,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,9,0,0,2,0,0,3,0,0,3,0,12],[12,92,0.1304,0.67857,0.26726,0.42857,0.64286,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,11,0,0,4,0,0,2,0,0,5,0,9],[16,92,0.1739,0.68304,0.26422,0.42857,0.64286,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,3,0,0,2,0,11],[20,92,0.2174,0.59374,0.28596,0.42857,0.42857,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,0,0,0,16,0,0,3,0,0,1,0,0,2,0,8],[24,92,0.2609,0.56696,0.25376,0.42857,0.42857,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,20,0,0,1,0,0,1,0,0,1,0,7],[28,92,0.3043,0.64286,0.26245,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,12,0,0,0,0,0,7,0,0,2,0,8],[32,92,0.3478,0.75445,0.24285,0.53539,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,3,0,0,2,0,14],[36,92,0.3913,0.64286,0.22868,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,11,0,0,5,0,0,5,0,0,5,0,5],[40,92,0.4348,0.58025,0.31944,0.42857,0.42859,1.0,0.0,1.0,2,10,0,2,0,2,0,0,2,0,0,11,0,0,4,0,0,1,0,0,0,0,10],[44,92,0.4783,0.53125,0.22084,0.42857,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,18,0,0,5,0,0,0,0,0,2,0,4],[48,92,0.5217,0.58032,0.23402,0.42857,0.571,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,13,0,0,7,0,0,4,0,0,1,0,5],[52,92,0.5652,0.44196,0.15303,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,2,0,0,24,0,0,2,0,0,0,0,0,1,0,1],[56,92,0.6087,0.4375,0.13333,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,26,0,0,1,0,0,0,0,0,2,0,0],[60,92,0.6522,0.46429,0.22016,0.42857,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,3,0,0,2,0,0,23,0,0,0,0,0,0,0,0,0,0,4],[64,92,0.6957,0.47768,0.19758,0.42857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,3,0,0,1,0,0,21,0,0,1,0,0,2,0,0,3,0,1],[68,92,0.7391,0.44643,0.13716,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,28,0,0,0,0,0,0,0,0,1,0,1],[72,92,0.7826,0.54911,0.2372,0.42857,0.42857,0.75,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,19,0,0,1,0,0,1,0,0,4,0,4],[76,92,0.8261,0.48661,0.16311,0.42857,0.42857,0.46429,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,22,0,0,4,0,0,1,0,0,2,0,1],[80,92,0.8696,0.46875,0.12492,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,26,0,0,2,0,0,2,0,0,0,0,1],[84,92,0.913,0.55804,0.22689,0.42857,0.42857,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,2,0,0,2,0,5],[88,92,0.9565,0.45536,0.1448,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,27,0,0,0,0,0,1,0,0,1,0,1],[92,92,1.0,0.41518,0.04164,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7901fe9fb1be0fb3","q":"10. (SWE 4) ${ }^{\\mathrm{IMO} 3}$ Let $1=a_{0} \\leq a_{1} \\leq a_{2} \\leq \\cdots \\leq a_{n} \\leq \\cdots$ be a sequence of real numbers. Consider the sequence $b_{1}, b_{2}, \\ldots$ defined by:  $$ b_{n}=\\sum_{k=1}^{n}\\left(1-\\frac{a_{k-1}}{a_{k}}\\right) \\frac{1}{\\sqrt{a_{k}}} $$  Prove that: (a) For all natural numbers $n, 0 \\leq b_{n}<2$. (b) Given an arbitrary $0 \\leq b<2$, there is a sequence $a_{0}, a_{1}, \\ldots, a_{n}, \\ldots$ of the above type such that $b_{n}>b$ is true for infinitely many natural numbers $n$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.92855,"x":0.99554,"p":[[0,32,0.0,0.95982,0.11971,1.0,1.0,1.0,0.4286,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[4,32,0.125,0.92855,0.16754,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,26],[8,32,0.25,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[12,32,0.375,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,32,0.5,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,32,0.625,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,32,0.75,0.9375,0.14698,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,26],[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],[32,32,1.0,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25]]},{"b":7,"e":1.0,"k":"flat","v":0.86159,"x":0.99554,"p":[[0,28,0.0,0.96428,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[4,28,0.1429,0.96874,0.1056,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[8,28,0.2857,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[12,28,0.4286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,28,0.5714,0.91518,0.15509,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,22],[20,28,0.7143,0.86159,0.20974,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,4,0,0,1,0,21],[24,28,0.8571,0.9464,0.13251,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,27],[28,28,1.0,0.93749,0.13335,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,25]]}]},{"i":"64faf55dd054324a","q":"Find the smallest prime which is not the difference (in some order) of a power of  $2$  and a power of  $3$ .","t":[{"b":5,"e":0.2857,"k":"falling","v":0.30357,"x":0.90179,"p":[[0,71,0.0,0.72767,0.22689,0.57143,0.71429,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,1,0,0,11,0,0,8,0,0,1,0,10],[4,71,0.0563,0.90179,0.19045,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,1,0,24],[8,71,0.1127,0.89732,0.17215,0.71429,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,1,0,22],[12,71,0.169,0.83929,0.22232,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,10,0,0,1,0,18],[16,71,0.2254,0.79462,0.23131,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,7,0,0,2,0,15],[20,71,0.2817,0.85711,0.15157,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,5,0,15],[24,71,0.338,0.80802,0.23855,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,5,0,0,2,0,17],[28,71,0.3944,0.72768,0.2212,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,12,0,0,0,0,10],[32,71,0.4507,0.72768,0.25595,0.42859,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,7,0,0,2,0,12],[36,71,0.507,0.61158,0.23484,0.53539,0.57143,0.71429,0.28571,1.0,0,6,0,0,0,0,0,0,7,0,0,1,0,0,12,0,0,6,0,0,0,0,6],[40,71,0.5634,0.65624,0.27167,0.42857,0.57143,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,5,0,0,3,0,0,8,0,0,5,0,0,0,0,10],[44,71,0.6197,0.73213,0.26905,0.57143,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,4,0,0,0,0,0,4,0,0,10,0,0,1,0,12],[48,71,0.6761,0.62944,0.25966,0.42859,0.57143,0.75,0.0,1.0,1,7,0,1,0,0,0,0,4,0,0,5,0,0,7,0,0,7,0,0,1,0,7],[52,71,0.7324,0.58482,0.27516,0.28571,0.57143,0.71429,0.0,1.0,1,7,0,1,0,0,0,0,8,0,0,3,0,0,8,0,0,5,0,0,0,0,7],[56,71,0.7887,0.57366,0.26755,0.28571,0.57143,0.71429,0.07143,1.0,0,6,0,0,1,0,0,0,8,0,0,5,0,0,6,0,0,5,0,0,1,0,6],[60,71,0.8451,0.62945,0.29637,0.42857,0.57143,1.0,0.0,1.0,1,10,0,1,0,1,0,0,5,0,0,4,0,0,7,0,0,4,0,0,0,0,10],[64,71,0.9014,0.56696,0.2461,0.39286,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,8,0,0,6,0,0,9,0,0,3,0,0,0,0,6],[68,71,0.9577,0.37053,0.18509,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,21,0,0,6,0,0,2,0,0,0,0,0,0,0,2],[71,71,1.0,0.30357,0.09943,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,1,0,0,25,0,0,3,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"falling","v":0.55795,"x":0.91518,"p":[[0,99,0.0,0.71427,0.31136,0.57143,0.71429,1.0,0.0,1.0,3,13,1,3,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,0,0,0,13],[4,99,0.0404,0.91518,0.13767,0.71429,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,23],[8,99,0.0808,0.87054,0.23244,0.71429,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,0,0,22],[12,99,0.1212,0.83034,0.18709,0.67857,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,7,0,0,0,0,17],[16,99,0.1616,0.83482,0.17536,0.71429,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,0,0,16],[20,99,0.202,0.85267,0.20357,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,9,0,0,0,0,19],[24,99,0.2424,0.86161,0.19719,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,9,0,0,1,0,19],[28,99,0.2828,0.84375,0.18336,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,0,0,0,17],[32,99,0.3232,0.84821,0.16728,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,0,0,17],[36,99,0.3636,0.83034,0.16538,0.71429,0.71429,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,0,0,15],[40,99,0.404,0.83034,0.19705,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,10,0,0,0,0,17],[44,99,0.4444,0.85268,0.1838,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,11,0,0,0,0,18],[48,99,0.4848,0.83482,0.17169,0.71429,0.85714,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,0,0,16],[52,99,0.5253,0.86607,0.2111,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,5,0,0,1,0,21],[56,99,0.5657,0.86604,0.20809,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,6,0,0,0,0,21],[60,99,0.6061,0.83482,0.15198,0.71429,0.71429,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,1,0,14],[64,99,0.6465,0.79909,0.20783,0.67857,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,9,0,0,0,0,15],[68,99,0.6869,0.91517,0.20159,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,25],[72,99,0.7273,0.82143,0.22303,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,7,0,0,0,0,18],[76,99,0.7677,0.875,0.17405,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,1,0,20],[80,99,0.8081,0.82588,0.17031,0.71429,0.71429,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,0,0,15],[84,99,0.8485,0.78124,0.20513,0.71429,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,12,0,0,0,0,13],[88,99,0.8889,0.83927,0.18816,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,1,0,17],[92,99,0.9293,0.65622,0.14224,0.57143,0.64286,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,12,0,0,2,0,2],[96,99,0.9697,0.56246,0.11811,0.5713,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,24,0,0,2,0,0,0,0,1],[99,99,1.0,0.55795,0.1488,0.571,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,4,0,0,22,0,0,1,0,0,0,0,2]]}]},{"i":"02f2af0e624e1f82","q":"Given an acute triangle  $ABC$  with  $O$  as its circumcenter.  Line  $AO$  intersects  $BC$  at  $D$ .  Points  $E$ ,  $F$  are on  $AB$ ,  $AC$  respectively such that  $A$ ,  $E$ ,  $D$ ,  $F$  are concyclic. Prove that the length of the projection of line segment  $EF$  on side  $BC$  does not depend on the positions of  $E$  and  $F$ 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 $n>3$  be an integer. If  $x_1<x_2<\\ldots<x_{n+2}$  are reals with  $x_1=0$ ,  $x_2=1$  and  $x_3>2$ , what is the maximal value of   $$ (\\frac{x_{n+1}+x_{n+2}-1}{x_{n+1}(x_{n+2}-1)})\\cdot (\\sum_{i=1}^{n}\\frac{(x_{i+2}-x_{i+1})(x_{i+1}-x_i)}{x_{i+2}-x_i})? $$","t":[{"b":6,"e":0.42857,"k":"rising","v":0.32143,"x":0.81696,"p":[[0,57,0.0,0.32143,0.24744,0.24999,0.28571,0.28571,0.0,1.0,2,3,1,2,0,6,0,0,20,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[4,57,0.0702,0.81696,0.29931,0.57143,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,2,0,21],[8,57,0.1404,0.72768,0.33571,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,5,0,0,4,0,0,2,0,0,0,0,0,1,0,18],[12,57,0.2105,0.74553,0.3169,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,4,0,0,6,0,0,1,0,0,1,0,0,1,0,18],[16,57,0.2807,0.75,0.3093,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,5,0,0,4,0,0,2,0,0,1,0,0,1,0,18],[20,57,0.3509,0.67411,0.30354,0.42857,0.64286,1.0,0.0,1.0,1,13,0,1,0,0,0,0,3,0,0,10,0,0,2,0,0,2,0,0,1,0,13],[24,57,0.4211,0.78571,0.28347,0.53571,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,5,0,0,3,0,0,2,0,0,1,0,0,3,0,18],[28,57,0.4912,0.70089,0.26088,0.42857,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,10,0,0,2,0,0,4,0,0,3,0,11],[32,57,0.5614,0.67411,0.27254,0.42857,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,12,0,0,4,0,0,1,0,0,1,0,12],[36,57,0.6316,0.63838,0.28118,0.42857,0.42857,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,14,0,0,1,0,0,2,0,0,2,0,10],[40,57,0.7018,0.62945,0.30485,0.42857,0.57143,1.0,0.0,1.0,1,11,0,1,0,0,0,0,6,0,0,7,0,0,5,0,0,1,0,0,1,0,11],[44,57,0.7719,0.70534,0.25986,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,11,0,0,3,0,0,3,0,0,2,0,12],[48,57,0.8421,0.73201,0.27376,0.42857,0.78571,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,4,0,0,2,0,14],[52,57,0.9123,0.62052,0.26871,0.42857,0.42859,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,13,0,0,3,0,0,0,0,0,4,0,8],[56,57,0.9825,0.60714,0.27894,0.42857,0.4286,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,13,0,0,5,0,0,0,0,0,0,0,10],[57,57,1.0,0.63393,0.2878,0.42857,0.5,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,6,0,0,10,0,0,3,0,0,0,0,0,3,0,10]]},{"b":7,"e":0.57143,"k":"rising","v":0.28125,"x":0.88839,"p":[[0,147,0.0,0.28125,0.15765,0.25,0.28571,0.28571,0.0,1.0,1,1,0,1,0,7,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[4,147,0.0272,0.88839,0.22794,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,3,0,24],[8,147,0.0544,0.83036,0.25862,0.67857,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,1,0,0,3,0,20],[12,147,0.0816,0.87946,0.25532,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,0,0,25],[16,147,0.1088,0.88839,0.21646,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,3,0,23],[20,147,0.1361,0.85714,0.25505,0.89286,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,0,0,0,0,0,24],[24,147,0.1633,0.78571,0.28794,0.42857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,20],[28,147,0.1905,0.76339,0.27804,0.4286,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,4,0,0,5,0,0,2,0,0,2,0,0,3,0,16],[32,147,0.2177,0.8125,0.26592,0.53571,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,0,0,0,2,0,20],[36,147,0.2449,0.82143,0.2369,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,2,0,0,3,0,18],[40,147,0.2721,0.67857,0.29234,0.42857,0.64286,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,3,0,0,10,0,0,2,0,0,1,0,0,3,0,12],[44,147,0.2993,0.77232,0.28763,0.53571,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,2,0,0,5,0,0,2,0,0,3,0,0,2,0,17],[48,147,0.3265,0.74999,0.28122,0.53539,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,1,0,0,0,0,17],[52,147,0.3537,0.77232,0.28316,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,3,0,0,5,0,0,0,0,0,3,0,17],[56,147,0.381,0.76784,0.27375,0.42857,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,3,0,0,1,0,17],[60,147,0.4082,0.79018,0.29447,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,2,0,0,6,0,0,0,0,0,2,0,0,2,0,19],[64,147,0.4354,0.72768,0.28203,0.42857,0.71429,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,4,0,0,0,0,15],[68,147,0.4626,0.73661,0.27689,0.42857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,7,0,0,4,0,0,1,0,0,2,0,15],[72,147,0.4898,0.74554,0.29609,0.42859,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,3,0,0,6,0,0,3,0,0,1,0,0,1,0,17],[76,147,0.517,0.77232,0.30485,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,3,0,0,6,0,0,0,0,0,1,0,0,3,0,18],[80,147,0.5442,0.70982,0.30823,0.42857,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,5,0,0,5,0,0,4,0,0,1,0,0,0,0,16],[84,147,0.5714,0.77677,0.24207,0.57132,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,3,0,0,1,0,16],[88,147,0.5986,0.77232,0.24185,0.57143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,3,0,0,4,0,14],[92,147,0.6259,0.8125,0.25364,0.53571,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,2,0,0,2,0,19],[96,147,0.6531,0.66072,0.27374,0.42857,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,2,0,0,13,0,0,4,0,0,1,0,0,0,0,12],[100,147,0.6803,0.67187,0.2862,0.42857,0.64286,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,5,0,1,7,0,0,3,0,0,3,0,0,1,0,12],[104,147,0.7075,0.74999,0.25506,0.42859,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,2,0,0,3,0,14],[108,147,0.7347,0.70982,0.27313,0.42857,0.78571,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,9,0,0,2,0,0,2,0,0,4,0,12],[112,147,0.7619,0.69195,0.26513,0.42857,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,7,0,0,7,0,0,2,0,0,1,0,12],[116,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Occidentalia there are  $20$  different companies, each looking to hire  $15$  new employees. A group of  $300$  applicants interview each of the companies. Each company qualifies each applicant as suitable or not suitable to work in it, in such a way that each of them finds exactly  $p$  suitable applicants, with  $p > 15$ . and each applicant is found suitable by at least one company. What is the smallest of  $p $ f or which it is always possible to assign  $15$  applicants to each company, given that each company is assigned only applicants that it considers appropriate, and that each of the  $300$  applicants is assigned to a company?","t":[{"b":4,"e":0.14286,"k":"falling","v":0.07143,"x":0.95536,"p":[[0,119,0.0,0.59819,0.28669,0.39286,0.71429,0.85714,0.0,1.0,2,3,0,2,0,2,0,0,4,0,0,3,0,0,2,0,0,10,0,0,6,0,3],[4,119,0.0336,0.87053,0.24317,0.85714,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,18],[8,119,0.0672,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,119,0.1008,0.91963,0.17476,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,23],[16,119,0.1345,0.94643,0.09942,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[20,119,0.1681,0.79464,0.29653,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,3,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,17],[24,119,0.2017,0.85713,0.26,0.857,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,21],[28,119,0.2353,0.87943,0.21166,0.82132,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,0,2,0,22],[32,119,0.2689,0.67848,0.36435,0.24999,0.85714,1.0,0.0,1.0,2,12,0,2,0,6,0,0,1,0,0,0,0,0,1,0,0,4,0,0,6,0,12],[36,119,0.3025,0.76783,0.3288,0.57132,1.0,1.0,0.0,1.0,3,17,0,3,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,0,4,0,17],[40,119,0.3361,0.71872,0.34716,0.571,0.85714,1.0,0.0,1.0,2,14,0,2,0,4,0,0,1,0,0,0,0,0,4,0,0,1,0,0,6,0,14],[44,119,0.3697,0.88839,0.23347,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,21],[48,119,0.4034,0.77232,0.32705,0.71429,0.92857,1.0,0.0,1.0,3,16,0,3,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,6,0,16],[52,119,0.437,0.74998,0.31944,0.71429,0.85714,1.0,0.0,1.0,4,10,0,4,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,12,0,10],[56,119,0.4706,0.72767,0.36309,0.57143,0.85714,1.0,0.0,1.0,3,15,0,3,0,4,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,15],[60,119,0.5042,0.84821,0.31529,0.85714,1.0,1.0,0.0,1.0,3,22,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,22],[64,119,0.5378,0.69197,0.38814,0.35716,0.85714,1.0,0.0,1.0,5,13,0,5,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,13],[68,119,0.5714,0.71429,0.39769,0.25,1.0,1.0,0.0,1.0,4,18,0,4,0,4,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,18],[72,119,0.605,0.63839,0.4103,0.14286,0.85714,1.0,0.0,1.0,6,13,0,6,0,4,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,13],[76,119,0.6387,0.54464,0.41869,0.14286,0.64286,1.0,0.0,1.0,6,11,0,6,0,7,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,11],[80,119,0.6723,0.56248,0.38785,0.14286,0.71429,0.85714,0.0,1.0,6,6,0,6,0,5,0,0,1,0,0,0,0,0,1,0,0,5,0,0,8,0,6],[84,119,0.7059,0.5982,0.44095,0.14286,0.85714,1.0,0.0,1.0,6,15,0,6,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,15],[88,119,0.7395,0.70087,0.37689,0.46397,0.85714,1.0,0.0,1.0,3,15,0,3,0,5,0,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,15],[92,119,0.7731,0.56696,0.40638,0.14286,0.78564,1.0,0.0,1.0,6,9,0,6,0,5,0,0,2,0,0,0,0,0,2,0,0,1,0,0,7,0,9],[96,119,0.8067,0.55357,0.38258,0.14286,0.71429,0.85714,0.0,1.0,5,7,0,5,0,5,0,0,2,0,0,3,0,0,0,0,0,3,0,0,7,0,7],[100,119,0.8403,0.52677,0.39356,0.14286,0.64286,0.85714,0.0,1.0,5,7,0,5,0,8,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,7],[104,119,0.8739,0.53572,0.43301,0.10714,0.71429,1.0,0.0,1.0,8,11,0,8,0,6,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,11],[108,119,0.9076,0.39285,0.39285,0.10714,0.14286,0.85704,0.0,1.0,8,6,0,8,0,10,0,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,6],[112,119,0.9412,0.14277,0.22868,0.0,0.14143,0.14286,0.0,0.85714,15,0,0,15,0,13,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[116,119,0.9748,0.08465,0.07002,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[119,119,1.0,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"falling","v":0.37495,"x":0.96429,"p":[[0,51,0.0,0.59816,0.24338,0.42857,0.57143,0.75,0.0,1.0,1,3,1,1,0,1,0,0,3,0,0,5,0,0,9,0,0,5,0,0,5,0,3],[4,51,0.0784,0.94196,0.09354,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,21],[8,51,0.1569,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[12,51,0.2353,0.93302,0.1071,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[16,51,0.3137,0.96429,0.15152,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[20,51,0.3922,0.94642,0.09943,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[24,51,0.4706,0.88392,0.19377,0.85711,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,0,0,5,0,20],[28,51,0.549,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[32,51,0.6275,0.94642,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[36,51,0.7059,0.91071,0.18123,0.85714,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,19],[40,51,0.7843,0.91517,0.14667,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,7,0,21],[44,51,0.8627,0.94642,0.09943,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[48,51,0.9412,0.76338,0.29367,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,1,0,0,2,0,0,3,0,0,3,0,0,6,0,14],[51,51,1.0,0.37495,0.23072,0.1429,0.35714,0.571,0.0,1.0,2,1,0,2,0,7,0,0,7,0,0,7,0,0,6,0,0,1,0,0,1,0,1]]}]},{"i":"c2035ec78ca17fdc","q":"Solve in integers the equation\n\\[ x^2+xy+y^2 = \\left(\\frac{x+y}{3}+1\\right)^3. \\]","t":[{"b":5,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,14,0.0,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[4,14,0.2857,0.95089,0.17717,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[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],[12,14,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[14,14,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"flat","v":0.88839,"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],[4,34,0.1176,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,34,0.2353,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[12,34,0.3529,0.94643,0.16656,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[16,34,0.4706,0.88839,0.198,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,5,0,21],[20,34,0.5882,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[24,34,0.7059,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[28,34,0.8235,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[32,34,0.9412,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[34,34,1.0,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]}]},{"i":"b0183f65a8c89e8f","q":"1. (AUS 2) Prove that for any positive integer $m$ there exist an infinite number of pairs of integers $(x, y)$ such that (i) $x$ and $y$ are relatively prime; (ii) $y$ divides $x^{2}+m$; (iii) $x$ divides $y^{2}+m$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.91071,"x":0.97768,"p":[[0,41,0.0,0.94196,0.10631,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[4,41,0.0976,0.92857,0.13832,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,25],[8,41,0.1951,0.94196,0.14664,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[12,41,0.2927,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[16,41,0.3902,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[20,41,0.4878,0.91964,0.15947,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,24],[24,41,0.5854,0.95089,0.14555,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[28,41,0.6829,0.91071,0.14617,0.82132,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,22],[32,41,0.7805,0.91071,0.1915,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,1,0,25],[36,41,0.878,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[40,41,0.9756,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[41,41,1.0,0.95089,0.11633,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,27]]},{"b":6,"e":0.71429,"k":"flat","v":0.76783,"x":0.98214,"p":[[0,48,0.0,0.91516,0.14666,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,22],[4,48,0.0833,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[8,48,0.1667,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[12,48,0.25,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,48,0.3333,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,26],[20,48,0.4167,0.93304,0.15561,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,26],[24,48,0.5,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[28,48,0.5833,0.94197,0.14222,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,27],[32,48,0.6667,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,48,0.75,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[40,48,0.8333,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[44,48,0.9167,0.84375,0.17261,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,13,0,0,1,0,16],[48,48,1.0,0.76783,0.12247,0.71429,0.71429,0.75,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,22,0,0,2,0,6]]}]},{"i":"69a33da4a9b35ab4","q":"Let  $ABC,AA_1A_2,BB_1B_2, CC_1C_2$  be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments  $A_2B_1,B_2C_1, C_2A_1$  by  $P,Q,R$  in this order. Prove that the triangle  $PQR$  is 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0,0,2,0,0,4,0,0,2,0,0,5,0,15],[28,108,0.2593,0.71874,0.31438,0.42859,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,4,0,0,2,0,0,3,0,0,2,0,0,4,0,14],[32,108,0.2963,0.77677,0.26472,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,2,0,0,8,0,13],[36,108,0.3333,0.7455,0.28957,0.57132,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,3,0,0,1,0,0,4,0,0,4,0,0,5,0,13],[40,108,0.3704,0.68299,0.23888,0.4286,0.71429,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,1,0,0,7,0,0,5,0,0,6,0,0,5,0,7],[44,108,0.4074,0.79909,0.18852,0.71429,0.85707,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,7,0,11],[48,108,0.4444,0.80354,0.24159,0.67846,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,4,0,0,4,0,16],[52,108,0.4815,0.70534,0.29437,0.42857,0.857,1.0,0.0,1.0,1,10,0,1,0,2,0,0,1,0,0,5,0,0,3,0,0,3,0,0,7,0,10],[56,108,0.5185,0.80801,0.20706,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,5,0,0,7,0,13],[60,108,0.5556,0.71424,0.26965,0.5354,0.78564,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,4,0,0,3,0,0,4,0,0,4,0,0,6,0,10],[64,108,0.5926,0.77677,0.21411,0.67857,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,7,0,0,6,0,11],[68,108,0.6296,0.79911,0.2693,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,1,0,0,3,0,0,1,0,0,5,0,0,3,0,17],[72,108,0.6667,0.73658,0.2262,0.57132,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,6,0,0,6,0,9],[76,108,0.7037,0.67404,0.27949,0.571,0.71429,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,2,0,0,1,0,0,6,0,0,7,0,0,7,0,6],[80,108,0.7407,0.70085,0.21832,0.571,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,8,0,0,6,0,6],[84,108,0.7778,0.71871,0.22726,0.571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,6,0,0,4,0,9],[88,108,0.8148,0.90177,0.1448,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,10,0,18],[92,108,0.8519,0.75891,0.21559,0.57143,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,9,0,0,3,0,11],[96,108,0.8889,0.78122,0.24742,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,5,0,0,4,0,14],[100,108,0.9259,0.82141,0.19234,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,7,0,0,7,0,13],[104,108,0.963,0.81247,0.21263,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,7,0,14],[108,108,1.0,0.74099,0.22148,0.571,0.78564,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,2,0,0,6,0,10]]}]},{"i":"2c24e41d6fbfe225","q":"14. C2 (AUS 4) Let $A B C D$ be a convex plane quadrilateral and let $A_{1}$ denote the circumcenter of $\\triangle B C D$. Define $B_{1}, C_{1}, D_{1}$ in a corresponding way. (a) Prove that either all of $A_{1}, B_{1}, C_{1}, D_{1}$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_{1}, C_{1}$ are on opposite sides of the line $B_{1} D_{1}$, and similarly, $B_{1}, D_{1}$ are on opposite sides of the line $A_{1} C_{1}$. (This establishes the convexity of the quadrilateral $A_{1} B_{1} C_{1} D_{1}$.) (b) Denote by $A_{2}$ the circumcenter of $B_{1} C_{1} D_{1}$, and define $B_{2}, C_{2}, D_{2}$ in an analogous way. Show that the quadrilateral $A_{2} B_{2} C_{2} D_{2}$ is similar to the quadrilateral $A B C D$.","t":[{"b":3,"e":0.57143,"k":"flat","v":0.51784,"x":0.67853,"p":[[0,45,0.0,0.52232,0.26392,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,9,0,0,6,0,0,0,0,0,9,0,0,2,0,3],[4,45,0.0889,0.67853,0.189,0.57142,0.57143,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,4,0,0,3,0,6],[8,45,0.1778,0.62942,0.21684,0.42859,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,8,0,0,10,0,0,5,0,0,1,0,6],[12,45,0.2667,0.58481,0.15303,0.5354,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,17,0,0,4,0,0,1,0,2],[16,45,0.3556,0.61157,0.212,0.4286,0.57143,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,5,0,0,11,0,0,7,0,0,0,0,5],[20,45,0.4444,0.62948,0.18154,0.571,0.57143,0.60714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,18,0,0,2,0,0,1,0,5],[24,45,0.5333,0.60713,0.14726,0.5714,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,5,0,0,16,0,0,6,0,0,3,0,1],[28,45,0.6222,0.55804,0.21535,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,9,0,0,10,0,0,5,0,0,1,0,3],[32,45,0.7111,0.66067,0.21652,0.571,0.57143,0.85704,0.2857,1.0,0,5,0,0,0,0,0,0,3,0,0,4,0,0,10,0,0,5,0,0,5,0,5],[36,45,0.8,0.51784,0.12239,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,8,0,0,16,0,0,4,0,0,0,0,0],[40,45,0.8889,0.55797,0.10925,0.57075,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,5,0,0,19,0,0,6,0,0,0,0,0],[44,45,0.9778,0.54018,0.10549,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,10,0,0,16,0,0,5,0,0,0,0,0],[45,45,1.0,0.58914,0.09261,0.57142,0.57143,0.71107,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,18,0,0,9,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.55349,"x":0.76784,"p":[[0,45,0.0,0.58035,0.26471,0.28571,0.50001,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,7,0,0,2,0,0,3,0,0,8,0,3],[4,45,0.0889,0.69636,0.23626,0.53465,0.57143,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,7,0,0,9,0,0,3,0,0,2,0,10],[8,45,0.1778,0.65623,0.20159,0.5354,0.57143,0.74996,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,7,0,0,9,0,0,7,0,0,3,0,5],[12,45,0.2667,0.6116,0.25812,0.4286,0.57143,0.75,0.0,1.0,2,6,0,2,0,0,0,0,1,0,0,6,0,0,12,0,0,3,0,0,2,0,6],[16,45,0.3556,0.76784,0.18815,0.67857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,10,0,0,5,0,9],[20,45,0.4444,0.63836,0.2172,0.42857,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,9,0,0,7,0,0,4,0,0,6,0,4],[24,45,0.5333,0.63839,0.2525,0.42857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,8,0,0,2,0,0,8,0,0,2,0,7],[28,45,0.6222,0.61606,0.18707,0.4286,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,10,0,0,8,0,0,6,0,0,5,0,2],[32,45,0.7111,0.55349,0.15046,0.42859,0.57121,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,7,0,0,16,0,0,4,0,0,1,0,1],[36,45,0.8,0.70088,0.25092,0.5713,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,4,0,0,5,0,0,6,0,0,7,0,7],[40,45,0.8889,0.62725,0.1827,0.42965,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,1,10,0,0,5,0,0,4,0,3],[44,45,0.9778,0.62944,0.18851,0.571,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,6,0,0,12,0,0,7,0,0,3,0,3],[45,45,1.0,0.5848,0.16115,0.4286,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,7,0,0,14,0,0,5,0,0,3,0,1]]}]},{"i":"cad302646db2fd5e","q":"A country has $n$ cities, labelled $1,2,3, \\ldots, n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly 1 , and it is also not permitted to put a road between cities 1 and $n$. Let $T_{n}$ be the total number of possible ways to build these roads. (a) For all odd $n$, prove that $T_{n}$ is divisible by $n$. (b) For all even $n$, prove that $T_{n}$ is divisible by $n / 2$.","t":[{"b":5,"e":0.71429,"k":"rising","v":0.1875,"x":0.88839,"p":[[0,45,0.0,0.1875,0.31224,0.0,0.0,0.42857,0.0,1.0,23,1,0,23,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,0,0,1],[4,45,0.0889,0.63392,0.39113,0.42825,0.71429,1.0,0.0,1.0,8,11,0,8,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,3,0,11],[8,45,0.1778,0.66071,0.40208,0.24999,0.78571,1.0,0.0,1.0,7,14,0,7,0,1,0,0,1,0,0,0,0,0,0,0,0,7,0,0,2,0,14],[12,45,0.2667,0.64732,0.3589,0.57143,0.71429,1.0,0.0,1.0,6,10,0,6,0,1,0,0,0,0,0,0,0,0,2,0,0,12,0,0,1,0,10],[16,45,0.3556,0.43304,0.4439,0.0,0.35714,0.89286,0.0,1.0,16,8,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,8],[20,45,0.4444,0.54018,0.42368,0.0,0.71429,1.0,0.0,1.0,11,10,0,11,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,2,0,10],[24,45,0.5333,0.69643,0.39082,0.67857,0.85714,1.0,0.0,1.0,7,16,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,16],[28,45,0.6222,0.3704,0.38596,0.0,0.28571,0.71429,0.0,1.0,15,3,0,15,0,0,0,0,3,0,0,0,0,0,2,0,0,6,0,0,3,0,3],[32,45,0.7111,0.48661,0.42687,0.0,0.71429,0.85714,0.0,1.0,13,7,0,13,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,4,0,7],[36,45,0.8,0.72321,0.36932,0.71429,0.85714,1.0,0.0,1.0,6,16,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,16],[40,45,0.8889,0.88839,0.198,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,20],[44,45,0.9778,0.83482,0.17896,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0,5,0,14],[45,45,1.0,0.85714,0.14725,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,7,0,14]]},{"b":6,"e":0.71429,"k":"rising","v":0.07589,"x":0.87946,"p":[[0,38,0.0,0.07589,0.20511,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[4,38,0.1053,0.83482,0.20238,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,2,0,15],[8,38,0.2105,0.76339,0.28259,0.71429,0.85714,1.0,0.0,1.0,2,13,0,2,0,0,0,0,2,0,0,1,0,0,1,0,0,9,0,0,4,0,13],[12,38,0.3158,0.57589,0.41108,0.0,0.71429,1.0,0.0,1.0,10,10,0,10,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,2,0,10],[16,38,0.4211,0.67411,0.38172,0.57143,0.71429,1.0,0.0,1.0,7,13,0,7,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,2,0,13],[20,38,0.5263,0.59807,0.3561,0.39286,0.71429,1.0,0.0,1.0,6,9,0,6,0,1,0,0,1,0,0,1,0,0,5,0,0,9,0,0,0,0,9],[24,38,0.6316,0.67411,0.38172,0.57143,0.71429,1.0,0.0,1.0,7,13,0,7,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,2,0,13],[28,38,0.7368,0.80804,0.20705,0.71429,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,2,0,13],[32,38,0.8421,0.84821,0.14258,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,3,0,14],[36,38,0.9474,0.87946,0.13415,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,6,0,16],[38,38,1.0,0.81683,0.1395,0.71429,0.71429,1.0,0.5714,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,2,0,11]]}]},{"i":"30782a422c0ecbe9","q":"There is a  $20\\times 25$  grid, each cell with blank. Tomita played the game using this grid. The game has several turns, and in  $n$ th turn, the following operation are performed: \n\nChoose a positive integer  $k$ , and cells  $A_1, A_2, \\dots, A_k$  such that  $A_{i+1}$  is adjacent to  $A_i$ , and is right, or above  $A_{i}$ , for  $1\\le i\\le n-1$ . We write down the number  $n$  for each cell  $A_1, A_2, \\dots, A_{k}$ . \n\nThe game terminates when all the cells have written a number. If Tomita play the game in such the way that the number of turns of the game achieves the minimum, find the number of possible configurations of the grid.","t":[{"b":3,"e":0.42857,"k":"rising","v":0.09375,"x":0.50445,"p":[[0,42,0.0,0.09375,0.16602,0.0,0.0,0.07143,0.0,0.42857,24,0,24,24,0,0,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.43302,0.14052,0.42857,0.42857,0.4642,0.0,0.71429,1,0,0,1,0,1,0,0,4,0,0,18,0,0,6,0,0,2,0,0,0,0,0],[8,42,0.1905,0.50445,0.10091,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,19,0,0,9,0,0,4,0,0,0,0,0],[12,42,0.2857,0.48215,0.09279,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,20,0,0,9,0,0,2,0,0,0,0,0],[16,42,0.381,0.45089,0.09523,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,23,0,0,4,0,0,2,0,0,0,0,0],[20,42,0.4762,0.45983,0.11143,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,20,0,0,7,0,0,2,0,0,0,0,0],[24,42,0.5714,0.46426,0.10099,0.42857,0.42857,0.571,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,20,0,0,7,0,0,2,0,0,0,0,0],[28,42,0.6667,0.48657,0.10009,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,17,0,0,11,0,0,2,0,0,0,0,0],[32,42,0.7619,0.45981,0.12233,0.42857,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,19,0,0,6,0,0,3,0,0,0,0,0],[36,42,0.8571,0.43302,0.0836,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,21,0,0,6,0,0,0,0,0,0,0,0],[40,42,0.9524,0.45533,0.10969,0.42857,0.42857,0.4642,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,23,0,0,7,0,0,1,0,0,0,0,0],[42,42,1.0,0.45987,0.05903,0.42857,0.42857,0.42895,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,25,0,0,7,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"rising","v":0.07589,"x":0.50003,"p":[[0,26,0.0,0.07589,0.15966,0.0,0.0,0.0,0.0,0.4286,26,0,26,26,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.50003,0.14283,0.42857,0.57121,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,12,0,0,13,0,0,4,0,0,0,0,0],[8,26,0.3077,0.47322,0.1448,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,17,0,0,6,0,0,5,0,0,0,0,0],[12,26,0.4615,0.45087,0.12929,0.42857,0.42857,0.571,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,19,0,0,7,0,0,2,0,0,0,0,0],[16,26,0.6154,0.47768,0.10479,0.42857,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,17,0,0,12,0,0,1,0,0,0,0,0],[20,26,0.7692,0.47542,0.13073,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,1,0,0,0,15,0,0,12,0,0,2,0,0,0,0,0],[24,26,0.9231,0.49106,0.13802,0.42857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,19,0,0,7,0,0,5,0,0,0,0,0],[26,26,1.0,0.45085,0.15609,0.42857,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,1,0,0,13,0,0,13,0,0,1,0,0,0,0,0]]}]},{"i":"a6007d26d073af1c","q":"Given a triangle  $ABC$ . Let  $\\Omega$  be the circumscribed circle of this triangle, and  $\\omega$  be the inscribed circle of this triangle. Let  $\\delta$  be a circle that touches the sides  $AB$  and  $AC$ , and also touches the circle  $\\Omega$   internally at point  $D$ . The line  $AD$  intersects the circle  $\\Omega$  at two points  $P$  and  $Q$  ( $P$  lies between  $A$  and  $Q$ ). Let  $O$  and  $I$  be the centers of the circles  $\\Omega$  and  $\\omega$ . Prove that  $OD \\parallel IQ$ .","t":[{"b":0,"e":0.14286,"k":"falling","v":0.03125,"x":0.46865,"p":[[0,52,0.0,0.41963,0.37616,0.14286,0.14288,0.89286,0.0,1.0,3,8,1,3,0,14,0,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,8],[4,52,0.0769,0.46865,0.30152,0.14286,0.42857,0.71429,0.0,1.0,2,3,0,2,0,8,0,0,3,0,0,4,0,0,3,0,0,8,0,0,1,0,3],[8,52,0.1538,0.14732,0.22442,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,9,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[12,52,0.2308,0.10268,0.17941,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,4,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,52,0.3077,0.15625,0.24053,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,9,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[20,52,0.3846,0.16963,0.27992,0.0,0.0,0.21429,0.0,1.0,21,1,0,21,0,3,0,0,0,0,0,2,0,0,3,0,0,2,0,0,0,0,1],[24,52,0.4615,0.06696,0.15966,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[28,52,0.5385,0.09375,0.15407,0.0,0.0,0.14287,0.0,0.57143,21,0,0,21,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[32,52,0.6154,0.13838,0.25624,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,7,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[36,52,0.6923,0.12946,0.21829,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,10,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,52,0.7692,0.12946,0.21829,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,10,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[44,52,0.8462,0.12946,0.17627,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,12,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[48,52,0.9231,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.0892,"x":0.36158,"p":[[0,51,0.0,0.2857,0.35891,0.10714,0.14286,0.1786,0.0,1.0,8,6,2,8,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,6],[4,51,0.0784,0.33481,0.27573,0.14286,0.35714,0.46418,0.0,1.0,7,1,0,7,0,7,0,0,2,0,0,8,0,0,3,0,0,3,0,0,1,0,1],[8,51,0.1569,0.36158,0.30088,0.10714,0.28571,0.60714,0.0,1.0,8,1,0,8,0,5,0,0,4,0,0,3,0,0,4,0,0,6,0,0,1,0,1],[12,51,0.2353,0.25444,0.27134,0.0,0.14286,0.46418,0.0,0.85714,12,0,0,12,0,7,0,0,2,0,0,3,0,0,4,0,0,3,0,0,1,0,0],[16,51,0.3137,0.2857,0.24741,0.10714,0.21428,0.42858,0.0,0.85714,8,0,0,8,0,8,0,0,2,0,0,8,0,0,4,0,0,0,0,0,2,0,0],[20,51,0.3922,0.20527,0.17108,0.14214,0.14286,0.32143,0.0,0.57143,7,0,0,7,0,14,0,0,3,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[24,51,0.4706,0.18741,0.2354,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,9,0,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[28,51,0.549,0.20536,0.23128,0.0,0.14286,0.1786,0.0,1.0,9,1,0,9,0,15,0,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,1],[32,51,0.6275,0.19634,0.23625,0.0,0.14286,0.28571,0.0,0.85714,13,0,0,13,0,9,0,0,3,0,0,2,0,0,3,0,0,1,0,0,1,0,0],[36,51,0.7059,0.12054,0.11904,0.0,0.14286,0.14286,0.0,0.57143,11,0,0,11,0,17,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,51,0.7843,0.17402,0.1881,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,15,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[44,51,0.8627,0.13393,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.143,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b837f506c4c463da","q":"Let $\\Gamma_{1}$ and $\\Gamma_{2}$ be two intersecting circles with centers $O_{1}$ and $O_{2}$ respectively, such that $\\Gamma_{2}$ intersects the line segment $O_{1} O_{2}$ at a point $A$. The intersection points of $\\Gamma_{1}$ and $\\Gamma_{2}$ are $C$ and $D$. The line $A D$ intersects $\\Gamma_{1}$ again at $S$. The line $C S$ intersects $O_{1} O_{2}$ at $F$. Let $\\Gamma_{3}$ be the circumcircle of triangle $A D F$. Denote $E$ as the second intersection point of $\\Gamma_{1}$ and $\\Gamma_{3}$.\nProve that $O_{1} E$ is tangent to $\\Gamma_{3}$.","t":[{"b":3,"e":0.0,"k":"falling","v":0.00893,"x":0.33472,"p":[[0,42,0.0,0.28572,0.11845,0.2857,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,5,0,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.33472,0.14996,0.2857,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,3,0,0,8,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[8,42,0.1905,0.26339,0.17896,0.10714,0.28571,0.42857,0.0,0.57143,8,0,0,8,0,2,0,0,11,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[12,42,0.2857,0.2634,0.17169,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,4,0,0,9,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[16,42,0.381,0.25437,0.17035,0.105,0.28571,0.42857,0.0,0.4286,8,0,0,8,0,3,0,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.16063,0.19151,0.0,0.0,0.32143,0.0,0.57143,17,0,0,17,0,3,0,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[24,42,0.5714,0.13839,0.16164,0.0,0.07143,0.2857,0.0,0.4286,16,0,0,16,0,6,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.10705,0.15566,0.0,0.0,0.17857,0.0,0.4286,20,0,0,20,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.03563,0.0713,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.26786,"x":0.37054,"p":[[0,65,0.0,0.28125,0.16935,0.14286,0.28571,0.42857,0.0,0.4286,7,0,0,7,0,2,0,0,8,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.35268,0.07974,0.28571,0.35714,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,15,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.36161,0.10705,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,11,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[12,65,0.1846,0.29469,0.14262,0.2857,0.28571,0.42857,0.0,0.43,4,0,0,4,0,3,0,0,12,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[16,65,0.2462,0.29911,0.1488,0.28571,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,1,0,0,12,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.29465,0.15541,0.25002,0.28571,0.42857,0.0,0.4286,5,0,0,5,0,3,0,0,9,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.34375,0.11769,0.28571,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,1,0,0,11,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.26786,0.12752,0.14289,0.28571,0.32143,0.0,0.4286,3,0,0,3,0,6,0,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.34822,0.11811,0.2857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,10,0,0,17,0,0,1,0,0,0,0,0,0,0,0],[36,65,0.5538,0.34821,0.1333,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,2,0,0,10,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[40,65,0.6154,0.37054,0.07017,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,13,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.33036,0.11539,0.28571,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,0,0,0,17,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[48,65,0.7385,0.33036,0.10972,0.28571,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,3,0,0,13,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.32143,0.11294,0.2857,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,1,0,0,16,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.33929,0.1171,0.2857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,4,0,0,9,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.35713,0.13361,0.28571,0.42857,0.42857,0.0,0.571,3,0,0,3,0,0,0,0,8,0,0,20,0,0,1,0,0,0,0,0,0,0,0],[64,65,0.9846,0.32141,0.12369,0.2857,0.28571,0.42857,0.0,0.571,1,0,0,1,0,5,0,0,12,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[65,65,1.0,0.29465,0.15947,0.14286,0.35714,0.42857,0.0,0.4286,5,0,0,5,0,4,0,0,7,0,0,16,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce5a86455adae18d","q":"Points  $A,B,C,D,E$  lie on a circle  $\\omega$  and point  $P$  lies outside the circle. The given points are such that (i) lines  $PB$  and  $PD$  are tangent to  $\\omega$ , (ii)  $P, A, C$  are collinear, and (iii)  $DE \\parallel AC$ . Prove that  $BE$  bisects  $AC$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"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],[4,64,0.0625,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,64,0.1875,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,64,0.3125,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,68,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,68,0.0588,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,68,0.1176,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,68,0.1765,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,68,0.2353,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,68,0.2941,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,68,0.3529,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,68,0.4118,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,68,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,68,0.5294,0.00875,0.03389,0.0,0.0,0.0,0.0,0.14,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,68,0.5882,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,68,0.6471,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,68,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,68,0.7647,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,68,0.8824,0.00884,0.03424,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,68,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bd699d8f0b67932c","q":"Let $k$ be a positive integer and denote the sum of the digits of a positive integer $n$ by $s(n)$. Prove that among the positive integers with $k$ digits, there are as many numbers $n$ that satisfy $s(n)<s(2 n)$ as there are numbers $n$ that satisfy $s(n)>s(2 n)$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.66515,"x":0.94196,"p":[[0,43,0.0,0.79463,0.21111,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,6,0,0,3,0,14],[4,43,0.093,0.91963,0.17476,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,25],[8,43,0.186,0.86159,0.24351,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,2,0,0,2,0,22],[12,43,0.2791,0.88839,0.18808,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,2,0,22],[16,43,0.3721,0.90625,0.17717,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,3,0,23],[20,43,0.4651,0.94196,0.14223,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,26],[24,43,0.5581,0.88839,0.16263,0.85711,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,6,0,19],[28,43,0.6512,0.83034,0.2156,0.57143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,3,0,18],[32,43,0.7442,0.90179,0.15746,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,22],[36,43,0.8372,0.82589,0.19475,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,4,0,0,3,0,16],[40,43,0.9302,0.83033,0.19706,0.57143,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,3,0,17],[43,43,1.0,0.66515,0.21903,0.57143,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,2,0,0,12,0,0,5,0,0,5,0,5]]},{"b":5,"e":1.0,"k":"flat","v":0.83036,"x":0.9375,"p":[[0,34,0.0,0.84821,0.21997,0.71429,1.0,1.0,0.0,1.0,1,17,1,1,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,5,0,17],[4,34,0.1176,0.87051,0.18338,0.67857,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,3,0,20],[8,34,0.2353,0.87052,0.15305,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,7,0,16],[12,34,0.3529,0.85268,0.16935,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,5,0,16],[16,34,0.4706,0.91515,0.13774,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,21],[20,34,0.5882,0.87946,0.17169,0.71429,1.0,1.0,0.5714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,3,0,20],[24,34,0.7059,0.9375,0.13333,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,24],[28,34,0.8235,0.83036,0.19377,0.67857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,1,0,17],[32,34,0.9412,0.8973,0.1483,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,7,0,19],[34,34,1.0,0.89286,0.11845,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,11,0,15]]}]},{"i":"ab2bc41224203d92","q":"Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a, b, c, d \\in S$ with $\\operatorname{gcd}(a, b) \\neq \\operatorname{gcd}(c, d)$. Prove that there exist three pairwise distinct $x, y, z \\in S$ such that $\\operatorname{gcd}(x, y)=\\operatorname{gcd}(y, z) \\neq \\operatorname{gcd}(z, x)$.","t":[{"b":0,"e":1.0,"k":"rising","v":0.31695,"x":0.9375,"p":[[0,64,0.0,0.31695,0.35125,0.0,0.28571,0.57111,0.0,1.0,14,4,4,14,0,0,0,0,7,0,0,2,0,0,3,0,0,1,0,0,1,0,4],[4,64,0.0625,0.64732,0.3499,0.39286,0.78571,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,0,8,0,8],[8,64,0.125,0.46428,0.36596,0.0,0.42857,0.85714,0.0,1.0,9,2,0,9,0,0,0,0,6,0,0,3,0,0,0,0,0,3,0,0,9,0,2],[12,64,0.1875,0.66072,0.33455,0.42857,0.71429,1.0,0.0,1.0,4,10,0,4,0,0,0,0,1,0,0,6,0,0,2,0,0,4,0,0,5,0,10],[16,64,0.25,0.60267,0.35126,0.42857,0.71429,0.85714,0.0,1.0,6,7,0,6,0,1,0,0,0,0,0,2,0,0,6,0,0,5,0,0,5,0,7],[20,64,0.3125,0.58482,0.32214,0.28571,0.64286,0.85714,0.0,1.0,3,4,0,3,0,0,0,0,9,0,0,1,0,0,3,0,0,2,0,0,10,0,4],[24,64,0.375,0.62053,0.35465,0.28571,0.85707,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,5,0,0,3,0,0,0,0,0,2,0,0,10,0,7],[28,64,0.4375,0.71874,0.27775,0.42859,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,2,0,0,0,0,0,4,0,0,11,0,8],[32,64,0.5,0.69643,0.28516,0.42857,0.78571,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,5,0,0,5,0,0,1,0,0,4,0,0,6,0,10],[36,64,0.5625,0.6339,0.30502,0.28571,0.71429,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,6,0,0,0,0,0,6,0,0,2,0,0,10,0,5],[40,64,0.625,0.64286,0.31944,0.42857,0.71429,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,4,0,0,4,0,0,3,0,0,4,0,0,6,0,8],[44,64,0.6875,0.80802,0.2276,0.67857,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,1,0,0,10,0,13],[48,64,0.75,0.72321,0.2257,0.57143,0.85714,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,3,0,0,5,0,0,5,0,0,13,0,4],[52,64,0.8125,0.72321,0.24984,0.57143,0.85714,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,3,0,0,11,0,7],[56,64,0.875,0.83027,0.20682,0.71429,0.85714,1.0,0.14,1.0,0,13,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,6,0,0,9,0,13],[60,64,0.9375,0.92411,0.15146,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,22],[64,64,1.0,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19]]},{"b":3,"e":0.5714,"k":"flat","v":0.29911,"x":0.70087,"p":[[0,49,0.0,0.29911,0.32214,0.0,0.28571,0.42857,0.0,1.0,13,1,3,13,0,0,0,0,10,0,0,2,0,0,1,0,0,0,0,0,5,0,1],[4,49,0.0816,0.54911,0.40894,0.0,0.71429,0.89286,0.0,1.0,9,8,0,9,0,0,0,0,4,0,0,1,0,0,1,0,0,2,0,0,7,0,8],[8,49,0.1633,0.49999,0.37457,0.0,0.57143,0.85714,0.0,1.0,9,4,0,9,0,0,0,0,4,0,0,1,0,0,4,0,0,3,0,0,7,0,4],[12,49,0.2449,0.55345,0.2785,0.28571,0.57121,0.75,0.0,1.0,2,2,0,2,0,1,0,0,7,0,0,4,0,0,3,0,0,7,0,0,6,0,2],[16,49,0.3265,0.5134,0.2942,0.28571,0.42857,0.85714,0.0,1.0,2,1,0,2,0,2,0,0,10,0,0,3,0,0,2,0,0,3,0,0,9,0,1],[20,49,0.4082,0.52676,0.32031,0.28571,0.57121,0.75,0.0,1.0,5,4,0,5,0,0,0,0,6,0,0,3,0,0,5,0,0,5,0,0,4,0,4],[24,49,0.4898,0.70087,0.2609,0.571,0.78571,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,3,0,0,3,0,0,6,0,0,3,0,0,9,0,7],[28,49,0.5714,0.5446,0.25364,0.28571,0.571,0.75,0.0,1.0,1,1,0,1,0,0,0,0,10,0,0,4,0,0,4,0,0,5,0,0,7,0,1],[32,49,0.6531,0.58916,0.26199,0.4286,0.57143,0.75,0.0,1.0,2,3,0,2,0,1,0,0,3,0,0,3,0,0,10,0,0,5,0,0,5,0,3],[36,49,0.7347,0.56247,0.31731,0.28571,0.571,0.85714,0.0,1.0,3,5,0,3,0,1,0,0,6,0,0,5,0,0,3,0,0,3,0,0,6,0,5],[40,49,0.8163,0.39276,0.24231,0.28571,0.42857,0.57143,0.0,1.0,4,1,0,4,0,2,0,0,9,0,0,7,0,0,7,0,0,0,0,0,2,0,1],[44,49,0.898,0.45978,0.30457,0.28571,0.571,0.71429,0.0,1.0,6,1,0,6,0,1,0,0,7,0,0,1,0,0,6,0,0,6,0,0,4,0,1],[48,49,0.9796,0.3973,0.20431,0.28571,0.28571,0.571,0.0,0.85714,2,0,0,2,0,1,0,0,14,0,0,6,0,0,5,0,0,2,0,0,2,0,0],[49,49,1.0,0.42858,0.16751,0.28571,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,10,0,0,6,0,0,10,0,0,3,0,0,0,0,0]]}]},{"i":"35d3a499a83abd2a","q":"There are  $13$  stones each of which weighs an integer number of grams. It is known that any  $12$  of them can be put on two pans of a balance scale, six on each pan, so that they are in equilibrium (i.e., each pan will carry an equal total weight). Prove that all stones weigh the same number of grams.","t":[{"b":1,"e":0.57143,"k":"falling","v":0.41964,"x":0.88393,"p":[[0,52,0.0,0.81696,0.29931,0.75,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0,2,0,22],[4,52,0.0769,0.73214,0.41764,0.5,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,22],[8,52,0.1538,0.88393,0.2911,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,25],[12,52,0.2308,0.79464,0.37105,0.89286,1.0,1.0,0.0,1.0,4,24,0,4,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,24],[16,52,0.3077,0.61161,0.45769,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,17],[20,52,0.3846,0.70982,0.42028,0.25,1.0,1.0,0.0,1.0,6,20,0,6,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,20],[24,52,0.4615,0.51339,0.46546,0.0,0.28571,1.0,0.0,1.0,11,15,0,11,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[28,52,0.5385,0.56696,0.46082,0.0,0.92857,1.0,0.0,1.0,9,16,0,9,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,16],[32,52,0.6154,0.5625,0.46006,0.0,0.78564,1.0,0.0,1.0,12,14,0,12,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,14],[36,52,0.6923,0.63392,0.45867,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,18],[40,52,0.7692,0.60268,0.46803,0.0,1.0,1.0,0.0,1.0,11,17,0,11,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,17],[44,52,0.8462,0.64284,0.43301,0.10714,0.92857,1.0,0.0,1.0,8,16,0,8,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,16],[48,52,0.9231,0.44643,0.43264,0.0,0.28571,1.0,0.0,1.0,12,11,0,12,0,0,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,11],[52,52,1.0,0.41964,0.41178,0.0,0.28571,1.0,0.0,1.0,12,9,0,12,0,1,0,0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,9]]},{"b":6,"e":0.28571,"k":"falling","v":0.42854,"x":0.90624,"p":[[0,64,0.0,0.85267,0.27776,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,3,0,23],[4,64,0.0625,0.90624,0.26394,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,28],[8,64,0.125,0.62946,0.45014,0.0,1.0,1.0,0.0,1.0,9,18,0,9,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,18],[12,64,0.1875,0.63391,0.45448,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,18],[16,64,0.25,0.67411,0.44065,0.10714,1.0,1.0,0.0,1.0,8,19,0,8,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,19],[20,64,0.3125,0.64732,0.41647,0.2857,1.0,1.0,0.0,1.0,5,18,0,5,0,2,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,18],[24,64,0.375,0.62054,0.43096,0.21427,0.92857,1.0,0.0,1.0,8,16,0,8,0,0,0,0,4,0,0,1,0,0,0,0,0,2,0,0,1,0,16],[28,64,0.4375,0.53125,0.46047,0.0,0.57143,1.0,0.0,1.0,10,15,0,10,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,15],[32,64,0.5,0.54464,0.46898,0.0,0.71429,1.0,0.0,1.0,12,16,0,12,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,16],[36,64,0.5625,0.61607,0.44526,0.0,0.9285,1.0,0.0,1.0,9,16,0,9,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,16],[40,64,0.625,0.59822,0.43512,0.10714,0.85714,1.0,0.0,1.0,8,15,0,8,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,0,2,0,15],[44,64,0.6875,0.58929,0.41304,0.25,0.71429,1.0,0.0,1.0,7,14,0,7,0,1,0,0,4,0,0,2,0,0,1,0,0,3,0,0,0,0,14],[48,64,0.75,0.50446,0.44318,0.0,0.35714,1.0,0.0,1.0,11,13,0,11,0,0,0,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,13],[52,64,0.8125,0.54018,0.46666,0.0,0.71429,1.0,0.0,1.0,12,15,0,12,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,15],[56,64,0.875,0.50446,0.41494,0.10714,0.35714,1.0,0.0,1.0,8,12,0,8,0,1,0,0,7,0,0,3,0,0,0,0,0,1,0,0,0,0,12],[60,64,0.9375,0.44195,0.3357,0.28571,0.28571,0.64282,0.0,1.0,5,6,0,5,0,0,0,0,15,0,0,1,0,0,3,0,0,0,0,0,2,0,6],[64,64,1.0,0.42854,0.25753,0.28571,0.28571,0.4642,0.0,1.0,1,4,0,1,0,0,0,0,19,0,0,4,0,0,3,0,0,0,0,0,1,0,4]]}]},{"i":"b284c7fa4bb7cfa4","q":"Let  $M$  be the midpoint of cathetus  $AB$  of triangle  $ABC$  with right angle  $A$ . Point  $D$  lies on the median  $AN$  of triangle  $AMC$  in such a way that the angles  $ACD$  and  $BCM$  are equal. Prove that the angle  $DBC$  is also equal to these angles.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.01777,"x":0.0758,"p":[[0,64,0.0,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,64,0.0625,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.143,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,64,0.1875,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,64,0.3125,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.0758,0.07966,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.07589,"p":[[0,138,0.0,0.06241,0.08694,0.0,0.0,0.14286,0.0,0.2857,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,138,0.029,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,138,0.058,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,138,0.087,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,138,0.1159,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,138,0.1449,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,138,0.1739,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,138,0.2029,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,138,0.2319,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,138,0.2609,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,138,0.2899,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,138,0.3188,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,138,0.3478,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,138,0.3768,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,138,0.4058,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,138,0.4348,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,138,0.4638,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,138,0.4928,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,138,0.5217,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,138,0.5507,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,138,0.5797,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,138,0.6087,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,138,0.6377,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,138,0.6667,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,138,0.6957,0.04018,0.06424,0.0,0.0,0.14286,0.0,0.143,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,138,0.7246,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,138,0.7536,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,138,0.7826,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,138,0.8116,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,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$n \\geq 2$ be an integer. An $n$-tuple $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ of positive integers is expensive if there exists a positive integer $k$ such that\n\n$$\n\\left(a_{1}+a_{2}\\right)\\left(a_{2}+a_{3}\\right) \\cdots \\cdots\\left(a_{n-1}+a_{n}\\right)\\left(a_{n}+a_{1}\\right)=2^{2 k-1} .\n$$\n\na) Find all positive integers $n \\geq 2$ for which there exists an expensive $n$-tuple.\nb) Prove that for every positive integer $m$ there exists an integer $n \\geq 2$ such that $m$ belongs to an expensive $n$-tuple.\n\nThere are exactly $n$ factors in the product on the left hand side.\nHarun Hindija, Bosnia and Herzegovina","t":[{"b":1,"e":0.85714,"k":"rising","v":0.2231,"x":0.76783,"p":[[0,72,0.0,0.2231,0.17103,0.14286,0.2143,0.28571,0.0,0.71429,6,0,4,6,0,10,0,0,12,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[4,72,0.0556,0.6784,0.23957,0.5354,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,2,0,0,2,0,0,7,0,0,13,0,2],[8,72,0.1111,0.66516,0.23585,0.5354,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,2,0,0,4,0,0,5,0,0,14,0,1],[12,72,0.1667,0.63824,0.22007,0.42857,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,4,0,0,3,0,0,7,0,0,12,0,0],[16,72,0.2222,0.65177,0.24727,0.42859,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,3,0,0,2,0,0,5,0,0,14,0,1],[20,72,0.2778,0.63392,0.23127,0.4286,0.71429,0.85714,0.1429,0.85714,0,0,0,0,0,2,0,0,3,0,0,5,0,0,3,0,0,7,0,0,12,0,0],[24,72,0.3333,0.60711,0.2369,0.42857,0.57143,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,6,0,0,4,0,0,3,0,0,11,0,1],[28,72,0.3889,0.71872,0.19395,0.71429,0.85707,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,8,0,0,17,0,0],[32,72,0.4444,0.62048,0.21902,0.42859,0.64286,0.85704,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,5,0,0,6,0,0,5,0,0,11,0,0],[36,72,0.5,0.74537,0.15046,0.71429,0.85707,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,9,0,0,17,0,0],[40,72,0.5556,0.76783,0.18472,0.71429,0.85714,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,23,0,0],[44,72,0.6111,0.76348,0.15412,0.71429,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,18,0,1],[48,72,0.6667,0.71873,0.1838,0.67857,0.85707,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,7,0,0,17,0,0],[52,72,0.7222,0.75004,0.17119,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,8,0,0,19,0,0],[56,72,0.7778,0.71425,0.18559,0.57132,0.857,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,4,0,0,3,0,0,6,0,0,17,0,0],[60,72,0.8333,0.70535,0.18536,0.67857,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,9,0,0,15,0,0],[64,72,0.8889,0.72317,0.17836,0.57132,0.85707,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,5,0,0,18,0,0],[68,72,0.9444,0.74551,0.15043,0.67857,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,6,0,0,18,0,0],[72,72,1.0,0.72764,0.1763,0.67857,0.85707,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,7,0,0,17,0,0]]},{"b":2,"e":0.85714,"k":"rising","v":0.17848,"x":0.79017,"p":[[0,90,0.0,0.17848,0.14288,0.0,0.14286,0.28571,0.0,0.57143,9,0,5,9,0,9,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,90,0.0444,0.59819,0.27764,0.28571,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,7,0,0,5,0,0,1,0,0,3,0,0,13,0,1],[8,90,0.0889,0.6339,0.25488,0.39286,0.78564,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,4,0,0,1,0,0,3,0,0,16,0,0],[12,90,0.1333,0.65177,0.22571,0.42857,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,6,0,0,2,0,0,4,0,0,15,0,0],[16,90,0.1778,0.58477,0.27283,0.28571,0.64264,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,10,0,0,2,0,0,3,0,0,5,0,0,9,0,2],[20,90,0.2222,0.62498,0.27836,0.42859,0.71429,0.85714,0.0,0.85714,3,0,0,3,0,0,0,0,4,0,0,2,0,0,2,0,0,8,0,0,13,0,0],[24,90,0.2667,0.54908,0.27689,0.28571,0.57143,0.75,0.0,0.85714,3,0,0,3,0,1,0,0,6,0,0,1,0,0,6,0,0,7,0,0,8,0,0],[28,90,0.3111,0.63838,0.26239,0.42857,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,3,0,0,4,0,0,2,0,0,5,0,0,15,0,0],[32,90,0.3556,0.57139,0.27664,0.28571,0.64286,0.85714,0.0,0.85714,2,0,0,2,0,1,0,0,7,0,0,2,0,0,4,0,0,5,0,0,11,0,0],[36,90,0.4,0.55797,0.26088,0.42857,0.571,0.75,0.0,0.85714,3,0,1,3,0,0,0,0,4,0,0,5,0,0,6,0,0,6,0,0,8,0,0],[40,90,0.4444,0.52675,0.29327,0.2857,0.4286,0.85714,0.0,1.0,1,1,0,1,0,4,0,0,7,0,0,5,0,0,3,0,0,0,0,0,11,0,1],[44,90,0.4889,0.61159,0.23211,0.39286,0.64286,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,3,0,0,5,0,0,4,0,0,12,0,0],[48,90,0.5333,0.65622,0.24706,0.5354,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,2,0,0,1,0,0,4,0,0,4,0,0,5,0,0,15,0,0],[52,90,0.5778,0.70096,0.19358,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,9,0,0,13,0,1],[56,90,0.6222,0.65175,0.23674,0.53539,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,4,0,0,2,0,0,4,0,0,6,0,0,14,0,0],[60,90,0.6667,0.70087,0.21829,0.71429,0.78564,0.85714,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,0,9,0,0,16,0,0],[64,90,0.7111,0.74549,0.14171,0.67857,0.78564,0.85714,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,8,0,0,15,0,1],[68,90,0.7556,0.76333,0.18426,0.71429,0.85707,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,20,0,1],[72,90,0.8,0.76784,0.15466,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,5,0,0,20,0,1],[76,90,0.8444,0.76336,0.14557,0.71429,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,9,0,0,19,0,0],[80,90,0.8889,0.72767,0.22119,0.71429,0.85714,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,0,0,0,3,0,0,2,0,0,6,0,0,18,0,1],[84,90,0.9333,0.74104,0.20959,0.71429,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,4,0,0,20,0,1],[88,90,0.9778,0.79017,0.13355,0.71429,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,7,0,0,21,0,1],[90,90,1.0,0.76333,0.12177,0.71429,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,18,0,0]]}]},{"i":"b2e1108c6e487379","q":"Let  $ABC$  be an isosceles triangle with  $AB = AC$  and  $\\angle A = 45^o$ . Its circumcircle  $(c)$  has center  $O, M$  is the midpoint of  $BC$  and  $D$  is the foot of the perpendicular from  $C$  to  $AB$ . With center  $C$  and radius  $CD$  we draw a circle which internally intersects  $AC$  at the point  $F$  and the circle  $(c)$  at the points  $Z$  and  $E$ , such that  $Z$  lies on the small arc  $BC$  and  $E$  on the small arc  $AC$ . Prove that the lines  $ZE$ ,  $CO$ ,  $FM$  are concurrent.\n\n*Brazitikos Silouanos, Greece*","t":[{"b":0,"e":0.42857,"k":"rising","v":0.22322,"x":0.43749,"p":[[0,23,0.0,0.22322,0.14698,0.14286,0.14286,0.42857,0.0,0.4286,3,0,0,3,0,18,0,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.22767,0.16309,0.14286,0.14286,0.42857,0.0,0.571,4,0,0,4,0,17,0,0,0,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[8,23,0.3478,0.41963,0.06118,0.42857,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,1,0,0,1,0,0,29,0,0,1,0,0,0,0,0,0,0,0],[12,23,0.5217,0.41076,0.04727,0.42857,0.42857,0.42857,0.2857,0.43,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.41522,0.05487,0.42857,0.42857,0.42857,0.14286,0.43,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.41965,0.03458,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.43749,0.03453,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.14277,"x":0.26338,"p":[[0,34,0.0,0.26338,0.1561,0.14286,0.14286,0.42857,0.0,0.571,2,0,0,2,0,15,0,0,2,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[4,34,0.1176,0.24107,0.14032,0.14286,0.14286,0.42857,0.0,0.4286,1,0,0,1,0,19,0,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.23661,0.15407,0.14286,0.14286,0.42857,0.0,0.57143,2,0,0,2,0,19,0,0,0,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[12,34,0.3529,0.24108,0.17291,0.14286,0.14286,0.42857,0.0,0.4286,6,0,0,6,0,12,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.16509,0.10781,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,27,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,34,0.5882,0.19196,0.11633,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,27,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,34,0.7059,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.14733,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"124723cb19384eb7","q":"Let ABCDA'B'C'D' be a rectangular parallelipiped, where ABCD is the lower face and A, B, C and D' are below A', B', C' and D', respectively. The parallelipiped is divided into eight parts by three planes parallel to its faces. For each vertex P, let V P denote the volume of the part containing P. Given that V A= 40, V C = 300 , V B' = 360 and V C'= 90, find the volume of ABCDA'B'C'D'.","t":[{"b":2,"e":0.42857,"k":"falling","v":0.39286,"x":0.59822,"p":[[0,26,0.0,0.59822,0.27765,0.39286,0.4286,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,8,0,0,9,0,0,2,0,0,2,0,0,4,0,7],[4,26,0.1538,0.47768,0.2008,0.28571,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,10,0,0,4,0,0,5,0,0,0,0,2],[8,26,0.3077,0.50446,0.2448,0.28571,0.42857,0.60714,0.2857,1.0,0,5,0,0,0,0,0,0,10,0,0,13,0,0,1,0,0,3,0,0,0,0,5],[12,26,0.4615,0.39286,0.09449,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,19,0,0,1,0,0,1,0,0,0,0,0],[16,26,0.6154,0.42411,0.11564,0.39286,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,20,0,0,1,0,0,3,0,0,0,0,0],[20,26,0.7692,0.45089,0.13415,0.42857,0.42857,0.42858,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,20,0,0,2,0,0,3,0,0,1,0,0],[24,26,0.9231,0.43301,0.09091,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,22,0,0,4,0,0,1,0,0,0,0,0],[26,26,1.0,0.41963,0.1006,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,22,0,0,1,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.429,"k":"flat","v":0.4464,"x":0.60267,"p":[[0,26,0.0,0.55808,0.24051,0.42857,0.42857,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,6,0,0,13,0,0,2,0,0,5,0,0,1,0,5],[4,26,0.1538,0.48213,0.18471,0.42857,0.42857,0.42858,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,22,0,0,2,0,0,1,0,0,0,0,3],[8,26,0.3077,0.60267,0.27137,0.42857,0.4286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,5,0,0,13,0,0,2,0,0,3,0,0,0,0,9],[12,26,0.4615,0.4464,0.17403,0.42857,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,17,0,0,4,0,0,3,0,0,0,0,1],[16,26,0.6154,0.5,0.21129,0.39286,0.42857,0.60714,0.2857,1.0,0,3,0,0,0,0,0,0,8,0,0,14,0,0,2,0,0,5,0,0,0,0,3],[20,26,0.7692,0.54911,0.23449,0.42857,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,16,0,0,4,0,0,4,0,0,0,0,5],[24,26,0.9231,0.51784,0.20124,0.42857,0.42857,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,7,0,0,12,0,0,4,0,0,6,0,0,1,0,2],[26,26,1.0,0.52676,0.17654,0.42857,0.4286,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,15,0,0,5,0,0,7,0,0,1,0,1]]}]},{"i":"e2f5c2ce76251eda","q":"A cactus is a finite simple connected graph where no two cycles share an edge. Show that in a nonempty cactus, there must exist a vertex which is part of at most one cycle.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,31,0.0,0.10714,0.24223,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[4,31,0.129,0.05357,0.14174,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,31,0.2581,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[31,31,1.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.02679,"x":0.39284,"p":[[0,21,0.0,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.07588,0.16742,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,21,0.381,0.05804,0.13767,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,21,0.5714,0.24552,0.28173,0.0,0.14286,0.57111,0.0,0.71429,14,0,0,14,0,5,0,0,4,0,0,0,0,0,3,0,0,6,0,0,0,0,0],[16,21,0.7619,0.27661,0.24216,0.14214,0.14288,0.35714,0.0,0.71429,6,0,0,6,0,11,0,0,7,0,0,0,0,0,3,0,0,5,0,0,0,0,0],[20,21,0.9524,0.39284,0.29233,0.14286,0.28571,0.71429,0.0,0.71429,7,0,0,7,0,4,0,0,6,0,0,0,0,0,3,0,0,12,0,0,0,0,0],[21,21,1.0,0.21874,0.24216,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,7,0,0,7,0,0,0,0,0,2,0,0,4,0,0,0,0,0]]}]},{"i":"394598189274246e","q":"Let  $x_1, x_2, x_3, x_4, x_5\\in\\mathbb{R}^+$  such that  $$ x_1^2-x_1x_2+x_2^2=x_2^2-x_2x_3+x_3^2=x_3^2-x_3x_4+x_4^2=x_4^2-x_4x_5+x_5^2=x_5^2-x_5x_1+x_1^2 $$  Prove that  $x_1=x_2=x_3=x_4=x_5$ .","t":[{"b":3,"e":0.71429,"k":"falling","v":0.4241,"x":0.87946,"p":[[0,19,0.0,0.87946,0.29038,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,24],[4,19,0.2105,0.84375,0.31412,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,24],[8,19,0.4211,0.45533,0.2659,0.28571,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,10,0,0,0,0,0,5,0,0,7,0,0,1,0,2],[12,19,0.6316,0.4241,0.24349,0.14286,0.42859,0.71429,0.14286,0.71429,0,0,0,0,0,11,0,0,4,0,0,2,0,0,5,0,0,10,0,0,0,0,0],[16,19,0.8421,0.50879,0.25501,0.25,0.57143,0.71429,0.14,1.0,0,1,0,0,0,8,0,0,3,0,0,0,0,0,8,0,0,11,0,0,1,0,1],[19,19,1.0,0.46426,0.24222,0.2857,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,7,0,0,8,0,0,0,0,0,5,0,0,11,0,0,1,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.7633,"x":1.0,"p":[[0,49,0.0,0.7633,0.34937,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,5,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,20],[4,49,0.0816,0.98213,0.04727,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,49,0.1633,0.94197,0.19185,1.0,1.0,1.0,0.1429,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[12,49,0.2449,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,49,0.3265,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[20,49,0.4082,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[24,49,0.4898,0.92857,0.20203,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,27],[28,49,0.5714,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,49,0.6531,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[36,49,0.7347,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,49,0.8163,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[44,49,0.898,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,49,0.9796,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[49,49,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"ded0b4014acb7c4f","q":"Given points  $O, A_1, A_2, ..., A_n$   on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors  $\\vec{x}$  and  $\\vec{y}$ , such that for any point  $A_i$ ,  $\\vec{OA_i }= k\\vec{x}+l \\vec{y}$ , where  $k$  and  $l$  are some integer numbers.\n\n(A.Glazyrin)","t":[{"b":1,"e":1.0,"k":"rising","v":0.79907,"x":1.0,"p":[[0,58,0.0,0.79907,0.27402,0.67846,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,2,0,0,5,0,17],[4,58,0.069,0.94196,0.16312,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[8,58,0.1379,0.93749,0.15129,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,25],[12,58,0.2069,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[16,58,0.2759,0.90625,0.18766,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,6,0,22],[20,58,0.3448,0.87499,0.25939,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,6,0,22],[24,58,0.4138,0.92411,0.15146,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,7,0,22],[28,58,0.4828,0.87275,0.21853,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,0,1,22],[32,58,0.5517,0.90625,0.14555,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,10,0,18],[36,58,0.6207,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[40,58,0.6897,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[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],[48,58,0.8276,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,58,0.8966,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[56,58,0.9655,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]},{"b":6,"e":1.0,"k":"flat","v":0.80357,"x":0.95535,"p":[[0,39,0.0,0.80357,0.3067,0.64286,1.0,1.0,0.1429,1.0,0,21,0,0,0,1,0,0,6,0,0,1,0,0,0,0,0,1,0,0,2,0,21],[4,39,0.1026,0.95535,0.11539,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[8,39,0.2051,0.94643,0.16269,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[12,39,0.3077,0.91518,0.19516,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,5,0,24],[16,39,0.4103,0.91963,0.15129,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[20,39,0.5128,0.88838,0.22514,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,4,0,23],[24,39,0.6154,0.91963,0.1595,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,3,0,24],[28,39,0.7179,0.91071,0.20439,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,24],[32,39,0.8205,0.83034,0.21262,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,5,0,0,5,0,16],[36,39,0.9231,0.90177,0.16148,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,9,0,19],[39,39,1.0,0.95312,0.13317,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,4,0,26]]}]},{"i":"cf23f9a51177e138","q":"There are some cities in a country; one of them is the capital. For any two cities  $A$  and  $B$  there is a direct flight from  $A$  to  $B$  and a direct flight from  $B$  to  $A$ , both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.","t":[{"b":1,"e":1.0,"k":"flat","v":0.8125,"x":0.90179,"p":[[0,17,0.0,0.8125,0.29329,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,0,4,0,19],[4,17,0.2353,0.8705,0.17631,0.82132,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,7,0,17],[8,17,0.4706,0.83925,0.18127,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,7,0,14],[12,17,0.7059,0.87054,0.17627,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,10,0,16],[16,17,0.9412,0.86158,0.19721,0.857,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,10,0,16],[17,17,1.0,0.90179,0.19045,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,5,0,22]]},{"b":7,"e":1.0,"k":"rising","v":0.72321,"x":1.0,"p":[[0,22,0.0,0.83482,0.22899,0.67857,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,3,0,0,2,0,19],[4,22,0.1818,0.72321,0.32525,0.53571,0.85714,1.0,0.0,1.0,2,14,0,2,0,1,0,0,4,0,0,1,0,0,1,0,0,6,0,0,3,0,14],[8,22,0.3636,0.92411,0.15966,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,6,0,23],[12,22,0.5455,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"178334f4cfcf2eb5","q":"The function  $f:\\mathbb{N}\\rightarrow \\mathbb{N}$  is **peruvian** if it satifies the following two properties: $\\triangleright f$  is strictly increasing. $\\triangleright$  The numbers  $a_1,a_2,a_3,\\dots$  where  $a_1=f(1)$  and  $a_{n+1}=f(a_n)$  for every  $n\\geq 1$ , are in arithmetic progression.\nDetermine all peruvian functions  $f:\\mathbb{N}\\rightarrow \\mathbb{N}$  such that  $f(1)=3$ .","t":[{"b":4,"e":0.571,"k":"falling","v":0.51783,"x":0.9241,"p":[[0,36,0.0,0.87945,0.18249,0.85714,0.85714,1.0,0.0,1.0,1,13,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,17,0,13],[4,36,0.1111,0.9241,0.07976,0.85714,0.92857,1.0,0.714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[8,36,0.2222,0.90624,0.06786,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[12,36,0.3333,0.90624,0.06786,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11],[16,36,0.4444,0.91517,0.07874,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,14],[20,36,0.5556,0.88839,0.08553,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,22,0,9],[24,36,0.6667,0.89285,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[28,36,0.7778,0.79909,0.19515,0.85711,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,24,0,3],[32,36,0.8889,0.74552,0.20433,0.57143,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,8,0,0,1,0,0,16,0,4],[36,36,1.0,0.51783,0.15871,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,5,0,0,17,0,0,2,0,0,2,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.4375,"x":0.89732,"p":[[0,30,0.0,0.84821,0.19541,0.85714,0.85714,1.0,0.0,1.0,1,10,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,18,0,10],[4,30,0.1333,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[8,30,0.2667,0.89732,0.06423,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[12,30,0.4,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[16,30,0.5333,0.89731,0.06424,0.85714,0.85714,1.0,0.857,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,9],[20,30,0.6667,0.87054,0.13533,0.85714,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,21,0,9],[24,30,0.8,0.62497,0.26905,0.28571,0.64286,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,10,0,0,2,0,0,4,0,0,1,0,0,12,0,3],[28,30,0.9333,0.49552,0.24739,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,14,0,0,4,0,0,3,0,0,3,0,0,6,0,1],[30,30,1.0,0.4375,0.25238,0.2857,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,13,0,0,3,0,0,2,0,0,6,0,0,1,0,2]]}]},{"i":"30100da32dc2db42","q":"Let $n, m$ be integers greater than 1 , and let $a_{1}, a_{2}, \\ldots, a_{m}$ be positive integers not greater than $n^{m}$. Prove that there exist positive integers $b_{1}, b_{2}, \\ldots, b_{m}$ not greater than $n$, such that\n\n$$\n\\operatorname{gcd}\\left(a_{1}+b_{1}, a_{2}+b_{2}, \\ldots, a_{m}+b_{m}\\right)<n\n$$\n\nwhere $\\operatorname{gcd}\\left(x_{1}, x_{2}, \\ldots, x_{m}\\right)$ denotes the greatest common divisor of $x_{1}, x_{2}, \\ldots, x_{m}$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.02232,"x":0.58474,"p":[[0,67,0.0,0.58474,0.22969,0.571,0.57143,0.71429,0.0,0.85714,3,0,3,3,0,0,0,0,1,0,0,0,0,0,18,0,0,3,0,0,7,0,0],[4,67,0.0597,0.46429,0.3234,0.14286,0.5,0.71429,0.0,1.0,6,1,0,6,0,4,0,0,2,0,0,4,0,0,4,0,0,5,0,0,6,0,1],[8,67,0.1194,0.50893,0.32328,0.14286,0.57143,0.85714,0.0,0.85714,6,0,0,6,0,3,0,0,0,0,0,5,0,0,3,0,0,6,0,0,9,0,0],[12,67,0.1791,0.48659,0.32313,0.14286,0.49979,0.75,0.0,0.85714,6,0,0,6,0,3,0,0,2,0,0,5,0,0,1,0,0,7,0,0,8,0,0],[16,67,0.2388,0.57133,0.26243,0.42859,0.571,0.85704,0.0,0.85714,3,0,0,3,0,1,0,0,1,0,0,5,0,0,10,0,0,2,0,0,10,0,0],[20,67,0.2985,0.37496,0.33832,0.0,0.42859,0.71429,0.0,0.85714,13,0,0,13,0,0,0,0,1,0,0,4,0,0,5,0,0,4,0,0,5,0,0],[24,67,0.3582,0.4553,0.29971,0.14286,0.571,0.71429,0.0,0.85714,6,0,0,6,0,4,0,0,1,0,0,3,0,0,7,0,0,7,0,0,4,0,0],[28,67,0.4179,0.45533,0.28444,0.14289,0.57121,0.71429,0.0,0.85714,5,0,0,5,0,4,0,0,2,0,0,3,0,0,9,0,0,5,0,0,4,0,0],[32,67,0.4776,0.37053,0.32509,0.0,0.28571,0.71429,0.0,0.85714,9,0,0,9,0,4,0,0,4,0,0,5,0,0,0,0,0,4,0,0,6,0,0],[36,67,0.5373,0.47767,0.3001,0.25,0.49979,0.71429,0.0,1.0,4,1,0,4,0,4,0,0,4,0,0,4,0,0,4,0,0,6,0,0,5,0,1],[40,67,0.597,0.4107,0.3149,0.14286,0.42857,0.71429,0.0,0.85714,7,0,0,7,0,4,0,0,4,0,0,4,0,0,3,0,0,4,0,0,6,0,0],[44,67,0.6567,0.50446,0.30301,0.24999,0.57143,0.71429,0.0,0.85714,5,0,0,5,0,3,0,0,2,0,0,3,0,0,4,0,0,9,0,0,6,0,0],[48,67,0.7164,0.47319,0.32622,0.14286,0.571,0.75,0.0,0.85714,7,0,0,7,0,2,0,0,3,0,0,3,0,0,4,0,0,5,0,0,8,0,0],[52,67,0.7761,0.47758,0.31257,0.14286,0.571,0.75,0.0,0.85714,5,0,0,5,0,5,0,0,1,0,0,3,0,0,7,0,0,3,0,0,8,0,0],[56,67,0.8358,0.45088,0.33902,0.0,0.57143,0.71429,0.0,0.85714,9,0,0,9,0,2,0,0,2,0,0,1,0,0,4,0,0,8,0,0,6,0,0],[60,67,0.8955,0.08482,0.21086,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[64,67,0.9552,0.05357,0.15872,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[67,67,1.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.02232,"x":0.60714,"p":[[0,64,0.0,0.58479,0.30589,0.5713,0.57143,0.74996,0.0,1.0,5,4,3,5,0,1,0,0,0,0,0,0,0,0,12,0,0,6,0,0,4,0,4],[4,64,0.0625,0.48212,0.3549,0.10714,0.57143,0.85714,0.0,1.0,8,1,0,8,0,3,0,0,1,0,0,2,0,0,4,0,0,4,0,0,9,0,1],[8,64,0.125,0.53572,0.34993,0.14286,0.71429,0.85714,0.0,0.85714,7,0,0,7,0,3,0,0,0,0,0,1,0,0,3,0,0,6,0,0,12,0,0],[12,64,0.1875,0.60714,0.24223,0.42857,0.64286,0.85714,0.0,0.85714,2,0,0,2,0,1,0,0,0,0,0,7,0,0,6,0,0,6,0,0,10,0,0],[16,64,0.25,0.51783,0.3004,0.39286,0.57143,0.71429,0.0,1.0,6,1,0,6,0,1,0,0,1,0,0,3,0,0,7,0,0,9,0,0,4,0,1],[20,64,0.3125,0.45536,0.29329,0.14286,0.57141,0.71429,0.0,0.85714,5,0,0,5,0,4,0,0,3,0,0,3,0,0,7,0,0,5,0,0,5,0,0],[24,64,0.375,0.37944,0.32655,0.0,0.42857,0.60714,0.0,1.0,9,1,0,9,0,6,0,0,0,0,0,2,0,0,7,0,0,4,0,0,3,0,1],[28,64,0.4375,0.31696,0.29392,0.0,0.2857,0.57143,0.0,0.85714,9,0,0,9,0,6,0,0,5,0,0,3,0,0,2,0,0,4,0,0,3,0,0],[32,64,0.5,0.42856,0.34809,0.0,0.57121,0.71429,0.0,0.85714,10,0,0,10,0,3,0,0,0,0,0,2,0,0,5,0,0,5,0,0,7,0,0],[36,64,0.5625,0.33929,0.30462,0.0,0.28571,0.60714,0.0,0.85714,9,0,0,9,0,6,0,0,3,0,0,3,0,0,3,0,0,5,0,0,3,0,0],[40,64,0.625,0.28124,0.29554,0.0,0.14286,0.57111,0.0,0.85714,11,0,0,11,0,8,0,0,1,0,0,3,0,0,4,0,0,2,0,0,3,0,0],[44,64,0.6875,0.37944,0.28258,0.10714,0.4286,0.57143,0.0,0.85714,8,0,0,8,0,4,0,0,1,0,0,4,0,0,10,0,0,3,0,0,2,0,0],[48,64,0.75,0.35259,0.31545,0.105,0.21429,0.60714,0.0,0.85714,8,0,0,8,0,8,0,0,1,0,0,4,0,0,3,0,0,3,0,0,5,0,0],[52,64,0.8125,0.24107,0.24856,0.0,0.14286,0.42857,0.0,0.85714,10,0,0,10,0,10,0,0,2,0,0,4,0,0,3,0,0,2,0,0,1,0,0],[56,64,0.875,0.1607,0.17031,0.0,0.14286,0.28571,0.0,0.571,13,0,0,13,0,9,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[60,64,0.9375,0.10714,0.16367,0.0,0.0,0.17857,0.0,0.4286,21,0,0,21,0,3,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fad4f75bb24fd47b","q":"Find the least possible value for the fraction  $$ \\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)} $$  over all distinct positive integers  $a, b, c$ . \nBy  $lcm(x, y)$  we mean the least common multiple of  $x, y$  and by  $gcd(x, y)$  we mean the greatest common divisor of  $x, y$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.48214,"x":0.88839,"p":[[0,143,0.0,0.48214,0.22232,0.28571,0.57141,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,3,0,0,6,0,0,10,0,0,5,0,0,1,0,1],[4,143,0.028,0.80357,0.15872,0.71429,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,0,0,12],[8,143,0.0559,0.81696,0.13474,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,1,0,11],[12,143,0.0839,0.77232,0.18161,0.71429,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,14,0,0,5,0,8],[16,143,0.1119,0.81696,0.15251,0.71429,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,2,0,12],[20,143,0.1399,0.77232,0.14664,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,17,0,0,2,0,8],[24,143,0.1678,0.72766,0.11496,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,23,0,0,2,0,3],[28,143,0.1958,0.73214,0.18472,0.71429,0.71429,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,17,0,0,1,0,7],[32,143,0.2238,0.77228,0.13773,0.71429,0.71429,0.78571,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,21,0,0,0,0,8],[36,143,0.2517,0.71873,0.09096,0.71429,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,0,1,0,2],[40,143,0.2797,0.75,0.14725,0.71429,0.71429,0.75,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,17,0,0,1,0,7],[44,143,0.3077,0.76338,0.14989,0.71429,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,20,0,0,0,0,8],[48,143,0.3357,0.74998,0.13833,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,21,0,0,1,0,6],[52,143,0.3636,0.83036,0.13092,0.71429,0.71429,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,4,0,11],[56,143,0.3916,0.79015,0.18209,0.57143,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,10,0,0,0,0,13],[60,143,0.4196,0.79018,0.12869,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,0,2,0,8],[64,143,0.4476,0.76784,0.15048,0.71429,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,16,0,0,2,0,8],[68,143,0.4755,0.73213,0.13245,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,19,0,0,3,0,4],[72,143,0.5035,0.79463,0.13805,0.71429,0.71429,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,2,0,9],[76,143,0.5315,0.78125,0.15561,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,14,0,0,3,0,9],[80,143,0.5594,0.73214,0.14617,0.71429,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,24,0,0,0,0,5],[84,143,0.5874,0.78572,0.16751,0.71429,0.71429,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,16,0,0,0,0,11],[88,143,0.6154,0.79462,0.1513,0.71429,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,0,2,0,10],[92,143,0.6434,0.82589,0.14167,0.71429,0.78564,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,5,0,11],[96,143,0.6713,0.85268,0.13592,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,3,0,14],[100,143,0.6993,0.82588,0.14612,0.71429,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,0,0,13],[104,143,0.7273,0.83928,0.15872,0.71429,0.85707,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,0,3,0,14],[108,143,0.7552,0.81695,0.14829,0.71429,0.85707,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,7,0,10],[112,143,0.7832,0.79911,0.14223,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,19,0,0,3,0,9],[116,143,0.8112,0.78125,0.17851,0.71429,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,13,0,0,5,0,9],[120,143,0.8392,0.85268,0.17307,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,9,0,0,2,0,17],[124,143,0.8671,0.88839,0.1504,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,1,0,20],[128,143,0.8951,0.84375,0.16506,0.71429,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,3,0,15],[132,143,0.9231,0.81247,0.15339,0.71429,0.71429,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,4,0,11],[136,143,0.951,0.83927,0.16271,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,2,0,15],[140,143,0.979,0.86605,0.16344,0.71429,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,5,0,17],[143,143,1.0,0.86161,0.16164,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,9,0,14]]},{"b":4,"e":0.71429,"k":"rising","v":0.44643,"x":0.89732,"p":[[0,94,0.0,0.44643,0.26666,0.14286,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,9,0,0,5,0,0,4,0,0,7,0,0,4,0,0,0,0,3],[4,94,0.0426,0.79464,0.17105,0.71429,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,17,0,0,1,0,11],[8,94,0.0851,0.81695,0.13942,0.71429,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,5,0,10],[12,94,0.1277,0.77679,0.14258,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,18,0,0,4,0,7],[16,94,0.1702,0.80357,0.14173,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,2,0,10],[20,94,0.2128,0.75446,0.15251,0.71429,0.71429,0.78571,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,17,0,0,0,0,8],[24,94,0.2553,0.79463,0.18879,0.71429,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,13,0,0,2,0,12],[28,94,0.2979,0.74107,0.16917,0.71429,0.71429,0.74999,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,19,0,0,2,0,6],[32,94,0.3404,0.76339,0.1365,0.71429,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,17,0,0,6,0,5],[36,94,0.383,0.78125,0.15561,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,17,0,0,2,0,9],[40,94,0.4255,0.83036,0.14032,0.71429,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,3,0,12],[44,94,0.4681,0.77231,0.16699,0.71429,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,18,0,0,3,0,8],[48,94,0.5106,0.78125,0.13825,0.71429,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,19,0,0,4,0,7],[52,94,0.5532,0.78571,0.14725,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,0,2,0,9],[56,94,0.5957,0.78571,0.22016,0.71429,0.78571,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,12,0,0,5,0,11],[60,94,0.6383,0.83034,0.20342,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,9,0,0,2,0,16],[64,94,0.6809,0.83928,0.13716,0.71429,0.85707,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,5,0,12],[68,94,0.7234,0.875,0.18814,0.82132,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,5,0,19],[72,94,0.766,0.84821,0.21998,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,5,0,17],[76,94,0.8085,0.88392,0.18708,0.85711,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,5,0,20],[80,94,0.8511,0.89732,0.11971,0.82132,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,7,0,17],[84,94,0.8936,0.89286,0.14726,0.71429,1.0,1.0,0.5714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,2,0,20],[88,94,0.9362,0.87053,0.17261,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,4,0,18],[92,94,0.9787,0.85714,0.12372,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,8,0,12],[94,94,1.0,0.86159,0.13592,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,6,0,14]]}]},{"i":"c8b681febbde9ae9","q":"We are given 2021 points on a plane, no three of which are collinear. Among any 5 of these points, at least 4 lie on the same circle. Is it necessarily true that at least 2020 of the points lie on the same circle?","t":[{"b":2,"e":0.0,"k":"falling","v":0.07142,"x":0.5088,"p":[[0,41,0.0,0.29015,0.27542,0.0,0.28571,0.57141,0.0,0.85714,13,0,4,13,0,2,0,0,2,0,0,4,0,0,9,0,0,1,0,0,1,0,0],[4,41,0.0976,0.5088,0.26483,0.2857,0.4286,0.60714,0.14,1.0,0,4,0,0,0,4,0,0,7,0,0,6,0,0,7,0,0,2,0,0,2,0,4],[8,41,0.1951,0.49094,0.33499,0.2857,0.4998,0.85714,0.0,1.0,6,4,0,6,0,1,0,0,5,0,0,4,0,0,6,0,0,1,0,0,5,0,4],[12,41,0.2927,0.40168,0.28452,0.2857,0.28571,0.57143,0.0,1.0,6,2,0,6,0,1,0,0,10,0,0,2,0,0,7,0,0,3,0,0,1,0,2],[16,41,0.3902,0.43289,0.29126,0.2857,0.28571,0.57143,0.0,1.0,4,3,0,4,0,2,0,0,11,0,0,1,0,0,8,0,0,1,0,0,2,0,3],[20,41,0.4878,0.37496,0.31691,0.0,0.28571,0.57143,0.0,1.0,9,3,0,9,0,2,0,0,6,0,0,1,0,0,10,0,0,0,0,0,1,0,3],[24,41,0.5854,0.37943,0.31663,0.0,0.28571,0.57143,0.0,1.0,9,2,0,9,0,1,0,0,8,0,0,1,0,0,6,0,0,3,0,0,2,0,2],[28,41,0.6829,0.41058,0.34027,0.0,0.42857,0.57143,0.0,1.0,9,4,0,9,0,1,0,0,5,0,0,4,0,0,6,0,0,1,0,0,2,0,4],[32,41,0.7805,0.20533,0.23124,0.0,0.07143,0.32143,0.0,0.57143,16,0,0,16,0,1,0,0,7,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[36,41,0.878,0.24094,0.26106,0.0,0.07,0.57111,0.0,0.57143,16,0,0,16,0,1,0,0,3,0,0,1,0,0,11,0,0,0,0,0,0,0,0],[40,41,0.9756,0.07142,0.21126,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[41,41,1.0,0.08482,0.25218,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2]]},{"b":6,"e":0.4286,"k":"flat","v":0.3705,"x":0.72316,"p":[[0,48,0.0,0.3705,0.32506,0.0,0.42857,0.57143,0.0,1.0,10,2,3,10,0,4,0,0,1,0,0,3,0,0,7,0,0,4,0,0,1,0,2],[4,48,0.0833,0.68295,0.31082,0.53539,0.64286,1.0,0.0,1.0,2,12,0,2,0,0,0,0,4,0,0,2,0,0,8,0,0,1,0,0,3,0,12],[8,48,0.1667,0.62496,0.26667,0.4286,0.57143,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,5,0,0,3,0,0,11,0,0,2,0,0,3,0,7],[12,48,0.25,0.60262,0.27603,0.39286,0.57143,0.89275,0.14286,1.0,0,8,0,0,0,2,0,0,6,0,0,2,0,0,12,0,0,1,0,0,1,0,8],[16,48,0.3333,0.58023,0.19541,0.53465,0.57143,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,3,0,0,17,0,0,2,0,0,2,0,3],[20,48,0.4167,0.57128,0.28135,0.42859,0.57143,0.74996,0.0,1.0,3,3,0,3,0,1,0,0,3,0,0,3,0,0,9,0,0,5,0,0,5,0,3],[24,48,0.5,0.48658,0.26692,0.2857,0.4286,0.57143,0.0,1.0,1,3,0,1,0,3,0,0,9,0,0,4,0,0,8,0,0,1,0,0,3,0,3],[28,48,0.5833,0.49103,0.24205,0.28571,0.5712,0.57143,0.0,1.0,3,1,0,3,0,1,0,0,5,0,0,4,0,0,12,0,0,4,0,0,2,0,1],[32,48,0.6667,0.58032,0.22286,0.5354,0.57143,0.60714,0.0,1.0,1,4,0,1,0,0,0,0,4,0,0,3,0,0,16,0,0,3,0,0,1,0,4],[36,48,0.75,0.70974,0.2563,0.571,0.57143,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,2,0,0,13,0,0,3,0,0,0,0,12],[40,48,0.8333,0.72316,0.31125,0.5354,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,3,0,0,7,0,0,0,0,0,1,0,16],[44,48,0.9167,0.63837,0.28341,0.53575,0.57143,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,3,0,0,3,0,0,11,0,0,2,0,0,3,0,8],[48,48,1.0,0.49982,0.27673,0.2857,0.571,0.57143,0.0,1.0,3,4,0,3,0,1,0,0,7,0,0,2,0,0,13,0,0,1,0,0,1,0,4]]}]},{"i":"52025fc5352ef3c9","q":"6. (CAN 2) Suppose that $\\left\\{x_{1}, x_{2}, \\ldots, x_{n}\\right\\}$ are positive integers for which $x_{1}+x_{2}+\\cdots+x_{n}=2(n+1)$. Show that there exists an integer $r$ with $0 \\leq r \\leq n-1$ for which the following $n-1$ inequalities hold:  $$ \\begin{aligned} x_{r+1}+\\cdots+x_{r+i} & \\leq 2 i+1 & & \\forall i, 1 \\leq i \\leq n-r ; \\\\ x_{r+1}+\\cdots+x_{n}+x_{1}+\\cdots+x_{i} & \\leq 2(n-r+i)+1 & & \\forall i, 1 \\leq i \\leq r-1 . \\end{aligned} $$  Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r$.","t":[{"b":4,"e":1.0,"k":"rising","v":0.433,"x":0.91963,"p":[[0,40,0.0,0.433,0.26116,0.25,0.42857,0.57143,0.0,1.0,3,2,2,3,0,5,0,0,4,0,0,5,0,0,11,0,0,1,0,0,1,0,2],[4,40,0.1,0.74991,0.31356,0.4286,0.92857,1.0,0.14,1.0,0,16,0,0,0,4,0,0,1,0,0,4,0,0,0,0,0,4,0,0,3,0,16],[8,40,0.2,0.75445,0.32973,0.57132,1.0,1.0,0.0,1.0,2,17,0,2,0,2,0,0,1,0,0,2,0,0,3,0,0,2,0,0,3,0,17],[12,40,0.3,0.79014,0.26243,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,3,0,0,2,0,17],[16,40,0.4,0.75891,0.31429,0.5354,0.92857,1.0,0.0,1.0,1,16,0,1,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,0,5,0,16],[20,40,0.5,0.85708,0.20211,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,3,0,19],[24,40,0.6,0.81694,0.29719,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,20],[28,40,0.7,0.91963,0.17837,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,24],[32,40,0.8,0.91517,0.16314,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,24],[36,40,0.9,0.90178,0.14914,0.85708,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,6,0,20],[40,40,1.0,0.90176,0.12079,0.857,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,12,0,16]]},{"b":5,"e":0.857,"k":"rising","v":0.42855,"x":0.97768,"p":[[0,53,0.0,0.42855,0.29232,0.14286,0.35714,0.57143,0.0,1.0,1,4,1,1,0,8,0,0,7,0,0,6,0,0,4,0,0,0,0,0,2,0,4],[4,53,0.0755,0.85267,0.26603,0.85711,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,19],[8,53,0.1509,0.75446,0.28845,0.57143,0.85707,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,5,0,0,4,0,14],[12,53,0.2264,0.82362,0.2416,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,4,1,0,2,0,18],[16,53,0.3019,0.87944,0.20553,0.82143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,21],[20,53,0.3774,0.91964,0.15947,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,24],[24,53,0.4528,0.84373,0.21238,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,4,0,17],[28,53,0.5283,0.89284,0.19887,0.85711,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,3,0,22],[32,53,0.6038,0.9598,0.10256,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,27],[36,53,0.6792,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[40,53,0.7547,0.93749,0.16342,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[44,53,0.8302,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[48,53,0.9057,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[52,53,0.9811,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[53,53,1.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25]]}]},{"i":"05a842d180116fef","q":"For a positive integer  $ n,$   $ a_{n}\\equal{}n\\sqrt{5}\\minus{} \\lfloor n\\sqrt{5}\\rfloor$ . Compute the maximum value and the minimum value of  $ a_{1},a_{2},\\ldots ,a_{2009}.$","t":[{"b":0,"e":0.28571,"k":"falling","v":0.41964,"x":0.79464,"p":[[0,27,0.0,0.65622,0.14224,0.57143,0.57143,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,7,0,0,0,0,4],[4,27,0.1481,0.79464,0.2111,0.67857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,9,0,0,1,0,14],[8,27,0.2963,0.75,0.20203,0.57143,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,12,0,0,7,0,0,1,0,11],[12,27,0.4444,0.59821,0.18708,0.57143,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,8,0,0,0,0,3],[16,27,0.5926,0.63393,0.22851,0.57143,0.57143,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,4,0,0,0,0,7],[20,27,0.7407,0.42411,0.19393,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,9,0,0,9,0,0,6,0,0,3,0,0,0,0,1],[24,27,0.8889,0.47768,0.14987,0.39286,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,11,0,0,7,0,0,6,0,0,0,0,0],[27,27,1.0,0.41964,0.17474,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,14,0,0,8,0,0,6,0,0,2,0,0,0,0,1]]},{"b":6,"e":0.71429,"k":"flat","v":0.58929,"x":0.7232,"p":[[0,22,0.0,0.7232,0.22571,0.57143,0.71429,1.0,0.0,1.0,1,10,1,1,0,0,0,0,0,0,0,1,0,0,11,0,0,9,0,0,0,0,10],[4,22,0.1818,0.62945,0.21087,0.57143,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,1,0,0,14,0,0,8,0,0,0,0,5],[8,22,0.3636,0.65624,0.14665,0.57143,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,8,0,0,0,0,4],[12,22,0.5455,0.60268,0.18118,0.57143,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,2,0,0,15,0,0,8,0,0,0,0,3],[16,22,0.7273,0.62054,0.21902,0.57143,0.57143,0.71429,0.0,1.0,1,4,1,1,0,1,0,0,2,0,0,0,0,0,14,0,0,10,0,0,0,0,4],[20,22,0.9091,0.58929,0.12753,0.57143,0.57143,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,2,0,0,20,0,0,7,0,0,0,0,1],[22,22,1.0,0.61606,0.09741,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,8,0,0,0,0,1]]}]},{"i":"79023d7d7a6190eb","q":"Let $a_{i j}, i=1,2, \\ldots, m$ and $j=1,2, \\ldots, n$, be positive real numbers. Prove that\n\n$$\n\\sum_{i=1}^{m}\\left(\\sum_{j=1}^{n} \\frac{1}{a_{i j}}\\right)^{-1} \\leq\\left(\\sum_{j=1}^{n}\\left(\\sum_{i=1}^{m} a_{i j}\\right)^{-1}\\right)^{-1} .\n$$\n\nWhen does the equality hold?","t":[{"b":0,"e":0.0,"k":"flat","v":0.07141,"x":0.41518,"p":[[0,104,0.0,0.21866,0.17855,0.14286,0.14286,0.2857,0.14,1.0,0,1,0,0,0,23,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[4,104,0.0385,0.40624,0.27224,0.2857,0.28571,0.42857,0.0,1.0,1,2,0,1,0,3,0,0,19,0,0,2,0,0,0,0,0,0,0,0,5,0,2],[8,104,0.0769,0.35715,0.21428,0.2857,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,1,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[12,104,0.1154,0.31696,0.14608,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,2,0,0,27,0,0,1,0,0,0,0,0,0,0,0,2,0,0],[16,104,0.1538,0.3125,0.05576,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[20,104,0.1923,0.41518,0.27747,0.28571,0.28571,0.32143,0.0,1.0,1,4,0,1,0,1,0,0,22,0,0,1,0,0,0,0,0,1,0,0,2,0,4],[24,104,0.2308,0.3125,0.14914,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[28,104,0.2692,0.35268,0.18205,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,27,0,0,2,0,0,0,0,0,0,0,0,2,0,1],[32,104,0.3077,0.34375,0.20159,0.28571,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,2,0,0,26,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[36,104,0.3462,0.37946,0.22759,0.28571,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,1,0,0,24,0,0,3,0,0,0,0,0,0,0,0,1,0,3],[40,104,0.3846,0.29911,0.15714,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,4,0,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[44,104,0.4231,0.34821,0.2111,0.28571,0.28571,0.28571,0.14286,1.0,0,3,0,0,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[48,104,0.4615,0.29017,0.13592,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,4,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,104,0.5,0.33035,0.17655,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,1,0,0,25,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[56,104,0.5385,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,3,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,104,0.5769,0.29018,0.13592,0.2857,0.28571,0.28571,0.0,0.85714,2,0,0,2,0,2,0,0,24,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[64,104,0.6154,0.30355,0.1587,0.2857,0.28571,0.28571,0.0,1.0,2,1,0,2,0,1,0,0,25,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[68,104,0.6538,0.27901,0.06767,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,1,0,1,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,104,0.6923,0.25893,0.12595,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,4,0,0,23,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[76,104,0.7308,0.25446,0.07771,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,3,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,104,0.7692,0.25446,0.08552,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,104,0.8077,0.2723,0.17625,0.2857,0.2857,0.28571,0.0,0.71429,6,0,0,6,0,1,0,0,20,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[88,104,0.8462,0.20536,0.12846,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,3,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,104,0.8846,0.22321,0.11258,0.24999,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,2,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,104,0.9231,0.25446,0.09268,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,2,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,104,0.9615,0.17857,0.12371,0.0,0.2857,0.2857,0.0,0.28571,9,0,0,9,0,6,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,104,1.0,0.07141,0.13827,0.0,0.0,0.03571,0.0,0.571,24,0,0,24,0,2,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.22321,"x":0.33928,"p":[[0,100,0.0,0.24545,0.19315,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,19,0,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[4,100,0.04,0.33482,0.19103,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,3,0,0,25,0,0,1,0,0,0,0,0,0,0,0,2,0,1],[8,100,0.08,0.33481,0.16212,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,2,0,0,25,0,0,2,0,0,0,0,0,1,0,0,2,0,0],[12,100,0.12,0.25446,0.1504,0.25,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,4,0,0,22,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,100,0.16,0.29018,0.16554,0.2857,0.28571,0.28571,0.0,1.0,3,1,0,3,0,1,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[20,100,0.2,0.3125,0.19704,0.2857,0.28571,0.28571,0.0,0.85714,3,0,0,3,0,1,0,0,24,0,0,1,0,0,0,0,0,0,0,0,3,0,0],[24,100,0.24,0.2366,0.10479,0.24999,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,4,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,100,0.28,0.33928,0.24157,0.28571,0.28571,0.28571,0.0,1.0,3,2,0,3,0,1,0,0,23,0,0,1,0,0,0,0,0,0,0,0,2,0,2],[32,100,0.32,0.26339,0.06297,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,100,0.36,0.22321,0.14258,0.14286,0.28571,0.28571,0.0,0.71429,6,0,0,6,0,5,0,0,20,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,100,0.4,0.27231,0.14878,0.2857,0.28571,0.28571,0.0,0.857,3,0,0,3,0,4,0,0,21,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[44,100,0.44,0.27678,0.08702,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,3,0,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[48,100,0.48,0.22768,0.1063,0.14286,0.2857,0.28571,0.0,0.42857,4,0,0,4,0,6,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,100,0.52,0.24107,0.12595,0.14289,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,7,0,0,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[56,100,0.56,0.25893,0.16536,0.2857,0.28571,0.28571,0.0,1.0,4,1,0,4,0,3,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,100,0.6,0.2857,0.1515,0.2857,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,1,0,0,22,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[64,100,0.64,0.26784,0.14171,0.2857,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,2,0,0,23,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[68,100,0.68,0.28571,0.15152,0.2857,0.28571,0.28571,0.0,1.0,2,1,0,2,0,2,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[72,100,0.72,0.31696,0.13236,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[76,100,0.76,0.28571,0.09449,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,4,0,0,26,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[80,100,0.8,0.31473,0.15958,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,4,0,1,23,0,0,1,0,0,1,0,0,0,0,0,2,0,0],[84,100,0.84,0.2991,0.06546,0.2857,0.28571,0.28571,0.1429,0.57143,0,0,0,0,0,1,0,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[88,100,0.88,0.29017,0.09094,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,3,0,0,27,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[92,100,0.92,0.2991,0.14445,0.2857,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,3,0,0,25,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[96,100,0.96,0.26785,0.05923,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,5,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,100,1.0,0.27232,0.06546,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,5,0,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6b77e5ad91cadbcc","q":"Positive integer  $n$  is given. We will call every injective function  $f:\\{1, \\dots, n\\}\\rightarrow\\{1, \\dots, n\\}$  a permutation. Pairwise distinct permutations  $f_1, \\dots, f_N$  are given. Assume that for every pair of indices (not necessarily distinct)  $i, j$ , there exists  $k$  such that  $f_i\\circ f_j=f_k$ . For  $1\\leq i<j\\leq n$  let  $Z(i, j)$  denote the number of indices  $l$  such that  $f_l(i)>f_l(j)$ . Prove that  $$ \\frac{N}{2}\\cdot \\sum_{1\\leq i<j\\leq n}Z(i, j)=\\sum_{1\\leq i<j\\leq n}Z(i, j)^2. $$","t":[{"b":4,"e":0.28571,"k":"flat","v":0.27678,"x":0.69642,"p":[[0,55,0.0,0.27678,0.23128,0.14286,0.28571,0.28571,0.0,0.85714,7,0,6,7,0,4,0,0,15,0,0,2,0,0,0,0,0,2,0,0,2,0,0],[4,55,0.0727,0.56696,0.27078,0.28571,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,12,0,0,1,0,0,3,0,0,6,0,0,6,0,3],[8,55,0.1455,0.62054,0.30222,0.28571,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,7,0,0,5,0,0,3,0,0,3,0,0,3,0,9],[12,55,0.2182,0.62944,0.28316,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,7,0,0,4,0,0,5,0,0,1,0,9],[16,55,0.2909,0.58478,0.24054,0.39286,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,7,0,0,2,0,0,11,0,0,4,0,0,3,0,4],[20,55,0.3636,0.65174,0.26473,0.571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,0,0,0,11,0,0,3,0,0,3,0,8],[24,55,0.4364,0.69642,0.27375,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,7,0,0,3,0,0,1,0,0,6,0,0,6,0,9],[28,55,0.5091,0.63393,0.31731,0.2857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,6,0,0,2,0,0,2,0,0,5,0,0,4,0,9],[32,55,0.5818,0.63833,0.23688,0.42859,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,4,0,0,9,0,0,5,0,0,3,0,6],[36,55,0.6545,0.66518,0.29366,0.39286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,6,0,0,2,0,0,3,0,0,7,0,0,2,0,10],[40,55,0.7273,0.53569,0.22015,0.28571,0.57121,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,9,0,0,5,0,0,11,0,0,2,0,0,2,0,3],[44,55,0.8,0.66962,0.23267,0.53539,0.64286,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,5,0,0,8,0,0,7,0,0,1,0,8],[48,55,0.8727,0.65622,0.24449,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,6,0,0,9,0,0,3,0,0,4,0,7],[52,55,0.9455,0.58036,0.23673,0.39286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,7,0,0,5,0,0,4,0,0,8,0,0,5,0,2],[55,55,1.0,0.42408,0.27774,0.14286,0.35714,0.57143,0.14286,1.0,0,3,0,0,0,10,0,0,6,0,0,5,0,0,5,0,0,1,0,0,2,0,3]]},{"b":7,"e":0.85714,"k":"flat","v":0.35267,"x":0.65173,"p":[[0,39,0.0,0.37052,0.26929,0.2857,0.28571,0.42857,0.0,1.0,3,2,3,3,0,4,0,0,15,0,0,3,0,0,2,0,0,0,0,0,3,0,2],[4,39,0.1026,0.62945,0.2693,0.39286,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,7,0,0,3,0,0,6,0,0,2,0,0,8,0,5],[8,39,0.2051,0.65173,0.27881,0.42857,0.57121,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,4,0,0,7,0,0,6,0,0,2,0,0,2,0,10],[12,39,0.3077,0.64284,0.27894,0.42857,0.57143,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,7,0,0,6,0,0,4,0,0,3,0,0,3,0,9],[16,39,0.4103,0.58926,0.29397,0.28571,0.49979,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,12,0,0,4,0,0,3,0,0,1,0,0,5,0,7],[20,39,0.5128,0.53125,0.30771,0.28571,0.42857,0.85714,0.14286,1.0,0,6,0,0,0,4,0,0,10,0,0,4,0,0,3,0,0,1,0,0,4,0,6],[24,39,0.6154,0.51338,0.2854,0.28571,0.28571,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,16,0,0,1,0,0,4,0,0,2,0,0,3,0,5],[28,39,0.7179,0.47768,0.26392,0.28571,0.35714,0.57143,0.14286,1.0,0,4,0,0,0,2,0,0,14,0,0,4,0,0,5,0,0,1,0,0,2,0,4],[32,39,0.8205,0.39284,0.20202,0.2857,0.28571,0.42858,0.14286,1.0,0,1,0,0,0,4,0,0,14,0,0,7,0,0,3,0,0,2,0,0,1,0,1],[36,39,0.9231,0.41071,0.23351,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,1,0,0,20,0,0,5,0,0,1,0,0,1,0,0,1,0,3],[39,39,1.0,0.35267,0.19228,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,3,0,0,21,0,0,5,0,0,0,0,0,0,0,0,2,0,1]]}]},{"i":"09d6032de965a8b6","q":"Let $A B C$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\\Omega$. Let $M$ be the midpoint of $A H$ and $N$ the midpoint of $B H$. Assume the points $M, N, O, H$ are distinct and lie on a circle $\\omega$. Prove that the circles $\\omega$ and $\\Omega$ are internally tangent to each other.","t":[{"b":3,"e":0.0,"k":"falling","v":0.12052,"x":0.57143,"p":[[0,52,0.0,0.56696,0.39202,0.21427,0.71429,1.0,0.0,1.0,8,10,4,8,0,0,0,0,3,0,0,2,0,0,0,0,0,9,0,0,0,0,10],[4,52,0.0769,0.57143,0.27199,0.5,0.71429,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,3,0,0,0,0,0,1,0,0,22,0,0,0,0,1],[8,52,0.1538,0.48213,0.34764,0.0,0.71429,0.71429,0.0,1.0,9,3,0,9,0,0,0,0,4,0,0,0,0,0,1,0,0,15,0,0,0,0,3],[12,52,0.2308,0.51772,0.29388,0.2857,0.71429,0.71429,0.0,1.0,6,1,0,6,0,0,0,0,4,0,0,1,0,0,2,0,0,18,0,0,0,0,1],[16,52,0.3077,0.45536,0.30813,0.14286,0.71429,0.71429,0.0,0.71429,7,0,0,7,0,3,0,0,3,0,0,1,0,0,0,0,0,18,0,0,0,0,0],[20,52,0.3846,0.46872,0.31589,0.10714,0.64286,0.71429,0.0,1.0,8,1,0,8,0,1,0,0,3,0,0,0,0,0,4,0,0,15,0,0,0,0,1],[24,52,0.4615,0.49106,0.30916,0.21427,0.71429,0.71429,0.0,0.71429,8,0,0,8,0,0,0,0,3,0,0,0,0,0,1,0,0,20,0,0,0,0,0],[28,52,0.5385,0.48661,0.31311,0.24999,0.71429,0.71429,0.0,1.0,7,1,0,7,0,1,0,0,4,0,0,1,0,0,0,0,0,18,0,0,0,0,1],[32,52,0.6154,0.48661,0.33476,0.0,0.71429,0.71429,0.0,1.0,9,2,0,9,0,0,0,0,2,0,0,1,0,0,2,0,0,16,0,0,0,0,2],[36,52,0.6923,0.37053,0.28763,0.0,0.28571,0.71429,0.0,0.71429,9,0,0,9,0,1,0,0,7,0,0,2,0,0,3,0,0,10,0,0,0,0,0],[40,52,0.7692,0.47321,0.2976,0.28571,0.71429,0.71429,0.0,0.71429,7,0,0,7,0,0,0,0,6,0,0,0,0,0,1,0,0,18,0,0,0,0,0],[44,52,0.8462,0.34375,0.31916,0.0,0.28571,0.71429,0.0,0.71429,13,0,0,13,0,0,0,0,5,0,0,1,0,0,1,0,0,12,0,0,0,0,0],[48,52,0.9231,0.29016,0.31233,0.0,0.2857,0.60714,0.0,1.0,14,1,0,14,0,1,0,0,7,0,0,0,0,0,2,0,0,7,0,0,0,0,1],[52,52,1.0,0.12052,0.23173,0.0,0.0,0.07143,0.0,0.71429,24,0,0,24,0,0,0,0,4,0,0,0,0,0,1,0,0,3,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.04911,"x":0.70088,"p":[[0,68,0.0,0.44643,0.4282,0.0,0.5,0.78571,0.0,1.0,14,8,3,14,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,0,0,8],[4,68,0.0588,0.68303,0.28288,0.67857,0.71429,1.0,0.0,1.0,2,9,0,2,0,0,0,0,4,0,0,1,0,0,1,0,0,15,0,0,0,0,9],[8,68,0.1176,0.61607,0.29329,0.71429,0.71429,0.71429,0.0,1.0,4,4,0,4,0,1,0,0,2,0,0,0,0,0,0,0,0,21,0,0,0,0,4],[12,68,0.1765,0.63839,0.34623,0.57143,0.71429,1.0,0.0,1.0,6,9,0,6,0,0,0,0,1,0,0,0,0,0,2,0,0,14,0,0,0,0,9],[16,68,0.2353,0.53125,0.35577,0.21427,0.71429,0.71429,0.0,1.0,8,5,0,8,0,0,0,0,3,0,0,1,0,0,0,0,0,15,0,0,0,0,5],[20,68,0.2941,0.46875,0.35577,0.10714,0.71429,0.71429,0.0,1.0,8,4,0,8,0,2,0,0,5,0,0,0,0,0,0,0,0,13,0,0,0,0,4],[24,68,0.3529,0.49553,0.39445,0.0,0.71429,0.74996,0.0,1.0,10,7,0,10,0,0,0,0,4,0,0,1,0,0,0,0,0,9,0,0,1,0,7],[28,68,0.4118,0.57588,0.31841,0.39286,0.71429,0.71429,0.0,1.0,6,4,0,6,0,0,0,0,2,0,0,1,0,0,1,0,0,18,0,0,0,0,4],[32,68,0.4706,0.70088,0.26574,0.71429,0.71429,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,3,0,0,0,0,0,1,0,0,17,0,0,1,0,8],[36,68,0.5294,0.63839,0.26722,0.71429,0.71429,0.71429,0.0,1.0,4,3,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,24,0,0,0,0,3],[40,68,0.5882,0.46872,0.33736,0.0,0.71429,0.71429,0.0,1.0,9,2,0,9,0,1,0,0,2,0,0,1,0,0,2,0,0,15,0,0,0,0,2],[44,68,0.6471,0.29464,0.29653,0.0,0.2857,0.71429,0.0,0.71429,13,0,0,13,0,1,0,0,8,0,0,0,0,0,1,0,0,9,0,0,0,0,0],[48,68,0.7059,0.34821,0.34615,0.0,0.28571,0.71429,0.0,1.0,13,2,0,13,0,1,0,0,5,0,0,1,0,0,0,0,0,10,0,0,0,0,2],[52,68,0.7647,0.35714,0.3312,0.0,0.28571,0.71429,0.0,0.71429,13,0,0,13,0,1,0,0,3,0,0,1,0,0,0,0,0,14,0,0,0,0,0],[56,68,0.8235,0.49554,0.37625,0.0,0.71429,0.71429,0.0,1.0,10,5,0,10,0,0,0,0,3,0,0,0,0,0,0,0,0,14,0,0,0,0,5],[60,68,0.8824,0.33928,0.35848,0.0,0.14285,0.71429,0.0,1.0,16,1,0,16,0,0,0,0,2,0,0,0,0,0,0,0,0,13,0,0,0,0,1],[64,68,0.9412,0.09375,0.21609,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[68,68,1.0,0.04911,0.14987,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"5c028fb891e4b425","q":"Nine skiers left the start line in turn and covered the distance, each at their own constant speed. Could it turn out that each skier participated in exactly four overtakes? (In each overtaking, exactly two skiers participate - the one\nwho is overtaking, and the one who is being overtaken.)","t":[{"b":4,"e":1.0,"k":"rising","v":0.59375,"x":0.98214,"p":[[0,31,0.0,0.59375,0.38151,0.39286,0.5,1.0,0.0,1.0,5,13,1,5,0,2,0,0,1,0,0,8,0,0,1,0,0,2,0,0,0,0,13],[4,31,0.129,0.66965,0.35613,0.42857,0.85714,1.0,0.0,1.0,1,16,0,1,0,5,0,0,0,0,0,8,0,0,1,0,0,1,0,0,0,0,16],[8,31,0.2581,0.81697,0.3179,0.75001,1.0,1.0,0.0,1.0,1,23,0,1,0,2,0,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,23],[12,31,0.3871,0.75893,0.3489,0.42857,1.0,1.0,0.0,1.0,2,21,0,2,0,2,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,21],[16,31,0.5161,0.87054,0.25092,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,24],[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],[24,31,0.7742,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,31,0.9032,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[31,31,1.0,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30]]},{"b":6,"e":1.0,"k":"rising","v":0.3258,"x":0.99554,"p":[[0,99,0.0,0.60715,0.37965,0.42857,0.4286,1.0,0.0,1.0,4,14,1,4,0,3,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,14],[4,99,0.0404,0.66964,0.42474,0.14286,1.0,1.0,0.0,1.0,6,19,0,6,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,19],[8,99,0.0808,0.57589,0.44246,0.10714,0.71429,1.0,0.0,1.0,8,16,0,8,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,16],[12,99,0.1212,0.5625,0.44598,0.0,0.64286,1.0,0.0,1.0,9,15,0,9,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,15],[16,99,0.1616,0.67411,0.4304,0.32143,1.0,1.0,0.0,1.0,8,19,0,8,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,19],[20,99,0.202,0.63393,0.43292,0.14286,1.0,1.0,0.0,1.0,7,18,0,7,0,2,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,18],[24,99,0.2424,0.78125,0.37625,0.64286,1.0,1.0,0.0,1.0,5,23,0,5,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,23],[28,99,0.2828,0.50447,0.41952,0.14286,0.42857,1.0,0.0,1.0,7,12,0,7,0,6,0,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,12],[32,99,0.3232,0.51786,0.43117,0.10714,0.42857,1.0,0.0,1.0,8,13,0,8,0,5,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,13],[36,99,0.3636,0.60268,0.43993,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,1,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,17],[40,99,0.404,0.625,0.43117,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,1,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,17],[44,99,0.4444,0.53572,0.4475,0.14286,0.42857,1.0,0.0,1.0,7,15,0,7,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,15],[48,99,0.4848,0.54464,0.45237,0.0,0.5,1.0,0.0,1.0,11,15,0,11,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,15],[52,99,0.5253,0.60715,0.41342,0.14286,0.7143,1.0,0.0,1.0,5,16,0,5,0,4,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,16],[56,99,0.5657,0.49991,0.42114,0.105,0.42857,1.0,0.0,1.0,8,12,0,8,0,4,0,0,2,0,0,5,0,0,0,0,0,1,0,0,0,0,12],[60,99,0.6061,0.52679,0.45937,0.0,0.42859,1.0,0.0,1.0,11,15,0,11,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,15],[64,99,0.6465,0.41964,0.39599,0.0,0.42857,1.0,0.0,1.0,9,9,0,9,0,5,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,9],[68,99,0.6869,0.58481,0.42612,0.14286,0.71407,1.0,0.0,1.0,6,15,0,6,0,4,0,0,3,0,0,2,0,0,1,0,0,0,0,0,1,0,15],[72,99,0.7273,0.61161,0.43921,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,1,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,17],[76,99,0.7677,0.3258,0.40447,0.0,0.14286,0.64286,0.0,1.0,14,7,0,14,0,6,0,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,7],[80,99,0.8081,0.59374,0.36789,0.28571,0.42859,1.0,0.0,1.0,3,13,0,3,0,3,0,0,3,0,0,8,0,0,1,0,0,1,0,0,0,0,13],[84,99,0.8485,0.53571,0.3993,0.24999,0.42857,1.0,0.0,1.0,7,12,0,7,0,1,0,0,4,0,0,6,0,0,1,0,0,1,0,0,0,0,12],[88,99,0.8889,0.44643,0.36553,0.14286,0.42857,0.78571,0.0,1.0,6,8,0,6,0,6,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,8],[92,99,0.9293,0.89286,0.26726,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[96,99,0.9697,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[99,99,1.0,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29]]}]},{"i":"c17298559951700a","q":"Let $n \\geq 2$ be a fixed positive integer. An integer will be called \" $n$-free\" if it is not a multiple of an $n$-th power of a prime. Let $M$ be an infinite set of rational numbers, such that the product of every $n$ elements of $M$ is an $n$-free integer. Prove that $M$ contains only integers.","t":[{"b":4,"e":1.0,"k":"flat","v":0.32589,"x":0.52228,"p":[[0,47,0.0,0.32589,0.2321,0.14286,0.28571,0.42858,0.0,1.0,4,1,0,4,0,6,0,0,11,0,0,5,0,0,2,0,0,3,0,0,0,0,1],[4,47,0.0851,0.39284,0.27663,0.24999,0.28571,0.57143,0.0,1.0,4,2,0,4,0,4,0,0,10,0,0,4,0,0,3,0,0,4,0,0,1,0,2],[8,47,0.1702,0.37499,0.20747,0.28571,0.35714,0.57143,0.0,0.71429,3,0,0,3,0,4,0,0,9,0,0,5,0,0,8,0,0,3,0,0,0,0,0],[12,47,0.2553,0.38839,0.26543,0.14286,0.35714,0.57143,0.0,1.0,5,1,0,5,0,4,0,0,7,0,0,4,0,0,5,0,0,6,0,0,0,0,1],[16,47,0.3404,0.43746,0.26469,0.2857,0.42857,0.60714,0.0,1.0,2,2,0,2,0,5,0,0,7,0,0,6,0,0,4,0,0,5,0,0,1,0,2],[20,47,0.4255,0.39729,0.22225,0.2857,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,5,0,0,10,0,0,1,0,0,10,0,0,3,0,0,1,0,0],[24,47,0.5106,0.40173,0.21554,0.28571,0.35714,0.57143,0.0,1.0,3,1,0,3,0,1,0,0,12,0,0,3,0,0,11,0,0,1,0,0,0,0,1],[28,47,0.5957,0.43301,0.25374,0.28571,0.42857,0.57143,0.0,1.0,1,3,0,1,0,5,0,0,9,0,0,5,0,0,7,0,0,2,0,0,0,0,3],[32,47,0.6809,0.42857,0.23146,0.28571,0.42859,0.57143,0.0,1.0,4,1,0,4,0,2,0,0,4,0,0,7,0,0,12,0,0,2,0,0,0,0,1],[36,47,0.766,0.42856,0.22015,0.28571,0.35714,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,12,0,0,3,0,0,7,0,0,5,0,0,0,0,1],[40,47,0.8511,0.40175,0.20337,0.2857,0.4998,0.57143,0.0,0.71429,3,0,0,3,0,3,0,0,8,0,0,2,0,0,15,0,0,1,0,0,0,0,0],[44,47,0.9362,0.52228,0.27573,0.28571,0.57121,0.71429,0.0,1.0,2,4,0,2,0,3,0,0,4,0,0,4,0,0,10,0,0,4,0,0,1,0,4],[47,47,1.0,0.45534,0.23265,0.28571,0.57121,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,10,0,0,0,0,0,12,0,0,3,0,0,0,0,2]]},{"b":5,"e":0.14286,"k":"rising","v":0.24997,"x":0.48658,"p":[[0,17,0.0,0.29463,0.19863,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,9,0,0,9,0,0,8,0,0,2,0,0,0,0,0,0,0,1],[4,17,0.2353,0.29911,0.20316,0.14286,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,6,0,0,10,0,0,5,0,0,4,0,0,2,0,0,0,0,0],[8,17,0.4706,0.25,0.17496,0.14286,0.2857,0.42857,0.0,0.57143,6,0,0,6,0,8,0,0,9,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[12,17,0.7059,0.24997,0.22584,0.0,0.21428,0.571,0.0,0.57143,10,0,0,10,0,6,0,0,7,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[16,17,0.9412,0.48658,0.2392,0.28571,0.57121,0.57143,0.0,1.0,1,3,0,1,0,2,0,0,9,0,0,2,0,0,13,0,0,2,0,0,0,0,3],[17,17,1.0,0.45088,0.26027,0.28571,0.57143,0.57143,0.0,1.0,4,1,0,4,0,3,0,0,5,0,0,1,0,0,13,0,0,4,0,0,1,0,1]]}]},{"i":"d4bbd0462c55944d","q":"Given a positive integer  $k$ , find the least integer  $n_k$  for which there exist five sets  $S_1, S_2, S_3, S_4, S_5$  with the following properties:\n\\[|S_j|=k \\text{ for } j=1, \\cdots , 5 , \\quad |\\bigcup_{j=1}^{5} S_j | = n_k ;\\]\n\\[|S_i \\cap S_{i+1}| = 0 = |S_5 \\cap S_1|, \\quad \\text{for } i=1,\\cdots ,4 \\]","t":[{"b":4,"e":1.0,"k":"flat","v":0.91518,"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],[4,22,0.1818,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,22,0.3636,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,22,0.5455,0.91518,0.18161,0.85714,1.0,1.0,0.0,1.0,1,20,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,20],[16,22,0.7273,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[20,22,0.9091,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[22,22,1.0,0.9375,0.08702,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20]]},{"b":5,"e":1.0,"k":"flat","v":0.89732,"x":0.99107,"p":[[0,42,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,42,0.0952,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[8,42,0.1905,0.94196,0.11214,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,24],[12,42,0.2857,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[16,42,0.381,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,42,0.4762,0.94642,0.12243,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,26],[24,42,0.5714,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,42,0.6667,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[32,42,0.7619,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[36,42,0.8571,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[40,42,0.9524,0.89732,0.10249,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,14],[42,42,1.0,0.90625,0.13651,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,20]]}]},{"i":"548d512937210c5e","q":"Let $w=w_{1} w_{2} \\ldots w_{n}$ be a word. Define a substring of $w$ to be a word of the form $w_{i} w_{i+1} \\ldots w_{j-1} w_{j}$, for some pair of positive integers $1 \\leq i \\leq j \\leq n$. Show that $w$ has at most $n$ distinct palindromic substrings.\nFor example, aaaaa has 5 distinct palindromic substrings, and abcata has 5 ( $a, b, c, t, a t a)$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.88391,"x":1.0,"p":[[0,10,0.0,0.88391,0.19048,0.85711,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,4,0,21],[4,10,0.4,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[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]]},{"b":4,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,12,0.0,0.93304,0.20198,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,28],[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]]}]},{"i":"bb90a0b654b8885b","q":"For every positive integer  $m$ , a  $m\\times 2018$  rectangle consists of unit squares (called \"cell\") is called *complete* if  the following conditions are met:\ni. In each cell is written either a \" $0$ \", a \" $1$ \" or nothing;\nii. For any binary string  $S$  with length  $2018$ , one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce  $S$  (In particular, if a row is already full and it produces  $S$  in the same manner then this condition ii. is satisfied).\nA *complete* rectangle is called *minimal*, if we remove any of its rows and then making it no longer *complete*.\n\na. Prove that for any positive integer  $k\\le 2018$  there exists a *minimal*  $2^k\\times 2018$  rectangle with exactly  $k$  columns containing both  $0$  and  $1$ .\nb. A *minimal*  $m\\times 2018$  rectangle has exactly  $k$  columns containing at least some  $0$  or  $1$  and the rest of columns are empty. Prove that  $m\\le 2^k$ .","t":[{"b":4,"e":0.571,"k":"flat","v":0.41517,"x":0.49996,"p":[[0,54,0.0,0.43303,0.08362,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,21,0,0,6,0,0,0,0,0,0,0,0],[4,54,0.0741,0.46872,0.13473,0.42857,0.42857,0.4642,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,20,0,0,5,0,0,1,0,0,2,0,0],[8,54,0.1481,0.47321,0.15741,0.42857,0.42857,0.46525,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,19,0,0,4,0,0,2,0,0,1,0,1],[12,54,0.2222,0.49105,0.15946,0.42857,0.42857,0.4642,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,22,0,0,4,0,0,2,0,0,0,0,2],[16,54,0.2963,0.47319,0.12076,0.42857,0.42857,0.4286,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,24,0,0,5,0,0,1,0,0,0,0,1],[20,54,0.3704,0.45091,0.10166,0.42857,0.42857,0.4286,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,23,0,0,5,0,0,0,0,0,1,0,0],[24,54,0.4444,0.47104,0.09239,0.42857,0.42857,0.4642,0.36,0.85714,0,0,0,0,0,0,0,0,0,0,1,23,0,0,7,0,0,0,0,0,1,0,0],[28,54,0.5185,0.41517,0.07454,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,23,0,0,3,0,0,0,0,0,0,0,0],[32,54,0.5926,0.47546,0.12573,0.42857,0.42857,0.51775,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,21,0,1,6,0,0,1,0,0,0,0,1],[36,54,0.6667,0.48213,0.12752,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,20,0,0,8,0,0,1,0,0,0,0,1],[40,54,0.7407,0.49996,0.13361,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,13,0,0,11,0,0,3,0,0,1,0,0],[44,54,0.8148,0.48213,0.15464,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,17,0,0,5,0,0,4,0,0,0,0,1],[48,54,0.8889,0.43304,0.13592,0.39286,0.42857,0.42858,0.2857,0.85714,0,0,0,0,0,0,0,0,8,0,0,19,0,0,3,0,0,0,0,0,2,0,0],[52,54,0.963,0.47305,0.13065,0.42857,0.4286,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,11,0,0,11,0,0,3,0,0,0,0,0],[54,54,1.0,0.48212,0.14172,0.42857,0.42857,0.571,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,20,0,0,5,0,0,3,0,0,0,0,1]]},{"b":7,"e":0.57143,"k":"flat","v":0.40176,"x":0.49553,"p":[[0,60,0.0,0.45533,0.13568,0.42857,0.42857,0.4286,0.28571,1.0,0,1,0,0,0,0,0,0,5,0,0,20,0,0,5,0,0,1,0,0,0,0,1],[4,60,0.0667,0.48217,0.15044,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,18,0,0,7,0,0,1,0,0,1,0,1],[8,60,0.1333,0.45087,0.08824,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,22,0,0,6,0,0,1,0,0,0,0,0],[12,60,0.2,0.4353,0.08441,0.42857,0.42857,0.42895,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,20,0,1,6,0,0,0,0,0,0,0,0],[16,60,0.2667,0.47764,0.12679,0.42857,0.42857,0.571,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,21,0,0,7,0,0,1,0,0,0,0,1],[20,60,0.3333,0.47325,0.10371,0.42857,0.42857,0.46525,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,23,0,0,6,0,0,1,0,0,1,0,0],[24,60,0.4,0.49553,0.21124,0.42857,0.42857,0.4642,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,18,0,0,3,0,0,1,0,0,0,0,4],[28,60,0.4667,0.44196,0.06543,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,25,0,0,5,0,0,0,0,0,0,0,0],[32,60,0.5333,0.45533,0.10371,0.42857,0.42857,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,17,0,0,9,0,0,1,0,0,0,0,0],[36,60,0.6,0.44195,0.09004,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,22,0,0,5,0,0,1,0,0,0,0,0],[40,60,0.6667,0.44639,0.07777,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,3,0,0,22,0,0,7,0,0,0,0,0,0,0,0],[44,60,0.7333,0.4598,0.07767,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,27,0,0,3,0,0,2,0,0,0,0,0],[48,60,0.8,0.40176,0.09057,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,18,0,0,4,0,0,0,0,0,0,0,0],[52,60,0.8667,0.4732,0.13091,0.42857,0.42857,0.57111,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,20,0,0,7,0,0,1,0,0,0,0,1],[56,60,0.9333,0.47319,0.16144,0.42857,0.42857,0.571,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,18,0,0,7,0,0,0,0,0,0,0,2],[60,60,1.0,0.45981,0.06899,0.42857,0.42857,0.4642,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,23,0,0,8,0,0,0,0,0,0,0,0]]}]},{"i":"5faa9725ad30cdd7","q":"A real sequence $\\left(a_{n}\\right)_{n=0}^{\\infty}$ is defined recursively by $a_{0}=2$ and the recursion formula\n\n$$\na_{n}= \\begin{cases}a_{n-1}^{2} & \\text { if } a_{n-1}<\\sqrt{3} \\\\ \\frac{a_{n-1}^{2}}{3} & \\text { if } a_{n-1} \\geqslant \\sqrt{3}\\end{cases}\n$$\n\nAnother real sequence $\\left(b_{n}\\right)_{n=1}^{\\infty}$ is defined in terms of the first by the formula\n\n$$\nb_{n}= \\begin{cases}0 & \\text { if } a_{n-1}<\\sqrt{3} \\\\ \\frac{1}{2^{n}} & \\text { if } a_{n-1} \\geqslant \\sqrt{3}\\end{cases}\n$$\n\nvalid for each $n \\geqslant 1$. Prove that\n\n$$\nb_{1}+b_{2}+\\cdots+b_{2020}<\\frac{2}{3}\n$$","t":[{"b":0,"e":0.42857,"k":"flat","v":0.38838,"x":0.49106,"p":[[0,47,0.0,0.48214,0.25443,0.28571,0.42857,0.57143,0.0,1.0,1,4,0,1,0,0,0,0,11,0,0,10,0,0,4,0,0,0,0,0,2,0,4],[4,47,0.0851,0.49106,0.19541,0.42857,0.42857,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,6,0,0,11,0,0,9,0,0,2,0,0,2,0,1],[8,47,0.1702,0.43748,0.305,0.2857,0.42857,0.57111,0.0,1.0,5,5,0,5,0,0,0,0,9,0,0,9,0,0,3,0,0,0,0,0,1,0,5],[12,47,0.2553,0.38838,0.34112,0.0,0.35714,0.57143,0.0,1.0,10,4,0,10,0,0,0,0,6,0,0,6,0,0,3,0,0,1,0,0,2,0,4],[16,47,0.3404,0.4375,0.03458,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0],[20,47,0.4255,0.48213,0.11151,0.42857,0.42857,0.57111,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,23,0,0,8,0,0,0,0,0,0,0,1],[24,47,0.5106,0.45537,0.10374,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,1],[28,47,0.5957,0.45088,0.05184,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0],[32,47,0.6809,0.45087,0.05181,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0],[36,47,0.766,0.42409,0.09769,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,2,0,0,25,0,0,4,0,0,0,0,0,0,0,0],[40,47,0.8511,0.45088,0.05184,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0],[44,47,0.9362,0.45534,0.06619,0.42857,0.42857,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,24,0,0,7,0,0,0,0,0,0,0,0],[47,47,1.0,0.45536,0.05576,0.42857,0.42857,0.4286,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.42857,"k":"falling","v":0.32141,"x":0.57142,"p":[[0,66,0.0,0.54464,0.25614,0.42857,0.42857,0.57143,0.0,1.0,1,6,0,1,0,0,0,0,4,0,0,14,0,0,6,0,0,0,0,0,1,0,6],[4,66,0.0606,0.47321,0.30397,0.28571,0.42857,0.60714,0.0,1.0,5,5,0,5,0,0,0,0,6,0,0,8,0,0,5,0,0,3,0,0,0,0,5],[8,66,0.1212,0.57142,0.25254,0.42857,0.42859,0.64286,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,13,0,0,6,0,0,0,0,0,1,0,7],[12,66,0.1818,0.50893,0.2111,0.42857,0.42857,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,7,0,0,12,0,0,9,0,0,0,0,0,0,0,4],[16,66,0.2424,0.49999,0.1821,0.42857,0.42857,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,4,0,0,17,0,0,8,0,0,0,0,0,0,0,3],[20,66,0.303,0.36607,0.15542,0.28571,0.28571,0.42857,0.0,1.0,1,1,0,1,0,0,0,0,17,0,0,11,0,0,2,0,0,0,0,0,0,0,1],[24,66,0.3636,0.38838,0.1758,0.28571,0.42857,0.42857,0.0,1.0,2,1,0,2,0,0,0,0,12,0,0,12,0,0,5,0,0,0,0,0,0,0,1],[28,66,0.4242,0.35713,0.14284,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,22,0,0,7,0,0,2,0,0,0,0,0,0,0,1],[32,66,0.4848,0.35268,0.09439,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,17,0,0,12,0,0,2,0,0,0,0,0,0,0,0],[36,66,0.5455,0.35714,0.09449,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,19,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[40,66,0.6061,0.36161,0.09439,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,18,0,0,11,0,0,3,0,0,0,0,0,0,0,0],[44,66,0.6667,0.33483,0.10479,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,20,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[48,66,0.7273,0.37946,0.09852,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,13,0,0,4,0,0,0,0,0,0,0,0],[52,66,0.7879,0.39284,0.10099,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,13,0,0,14,0,0,5,0,0,0,0,0,0,0,0],[56,66,0.8485,0.34821,0.08703,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,20,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[60,66,0.9091,0.35267,0.08734,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,19,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[64,66,0.9697,0.36159,0.10089,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,19,0,0,9,0,0,4,0,0,0,0,0,0,0,0],[66,66,1.0,0.32141,0.10098,0.28571,0.28571,0.32143,0.0,0.57143,1,0,0,1,0,0,0,0,23,0,0,6,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"6ec2ce2ceb22290b","q":"Can we put intengers  $1,2,\\cdots,12$  on a circle, number them  $a_1,a_2,\\cdots,a_{12}$  in order. For any  $1\\leq i<j\\leq12$ ,  $|a_i-a_j|\\neq|i-j|$ ?","t":[{"b":0,"e":1.0,"k":"rising","v":0.64286,"x":1.0,"p":[[0,55,0.0,0.71875,0.33973,0.28571,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,1,0,18],[4,55,0.0727,0.71875,0.36681,0.28571,1.0,1.0,0.0,1.0,3,18,0,3,0,0,0,0,7,0,0,0,0,0,0,0,0,3,0,0,1,0,18],[8,55,0.1455,0.80803,0.33237,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,23],[12,55,0.2182,0.67411,0.35934,0.28571,0.71429,1.0,0.0,1.0,2,15,0,2,0,2,0,0,7,0,0,0,0,0,0,0,0,6,0,0,0,0,15],[16,55,0.2909,0.72321,0.38621,0.28571,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,20],[20,55,0.3636,0.76339,0.35465,0.60714,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,5,0,0,0,0,0,0,0,0,3,0,0,1,0,20],[24,55,0.4364,0.73214,0.3549,0.28571,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,8,0,0,0,0,0,0,0,0,3,0,0,0,0,19],[28,55,0.5091,0.85268,0.27313,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,0,0,22],[32,55,0.5818,0.83482,0.33902,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[36,55,0.6545,0.73659,0.37135,0.28571,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,5,0,0,0,0,0,1,0,0,2,0,0,0,0,20],[40,55,0.7273,0.69196,0.41819,0.28571,1.0,1.0,0.0,1.0,6,20,0,6,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,20],[44,55,0.8,0.64286,0.41033,0.28571,1.0,1.0,0.0,1.0,5,17,0,5,0,1,0,0,7,0,0,0,0,0,0,0,0,2,0,0,0,0,17],[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":6,"e":1.0,"k":"rising","v":0.69196,"x":0.97321,"p":[[0,29,0.0,0.69196,0.37815,0.28571,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[4,29,0.1379,0.81696,0.33357,0.92857,1.0,1.0,0.0,1.0,2,24,0,2,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[8,29,0.2759,0.83482,0.29904,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,5,0,0,0,0,0,0,0,0,2,0,0,1,0,23],[12,29,0.4138,0.8125,0.29545,0.71429,1.0,1.0,0.0,1.0,2,19,1,2,0,1,0,0,1,0,0,0,0,0,0,0,0,8,0,0,1,0,19],[16,29,0.5517,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,29,0.6897,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[24,29,0.8276,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[28,29,0.9655,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[29,29,1.0,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22]]}]},{"i":"9e13613de1b0b830","q":"A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset  $S$  of dominoes is chosen. Is it true that at least one of the following 2 statements is false?\n(1) There are  $2022$  more horizontal dominoes than vertical dominoes in  $S$ .\n(2) The cells covered by the dominoes in  $S$  can be tiled completely and exactly by  $L$ -shaped tetrominoes.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.44191,"p":[[0,57,0.0,0.09375,0.20705,0.0,0.0,0.03571,0.0,1.0,24,1,2,24,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[4,57,0.0702,0.35711,0.30721,0.0,0.28571,0.57143,0.0,1.0,10,2,0,10,0,0,0,0,8,0,0,3,0,0,5,0,0,3,0,0,1,0,2],[8,57,0.1404,0.31686,0.30463,0.0,0.2857,0.57141,0.0,0.85714,13,0,0,13,0,1,0,0,3,0,0,5,0,0,3,0,0,5,0,0,2,0,0],[12,57,0.2105,0.34372,0.28313,0.0,0.28571,0.571,0.0,1.0,9,1,0,9,0,1,0,0,8,0,0,5,0,0,3,0,0,4,0,0,1,0,1],[16,57,0.2807,0.33482,0.2615,0.24999,0.28571,0.4286,0.0,1.0,7,1,0,7,0,1,0,0,13,0,0,5,0,0,0,0,0,4,0,0,1,0,1],[20,57,0.3509,0.44191,0.27045,0.2857,0.42857,0.71429,0.0,1.0,5,1,0,5,0,0,0,0,8,0,0,6,0,0,4,0,0,6,0,0,2,0,1],[24,57,0.4211,0.32137,0.28788,0.0,0.35714,0.571,0.0,0.85714,12,0,0,12,0,1,0,0,3,0,0,5,0,0,7,0,0,2,0,0,2,0,0],[28,57,0.4912,0.38838,0.28844,0.14286,0.35716,0.60714,0.0,0.85714,7,0,0,7,0,3,0,0,6,0,0,3,0,0,5,0,0,5,0,0,3,0,0],[32,57,0.5614,0.42409,0.32039,0.2857,0.28571,0.71429,0.0,1.0,7,3,0,7,0,0,0,0,10,0,0,3,0,0,2,0,0,5,0,0,2,0,3],[36,57,0.6316,0.43745,0.29217,0.14289,0.4998,0.71429,0.0,0.85714,6,0,0,6,0,3,0,0,4,0,0,3,0,0,5,0,0,8,0,0,3,0,0],[40,57,0.7018,0.38389,0.26348,0.14289,0.42857,0.57143,0.0,0.85714,6,0,0,6,0,3,0,0,6,0,0,5,0,0,6,0,0,4,0,0,2,0,0],[44,57,0.7719,0.38391,0.2635,0.14289,0.35714,0.57143,0.0,0.85714,6,0,0,6,0,3,0,0,7,0,0,3,0,0,6,0,0,6,0,0,1,0,0],[48,57,0.8421,0.33033,0.25109,0.10714,0.28571,0.4642,0.0,0.85714,8,0,0,8,0,1,0,0,10,0,0,5,0,0,3,0,0,4,0,0,1,0,0],[52,57,0.9123,0.3348,0.31663,0.0,0.2857,0.71429,0.0,0.85714,12,0,0,12,0,1,0,0,7,0,0,0,0,0,2,0,0,8,0,0,2,0,0],[56,57,0.9825,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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":4,"e":0.42857,"k":"rising","v":0.09812,"x":0.39732,"p":[[0,35,0.0,0.09812,0.15333,0.0,0.0,0.17857,0.0,0.57143,21,0,2,21,0,3,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,35,0.1143,0.39284,0.32142,0.10714,0.28571,0.60714,0.0,1.0,8,3,0,8,0,1,0,0,8,0,0,5,0,0,2,0,0,3,0,0,2,0,3],[8,35,0.2286,0.26336,0.23446,0.0,0.28571,0.42857,0.0,0.85714,11,0,0,11,0,2,0,0,8,0,0,5,0,0,5,0,0,0,0,0,1,0,0],[12,35,0.3429,0.3125,0.28669,0.0,0.28571,0.46428,0.0,1.0,11,1,0,11,0,1,0,0,7,0,0,5,0,0,3,0,0,3,0,0,1,0,1],[16,35,0.4571,0.26786,0.28738,0.0,0.2143,0.32143,0.0,0.85714,12,0,0,12,0,4,0,0,8,0,0,1,0,0,0,0,0,5,0,0,2,0,0],[20,35,0.5714,0.26334,0.29252,0.0,0.14286,0.571,0.0,0.85714,15,0,0,15,0,2,0,0,4,0,0,1,0,0,5,0,0,4,0,0,1,0,0],[24,35,0.6857,0.39732,0.35487,0.0,0.35714,0.71429,0.0,1.0,11,2,0,11,0,2,0,0,3,0,0,2,0,0,2,0,0,7,0,0,3,0,2],[28,35,0.8,0.2857,0.29879,0.0,0.2857,0.4642,0.0,1.0,13,1,0,13,0,1,0,0,8,0,0,2,0,0,2,0,0,4,0,0,1,0,1],[32,35,0.9143,0.26338,0.31562,0.0,0.2143,0.35704,0.0,1.0,15,2,0,15,0,1,0,0,8,0,0,0,0,0,3,0,0,2,0,0,1,0,2],[35,35,1.0,0.25427,0.20127,0.105,0.28571,0.28571,0.0,0.85714,8,0,0,8,0,3,0,0,15,0,0,2,0,0,3,0,0,0,0,0,1,0,0]]}]},{"i":"ce4185e81c61eb84","q":"4. (BEL 5) On the sides of the triangle $A B C$, three similar isosceles triangles $A B P(A P=P B), A Q C(A Q=Q C)$, and $B R C(B R=R C)$ are constructed. The first two are constructed externally to the triangle $A B C$, but the third is placed in the same half-plane determined by the line $B C$ as the triangle $A B C$. Prove that $A P R Q$ is a parallelogram.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.2232,"x":0.36161,"p":[[0,56,0.0,0.27669,0.18883,0.14286,0.2857,0.28571,0.14,1.0,0,1,0,0,0,13,0,0,15,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[4,56,0.0714,0.33482,0.23038,0.14286,0.28571,0.42857,0.0,1.0,1,2,0,1,0,12,0,0,4,0,0,12,0,0,0,0,0,1,0,0,0,0,2],[8,56,0.1429,0.31688,0.18126,0.14286,0.35714,0.42857,0.14,1.0,0,1,0,0,0,13,0,0,3,0,0,15,0,0,0,0,0,0,0,0,0,0,1],[12,56,0.2143,0.27679,0.13333,0.14286,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,15,0,0,4,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.34366,0.19195,0.14286,0.42857,0.42857,0.14,1.0,0,1,0,0,0,12,0,0,1,0,0,17,0,0,0,0,0,1,0,0,0,0,1],[20,56,0.3571,0.33027,0.14491,0.14286,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,10,0,0,4,0,0,17,0,0,0,0,0,1,0,0,0,0,0],[24,56,0.4286,0.33919,0.22523,0.14286,0.42857,0.42857,0.0,1.0,1,2,0,1,0,12,0,0,2,0,0,14,0,0,1,0,0,0,0,0,0,0,2],[28,56,0.5,0.28571,0.16752,0.14286,0.2857,0.42857,0.0,0.71429,2,0,0,2,0,12,0,0,5,0,0,11,0,0,1,0,0,1,0,0,0,0,0],[32,56,0.5714,0.33036,0.15335,0.14286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,3,0,0,18,0,0,0,0,0,1,0,0,0,0,0],[36,56,0.6429,0.34366,0.15521,0.14286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,0,0,0,21,0,0,0,0,0,1,0,0,0,0,0],[40,56,0.7143,0.32143,0.23958,0.14286,0.28571,0.42857,0.0,1.0,2,2,0,2,0,13,0,0,2,0,0,12,0,0,0,0,0,1,0,0,0,0,2],[44,56,0.7857,0.36161,0.22864,0.14286,0.42857,0.42857,0.0,1.0,1,2,0,1,0,11,0,0,0,0,0,17,0,0,0,0,0,1,0,0,0,0,2],[48,56,0.8571,0.29911,0.16115,0.14286,0.42857,0.42857,0.0,0.4286,3,0,0,3,0,10,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.2232,0.16725,0.14286,0.14286,0.42857,0.0,0.571,6,0,0,6,0,13,0,0,3,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[56,56,1.0,0.25,0.20825,0.14286,0.1429,0.42857,0.0,1.0,6,1,0,6,0,11,0,0,4,0,0,10,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.14286,"k":"flat","v":0.18304,"x":0.47759,"p":[[0,90,0.0,0.27231,0.14442,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,13,0,0,13,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[4,90,0.0444,0.37505,0.19151,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,9,0,0,2,0,0,18,0,0,0,0,0,2,0,0,0,0,1],[8,90,0.0889,0.35268,0.18893,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,11,0,0,1,0,0,18,0,0,0,0,0,1,0,0,0,0,1],[12,90,0.1333,0.42402,0.21584,0.39285,0.42857,0.42858,0.14,1.0,0,2,0,0,0,7,0,0,1,0,0,19,0,0,0,0,0,3,0,0,0,0,2],[16,90,0.1778,0.33929,0.14617,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,10,0,0,2,0,0,19,0,0,0,0,0,1,0,0,0,0,0],[20,90,0.2222,0.41964,0.24206,0.14286,0.42857,0.4286,0.14286,1.0,0,3,0,0,0,9,0,0,1,0,0,15,0,0,3,0,0,1,0,0,0,0,3],[24,90,0.2667,0.45536,0.2822,0.14286,0.42857,0.46431,0.14286,1.0,0,5,0,0,0,9,0,0,1,0,0,14,0,0,1,0,0,2,0,0,0,0,5],[28,90,0.3111,0.47759,0.23866,0.42857,0.42857,0.42858,0.14,1.0,0,4,0,0,0,5,0,0,0,0,0,20,0,0,1,0,0,2,0,0,0,0,4],[32,90,0.3556,0.4375,0.2111,0.42857,0.42857,0.4286,0.14286,1.0,0,2,0,0,0,6,0,0,1,0,0,19,0,0,1,0,0,3,0,0,0,0,2],[36,90,0.4,0.4554,0.21558,0.42857,0.42857,0.4286,0.14286,1.0,0,3,0,0,0,5,0,0,0,0,0,21,0,0,2,0,0,1,0,0,0,0,3],[40,90,0.4444,0.41521,0.24316,0.25001,0.42857,0.42858,0.0,1.0,2,2,0,2,0,6,0,0,1,0,0,17,0,0,1,0,0,2,0,0,1,0,2],[44,90,0.4889,0.33036,0.20024,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,14,0,0,1,0,0,14,0,0,1,0,0,1,0,0,0,0,1],[48,90,0.5333,0.2991,0.20626,0.14286,0.35714,0.42857,0.0,1.0,3,1,0,3,0,12,0,0,1,0,0,14,0,0,1,0,0,0,0,0,0,0,1],[52,90,0.5778,0.33035,0.22427,0.14286,0.21429,0.42857,0.14286,1.0,0,1,0,0,0,16,0,0,1,0,0,10,0,0,1,0,0,3,0,0,0,0,1],[56,90,0.6222,0.24999,0.21427,0.14286,0.14286,0.42857,0.0,1.0,3,1,0,3,0,18,0,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,1],[60,90,0.6667,0.28554,0.2144,0.14286,0.14286,0.42857,0.0,0.71429,3,0,0,3,0,15,0,0,1,0,0,9,0,0,0,0,0,4,0,0,0,0,0],[64,90,0.7111,0.32143,0.16366,0.14286,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,9,0,0,2,0,0,18,0,0,0,0,0,1,0,0,0,0,0],[68,90,0.7556,0.31697,0.25935,0.14286,0.21431,0.42857,0.0,1.0,4,2,0,4,0,12,0,0,1,0,0,11,0,0,0,0,0,2,0,0,0,0,2],[72,90,0.8,0.28572,0.15152,0.14286,0.35714,0.42857,0.0,0.4286,2,0,0,2,0,12,0,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[76,90,0.8444,0.23206,0.15877,0.14286,0.14286,0.42857,0.0,0.4286,4,0,0,4,0,16,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[80,90,0.8889,0.29018,0.1838,0.14286,0.35714,0.42857,0.0,0.71429,6,0,0,6,0,5,0,0,5,0,0,15,0,0,0,0,0,1,0,0,0,0,0],[84,90,0.9333,0.28125,0.17672,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,10,0,0,3,0,0,14,0,0,0,0,0,1,0,0,0,0,0],[88,90,0.9778,0.2634,0.17536,0.14286,0.35714,0.42857,0.0,0.4286,6,0,0,6,0,9,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[90,90,1.0,0.18304,0.1996,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,5,0,0,1,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"609f3a07b8adbca8","q":"Given  $ n$  points on the plane, which are the vertices of a convex polygon,  $ n > 3$ . There exists  $ k$  regular triangles with the side equal to  $ 1$  and the vertices at the given points.\r\n\n- Prove that  $ k < \\frac {2}{3}n$ .\n- Construct the configuration with  $ k > 0.666n$ .","t":[{"b":0,"e":0.71429,"k":"rising","v":0.12045,"x":0.55354,"p":[[0,42,0.0,0.12045,0.17168,0.0,0.0,0.14289,0.0,0.57143,18,0,17,18,0,7,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,42,0.0952,0.46426,0.19229,0.2857,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,5,0,0,5,0,0,11,0,0,6,0,0,0,0,0],[8,42,0.1905,0.46869,0.18289,0.28571,0.4998,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,8,0,0,6,0,0,10,0,0,6,0,0,0,0,0],[12,42,0.2857,0.49542,0.19243,0.2857,0.571,0.71429,0.14,0.71429,0,0,0,0,0,3,0,0,6,0,0,6,0,0,7,0,0,10,0,0,0,0,0],[16,42,0.381,0.47764,0.20703,0.28571,0.571,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,6,0,0,4,0,0,11,0,0,6,0,0,1,0,0],[20,42,0.4762,0.51337,0.19185,0.28571,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,4,0,0,9,0,0,9,0,0,1,0,0],[24,42,0.5714,0.43745,0.19861,0.28571,0.4286,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,6,0,0,7,0,0,10,0,0,3,0,0,1,0,0],[28,42,0.6667,0.54449,0.18031,0.42857,0.57143,0.71429,0.14,0.71429,0,0,0,0,0,2,0,0,4,0,0,5,0,0,8,0,0,13,0,0,0,0,0],[32,42,0.7619,0.55354,0.21649,0.42859,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,4,0,0,3,0,0,10,0,0,10,0,0,0,0,2],[36,42,0.8571,0.48212,0.20747,0.28571,0.57121,0.71407,0.0,0.85714,1,0,0,1,0,1,0,0,10,0,0,3,0,0,8,0,0,8,0,0,1,0,0],[40,42,0.9524,0.42851,0.19557,0.25,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,8,0,0,3,0,0,5,0,0,13,0,0,3,0,0,0,0,0],[42,42,1.0,0.45535,0.22142,0.2857,0.57141,0.71429,0.14286,0.71429,0,0,0,0,0,7,0,0,6,0,0,2,0,0,8,0,0,9,0,0,0,0,0]]},{"b":3,"e":0.571,"k":"rising","v":0.07143,"x":0.51785,"p":[[0,40,0.0,0.07143,0.13363,0.0,0.0,0.03571,0.0,0.4286,24,0,21,24,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.50435,0.24231,0.42857,0.57143,0.71429,0.0,0.85714,3,0,0,3,0,3,0,0,1,0,0,5,0,0,8,0,0,11,0,0,1,0,0],[8,40,0.2,0.49547,0.19553,0.42857,0.571,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,1,0,0,6,0,0,13,0,0,7,0,0,0,0,0],[12,40,0.3,0.47318,0.18705,0.39286,0.4286,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,5,0,0,9,0,0,8,0,0,7,0,0,0,0,0],[16,40,0.4,0.46426,0.22015,0.2857,0.42859,0.60714,0.14286,1.0,0,1,0,0,0,5,0,0,7,0,0,5,0,0,7,0,0,7,0,0,0,0,1],[20,40,0.5,0.41066,0.19798,0.25002,0.4286,0.5711,0.0,0.71429,1,0,0,1,0,7,0,0,3,0,0,8,0,0,10,0,0,3,0,0,0,0,0],[24,40,0.6,0.38387,0.2326,0.14289,0.4286,0.57143,0.0,0.71429,4,0,0,4,0,5,0,0,6,0,0,3,0,0,10,0,0,4,0,0,0,0,0],[28,40,0.7,0.51785,0.23076,0.39285,0.57143,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,3,0,0,4,0,0,8,0,0,11,0,0,0,0,1],[32,40,0.8,0.41062,0.21046,0.2857,0.42857,0.57111,0.0,0.85714,1,0,0,1,0,6,0,0,7,0,0,5,0,0,9,0,0,3,0,0,1,0,0],[36,40,0.9,0.34368,0.19178,0.1429,0.28571,0.571,0.0,0.57143,3,0,0,3,0,6,0,0,8,0,0,5,0,0,10,0,0,0,0,0,0,0,0],[40,40,1.0,0.29454,0.21415,0.14286,0.2857,0.4286,0.0,0.71429,5,0,0,5,0,8,0,0,8,0,0,5,0,0,3,0,0,3,0,0,0,0,0]]}]},{"i":"384df32b2bfa93f3","q":"In an isosceles triangle  $ABC$  in which  $AC = BC$  and  $\\angle ABC < 60^o$ ,  $I$  and  $O$  are the centers of the inscribed and circumscribed circles, respectively. For the triangle  $BIO$ , the circumscribed circle intersects the side  $BC$  again at  $D$ . Prove that:\ni) lines  $AC$  and  $DI$  are parallel,\nii) lines  $OD$  and  $IB$  are perpendicular.","t":[{"b":2,"e":1.0,"k":"flat","v":0.73661,"x":0.94197,"p":[[0,78,0.0,0.82142,0.18559,0.67857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,5,0,14],[4,78,0.0513,0.91518,0.16311,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,23],[8,78,0.1026,0.94197,0.12299,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,24],[12,78,0.1538,0.9375,0.18536,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,26],[16,78,0.2051,0.89732,0.22083,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,23],[20,78,0.2564,0.85714,0.19233,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,6,0,16],[24,78,0.3077,0.89731,0.15253,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,10,0,18],[28,78,0.359,0.80804,0.21609,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,6,0,0,7,0,13],[32,78,0.4103,0.85268,0.26362,0.85714,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,3,0,0,5,0,20],[36,78,0.4615,0.80357,0.22232,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,5,0,0,9,0,12],[40,78,0.5128,0.81248,0.1871,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,9,0,11],[44,78,0.5641,0.81696,0.2237,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,8,0,0,6,0,14],[48,78,0.6154,0.77677,0.22852,0.71429,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,7,0,0,7,0,11],[52,78,0.6667,0.76785,0.23351,0.71429,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,4,0,0,13,0,8],[56,78,0.7179,0.73661,0.2699,0.57143,0.78571,1.0,0.0,1.0,1,11,0,1,0,0,0,0,3,0,0,3,0,0,2,0,0,7,0,0,5,0,11],[60,78,0.7692,0.84375,0.24317,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,18],[64,78,0.8205,0.85267,0.16554,0.85711,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,13,0,12],[68,78,0.8718,0.81697,0.17581,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,6,0,12],[72,78,0.9231,0.90625,0.14987,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,8,0,19],[76,78,0.9744,0.88839,0.11701,0.82132,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,15],[78,78,1.0,0.87945,0.14337,0.82143,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,8,0,16]]},{"b":5,"e":0.71429,"k":"falling","v":0.49104,"x":0.875,"p":[[0,90,0.0,0.80802,0.27805,0.71429,0.85714,1.0,0.0,1.0,2,15,2,2,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,15],[4,90,0.0444,0.84372,0.21241,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,6,0,17],[8,90,0.0889,0.86161,0.20972,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,9,0,17],[12,90,0.1333,0.875,0.22232,0.85711,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,0,4,0,21],[16,90,0.1778,0.83482,0.23449,0.85714,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,10,0,15],[20,90,0.2222,0.83482,0.21756,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,3,0,0,7,0,16],[24,90,0.2667,0.83036,0.21852,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,7,0,0,6,0,15],[28,90,0.3111,0.86607,0.24206,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,0,6,0,20],[32,90,0.3556,0.81696,0.24284,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,0,0,0,0,0,0,3,0,0,7,0,0,5,0,15],[36,90,0.4,0.7366,0.28818,0.71429,0.85707,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,1,0,0,2,0,0,0,0,0,8,0,0,6,0,11],[40,90,0.4444,0.84375,0.22689,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,0,4,0,18],[44,90,0.4889,0.80804,0.26633,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,3,0,0,1,0,0,6,0,0,3,0,17],[48,90,0.5333,0.71427,0.29234,0.53539,0.71429,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,1,0,0,3,0,0,1,0,0,8,0,0,4,0,11],[52,90,0.5778,0.80802,0.22761,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,7,0,0,7,0,13],[56,90,0.6222,0.8213,0.2021,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,9,0,12],[60,90,0.6667,0.82143,0.20516,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,5,0,0,8,0,13],[64,90,0.7111,0.80802,0.23039,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,9,0,0,1,0,16],[68,90,0.7556,0.69643,0.25443,0.53571,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,2,0,0,4,0,0,3,0,0,6,0,0,9,0,6],[72,90,0.8,0.76784,0.2468,0.57143,0.85707,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,4,0,0,3,0,0,6,0,0,4,0,13],[76,90,0.8444,0.76784,0.2468,0.71429,0.78571,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,9,0,0,4,0,12],[80,90,0.8889,0.70089,0.28428,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,1,0,0,2,0,0,3,0,0,9,0,0,3,0,10],[84,90,0.9333,0.7365,0.28392,0.57143,0.78571,1.0,0.0,1.0,1,12,0,1,0,1,0,0,3,0,0,1,0,0,3,0,0,7,0,0,4,0,12],[88,90,0.9778,0.66069,0.26185,0.5354,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,4,0,0,2,0,0,5,0,0,9,0,0,3,0,7],[90,90,1.0,0.49104,0.24205,0.28571,0.4286,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,5,0,0,8,0,0,4,0,0,7,0,0,1,0,2]]}]},{"i":"3374e1d3343e6de5","q":"Let  $a_1,a_2,\\ldots,a_7, b_1,b_2,\\ldots,b_7\\geq 0$  be real numbers satisfying  $a_i+b_i\\le 2$  for all  $i=\\overline{1,7}$ . \nProve that there exist  $k\\ne m$  such that  $|a_k-a_m|+|b_k-b_m|\\le 1$ .\n\nThanks for show me the mistake typing","t":[{"b":0,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,37,0.0,0.91963,0.22,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[4,37,0.1081,0.91071,0.22517,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[8,37,0.2162,0.93303,0.18553,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,27],[12,37,0.3243,0.92857,0.18558,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,26],[16,37,0.4324,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,37,0.5405,0.93302,0.21722,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[24,37,0.6486,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,37,0.7568,0.9375,0.20805,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[32,37,0.8649,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,37,0.973,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[37,37,1.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.83927,"x":0.99554,"p":[[0,39,0.0,0.87053,0.25843,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,0,2,0,24],[4,39,0.1026,0.95982,0.15663,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[8,39,0.2051,0.93304,0.16746,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,2,0,26],[12,39,0.3077,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[16,39,0.4103,0.91071,0.18472,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,3,0,24],[20,39,0.5128,0.83927,0.25192,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,1,0,0,4,0,20],[24,39,0.6154,0.86607,0.24728,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,6,0,21],[28,39,0.7179,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[32,39,0.8205,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[36,39,0.9231,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[39,39,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"28bb9943e0c7f36e","q":"Find all positive integers  $n$  such that it is possible to split the numbers from  $1$  to  $2n$  in two groups  $(a_1,a_2,..,a_n)$ ,  $(b_1,b_2,...,b_n)$  in such a way that  $2n\\mid a_1a_2\\cdots a_n+b_1b_2\\cdots b_n-1$ .\n\n*Proposed by Alef Pineda*","t":[{"b":2,"e":0.85714,"k":"flat","v":0.80804,"x":0.98661,"p":[[0,84,0.0,0.94195,0.10635,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[4,84,0.0476,0.96875,0.05906,1.0,1.0,1.0,0.8571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[8,84,0.0952,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,84,0.1429,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[16,84,0.1905,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[20,84,0.2381,0.92857,0.07143,0.85714,0.92857,1.0,0.85714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[24,84,0.2857,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[28,84,0.3333,0.91071,0.15465,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,17],[32,84,0.381,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[36,84,0.4286,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[40,84,0.4762,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[44,84,0.5238,0.91964,0.07087,0.85714,0.85714,1.0,0.85714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,14],[48,84,0.5714,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[52,84,0.619,0.875,0.11151,0.85714,0.85714,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,20,0,9],[56,84,0.6667,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15],[60,84,0.7143,0.89285,0.06186,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[64,84,0.7619,0.89732,0.08171,0.85714,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,11],[68,84,0.8095,0.87946,0.06298,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,6],[72,84,0.8571,0.84821,0.16728,0.85714,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,6],[76,84,0.9048,0.85714,0.05051,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,2],[80,84,0.9524,0.85714,2e-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],[84,84,1.0,0.80804,0.13175,0.85714,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,27,0,0]]},{"b":3,"e":0.85714,"k":"flat","v":0.83925,"x":0.97768,"p":[[0,61,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,61,0.0656,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,61,0.1311,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[12,61,0.1967,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[16,61,0.2623,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[20,61,0.3279,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[24,61,0.3934,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[28,61,0.459,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[32,61,0.5246,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,61,0.5902,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[40,61,0.6557,0.92411,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17],[44,61,0.7213,0.92411,0.07129,0.85714,0.85714,1.0,0.85714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,15],[48,61,0.7869,0.91517,0.07017,0.85714,0.85714,1.0,0.857,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[52,61,0.8525,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[56,61,0.918,0.86607,0.07936,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,25,0,5],[60,61,0.9836,0.8616,0.06667,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,25,0,4],[61,61,1.0,0.83925,0.09286,0.85714,0.85714,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,24,0,3]]}]},{"i":"5c87b897cc1b2045","q":"Consider the sequence  $(x_n)_{n\\ge 1}$  where  $x_1=1,x_2=2011$  and  $x_{n+2}=4022x_{n+1}-x_n$  for all  $n\\in\\mathbb{N}$ . Prove that  $\\frac{x_{2012}+1}{2012}$  is a perfect square.","t":[{"b":1,"e":1.0,"k":"flat","v":0.59821,"x":0.91071,"p":[[0,57,0.0,0.75,0.26,0.57143,0.71429,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,4,0,0,4,0,0,8,0,0,0,0,14],[4,57,0.0702,0.69643,0.24679,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,2,0,0,4,0,0,12,0,0,2,0,8],[8,57,0.1404,0.875,0.20124,0.7143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,20],[12,57,0.2105,0.79911,0.23107,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,8,0,0,4,0,14],[16,57,0.2807,0.70089,0.27976,0.53571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,2,0,0,3,0,0,1,0,0,12,0,0,0,0,11],[20,57,0.3509,0.70534,0.30917,0.57143,0.71429,1.0,0.0,1.0,1,11,1,1,0,4,0,0,1,0,0,0,0,0,4,0,0,7,0,0,4,0,11],[24,57,0.4211,0.75,0.29233,0.67857,0.78571,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,2,0,0,2,0,0,1,0,0,8,0,0,1,0,15],[28,57,0.4912,0.75446,0.26058,0.71429,0.71429,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,1,0,0,2,0,0,11,0,0,1,0,13],[32,57,0.5614,0.66072,0.31693,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,6,0,0,1,0,0,3,0,0,2,0,0,7,0,0,3,0,10],[36,57,0.6316,0.84375,0.19351,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,2,0,17],[40,57,0.7018,0.71874,0.27776,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,2,0,0,1,0,0,4,0,0,8,0,0,3,0,11],[44,57,0.7719,0.84372,0.23247,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,4,0,0,3,0,19],[48,57,0.8421,0.75884,0.29348,0.71429,0.85714,1.0,0.14,1.0,0,16,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,9,0,0,0,0,16],[52,57,0.9123,0.59821,0.3489,0.24999,0.71429,1.0,0.14286,1.0,0,10,0,0,0,8,0,0,4,0,0,1,0,0,2,0,0,5,0,0,2,0,10],[56,57,0.9825,0.91071,0.14174,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,21],[57,57,1.0,0.84821,0.17473,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,10,0,0,3,0,16]]},{"b":3,"e":0.57143,"k":"falling","v":0.49106,"x":0.83036,"p":[[0,75,0.0,0.78125,0.25501,0.67857,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,3,0,0,0,0,0,4,0,0,7,0,0,2,0,15],[4,75,0.0533,0.66964,0.29329,0.53572,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,3,0,0,1,0,0,4,0,0,8,0,0,3,0,9],[8,75,0.1067,0.83036,0.20652,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,10,0,0,2,0,16],[12,75,0.16,0.76784,0.22518,0.67857,0.71429,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,10,0,0,2,0,12],[16,75,0.2133,0.72321,0.20806,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,1,0,0,6,0,0,14,0,0,1,0,8],[20,75,0.2667,0.70533,0.24985,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,2,0,0,8,0,0,8,0,0,1,0,10],[24,75,0.32,0.65177,0.22851,0.57132,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,5,0,0,1,0,0,3,0,0,17,0,0,0,0,5],[28,75,0.3733,0.54923,0.24237,0.42857,0.57143,0.71429,0.0,1.0,1,3,1,1,0,2,0,0,4,0,0,6,0,0,6,0,0,10,0,0,0,0,3],[32,75,0.4267,0.49106,0.25985,0.24999,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,8,0,0,5,0,0,1,0,0,3,0,0,14,0,0,0,0,1],[36,75,0.48,0.55802,0.27049,0.28571,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,6,0,0,3,0,0,2,0,0,6,0,0,11,0,0,0,0,4],[40,75,0.5333,0.64284,0.23691,0.57143,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,1,0,0,1,0,0,3,0,0,19,0,0,0,0,4],[44,75,0.5867,0.52675,0.26107,0.2857,0.57121,0.71429,0.14286,1.0,0,3,0,0,0,6,0,0,4,0,0,3,0,0,6,0,0,10,0,0,0,0,3],[48,75,0.64,0.66963,0.26831,0.57132,0.71429,0.78571,0.14286,1.0,0,8,0,0,0,4,0,0,1,0,0,2,0,0,3,0,0,14,0,0,0,0,8],[52,75,0.6933,0.62052,0.16982,0.57143,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,0,0,0,11,0,0,15,0,0,1,0,1],[56,75,0.7467,0.68749,0.23266,0.67857,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,1,0,0,0,0,0,4,0,0,17,0,0,1,0,6],[60,75,0.8,0.5758,0.24883,0.53571,0.71429,0.71429,0.14,1.0,0,1,0,0,0,7,0,0,0,0,0,1,0,0,5,0,0,16,0,0,2,0,1],[64,75,0.8533,0.64284,0.17497,0.57143,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,20,0,0,1,0,1],[68,75,0.9067,0.64286,0.21429,0.67857,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,0,0,0,1,0,0,3,0,0,21,0,0,1,0,2],[72,75,0.96,0.63837,0.20512,0.57143,0.71429,0.71429,0.0,1.0,1,2,1,1,0,2,0,0,0,0,0,1,0,0,6,0,0,20,0,0,0,0,2],[75,75,1.0,0.59374,0.15198,0.57143,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,1,0,0,14,0,0,14,0,0,0,0,0]]}]},{"i":"fa4edc51394d3f84","q":"Find all couples of non-zero integers  $(x,y)$  such that,  $x^2+y^2$  is a common divisor of  $x^5+y$  and  $y^5+x$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.55356,"x":0.99554,"p":[[0,68,0.0,0.65625,0.2942,0.39286,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,7,0,0,4,0,0,3,0,0,4,0,0,3,0,10],[4,68,0.0588,0.71428,0.26244,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,6,0,0,1,0,0,4,0,0,8,0,0,2,0,11],[8,68,0.1176,0.55356,0.25191,0.39286,0.42857,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,8,0,0,10,0,0,4,0,0,4,0,0,0,0,6],[12,68,0.1765,0.62499,0.25939,0.42857,0.71429,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,7,0,0,6,0,0,2,0,0,9,0,0,1,0,7],[16,68,0.2353,0.62946,0.2621,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,3,0,0,6,0,0,6,0,0,6,0,0,2,0,7],[20,68,0.2941,0.7857,0.19886,0.71429,0.78571,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,11,0,0,6,0,10],[24,68,0.3529,0.70987,0.25118,0.57142,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,1,0,0,4,0,0,3,0,0,10,0,0,3,0,9],[28,68,0.4118,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[32,68,0.4706,0.90179,0.21261,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,4,0,24],[36,68,0.5294,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,68,0.5882,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[48,68,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,68,0.7647,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[56,68,0.8235,0.96875,0.10555,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,28],[60,68,0.8824,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,68,0.9412,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[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":7,"e":0.28571,"k":"falling","v":0.34375,"x":0.7232,"p":[[0,12,0.0,0.7232,0.23942,0.57143,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,8,0,0,1,0,11],[4,12,0.3333,0.59375,0.25281,0.39286,0.57143,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,8,0,0,6,0,0,3,0,0,9,0,0,0,0,6],[8,12,0.6667,0.375,0.1171,0.28571,0.28571,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,18,0,0,9,0,0,4,0,0,1,0,0,0,0,0],[12,12,1.0,0.34375,0.10013,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,5,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"e9a5271f44e25dec","q":"Find the smallest value of the expression  $|3 \\cdot 5^m - 11 \\cdot 13^n|$  for all  $m,n \\in N$ . \n\n(Folklore)","t":[{"b":3,"e":1.0,"k":"flat","v":0.7991,"x":0.92857,"p":[[0,32,0.0,0.80803,0.21902,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,3,0,0,10,0,12],[4,32,0.125,0.86607,0.15542,0.85711,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,11,0,14],[8,32,0.25,0.7991,0.23107,0.71429,0.85714,1.0,0.0,1.0,1,9,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,14,0,9],[12,32,0.375,0.80803,0.16602,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,17,0,6],[16,32,0.5,0.87946,0.1017,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,21,0,9],[20,32,0.625,0.83035,0.08328,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,29,0,0],[24,32,0.75,0.90624,0.07669,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[28,32,0.875,0.90179,0.09062,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,19,0,12],[32,32,1.0,0.92857,0.10102,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19]]},{"b":6,"e":1.0,"k":"flat","v":0.58022,"x":0.82143,"p":[[0,65,0.0,0.8079,0.22197,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,6,0,0,7,0,13],[4,65,0.0615,0.82143,0.22588,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,6,0,0,9,0,13],[8,65,0.1231,0.6741,0.21497,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,1,0,0,4,0,0,10,0,0,11,0,1],[12,65,0.1846,0.64732,0.35171,0.28571,0.85714,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,5,0,0,0,0,0,1,0,0,3,0,0,11,0,7],[16,65,0.2462,0.59822,0.34522,0.25001,0.57143,0.89286,0.0,1.0,1,8,0,1,0,7,0,0,3,0,0,0,0,0,6,0,0,1,0,0,6,0,8],[20,65,0.3077,0.58022,0.33676,0.28571,0.64071,0.85714,0.0,1.0,4,5,0,4,0,3,0,0,2,0,0,2,0,0,5,0,0,4,0,0,7,0,5],[24,65,0.3692,0.73659,0.21163,0.57143,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,2,0,0,15,0,4],[28,65,0.4308,0.6875,0.21558,0.57143,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,0,0,0,13,0,0,4,0,0,7,0,5],[32,65,0.4923,0.62499,0.17768,0.57143,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,1,0,0,17,0,0,6,0,0,5,0,1],[36,65,0.5538,0.70088,0.16507,0.57143,0.57143,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,8,0,4],[40,65,0.6154,0.70534,0.15543,0.57143,0.57143,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,3,0,0,9,0,3],[44,65,0.6769,0.67406,0.18639,0.57143,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,1,0,0,17,0,0,2,0,0,8,0,3],[48,65,0.7385,0.66963,0.14033,0.57143,0.57143,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,1,0,0,9,0,1],[52,65,0.8,0.69638,0.15874,0.57143,0.57143,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,9,0,3],[56,65,0.8615,0.68749,0.15747,0.57143,0.57143,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,1,0,0,10,0,2],[60,65,0.9231,0.67409,0.15664,0.57143,0.57143,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,2,0,0,10,0,1],[64,65,0.9846,0.65625,0.18162,0.57143,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,18,0,0,1,0,0,10,0,1],[65,65,1.0,0.69195,0.16409,0.57143,0.57143,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,17,0,0,0,0,0,13,0,1]]}]},{"i":"b851eb10b511734a","q":"In a tournament, every team plays exactly once against every other team. One won match earns  $3$  points for the winner and  $0$  for the loser. With a draw both teams receive  $1$  point each. At the end of the tournament it appears that all teams together have achieved  $15$  points. The last team on the final list scored exactly  $1$  point. The second to last team has not lost a match. \na) How many teams participated in the tournament? \nb) How many points did the team score in second place in the final ranking?","t":[{"b":5,"e":0.85714,"k":"flat","v":0.88392,"x":1.0,"p":[[0,64,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,64,0.0625,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,64,0.125,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[12,64,0.1875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,64,0.25,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[20,64,0.3125,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[24,64,0.375,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[28,64,0.4375,0.95982,0.13475,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,28],[32,64,0.5,0.93304,0.13825,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,4,0,24],[36,64,0.5625,0.90625,0.13651,0.85711,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,20],[40,64,0.625,0.89732,0.1525,0.82143,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,4,0,20],[44,64,0.6875,0.90625,0.14555,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,10,0,18],[48,64,0.75,0.91518,0.14223,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,22],[52,64,0.8125,0.90625,0.13175,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[56,64,0.875,0.93303,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[60,64,0.9375,0.90622,0.15413,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,21],[64,64,1.0,0.88392,0.17657,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,4,0,20]]},{"b":7,"e":1.0,"k":"flat","v":0.89732,"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],[4,20,0.2,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[8,20,0.4,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,20,0.6,0.94195,0.15919,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,28],[16,20,0.8,0.89732,0.20277,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,24],[20,20,1.0,0.91961,0.15132,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23]]}]},{"i":"b11624e752b5f4cf","q":"Find the largest integer  $k$  with the following property: Whenever real numbers  $x_1,x_2,\\dots,x_{2024}$  satisfy\n\\[x_1^2=(x_1+x_2)^2=\\dots=(x_1+x_2+\\dots+x_{2024})^2,\\]\nat least  $k$  of them are equal.","t":[{"b":0,"e":1.0,"k":"flat","v":0.84821,"x":0.93304,"p":[[0,41,0.0,0.91071,0.13716,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,1,0,22],[4,41,0.0976,0.84821,0.21706,0.71429,1.0,1.0,0.0,1.0,1,18,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,2,0,18],[8,41,0.1951,0.92409,0.13828,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,24],[12,41,0.2927,0.89732,0.1394,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,20],[16,41,0.3902,0.88839,0.13236,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,3,0,18],[20,41,0.4878,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[24,41,0.5854,0.89732,0.1439,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,3,0,20],[28,41,0.6829,0.86161,0.13592,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,6,0,14],[32,41,0.7805,0.90625,0.13175,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,21],[36,41,0.878,0.91518,0.12807,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,22],[40,41,0.9756,0.91518,0.12807,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,22],[41,41,1.0,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23]]},{"b":1,"e":1.0,"k":"flat","v":0.81696,"x":0.91964,"p":[[0,44,0.0,0.88392,0.14481,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,4,0,18],[4,44,0.0909,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[8,44,0.1818,0.86161,0.145,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,2,0,16],[12,44,0.2727,0.83036,0.15335,0.71429,0.78571,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,3,0,13],[16,44,0.3636,0.84821,0.13333,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,4,0,13],[20,44,0.4545,0.88393,0.1357,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,5,0,17],[24,44,0.5455,0.875,0.14174,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,3,0,17],[28,44,0.6364,0.86607,0.14258,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,3,0,16],[32,44,0.7273,0.84375,0.13997,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,4,0,13],[36,44,0.8182,0.81696,0.13474,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,4,0,10],[40,44,0.9091,0.82142,0.14728,0.71429,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,2,0,12],[44,44,1.0,0.82143,0.14286,0.71429,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,4,0,11]]}]},{"i":"ab17fc15763c1dd3","q":"Show that for any integer $n \\geq 2$ the sum of the fractions $\\frac{1}{a b}$, where $a$ and $b$ are relatively prime positive integers such that $a<b \\leq n$ and $a+b>n$, equals $\\frac{1}{2}$.","t":[{"b":2,"e":1.0,"k":"rising","v":0.73661,"x":1.0,"p":[[0,41,0.0,0.75893,0.40475,0.75,1.0,1.0,0.0,1.0,6,22,0,6,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,22],[4,41,0.0976,0.7857,0.38796,0.82143,1.0,1.0,0.0,1.0,6,23,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,23],[8,41,0.1951,0.95089,0.18766,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[12,41,0.2927,0.80804,0.36876,0.96429,1.0,1.0,0.0,1.0,5,24,0,5,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,24],[16,41,0.3902,0.80804,0.38896,1.0,1.0,1.0,0.0,1.0,6,25,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[20,41,0.4878,0.73661,0.41971,0.39286,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,22],[24,41,0.5854,0.74106,0.41717,0.49968,1.0,1.0,0.0,1.0,7,22,0,7,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,22],[28,41,0.6829,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,41,0.7805,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[40,41,0.9756,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":0.67411,"x":1.0,"p":[[0,18,0.0,0.88839,0.29609,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,27],[4,18,0.2222,0.82143,0.35174,0.85714,1.0,1.0,0.0,1.0,4,23,0,4,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,23],[8,18,0.4444,0.67411,0.46047,0.0,1.0,1.0,0.0,1.0,10,21,0,10,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,21],[12,18,0.6667,0.82143,0.3677,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,25],[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":"2924bd32a459d966","q":"Let  $a,b,c>0$  such that  $a+b+c=3$ . Prove that :  $$ \\frac{ab}{ab+a+b}+\\frac{bc}{bc+b+c}+\\frac{ca}{ca+c+a}+\\frac{1}{9}\\left(\\frac{(a-b)^2}{ab+a+b}+\\frac{(b-c)^2}{bc+b+c}+\\frac{(c-a)^2}{ca+c+a}\\right)\\leq1. $$","t":[{"b":2,"e":0.14286,"k":"flat","v":0.22768,"x":0.49999,"p":[[0,28,0.0,0.22768,0.27632,0.14286,0.14286,0.14286,0.0,1.0,5,3,0,5,0,21,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[4,28,0.1429,0.28125,0.27312,0.14286,0.14286,0.28571,0.0,1.0,2,2,0,2,0,20,0,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,2],[8,28,0.2857,0.49999,0.34441,0.24999,0.28571,0.89275,0.0,1.0,1,8,0,1,0,7,0,0,9,0,0,1,0,0,3,0,0,2,0,0,1,0,8],[12,28,0.4286,0.23214,0.22232,0.0,0.2857,0.28571,0.0,0.71429,10,0,0,10,0,5,0,0,12,0,0,1,0,0,0,0,0,4,0,0,0,0,0],[16,28,0.5714,0.25446,0.19799,0.14286,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,4,0,0,17,0,0,0,0,0,1,0,0,3,0,0,0,0,0],[20,28,0.7143,0.29018,0.22442,0.14286,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,2,0,0,17,0,0,0,0,0,1,0,0,5,0,0,0,0,0],[24,28,0.8571,0.25893,0.25111,0.0,0.28571,0.28571,0.0,0.71429,11,0,0,11,0,2,0,0,13,0,0,0,0,0,0,0,0,6,0,0,0,0,0],[28,28,1.0,0.27232,0.25843,0.0,0.28571,0.28571,0.0,1.0,9,1,0,9,0,4,0,0,13,0,0,0,0,0,1,0,0,4,0,0,0,0,1]]},{"b":5,"e":1.0,"k":"volatile","v":0.16963,"x":0.98213,"p":[[0,15,0.0,0.17857,0.22588,0.14286,0.14286,0.14286,0.0,1.0,6,2,0,6,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,15,0.2667,0.16963,0.10967,0.14286,0.14286,0.14286,0.0,0.571,3,0,0,3,0,23,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.38839,0.38004,0.14286,0.14286,0.89286,0.0,1.0,2,8,0,2,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,8],[12,15,0.8,0.98213,0.07791,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[15,15,1.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"6016fcdcc9a2069c","q":"Find the number of ways a series of  $+$  and  $-$  signs can be inserted between the numbers  $0,1,2,\\cdots, 12$  such that the value of the resulting expression is divisible by 5.\n\n*Proposed by Matthew Lerner-Brecher*","t":[{"b":0,"e":0.57143,"k":"flat","v":0.83035,"x":0.95089,"p":[[0,8,0.0,0.93304,0.22583,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,28],[4,8,0.5,0.95089,0.18766,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[8,8,1.0,0.83035,0.1357,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,13,0,8]]},{"b":6,"e":1.0,"k":"rising","v":0.83929,"x":1.0,"p":[[0,9,0.0,0.83929,0.33455,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,25],[4,9,0.4444,0.94643,0.18123,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[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":"39ac646dbf573fdc","q":"Find all rational numbers  $a$ ,for which there exist infinitely many positive rational numbers  $q$  such that the equation  $[x^a].{x^a}=q$  has no solution in rational numbers.(A.Vasiliev)","t":[{"b":4,"e":0.857,"k":"rising","v":0.0625,"x":0.86607,"p":[[0,50,0.0,0.0625,0.13333,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.79908,0.20161,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,6,0,12],[8,50,0.16,0.86607,0.18536,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,5,0,18],[12,50,0.24,0.77232,0.18509,0.71429,0.85707,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,8,0,0,10,0,7],[16,50,0.32,0.70532,0.26473,0.571,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,0,4,0,0,4,0,0,8,0,0,4,0,9],[20,50,0.4,0.70085,0.2124,0.571,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,7,0,0,6,0,6],[24,50,0.48,0.68298,0.23889,0.5354,0.71429,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,8,0,0,4,0,7],[28,50,0.56,0.69639,0.23893,0.571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,9,0,0,1,0,9],[32,50,0.64,0.57587,0.27545,0.39286,0.57143,0.75,0.0,1.0,2,3,0,2,0,1,0,0,5,0,0,4,0,0,5,0,0,7,0,0,5,0,3],[36,50,0.72,0.61152,0.28415,0.42857,0.64286,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,2,0,0,5,0,0,5,0,0,7,0,0,3,0,6],[40,50,0.8,0.61147,0.21219,0.4286,0.64286,0.71429,0.14,1.0,0,1,0,0,0,2,0,0,2,0,0,5,0,0,7,0,0,9,0,0,6,0,1],[44,50,0.88,0.69642,0.28065,0.42859,0.85714,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,4,0,0,3,0,0,3,0,0,3,0,0,9,0,8],[48,50,0.96,0.66069,0.2442,0.4286,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,0,0,0,7,0,0,5,0,0,6,0,0,8,0,4],[50,50,1.0,0.58923,0.24679,0.5354,0.64286,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,1,0,0,3,0,0,8,0,0,10,0,0,5,0,1]]},{"b":7,"e":0.57143,"k":"rising","v":0.10705,"x":0.86164,"p":[[0,65,0.0,0.10705,0.22586,0.0,0.0,0.035,0.0,0.85714,24,0,0,24,0,3,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[4,65,0.0615,0.80803,0.18423,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,7,0,0,6,0,12],[8,65,0.1231,0.80357,0.19149,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,0,6,0,12],[12,65,0.1846,0.86164,0.17663,0.82143,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,8,0,16],[16,65,0.2462,0.81247,0.1654,0.71429,0.78564,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,4,0,12],[20,65,0.3077,0.73214,0.20748,0.67857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,11,0,0,7,0,6],[24,65,0.3692,0.79016,0.18554,0.71429,0.85707,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,8,0,0,7,0,10],[28,65,0.4308,0.78124,0.21425,0.71429,0.78569,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,10,0,0,5,0,11],[32,65,0.4923,0.73663,0.19914,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,7,0,0,9,0,6],[36,65,0.5538,0.71872,0.15767,0.71429,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,16,0,0,6,0,3],[40,65,0.6154,0.75445,0.18978,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,6,0,0,7,0,8],[44,65,0.6769,0.78125,0.22299,0.71429,0.78564,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,10,0,0,5,0,11],[48,65,0.7385,0.77238,0.17814,0.71429,0.71429,0.895,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,12,0,0,6,0,8],[52,65,0.8,0.77232,0.17807,0.71429,0.78564,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,9,0,0,9,0,7],[56,65,0.8615,0.69641,0.25692,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,3,0,0,0,0,0,8,0,0,6,0,0,5,0,8],[60,65,0.9231,0.72316,0.22289,0.571,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,7,0,0,4,0,9],[64,65,0.9846,0.63383,0.26728,0.42859,0.71429,0.85714,0.0,1.0,1,5,0,1,0,2,0,0,2,0,0,5,0,0,3,0,0,10,0,0,4,0,5],[65,65,1.0,0.47768,0.15815,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,12,0,0,8,0,0,4,0,0,1,0,0]]}]},{"i":"f1d66c6008820a68","q":"Let   $ABC$  be a triangle whose angles  $\\alpha=\\angle CAB$  and  $\\beta=\\angle CBA$  are greater than  $45^{\\circ}$ . Above the side   $AB$  a right isosceles triangle   $ABR$  is constructed with  $AB$  as the hypotenuse, such that   $R$  is inside the triangle   $ABC$ . Analogously we construct above the sides  $BC$  and   $AC$  the right isosceles triangles   $CBP$  and  $ACQ$ , right at  $P$  and in   $Q$ , but with these outside the triangle  $ABC$ . Prove that  $CQRP$  is a parallelogram.","t":[{"b":4,"e":0.71429,"k":"flat","v":0.85714,"x":1.0,"p":[[0,43,0.0,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,43,0.093,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[12,43,0.2791,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,43,0.3721,0.95536,0.16146,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[20,43,0.4651,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,43,0.5581,0.9375,0.14258,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,26],[28,43,0.6512,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[32,43,0.7442,0.95088,0.12173,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[36,43,0.8372,0.9107,0.20126,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,24],[40,43,0.9302,0.90625,0.21312,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,25],[43,43,1.0,0.85714,0.19562,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,6,0,0,2,0,19]]},{"b":5,"e":0.57143,"k":"flat","v":0.89286,"x":1.0,"p":[[0,44,0.0,0.95089,0.19759,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[4,44,0.0909,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,44,0.1818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,44,0.2727,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[16,44,0.3636,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[20,44,0.4545,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,44,0.5455,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,44,0.6364,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[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],[36,44,0.8182,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[40,44,0.9091,0.95089,0.16213,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,29],[44,44,1.0,0.89286,0.19885,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,24]]}]},{"i":"9283b98f164ed595","q":"Let  $ABCD$  be a convex quadrilateral with the line  $CD$  being tangent to the circle on diameter  $AB$ . Prove that the line  $AB$  is tangent to the circle on diameter  $CD$  if and only if the lines  $BC$  and  $AD$  are parallel.","t":[{"b":0,"e":0.1429,"k":"flat","v":0.09812,"x":0.50893,"p":[[0,100,0.0,0.16069,0.30248,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,1],[4,100,0.04,0.37946,0.43684,0.0,0.14286,0.89286,0.0,1.0,15,8,0,15,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,8],[8,100,0.08,0.30355,0.41147,0.0,0.0,0.85704,0.0,1.0,17,5,0,17,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,5],[12,100,0.12,0.37946,0.4339,0.0,0.14286,0.89286,0.0,1.0,15,8,0,15,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,8],[16,100,0.16,0.31688,0.41613,0.0,0.07,0.67857,0.0,1.0,16,8,0,16,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,8],[20,100,0.2,0.29455,0.41181,0.0,0.0,0.53571,0.0,1.0,18,7,0,18,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,7],[24,100,0.24,0.25445,0.3758,0.0,0.0,0.571,0.0,1.0,20,4,0,20,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,0,1,0,4],[28,100,0.28,0.50893,0.45867,0.0,0.42857,1.0,0.0,1.0,11,14,0,11,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,14],[32,100,0.32,0.41516,0.42611,0.0,0.28573,0.85714,0.0,1.0,14,7,0,14,0,2,0,0,0,0,0,2,0,0,3,0,0,0,0,0,4,0,7],[36,100,0.36,0.28571,0.39286,0.0,0.07143,0.28571,0.0,1.0,16,7,0,16,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[40,100,0.4,0.28125,0.42331,0.0,0.0,0.46429,0.0,1.0,20,8,0,20,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[44,100,0.44,0.39732,0.43993,0.0,0.14288,0.89286,0.0,1.0,14,8,0,14,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,8],[48,100,0.48,0.34374,0.39424,0.0,0.14288,0.71429,0.0,1.0,14,6,0,14,0,3,0,0,3,0,0,2,0,0,1,0,0,2,0,0,1,0,6],[52,100,0.52,0.30357,0.41148,0.0,0.0,0.74996,0.0,1.0,18,5,0,18,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,5],[56,100,0.56,0.42856,0.45034,0.0,0.2857,1.0,0.0,1.0,15,11,0,15,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,11],[60,100,0.6,0.30802,0.41358,0.0,0.0,0.74999,0.0,1.0,18,6,0,18,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,6],[64,100,0.64,0.25893,0.36498,0.0,0.14286,0.32143,0.0,1.0,15,5,0,15,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[68,100,0.68,0.22768,0.36221,0.0,0.0,0.2857,0.0,1.0,19,4,0,19,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[72,100,0.72,0.28112,0.37539,0.0,0.07,0.571,0.0,1.0,16,3,0,16,0,6,0,0,0,0,0,0,0,0,3,0,0,0,0,0,4,0,3],[76,100,0.76,0.33929,0.42521,0.0,0.0,0.78571,0.0,1.0,17,8,0,17,0,1,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,8],[80,100,0.8,0.09812,0.19702,0.0,0.0,0.14287,0.0,1.0,22,1,0,22,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[84,100,0.84,0.2366,0.3773,0.0,0.0,0.21432,0.0,1.0,19,5,0,19,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,5],[88,100,0.88,0.33024,0.3887,0.0,0.14286,0.571,0.0,1.0,13,7,0,13,0,4,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,7],[92,100,0.92,0.20534,0.3349,0.0,0.0,0.1786,0.0,1.0,18,4,0,18,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,4],[96,100,0.96,0.33481,0.36527,0.0,0.2857,0.57143,0.0,1.0,12,5,0,12,0,3,0,0,7,0,0,0,0,0,3,0,0,1,0,0,1,0,5],[100,100,1.0,0.15625,0.25595,0.0,0.0,0.2857,0.0,1.0,18,2,0,18,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,2]]},{"b":3,"e":0.0,"k":"rising","v":0.10705,"x":0.48214,"p":[[0,88,0.0,0.13839,0.2977,0.0,0.0,0.03571,0.0,1.0,24,3,0,24,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[4,88,0.0455,0.45979,0.42966,0.0,0.42836,1.0,0.0,1.0,12,9,0,12,0,2,0,0,2,0,0,0,0,0,3,0,0,2,0,0,2,0,9],[8,88,0.0909,0.28571,0.40564,0.0,0.0,0.39286,0.0,1.0,18,7,0,18,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,7],[12,88,0.1364,0.24553,0.36811,0.0,0.0,0.39286,0.0,1.0,20,3,0,20,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,2,0,3],[16,88,0.1818,0.35714,0.45175,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,9],[20,88,0.2273,0.3392,0.43268,0.0,0.14143,0.89286,0.0,1.0,15,8,0,15,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,8],[24,88,0.2727,0.25893,0.40159,0.0,0.0,0.2857,0.0,1.0,19,7,0,19,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[28,88,0.3182,0.34811,0.404,0.0,0.07,0.74996,0.0,1.0,16,5,0,16,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,0,3,0,5],[32,88,0.3636,0.34821,0.41793,0.0,0.07143,0.85714,0.0,1.0,16,6,0,16,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,4,0,6],[36,88,0.4091,0.33479,0.40501,0.0,0.14286,0.64254,0.0,1.0,15,7,0,15,0,2,0,0,5,0,0,0,0,0,2,0,0,0,0,0,1,0,7],[40,88,0.4545,0.33479,0.42348,0.0,0.07143,0.78571,0.0,1.0,16,8,0,16,0,4,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,8],[44,88,0.5,0.23211,0.35126,0.0,0.0,0.2857,0.0,1.0,17,4,0,17,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,4],[48,88,0.5455,0.21875,0.3694,0.0,0.0,0.2857,0.0,1.0,21,5,0,21,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,5],[52,88,0.5909,0.10705,0.23418,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[56,88,0.6364,0.39722,0.44859,0.0,0.14143,1.0,0.0,1.0,15,10,0,15,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,10],[60,88,0.6818,0.3482,0.43584,0.0,0.07143,0.89286,0.0,1.0,16,8,0,16,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,8],[64,88,0.7273,0.35705,0.42713,0.0,0.07,0.89286,0.0,1.0,16,8,0,16,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,0,1,0,8],[68,88,0.7727,0.30356,0.38754,0.0,0.14286,0.60682,0.0,1.0,15,6,0,15,0,4,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,6],[72,88,0.8182,0.48214,0.4414,0.0,0.28571,1.0,0.0,1.0,11,11,0,11,0,2,0,0,4,0,0,0,0,0,1,0,0,1,0,0,2,0,11],[76,88,0.8636,0.33036,0.44383,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,9],[80,88,0.9091,0.29902,0.42466,0.0,0.0,0.78571,0.0,1.0,17,8,0,17,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[84,88,0.9545,0.37052,0.42236,0.0,0.07143,0.85714,0.0,1.0,16,7,0,16,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,0,2,0,7],[88,88,1.0,0.4375,0.43439,0.0,0.42857,1.0,0.0,1.0,14,9,0,14,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,9]]}]},{"i":"76b39d151cb73b7d","q":"One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set { $1,2,3$ }, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly  $200$ ; else it is successful. Prove that the sages can create a strategy, by which they can win the 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any real numbers sequence  $\\{x_n\\}$  ,suppose that  $\\{y_n\\}$  is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\\sum\\limits_{i = 1}^{n} {x^2_i})^{ \\frac{1}{2}}$   ${(n \\ge 1})$   .\nFind the smallest positive number   $\\lambda$  such that  for any real numbers sequence  $\\{x_n\\}$  and all positive integers  $m$  , have  $\\frac{1}{m}\\sum\\limits_{i = 1}^{m} {x^2_i}\\le\\sum\\limits_{i = 1}^{m} {\\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )","t":[{"b":1,"e":0.42857,"k":"falling","v":0.31694,"x":0.98214,"p":[[0,159,0.0,0.93749,0.17474,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[4,159,0.0252,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,159,0.0503,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,159,0.0755,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,159,0.1006,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[20,159,0.1258,0.85714,0.28121,0.85714,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,21],[24,159,0.1509,0.97766,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[28,159,0.1761,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[32,159,0.2013,0.88392,0.23538,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,0,21],[36,159,0.2264,0.83926,0.21947,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,6,0,0,4,0,17],[40,159,0.2516,0.83472,0.27713,0.82132,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,4,0,20],[44,159,0.2767,0.8214,0.27896,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,0,4,0,19],[48,159,0.3019,0.88391,0.22992,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,5,0,22],[52,159,0.327,0.85266,0.27777,0.85711,1.0,1.0,0.0,1.0,1,22,0,1,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,22],[56,159,0.3522,0.84821,0.2765,0.8214,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,3,0,0,2,0,22],[60,159,0.3774,0.88839,0.21646,0.85711,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,0,2,0,23],[64,159,0.4025,0.80357,0.30877,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,4,0,19],[68,159,0.4277,0.8348,0.29475,0.857,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,20],[72,159,0.4528,0.86159,0.27079,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,2,0,23],[76,159,0.478,0.86158,0.22444,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,5,0,0,3,0,20],[80,159,0.5031,0.87267,0.21439,0.83918,1.0,1.0,0.14,1.0,0,20,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,1,0,4,0,20],[84,159,0.5283,0.85714,0.25505,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0,4,0,21],[88,159,0.5535,0.72767,0.28874,0.57132,0.78571,1.0,0.0,1.0,1,12,0,1,0,1,0,0,3,0,0,2,0,0,3,0,0,6,0,0,4,0,12],[92,159,0.5786,0.79254,0.27395,0.71429,0.85714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,1,0,0,1,0,0,2,0,0,5,1,0,6,0,14],[96,159,0.6038,0.83036,0.27067,0.82143,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,7,0,17],[100,159,0.6289,0.7008,0.35254,0.64286,0.85714,1.0,0.0,1.0,4,10,0,4,0,2,0,0,1,0,0,1,0,0,0,0,0,4,0,0,10,0,10],[104,159,0.6541,0.6696,0.37019,0.42857,0.78564,1.0,0.0,1.0,4,15,0,4,0,2,0,0,1,0,0,3,0,0,4,0,0,2,0,0,1,0,15],[108,159,0.6792,0.71873,0.25874,0.57143,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,0,2,0,0,6,0,0,7,0,0,5,0,9],[112,159,0.7044,0.6897,0.33342,0.41068,0.85712,1.0,0.0,1.0,2,12,0,2,0,2,0,0,3,0,1,1,0,0,3,0,0,3,0,0,5,0,12],[116,159,0.7296,0.58927,0.3458,0.28571,0.4998,1.0,0.0,1.0,2,10,0,2,0,4,0,0,3,0,0,7,0,0,1,0,0,3,0,0,2,0,10],[120,159,0.7547,0.60935,0.32043,0.39286,0.57143,0.89286,0.0,1.0,2,8,0,2,0,2,0,1,3,0,0,4,0,0,5,0,0,3,0,0,4,0,8],[124,159,0.7799,0.61605,0.30397,0.42857,0.57143,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,3,0,0,5,0,0,5,0,0,4,0,0,3,0,8],[128,159,0.805,0.66069,0.34764,0.42857,0.71429,1.0,0.0,1.0,4,11,0,4,0,1,0,0,2,0,0,3,0,0,1,0,0,7,0,0,3,0,11],[132,159,0.8302,0.52228,0.36702,0.14289,0.4998,1.0,0.0,1.0,5,9,0,5,0,4,0,0,2,0,0,5,0,0,5,0,0,1,0,0,1,0,9],[136,159,0.8553,0.45981,0.3383,0.14289,0.42859,0.71429,0.0,1.0,6,4,0,6,0,3,0,0,5,0,0,4,0,0,3,0,0,4,0,0,3,0,4],[140,159,0.8805,0.4464,0.32289,0.1429,0.42857,0.71407,0.0,1.0,5,4,0,5,0,4,0,0,5,0,0,5,0,0,4,0,0,3,0,0,2,0,4],[144,159,0.9057,0.47318,0.26348,0.28571,0.42857,0.60714,0.0,1.0,2,2,0,2,0,4,0,0,3,0,0,11,0,0,4,0,0,3,0,0,3,0,2],[148,159,0.9308,0.41516,0.30379,0.14289,0.35714,0.57143,0.0,1.0,4,4,0,4,0,5,0,0,7,0,0,5,0,0,5,0,0,1,0,0,1,0,4],[152,159,0.956,0.41959,0.26467,0.1429,0.42857,0.57143,0.0,1.0,4,1,0,4,0,5,0,0,3,0,0,7,0,0,6,0,0,5,0,0,1,0,1],[156,159,0.9811,0.33247,0.25232,0.14286,0.28571,0.57111,0.0,1.0,4,1,0,4,0,8,0,1,8,0,0,1,0,0,7,0,0,1,0,0,1,0,1],[159,159,1.0,0.31694,0.24737,0.14286,0.2857,0.571,0.0,0.85714,6,0,0,6,0,6,0,1,8,0,0,2,0,0,4,1,0,3,0,0,1,0,0]]},{"b":3,"e":0.71429,"k":"flat","v":0.90179,"x":0.99107,"p":[[0,97,0.0,0.93303,0.21124,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,27],[4,97,0.0412,0.96875,0.09269,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[8,97,0.0825,0.94196,0.15093,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,26],[12,97,0.1237,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,97,0.1649,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,97,0.2062,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,97,0.2474,0.93304,0.18893,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[28,97,0.2887,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[32,97,0.3299,0.92857,0.21129,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,28],[36,97,0.3711,0.90179,0.25862,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,26],[40,97,0.4124,0.9241,0.2172,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[44,97,0.4536,0.9375,0.16342,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[48,97,0.4948,0.95536,0.11538,1.0,1.0,1.0,0.4286,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[52,97,0.5361,0.96872,0.10565,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[56,97,0.5773,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[60,97,0.6186,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[64,97,0.6598,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[68,97,0.701,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[72,97,0.7423,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[76,97,0.7835,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[80,97,0.8247,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[84,97,0.866,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[88,97,0.9072,0.95981,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[92,97,0.9485,0.97768,0.05187,1.0,1.0,1.0,0.8571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[96,97,0.9897,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[97,97,1.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27]]}]},{"i":"622b02cc4286332d","q":"Let $a_{1}, a_{2}, \\ldots, a_{9}$ 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 $i, 2 \\leq i \\leq 8$, the inequality $a_{i-1}+a_{i+1}<2 a_{i}$ holds. Will the statement remain true if we change the number 2 in the last inequality to $1.9 ?$","t":[{"b":6,"e":0.14286,"k":"flat","v":0.29021,"x":0.52674,"p":[[0,59,0.0,0.47767,0.23037,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,4,0,0,5,0,0,10,0,0,9,0,0,0,0,0,1,0,3],[4,59,0.0678,0.50445,0.21124,0.42857,0.57143,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,4,0,0,5,0,0,17,0,0,0,0,0,1,0,2],[8,59,0.1356,0.48212,0.21942,0.28571,0.42859,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,9,0,0,6,0,0,11,0,0,1,0,0,0,0,3],[12,59,0.2034,0.50444,0.26959,0.28571,0.49979,0.60714,0.0,1.0,1,4,0,1,0,4,0,0,5,0,0,6,0,0,8,0,0,3,0,0,1,0,4],[16,59,0.2712,0.4687,0.27252,0.28571,0.42857,0.60714,0.0,1.0,1,2,0,1,0,4,0,0,9,0,0,6,0,0,4,0,0,1,0,0,5,0,2],[20,59,0.339,0.50889,0.24726,0.28571,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,3,0,0,6,0,0,10,0,0,5,0,0,2,0,0,3,0,3],[24,59,0.4068,0.44194,0.17259,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,10,0,0,7,0,0,11,0,0,0,0,0,2,0,0],[28,59,0.4746,0.52674,0.24336,0.42857,0.42859,0.60714,0.14286,1.0,0,4,0,0,0,3,0,0,4,0,0,10,0,0,7,0,0,3,0,0,1,0,4],[32,59,0.5424,0.39284,0.20202,0.28571,0.42857,0.4642,0.0,1.0,1,1,0,1,0,4,0,0,10,0,0,9,0,0,6,0,0,0,0,0,1,0,1],[36,59,0.6102,0.45536,0.20958,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,8,0,0,8,0,0,8,0,0,3,0,0,1,0,1],[40,59,0.678,0.37949,0.19432,0.2857,0.28571,0.42895,0.14286,1.0,0,1,0,0,0,6,0,0,11,0,0,8,0,0,4,0,0,2,0,0,0,0,1],[44,59,0.7458,0.38392,0.24855,0.14289,0.28571,0.57111,0.0,1.0,1,2,0,1,0,8,0,0,9,0,0,5,0,0,5,0,0,1,0,0,1,0,2],[48,59,0.8136,0.29021,0.14502,0.14286,0.28571,0.42857,0.0,0.571,2,0,0,2,0,9,0,0,8,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[52,59,0.8814,0.40623,0.23174,0.28571,0.42857,0.57111,0.0,1.0,2,1,0,2,0,5,0,0,6,0,0,9,0,0,7,0,0,0,0,0,2,0,1],[56,59,0.9492,0.30356,0.16267,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,13,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[59,59,1.0,0.39283,0.17854,0.28571,0.42857,0.42858,0.14286,0.8571,0,0,0,0,0,6,0,0,7,0,0,12,0,0,4,0,0,2,0,0,1,0,0]]},{"b":7,"e":0.2857,"k":"falling","v":0.25437,"x":0.57584,"p":[[0,124,0.0,0.45076,0.16423,0.39286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,6,0,0,11,0,0,11,0,0,1,0,0,1,0,0],[4,124,0.0323,0.50445,0.16745,0.42857,0.57121,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,8,0,0,13,0,0,2,0,0,1,0,1],[8,124,0.0645,0.47767,0.20704,0.28571,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,6,0,0,9,0,0,10,0,0,0,0,0,3,0,1],[12,124,0.0968,0.55357,0.24157,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,5,0,0,8,0,0,7,0,0,3,0,0,4,0,3],[16,124,0.129,0.54014,0.22227,0.42857,0.571,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,3,0,0,10,0,0,11,0,0,0,0,0,3,0,3],[20,124,0.1613,0.57584,0.20973,0.53539,0.57143,0.57143,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,4,0,0,17,0,0,2,0,0,2,0,3],[24,124,0.1935,0.5357,0.23419,0.42857,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,6,0,0,13,0,0,0,0,0,3,0,3],[28,124,0.2258,0.53125,0.20896,0.42857,0.50001,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,10,0,0,11,0,0,1,0,0,0,0,4],[32,124,0.2581,0.51339,0.24707,0.39286,0.49999,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,4,0,0,8,0,0,7,0,0,4,0,0,3,0,2],[36,124,0.2903,0.51339,0.24185,0.28571,0.50001,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,6,0,0,7,0,0,10,0,0,0,0,0,3,0,3],[40,124,0.3226,0.4732,0.20651,0.28571,0.4286,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,9,0,0,6,0,0,9,0,0,3,0,0,3,0,0],[44,124,0.3548,0.54463,0.20958,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,7,0,0,10,0,0,4,0,0,4,0,1],[48,124,0.3871,0.51338,0.22548,0.39286,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,8,0,0,9,0,0,1,0,0,5,0,1],[52,124,0.4194,0.5089,0.25738,0.28571,0.57141,0.57143,0.0,1.0,2,3,0,2,0,2,0,0,5,0,0,4,0,0,13,0,0,1,0,0,2,0,3],[56,124,0.4516,0.49997,0.18556,0.39286,0.57121,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,6,0,0,13,0,0,2,0,0,3,0,0],[60,124,0.4839,0.44195,0.19999,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,4,0,0,8,0,0,8,0,0,8,0,0,1,0,0,3,0,0],[64,124,0.5161,0.55801,0.2212,0.42857,0.57143,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,5,0,0,6,0,0,13,0,0,2,0,0,1,0,4],[68,124,0.5484,0.54016,0.21939,0.42857,0.57143,0.60714,0.0,1.0,1,2,0,1,0,1,0,0,4,0,0,6,0,0,12,0,0,4,0,0,2,0,2],[72,124,0.5806,0.52222,0.26405,0.28571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,4,0,0,5,0,0,9,0,0,2,0,0,5,0,2],[76,124,0.6129,0.45982,0.21049,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,2,0,0,11,0,0,5,0,0,11,0,0,0,0,0,1,0,2],[80,124,0.6452,0.42856,0.23145,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,6,0,0,5,0,0,8,0,0,8,0,0,1,0,0,2,0,1],[84,124,0.6774,0.53122,0.19959,0.42857,0.57143,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,5,0,0,7,0,0,12,0,0,2,0,0,5,0,0],[88,124,0.7097,0.46874,0.21792,0.28571,0.42859,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,8,0,0,6,0,0,10,0,0,2,0,0,2,0,1],[92,124,0.7419,0.44195,0.19677,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,10,0,0,8,0,0,9,0,0,1,0,0,1,0,1],[96,124,0.7742,0.50892,0.25489,0.28571,0.50001,0.60714,0.0,1.0,2,2,0,2,0,1,0,0,6,0,0,7,0,0,8,0,0,2,0,0,4,0,2],[100,124,0.8065,0.3482,0.14696,0.2857,0.28571,0.42858,0.14286,0.57143,0,0,0,0,0,7,0,0,10,0,0,9,0,0,6,0,0,0,0,0,0,0,0],[104,124,0.8387,0.45084,0.22896,0.28571,0.42859,0.57141,0.14286,1.0,0,2,0,0,0,6,0,0,5,0,0,8,0,0,9,0,0,1,0,0,1,0,2],[108,124,0.871,0.40159,0.16563,0.2857,0.42857,0.57143,0.14,0.71429,0,0,0,0,0,5,0,0,9,0,0,6,0,0,11,0,0,1,0,0,0,0,0],[112,124,0.9032,0.39731,0.19143,0.28571,0.28571,0.46418,0.14286,1.0,0,1,0,0,0,3,0,0,15,0,0,6,0,0,4,0,0,3,0,0,0,0,1],[116,124,0.9355,0.33918,0.18822,0.24999,0.28571,0.42857,0.0,1.0,1,1,0,1,0,7,0,0,11,0,0,8,0,0,4,0,0,0,0,0,0,0,1],[120,124,0.9677,0.32142,0.14723,0.1429,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,9,0,0,10,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[124,124,1.0,0.25437,0.11147,0.14286,0.2857,0.28571,0.14,0.57143,0,0,0,0,0,12,0,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"484c476fa3a7e774","q":"Find all positive integers  $a, b, c$  such that  $ab+1$  and  $c$  are coprimes and:  $$ a(ba+1)(ca^2+ba+1)=2021^{2021} $$","t":[{"b":4,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,9,0.0,0.86607,0.23128,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,2,0,22],[4,9,0.4444,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":7,"e":0.42857,"k":"falling","v":0.5625,"x":0.95536,"p":[[0,51,0.0,0.87947,0.22334,0.85714,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,0,3,0,22],[4,51,0.0784,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[8,51,0.1569,0.80804,0.27574,0.67857,1.0,1.0,0.0,1.0,1,19,0,1,0,1,0,0,0,0,0,3,0,0,3,0,0,4,0,0,1,0,19],[12,51,0.2353,0.74552,0.22795,0.57132,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,2,0,0,9,0,9],[16,51,0.3137,0.74091,0.23269,0.57132,0.85714,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,5,0,0,11,0,7],[20,51,0.3922,0.76339,0.19103,0.57143,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,4,0,0,11,0,7],[24,51,0.4706,0.60714,0.25505,0.42857,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,6,0,0,9,0,0,3,0,0,5,0,0,3,0,6],[28,51,0.549,0.58482,0.22968,0.39286,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,8,0,0,4,0,0,6,0,0,8,0,0,3,0,3],[32,51,0.6275,0.66963,0.19378,0.53539,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,7,0,0,5,0,0,10,0,2],[36,51,0.7059,0.58481,0.21237,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,5,0,0,9,0,0,7,0,0,2,0,3],[40,51,0.7843,0.70534,0.21411,0.57143,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,3,0,0,9,0,0,6,0,0,5,0,7],[44,51,0.8627,0.65177,0.2141,0.42857,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,0,5,0,0,6,0,0,4,0,0,12,0,1],[48,51,0.9412,0.58481,0.2212,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,7,0,0,5,0,0,7,0,0,6,0,0,5,0,2],[51,51,1.0,0.5625,0.2141,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,7,0,0,7,0,0,6,0,0,7,0,0,3,0,2]]}]},{"i":"f4512e5c586632a8","q":"Each of the integers from 1 to 4027 has been colored either green or red. Changing the color of a number is making it red  if it was green and making it green if it was red. Two positive integers  $m$  and  $n$  are said to be *cuates* if either  $\\frac{m}{n}$  or  $\\frac{n}{m}$  is a prime number. A *step* consists in choosing two numbers that are cuates and changing the color of each of them. Show it is possible to apply a sequence of steps such that every integer from 1 to 2014 is green.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.23213,"p":[[0,66,0.0,0.13839,0.27313,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[4,66,0.0606,0.23213,0.33071,0.0,0.0,0.42857,0.0,1.0,19,3,0,19,0,1,0,0,2,0,0,3,0,0,3,0,0,1,0,0,0,0,3],[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],[12,66,0.1818,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,66,0.2424,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,66,0.303,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,66,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[40,66,0.6061,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,66,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,66,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,66,0.7879,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[60,66,0.9091,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.18302,"p":[[0,45,0.0,0.18302,0.27485,0.0,0.0,0.42857,0.0,1.0,21,1,0,21,0,0,0,0,1,0,0,5,0,0,3,0,0,1,0,0,0,0,1],[4,45,0.0889,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.03125,0.10555,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[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],[20,45,0.4444,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[36,45,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"16bb0fc20f93abb2","q":"During a day  $2016$  customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer  $k$  that makes the following statement always true.\n\nWe can find  $k$  customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.","t":[{"b":3,"e":1.0,"k":"flat","v":0.99554,"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],[6,6,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.95982,"x":0.99554,"p":[[0,38,0.0,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[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],[8,38,0.2105,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,38,0.3158,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,38,0.4211,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,38,0.5263,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,38,0.6316,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,38,0.7368,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,38,0.8421,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,38,0.9474,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[38,38,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"34c9204cde72fc09","q":"18. (FRG 3) ${ }^{\\mathrm{IMO} 3}$ Let $a, b, c$ be positive integers satisfying $(a, b)=(b, c)=$ $(c, a)=1$. Show that $2 a b c-a b-b c-c a$ is the largest integer not representable as  $$ x b c+y c a+z a b $$  with nonnegative integers $x, y, z$.","t":[{"b":2,"e":1.0,"k":"rising","v":0.41956,"x":0.96875,"p":[[0,60,0.0,0.57588,0.31029,0.24999,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,8,0,0,1,0,0,3,0,0,6,0,0,3,0,0,6,0,5],[4,60,0.0667,0.55353,0.28738,0.35714,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,8,0,0,0,0,0,2,0,0,12,0,0,3,0,0,2,0,5],[8,60,0.1333,0.41956,0.31334,0.14286,0.2857,0.71429,0.0,1.0,1,2,0,1,0,13,0,0,3,0,0,4,0,0,1,0,0,3,0,0,5,0,2],[12,60,0.2,0.58927,0.33264,0.25,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,8,0,0,3,0,0,1,0,0,5,0,0,2,0,0,6,0,7],[16,60,0.2667,0.91963,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,12,0,18],[20,60,0.3333,0.85714,0.15972,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,14,0,12],[24,60,0.4,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[28,60,0.4667,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[32,60,0.5333,0.91964,0.13333,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,21],[36,60,0.6,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[40,60,0.6667,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[44,60,0.7333,0.95535,0.08329,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[48,60,0.8,0.95089,0.09182,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,23],[52,60,0.8667,0.89285,0.13363,0.85711,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,8,0,17],[56,60,0.9333,0.91071,0.10565,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17],[60,60,1.0,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21]]},{"b":4,"e":0.571,"k":"falling","v":0.24536,"x":0.73659,"p":[[0,30,0.0,0.73659,0.28819,0.57132,0.85712,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,3,0,0,2,0,0,5,0,0,3,0,0,3,0,14],[4,30,0.1333,0.43737,0.27657,0.14286,0.42857,0.60714,0.14,1.0,0,2,0,0,0,11,0,0,4,0,0,3,0,0,6,0,0,4,0,0,2,0,2],[8,30,0.2667,0.51784,0.36552,0.14286,0.49979,0.89286,0.14286,1.0,0,8,0,0,0,13,0,0,2,0,0,1,0,0,4,0,0,0,0,0,4,0,8],[12,30,0.4,0.36161,0.27195,0.14286,0.1429,0.57143,0.0,1.0,1,2,0,1,0,16,0,0,1,0,0,1,0,0,9,0,0,2,0,0,0,0,2],[16,30,0.5333,0.24536,0.14839,0.14286,0.14286,0.32143,0.14,0.57143,0,0,0,0,0,20,0,0,4,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[20,30,0.6667,0.35248,0.2343,0.14286,0.28571,0.57143,0.14,0.85714,0,0,0,0,0,13,0,0,8,0,0,1,0,0,5,0,0,3,0,0,2,0,0],[24,30,0.8,0.25883,0.22993,0.14286,0.14286,0.28571,0.14,1.0,0,2,0,0,0,21,0,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[28,30,0.9333,0.30348,0.19812,0.14286,0.2857,0.32143,0.14,1.0,0,1,0,0,0,13,0,0,11,0,0,3,0,0,3,0,0,1,0,0,0,0,1],[30,30,1.0,0.31241,0.22435,0.14286,0.2143,0.42857,0.14,1.0,0,1,0,0,0,16,0,0,6,0,0,3,0,0,5,0,0,0,0,0,1,0,1]]}]},{"i":"055dec286b3adb49","q":"The distinct nonzero real numbers  $a$ ,  $b$ , and  $c$  satisfy:  $$ \\frac{a}{b^2} + \\frac{b}{c^2} + \\frac{c}{a^2} = \\frac{a}{c^2} + \\frac{b}{a^2} + \\frac{c}{b^2}. $$  Find the minimum value of the expression:  $$ \\frac{(a + b)(a + c)}{a^2} + \\frac{(a + b)(b + c)}{b^2} + \\frac{(c + b)(a + c)}{c^2} $$","t":[{"b":0,"e":0.57143,"k":"flat","v":0.7098,"x":0.84373,"p":[[0,9,0.0,0.84373,0.1689,0.85711,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,14,0,11],[4,9,0.4444,0.78125,0.22299,0.71429,0.85714,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0,15,0,7],[8,9,0.8889,0.77231,0.17806,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,4,0,0,17,0,4],[9,9,1.0,0.7098,0.19393,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,9,0,0,10,0,3]]},{"b":1,"e":0.857,"k":"flat","v":0.66517,"x":0.83036,"p":[[0,46,0.0,0.77679,0.20183,0.67857,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,2,0,0,16,0,6],[4,46,0.087,0.66517,0.31259,0.53571,0.85707,0.85714,0.0,1.0,3,6,2,3,0,1,0,0,2,0,0,2,0,0,5,0,0,2,0,0,11,0,6],[8,46,0.1739,0.83036,0.14914,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,13,0,9],[12,46,0.2609,0.82143,0.17496,0.82143,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,15,0,9],[16,46,0.3478,0.82142,0.15972,0.82132,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,17,0,7],[20,46,0.4348,0.74999,0.19563,0.57143,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,3,0,0,14,0,5],[24,46,0.5217,0.72321,0.20183,0.57143,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,3,0,0,15,0,3],[28,46,0.6087,0.75,0.18557,0.57143,0.85707,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,6,0,0,12,0,5],[32,46,0.6957,0.77232,0.18509,0.67857,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,16,0,5],[36,46,0.7826,0.72767,0.19351,0.57143,0.78564,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,6,0,0,12,0,4],[40,46,0.8696,0.78571,0.18557,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,6,0,0,13,0,7],[44,46,0.9565,0.75446,0.18292,0.67857,0.85707,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,7,0,0,12,0,5],[46,46,1.0,0.82589,0.18466,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,15,0,10]]}]},{"i":"362c92a91533569d","q":"AUS   A social club has $n$ members. They have the membership numbers $1,2, \\ldots, n$, respectively. From time to time members send presents to other members, including items they have already received as presents from other members. In order to avoid the embarrassing situation that a member might receive a present that he or she has sent to other members, the club adds the following rule to its statutes at one of its annual general meetings: \"A member with membership number $a$ is permitted to send a present to a member with membership number $b$ if and only if $a(b-1)$ is a multiple of $n$.\" Prove that, if each member follows this rule, none will receive a present from another member that he or she has already sent to other members.  Alternative formulation: Let $G$ be a directed graph with $n$ vertices $v_{1}, v_{2}, \\ldots, v_{n}$, such that there is an edge going from $v_{a}$ to $v_{b}$ if and only if $a$ and $b$ are distinct and $a(b-1)$ is a multiple of $n$. Prove that this graph does not contain a directed cycle.","t":[{"b":4,"e":1.0,"k":"flat","v":0.58926,"x":0.77219,"p":[[0,13,0.0,0.6875,0.24856,0.57143,0.71429,0.85714,0.0,1.0,2,6,0,2,0,0,0,0,0,0,0,3,0,0,7,0,0,9,0,0,5,0,6],[4,13,0.3077,0.70534,0.2878,0.57143,0.71429,1.0,0.0,1.0,3,10,0,3,0,0,0,0,0,0,0,0,0,0,11,0,0,4,0,0,4,0,10],[8,13,0.6154,0.6339,0.23129,0.57143,0.57143,0.71429,0.0,1.0,1,4,0,1,0,1,0,0,2,0,0,1,0,0,12,0,0,8,0,0,3,0,4],[12,13,0.9231,0.58926,0.29179,0.57132,0.64286,0.71429,0.0,1.0,4,3,0,4,0,2,0,0,0,0,0,0,0,0,10,0,0,9,0,0,4,0,3],[13,13,1.0,0.77219,0.12306,0.71429,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,21,0,0,3,0,6]]},{"b":6,"e":0.71429,"k":"rising","v":0.59371,"x":0.89732,"p":[[0,54,0.0,0.66964,0.32818,0.53572,0.71429,1.0,0.0,1.0,3,10,0,3,0,2,0,0,1,0,0,2,0,0,4,0,0,6,0,0,4,0,10],[4,54,0.0741,0.71426,0.189,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,6,0,0,4,0,7],[8,54,0.1481,0.71428,0.23957,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,3,0,0,1,0,0,7,0,0,5,0,0,8,0,7],[12,54,0.2222,0.76786,0.20124,0.67857,0.78571,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,0,7,0,9],[16,54,0.2963,0.59371,0.28146,0.42857,0.57143,0.74996,0.0,1.0,1,5,0,1,0,4,0,0,2,0,0,2,0,0,9,0,0,6,0,0,3,0,5],[20,54,0.3704,0.60706,0.28808,0.42859,0.57143,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,3,0,0,2,0,0,10,0,0,4,0,0,2,0,7],[24,54,0.4444,0.79909,0.19187,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,4,0,0,12,0,9],[28,54,0.5185,0.89732,0.1439,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,11,0,17],[32,54,0.5926,0.8616,0.13115,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,17,0,10],[36,54,0.6667,0.86607,0.11811,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,19,0,9],[40,54,0.7407,0.85267,0.14054,0.857,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,16,0,10],[44,54,0.8148,0.87499,0.14617,0.85711,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,12,0,14],[48,54,0.8889,0.83928,0.14174,0.82132,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,15,0,9],[52,54,0.963,0.82589,0.15458,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,14,0,9],[54,54,1.0,0.82587,0.14169,0.71429,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,14,0,8]]}]},{"i":"fe577c06fe9a9d86","q":"Given triangle  $ABC$  with  $AB<AC$ . The line passing through  $B$  and parallel to  $AC$  meets the external angle bisector of  $\\angle BAC$  at  $D$ . The line passing through  $C$  and parallel to  $AB$  meets this bisector at  $E$ . Point  $F$  lies on the side  $AC$  and satisfies the equality  $FC=AB$ . Prove that  $DF=FE$ .","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,10,0.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.04464,"p":[[0,16,0.0,0.03116,0.0589,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04464,0.12078,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,16,0.5,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7d2d6e38d6a194fc","q":"Given an arbitrary set of  $2k+1$  integers  $\\{a_1,a_2,...,a_{2k+1}\\}$ .  We make a new set   $$  \\{(a_1+a_2)/2, (a_2+a_3)/2, (a_{2k}+a_{2k+1})/2, (a_{2k+1}+a_1)/2\\} $$   and a new one, according to the same rule, and so on...  Prove that if we obtain integers only, the initial set consisted of equal integers only.","t":[{"b":3,"e":0.85714,"k":"rising","v":0.58035,"x":0.85713,"p":[[0,36,0.0,0.58035,0.28779,0.42857,0.4998,0.85714,0.0,0.85714,2,0,0,2,0,3,0,0,0,0,0,11,0,0,1,0,0,0,0,0,15,0,0],[4,36,0.1111,0.7723,0.18161,0.857,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,25,0,0],[8,36,0.2222,0.7991,0.14223,0.85714,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,27,0,0],[12,36,0.3333,0.85267,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[16,36,0.4444,0.85712,5e-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],[20,36,0.5556,0.85713,4e-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],[24,36,0.6667,0.85713,3e-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],[28,36,0.7778,0.85267,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,1],[32,36,0.8889,0.84373,0.07456,0.85714,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0],[36,36,1.0,0.85267,0.02485,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0]]},{"b":5,"e":0.85714,"k":"rising","v":0.54463,"x":0.85714,"p":[[0,38,0.0,0.54463,0.31831,0.39285,0.49979,0.85714,0.0,0.85714,4,0,0,4,0,3,0,0,1,0,0,8,0,0,1,0,0,1,0,0,14,0,0],[4,38,0.1053,0.69642,0.24156,0.42859,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,0,0,0,6,0,0,0,0,0,3,0,0,20,0,0],[8,38,0.2105,0.82142,0.13363,0.85714,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,29,0,0],[12,38,0.3158,0.81249,0.10971,0.85711,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,26,0,0],[16,38,0.4211,0.79909,0.15092,0.857,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,26,0,0],[20,38,0.5263,0.73211,0.2044,0.571,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,6,0,0,2,0,0,2,0,0,20,0,1],[24,38,0.6316,0.85267,0.02485,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0],[28,38,0.7368,0.85714,0.03571,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,30,0,1],[32,38,0.8421,0.83478,0.07244,0.857,0.85714,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0],[36,38,0.9474,0.83481,0.07239,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,29,0,0],[38,38,1.0,0.84375,0.04164,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0]]}]},{"i":"27d477b1189bfc82","q":"An acute triangle  $\\triangle{ABC}$  is inscribed in a circle with centre  $O$ . The altitudes of the triangle are  $AD,BE$  and  $CF$ . The line  $EF$  cut the circumference on  $P$  and  $Q$ .\na) Show that  $OA$  is perpendicular to  $PQ$ .\nb) If  $M$  is the midpoint of  $BC$ , show that  $AP^2=2AD\\cdot{OM}$ .","t":[{"b":1,"e":0.28571,"k":"flat","v":0.24107,"x":0.30357,"p":[[0,37,0.0,0.26343,0.08836,0.2857,0.2857,0.28571,0.0,0.43,3,0,0,3,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.26787,0.1171,0.2857,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,1,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,37,0.2162,0.2857,0.07138,0.2857,0.28571,0.28571,0.0,0.571,1,0,0,1,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,37,0.3243,0.30357,0.16269,0.2857,0.2857,0.28571,0.0,1.0,2,1,0,2,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[16,37,0.4324,0.24107,0.10374,0.2857,0.2857,0.28571,0.0,0.286,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.25892,0.10971,0.2857,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,37,0.6486,0.24999,0.09449,0.2857,0.2857,0.28571,0.0,0.28571,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.28125,0.15355,0.2857,0.28571,0.28571,0.0,1.0,3,1,0,3,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,37,0.8649,0.26787,0.09943,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,37,0.973,0.28571,1e-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],[37,37,1.0,0.28124,0.10403,0.2857,0.2857,0.28571,0.0,0.71429,2,0,0,2,0,0,0,0,29,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.24105,"x":0.29464,"p":[[0,14,0.0,0.29464,0.13803,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,14,0.2857,0.24105,0.12592,0.2857,0.2857,0.28571,0.0,0.571,6,0,0,6,0,0,0,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,14,0.5714,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.25001,0.0945,0.2857,0.2857,0.28571,0.0,0.286,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.25891,0.10968,0.2857,0.28571,0.28571,0.0,0.571,4,0,0,4,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"db1e50da92f31457","q":"Find all functions  $ f: \\mathbb{R} \\to \\mathbb{R}$ , such that\r\n\\[ f(xf(y)) \\plus{} f(f(x) \\plus{} f(y)) \\equal{} yf(x) \\plus{} f(x \\plus{} f(y))\\]\r\nholds for all  $ x$ ,  $ y \\in \\mathbb{R}$ , where  $ \\mathbb{R}$  denotes the set of real numbers.","t":[{"b":5,"e":1.0,"k":"rising","v":0.56694,"x":0.97768,"p":[[0,70,0.0,0.7857,0.25755,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,3,0,0,3,0,16],[4,70,0.0571,0.76784,0.28292,0.57132,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,1,0,0,4,0,0,5,0,0,0,0,0,4,0,16],[8,70,0.1143,0.7232,0.2922,0.57143,0.85707,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,4,0,0,1,0,0,6,0,0,2,0,0,4,0,13],[12,70,0.1714,0.56694,0.31841,0.28571,0.571,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,4,0,0,5,0,0,5,0,0,3,0,0,0,0,9],[16,70,0.2286,0.58482,0.29957,0.42857,0.50001,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,4,0,0,9,0,0,4,0,0,2,0,0,2,0,8],[20,70,0.2857,0.62054,0.29366,0.42857,0.57143,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,5,0,0,7,0,0,5,0,0,1,0,0,3,0,9],[24,70,0.3429,0.92857,0.16751,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,23],[28,70,0.4,0.87054,0.20935,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,4,0,0,2,0,21],[32,70,0.4571,0.89286,0.21429,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,22],[36,70,0.5143,0.94643,0.15872,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,27],[40,70,0.5714,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,70,0.6286,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[48,70,0.6857,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[52,70,0.7429,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[56,70,0.8,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[60,70,0.8571,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[64,70,0.9143,0.94642,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[68,70,0.9714,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[70,70,1.0,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20]]},{"b":6,"e":1.0,"k":"rising","v":0.74551,"x":1.0,"p":[[0,17,0.0,0.78125,0.2923,0.53569,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,4,0,0,3,0,0,2,0,0,1,0,0,3,0,18],[4,17,0.2353,0.75443,0.27256,0.571,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,1,0,0,3,0,15],[8,17,0.4706,0.74551,0.23888,0.57143,0.85707,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,2,0,0,9,0,9],[12,17,0.7059,0.74998,0.26001,0.57143,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,4,0,0,2,0,0,7,0,0,1,0,0,5,0,13],[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]]}]},{"i":"7ef20ccb002fe99b","q":"For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_{1}, a_{2}, \\ldots, a_{n}$ satisfying the following conditions:\n\n$$\nS\\left(a_{1}\\right)<S\\left(a_{2}\\right)<\\cdots<S\\left(a_{n}\\right) \\text { and } S\\left(a_{i}\\right)=P\\left(a_{i+1}\\right) \\quad(i=1,2, \\ldots, n)\n$$\n\n(We let $\\left.a_{n+1}=a_{1}.\\right)$ (Problem Committee of the Japan Mathematical Olympiad Foundation)","t":[{"b":0,"e":1.0,"k":"flat","v":0.91964,"x":0.99554,"p":[[0,25,0.0,0.91964,0.15947,0.85714,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,4,0,23],[4,25,0.16,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,25,0.32,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[12,25,0.48,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,25,0.64,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,25,0.8,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,25,0.96,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[25,25,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,30,0.0,0.93304,0.1636,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,25],[4,30,0.1333,0.95536,0.12595,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,27],[8,30,0.2667,0.95536,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[12,30,0.4,0.95087,0.11076,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[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],[24,30,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"e1faa09e25b2a292","q":"Let  $A$  be an acute angle such that  $\\tan A = 2\\cos A$ . Find the number of positive integers  $n$  less than or equal to  $1000$  such that  $\\sec^n A + \\tan^n A$  is a positive integer whose units digit is  $9$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.93304,"x":0.99107,"p":[[0,8,0.0,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[4,8,0.5,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[8,8,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.9375,"x":0.99107,"p":[[0,38,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,38,0.1053,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[8,38,0.2105,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[12,38,0.3158,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[16,38,0.4211,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[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],[24,38,0.6316,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[28,38,0.7368,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,38,0.8421,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[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],[38,38,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]}]},{"i":"cf0e62b2d2faaf77","q":"Find all functions  $ f,g: \\mathbb Q \\to \\mathbb Q$  such that for all rational numbers  $ x,y$  we have\r\n\\[ f(f(x) \\plus{} g(y) ) \\equal{} g(f(x)) \\plus{} y .\r\n\\]","t":[{"b":1,"e":1.0,"k":"flat","v":0.78571,"x":0.94629,"p":[[0,25,0.0,0.81027,0.22132,0.71429,0.85714,1.0,0.21429,1.0,0,12,0,0,0,0,0,1,1,0,0,3,0,0,0,0,0,5,0,0,10,0,12],[4,25,0.16,0.78571,0.23419,0.67857,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,2,0,0,11,0,11],[8,25,0.32,0.80357,0.22798,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0,3,0,0,9,0,13],[12,25,0.48,0.90625,0.18074,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,8,0,21],[16,25,0.64,0.88393,0.16146,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,8,0,17],[20,25,0.8,0.94629,0.11736,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[24,25,0.96,0.90625,0.09182,0.85714,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,13],[25,25,1.0,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19]]},{"b":5,"e":1.0,"k":"flat","v":0.79241,"x":0.96875,"p":[[0,25,0.0,0.82588,0.22796,0.82143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,9,0,15],[4,25,0.16,0.79462,0.22287,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,3,0,0,9,0,12],[8,25,0.32,0.82812,0.23273,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,2,1,0,5,0,17],[12,25,0.48,0.83036,0.18013,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,7,0,0,11,0,11],[16,25,0.64,0.79241,0.23376,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,7,1,0,10,0,10],[20,25,0.8,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[24,25,0.96,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[25,25,1.0,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]}]},{"i":"096c93c8d2f13cfa","q":"A deck of  $ 2n\\plus{}1$  cards consists of a joker and, for each number between 1 and  $ n$  inclusive, two cards marked with that number. The  $ 2n\\plus{}1$  cards are placed in a row, with the joker in the middle. For each  $ k$  with  $ 1 \\leq k \\leq n,$  the two cards numbered  $ k$  have exactly  $ k\\minus{}1$  cards between them. Determine all the values of  $ n$  not exceeding 10 for which this arrangement is possible. For which values of  $ n$  is it impossible?","t":[{"b":0,"e":0.71429,"k":"falling","v":0.49999,"x":0.95982,"p":[[0,112,0.0,0.95982,0.17941,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[4,112,0.0357,0.95535,0.10375,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[8,112,0.0714,0.95087,0.13181,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,27],[12,112,0.1071,0.94642,0.18123,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[16,112,0.1429,0.90625,0.22191,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,25],[20,112,0.1786,0.92411,0.17122,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,25],[24,112,0.2143,0.88393,0.25112,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,25],[28,112,0.25,0.94196,0.16312,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,27],[32,112,0.2857,0.79018,0.31132,0.67857,1.0,1.0,0.0,1.0,1,20,1,1,0,2,0,0,2,0,0,1,0,0,2,0,0,4,0,0,0,0,20],[36,112,0.3214,0.69643,0.3549,0.4286,0.71429,1.0,0.0,1.0,4,15,0,4,0,1,0,0,1,0,0,3,0,0,1,0,0,7,0,0,0,0,15],[40,112,0.3571,0.82142,0.27895,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,4,0,0,1,0,0,3,0,0,2,0,20],[44,112,0.3929,0.85266,0.28006,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,0,23],[48,112,0.4286,0.83479,0.24254,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,0,1,0,20],[52,112,0.4643,0.85268,0.28456,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,24],[56,112,0.5,0.84376,0.2865,0.71429,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,2,0,0,1,0,0,0,0,0,4,0,0,0,0,23],[60,112,0.5357,0.76338,0.30642,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,1,0,0,3,0,0,3,0,0,3,0,0,2,0,17],[64,112,0.5714,0.86159,0.2487,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,4,0,21],[68,112,0.6071,0.77228,0.29639,0.571,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,4,0,0,1,0,0,5,0,0,2,0,0,1,0,18],[72,112,0.6429,0.83482,0.27458,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,2,0,21],[76,112,0.6786,0.90625,0.21011,0.96429,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,24],[80,112,0.7143,0.84374,0.27748,0.71429,1.0,1.0,0.0,1.0,2,21,1,2,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,2,0,21],[84,112,0.75,0.83929,0.26666,0.71429,1.0,1.0,0.1429,1.0,0,22,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,4,0,0,0,0,22],[88,112,0.7857,0.86606,0.21412,0.71429,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,4,0,0,2,0,21],[92,112,0.8214,0.88839,0.24675,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,25],[96,112,0.8571,0.79463,0.32721,0.71429,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,0,1,0,20],[100,112,0.8929,0.70981,0.28457,0.57143,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,3,0,0,1,0,0,6,0,0,7,0,0,1,0,12],[104,112,0.9286,0.66964,0.2976,0.42857,0.71429,1.0,0.0,1.0,1,11,0,1,0,2,0,0,2,0,0,5,0,0,3,0,0,8,0,0,0,0,11],[108,112,0.9643,0.56249,0.37447,0.14286,0.64286,1.0,0.0,1.0,5,10,0,5,0,4,0,0,1,0,0,4,0,0,2,0,0,6,0,0,0,0,10],[112,112,1.0,0.49999,0.24222,0.42857,0.57143,0.71429,0.0,0.71429,3,0,0,3,0,3,0,0,1,0,0,7,0,0,4,0,0,14,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"flat","v":0.76786,"x":1.0,"p":[[0,74,0.0,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[4,74,0.0541,0.91518,0.22829,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[8,74,0.1081,0.89732,0.24284,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,27],[12,74,0.1622,0.76786,0.36726,0.53571,1.0,1.0,0.0,1.0,3,21,2,3,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,21],[16,74,0.2162,0.87053,0.28203,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,25],[20,74,0.2703,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,74,0.3243,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[28,74,0.3784,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,74,0.4324,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,74,0.4865,0.9732,0.08334,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[40,74,0.5405,0.93304,0.18552,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[44,74,0.5946,0.94195,0.14668,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,27],[48,74,0.6486,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[52,74,0.7027,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,74,0.7568,0.9866,0.04167,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[60,74,0.8108,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[64,74,0.8649,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[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],[72,74,0.973,0.95535,0.13092,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,28],[74,74,1.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30]]}]},{"i":"54391795b7e1d7fe","q":"Let  ${AB}$  be a chord of a circle  ${(c)}$  centered at  ${O}$ , and let  ${K}$  be a point on the segment  ${AB}$   such that  ${AK<BK}$ . Two circles through  ${K}$ , internally tangent to  ${(c)}$  at  ${A}$  and  ${B}$ , respectively, meet again at  ${L}$ . Let  ${P}$  be one of the points of intersection of the line  ${KL}$  and the circle  ${(c)}$ ,  and let the lines  ${AB}$  and  ${LO}$  meet at  ${M}$ . Prove that the line  ${MP}$  is tangent to the circle  ${(c)}$ .\n\nTheoklitos Paragyiou (Cyprus)","t":[{"b":2,"e":0.14286,"k":"flat","v":0.06241,"x":0.2275,"p":[[0,80,0.0,0.13838,0.2097,0.0,0.0,0.2857,0.0,0.85714,19,0,2,19,0,4,0,0,4,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[4,80,0.05,0.1875,0.16146,0.0,0.14286,0.32143,0.0,0.4286,9,0,0,9,0,12,0,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,80,0.1,0.18302,0.16837,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,10,0,0,7,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,80,0.15,0.2275,0.2048,0.105,0.14286,0.42857,0.0,0.85714,8,0,0,8,0,11,0,0,3,0,0,8,0,0,1,0,0,0,0,0,1,0,0],[16,80,0.2,0.15616,0.13998,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,0,13,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,80,0.25,0.16072,0.12242,0.14286,0.14286,0.1429,0.0,0.42857,6,0,0,6,0,20,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,80,0.3,0.19197,0.14555,0.14286,0.14286,0.28571,0.0,0.4286,7,0,0,7,0,13,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[28,80,0.35,0.13831,0.13592,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,15,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,80,0.4,0.125,0.13716,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,16,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,80,0.45,0.17849,0.12879,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,18,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,80,0.5,0.12054,0.1017,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,18,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,80,0.55,0.14732,0.12103,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,80,0.6,0.12492,0.09279,0.0,0.14286,0.14287,0.0,0.286,9,0,0,9,0,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,80,0.65,0.125,0.09942,0.0,0.14286,0.14287,0.0,0.42857,9,0,0,9,0,19,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,80,0.7,0.13393,0.07936,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,80,0.75,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,80,0.8,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,80,0.85,0.0934,0.07646,0.0,0.14143,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,80,0.9,0.10715,0.07143,0.0,0.14286,0.14286,0.0,0.2857,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,80,0.95,0.08455,0.06995,0.0,0.14143,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,80,1.0,0.10268,0.10249,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.1429,"k":"flat","v":0.05348,"x":0.19634,"p":[[0,65,0.0,0.07142,0.15968,0.0,0.0,0.0,0.0,0.571,26,0,1,26,0,1,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,65,0.0615,0.17393,0.14171,0.14,0.14286,0.2857,0.0,0.57143,7,0,0,7,0,16,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,65,0.1231,0.18749,0.1491,0.14286,0.14286,0.28571,0.0,0.571,7,0,0,7,0,14,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,65,0.1846,0.18749,0.15332,0.14286,0.14286,0.2857,0.0,0.57143,6,0,0,6,0,17,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,65,0.2462,0.19196,0.20079,0.0,0.14286,0.2857,0.0,0.85714,10,0,0,10,0,11,0,0,6,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[20,65,0.3077,0.19634,0.19152,0.0,0.14286,0.42857,0.0,0.57143,12,0,1,12,0,7,0,0,4,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[24,65,0.3692,0.09822,0.14033,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,7,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.14713,0.17305,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,12,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[32,65,0.4923,0.11161,0.12745,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,11,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,65,0.5538,0.09374,0.12163,0.0,0.07143,0.14286,0.0,0.571,16,0,0,16,0,13,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,65,0.6154,0.14286,0.15568,0.0,0.14286,0.1786,0.0,0.57143,13,0,0,13,0,11,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[44,65,0.6769,0.14732,0.20355,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,6,0,0,3,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[48,65,0.7385,0.10268,0.10248,0.0,0.14286,0.14287,0.0,0.28571,14,0,0,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.07143,0.12372,0.0,0.0,0.14287,0.0,0.42857,23,0,0,23,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.05348,0.09934,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.10706,0.14284,0.0,0.14143,0.14286,0.0,0.71429,15,0,0,15,0,13,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[65,65,1.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"09ad5cc028348247","q":"Determine all pairs of integers  $(x, y)$  satisfying the equation\n\\[y(x + y) = x^3- 7x^2 + 11x - 3.\\]","t":[{"b":2,"e":0.28571,"k":"falling","v":0.23661,"x":0.84374,"p":[[0,37,0.0,0.60715,0.27664,0.39286,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,5,0,0,3,0,0,3,0,0,2,0,0,8,0,0,9,0,2],[4,37,0.1081,0.82142,0.13835,0.71429,0.85714,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,18,0,5],[8,37,0.2162,0.84374,0.1488,0.82132,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,15,0,9],[12,37,0.3243,0.81247,0.1654,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,6,0,0,13,0,8],[16,37,0.4324,0.78125,0.15561,0.71429,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,0,17,0,3],[20,37,0.5405,0.79463,0.17106,0.71429,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,15,0,6],[24,37,0.6486,0.77677,0.18878,0.71429,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,5,0,0,17,0,4],[28,37,0.7568,0.72758,0.22429,0.71429,0.85714,0.85714,0.14,1.0,0,2,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,8,0,0,16,0,2],[32,37,0.8649,0.56696,0.28679,0.28571,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,5,0,0,6,0,0,3,0,0,1,0,0,7,0,0,8,0,2],[36,37,0.973,0.23661,0.06785,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.26784,0.08559,0.2857,0.28571,0.28571,0.14286,0.571,0,0,0,0,0,7,0,0,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.24098,"x":0.85713,"p":[[0,83,0.0,0.62947,0.24446,0.42857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,5,0,0,2,0,0,4,0,0,9,0,0,8,0,2],[4,83,0.0482,0.85713,0.09453,0.85714,0.85714,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,21,0,6],[8,83,0.0964,0.84374,0.09006,0.85711,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,22,0,4],[12,83,0.1446,0.83035,0.12078,0.82132,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,19,0,5],[16,83,0.1928,0.80357,0.14174,0.71429,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,20,0,3],[20,83,0.241,0.84375,0.17261,0.85714,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,4,0,0,15,0,10],[24,83,0.2892,0.77678,0.20183,0.71429,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,5,0,0,18,0,4],[28,83,0.3373,0.75443,0.17941,0.71429,0.857,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,8,0,0,18,0,1],[32,83,0.3855,0.79022,0.18884,0.71429,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,5,0,0,19,0,4],[36,83,0.4337,0.78571,0.17128,0.71429,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,10,0,0,17,0,3],[40,83,0.4819,0.72768,0.20935,0.71429,0.78571,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,9,0,0,13,0,3],[44,83,0.5301,0.72767,0.17985,0.71429,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,12,0,0,12,0,2],[48,83,0.5783,0.73212,0.22232,0.71429,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,2,0,0,0,0,0,3,0,0,6,0,0,17,0,2],[52,83,0.6265,0.76348,0.13656,0.71429,0.85707,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,11,0,0,16,0,1],[56,83,0.6747,0.7857,0.14727,0.71429,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,10,0,0,14,0,4],[60,83,0.7229,0.72767,0.19351,0.71429,0.78564,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,9,0,0,14,0,2],[64,83,0.7711,0.75443,0.1864,0.71429,0.85714,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,7,0,0,17,0,2],[68,83,0.8193,0.75,0.17128,0.71429,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,9,0,0,15,0,2],[72,83,0.8675,0.7009,0.21827,0.67857,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,4,0,0,0,0,0,3,0,0,10,0,0,12,0,2],[76,83,0.9157,0.54018,0.26901,0.28571,0.57143,0.85714,0.14286,1.0,0,1,0,0,0,3,0,0,10,0,0,2,0,0,3,0,0,5,0,0,8,0,1],[80,83,0.9639,0.35714,0.24744,0.14289,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,9,0,0,16,0,0,0,0,0,1,0,0,2,0,0,3,0,1],[83,83,1.0,0.24098,0.06635,0.14286,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"39e976c6f52c97f0","q":"Let $A B C$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $B C$ such that $B E=E F=F C$. Points $X$ and $Y$ lie on lines $A B$ and $A C$, respectively, so that $X, Y$, and $G$ are not collinear. If the line through $E$ parallel to $X G$ and the line through $F$ parallel to $Y G$ intersect at $P \\neq G$, prove that $G P$ passes through the midpoint of $X Y$.","t":[{"b":1,"e":0.1429,"k":"flat","v":0.04455,"x":0.14286,"p":[[0,155,0.0,0.14286,0.23146,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,155,0.0258,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,155,0.0516,0.06679,0.0711,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,155,0.0774,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,155,0.1032,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,155,0.129,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,155,0.1548,0.07589,0.08737,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,155,0.1806,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,155,0.2065,0.11143,0.06894,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,155,0.2323,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,155,0.2581,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,155,0.2839,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,155,0.3097,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,155,0.3355,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,155,0.3613,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,155,0.3871,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,155,0.4129,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,155,0.4387,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,155,0.4645,0.06251,0.07087,0.0,0.0,0.14286,0.0,0.143,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,155,0.4903,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,155,0.5161,0.06687,0.0712,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,155,0.5419,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,155,0.5677,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.143,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,155,0.5935,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,155,0.6194,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,155,0.6452,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,155,0.671,0.07134,0.07978,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,155,0.6968,0.08473,0.08639,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,155,0.7226,0.06232,0.07067,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,155,0.7484,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,155,0.7742,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,155,0.8,0.0892,0.06909,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,155,0.8258,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,155,0.8516,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,155,0.8774,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,155,0.9032,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,155,0.929,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[148,155,0.9548,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[152,155,0.9806,0.09375,0.09852,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,13,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[155,155,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.04911,"x":0.12027,"p":[[0,96,0.0,0.12027,0.07232,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,96,0.0417,0.1025,0.0734,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,96,0.0833,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,96,0.125,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,96,0.1667,0.0758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 $n$   a positive integer and let   $x_1, \\ldots, x_n, y_1, \\ldots, y_n$  real positive numbers such that   $x_1+\\ldots+x_n=y_1+\\ldots+y_n=1$ . Prove that:  $$ |x_1-y_1|+\\ldots+|x_n-y_n|\\leq 2-\\underset{1\\leq i\\leq n}{min} \\;\\dfrac{x_i}{y_i}-\\underset{1\\leq i\\leq n}{min} \\;\\dfrac{y_i}{x_i} $$","t":[{"b":1,"e":0.14286,"k":"falling","v":0.08473,"x":0.58486,"p":[[0,16,0.0,0.56695,0.33404,0.28571,0.57143,0.85714,0.0,1.0,2,7,0,2,0,4,0,0,5,0,0,3,0,0,5,0,0,1,0,0,5,0,7],[4,16,0.25,0.56246,0.29867,0.39286,0.57121,0.74996,0.0,1.0,3,5,0,3,0,1,0,0,4,0,0,5,0,0,6,0,0,5,0,0,3,0,5],[8,16,0.5,0.58486,0.35056,0.2857,0.71429,0.85714,0.0,1.0,4,7,0,4,0,3,0,0,3,0,0,2,0,0,3,0,0,5,0,0,5,0,7],[12,16,0.75,0.09822,0.1448,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,11,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,16,1.0,0.08473,0.1121,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.11607,"x":0.49107,"p":[[0,35,0.0,0.49107,0.3387,0.24999,0.42857,0.85714,0.0,1.0,3,6,0,3,0,5,0,0,6,0,0,5,0,0,2,0,0,2,0,0,3,0,6],[4,35,0.1143,0.46875,0.3611,0.14286,0.35714,0.85704,0.0,1.0,5,5,0,5,0,6,0,0,5,0,0,2,0,0,1,0,0,4,0,0,4,0,5],[8,35,0.2286,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.13384,0.0612,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.125,0.11152,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,20,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,35,0.5714,0.12054,0.08827,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.12491,0.07782,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.125,0.05923,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.11607,0.06621,0.14286,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fd4b6f31c0b0d34e","q":"Distinct integers  $x,y,z{}$  verify the relation  $(x-y)(y-z)(z-x)=x+y+z$ . Find the smallest possibile value of  $|x+y+z|$ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.78125,"x":0.90178,"p":[[0,28,0.0,0.79017,0.11836,0.71429,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,18,0,1],[4,28,0.1429,0.7991,0.17445,0.71429,0.85714,0.85714,0.0,1.0,1,5,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,14,0,5],[8,28,0.2857,0.90178,0.0974,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[12,28,0.4286,0.89286,0.10714,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,14],[16,28,0.5714,0.78125,0.32924,0.71429,0.85714,1.0,0.0,1.0,4,15,4,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,8,0,15],[20,28,0.7143,0.88392,0.10374,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,17,0,11],[24,28,0.8571,0.85714,0.07986,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,5],[28,28,1.0,0.81696,0.07349,0.82132,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,24,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.8125,"x":1.0,"p":[[0,59,0.0,0.82589,0.09932,0.71429,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,18,0,4],[4,59,0.0678,0.8125,0.08328,0.71429,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,18,0,2],[8,59,0.1356,0.83929,0.07784,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,22,0,3],[12,59,0.2034,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[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],[20,59,0.339,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,59,0.4068,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,59,0.4746,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,59,0.5424,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[36,59,0.6102,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,59,0.678,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[48,59,0.8136,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,59,0.8814,0.91071,0.10565,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17],[56,59,0.9492,0.90179,0.0974,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,14],[59,59,1.0,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19]]}]},{"i":"dd398e715288c7e9","q":"There are two given different polynomials  $P(x),Q(x)$  with real coefficients such that  $P(Q(x))=Q(P(x))$ . Prove that  $\\forall n\\in \\mathbb{Z_{+}}$  polynomial:\n\\[\\underbrace{P(P(\\ldots P(P}_{n}(x))\\ldots))- \\underbrace{Q(Q(\\ldots Q(Q}_{n}(x))\\ldots))\\]\nis divisible by  $P(x)-Q(x)$ .","t":[{"b":4,"e":0.57143,"k":"falling","v":0.45533,"x":0.70536,"p":[[0,18,0.0,0.66518,0.19103,0.71429,0.71429,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,22,0,0,1,0,2],[4,18,0.2222,0.70536,0.15947,0.71429,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,23,0,0,0,0,4],[8,18,0.4444,0.607,0.1855,0.53539,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,5,0,0,5,0,0,18,0,0,0,0,1],[12,18,0.6667,0.52222,0.16622,0.42857,0.57121,0.71429,0.14,0.71429,0,0,0,0,0,2,0,0,2,0,0,11,0,0,7,0,0,10,0,0,0,0,0],[16,18,0.8889,0.45533,0.17653,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,7,0,0,9,0,0,7,0,0,6,0,0,0,0,0],[18,18,1.0,0.48659,0.1885,0.42857,0.57121,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,2,0,0,9,0,0,10,0,0,7,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.38393,"x":0.74999,"p":[[0,22,0.0,0.66964,0.18708,0.71429,0.71429,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,23,0,0,1,0,2],[4,22,0.1818,0.74999,0.11296,0.71429,0.71429,0.71429,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,0,0,0,5],[8,22,0.3636,0.71429,0.06186,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,0,0,1],[12,22,0.5455,0.71426,0.09453,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,27,0,0,0,0,2],[16,22,0.7273,0.67857,0.20825,0.71429,0.71429,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,3,0,0,0,0,0,2,0,0,22,0,0,0,0,4],[20,22,0.9091,0.50893,0.3071,0.14286,0.71429,0.71429,0.0,1.0,2,3,0,2,0,7,0,0,4,0,0,1,0,0,0,0,0,15,0,0,0,0,3],[22,22,1.0,0.38393,0.26592,0.25,0.28571,0.71429,0.0,1.0,4,1,0,4,0,4,0,0,12,0,0,1,0,0,2,0,0,8,0,0,0,0,1]]}]},{"i":"ccf17cfbae64575c","q":"$\\triangle{ABC}$  is equailateral.  $E$  is any point on  $\\overline{AC}$  produced and the equilateral  $\\triangle{ECD}$  is drawn. If  $M$  and  $N$  are the midpoints of  $\\overline{AD}$  and  $\\overline{EB}$  respectively then show that  $\\triangle{CMN}$  is equilateral.","t":[{"b":3,"e":1.0,"k":"flat","v":0.89286,"x":0.97321,"p":[[0,11,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,11,0.3636,0.91515,0.17812,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,2,0,25],[8,11,0.7273,0.89286,0.19885,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,1,0,24],[11,11,1.0,0.97321,0.10375,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29]]},{"b":7,"e":1.0,"k":"flat","v":0.86159,"x":0.97321,"p":[[0,9,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,9,0.4444,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,26],[8,9,0.8889,0.91071,0.17768,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,3,0,24],[9,9,1.0,0.86159,0.22443,0.78571,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,0,0,0,2,0,22]]}]},{"i":"69d1e63d3f69a27a","q":"Let  $n$  and  $k$  be positive integers. An  $n$ -tuple  $(a_1, a_2,\\ldots , a_n)$  is called a permutation if every number from the set  $\\{1, 2, . . . , n\\}$  occurs in it exactly once. For a permutation  $(p_1, p_2, . . . , p_n)$ , we define its  $k$ -mutation to be the  $n$ -tuple  $$ (p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}), $$  where indices are taken modulo  $n$ . Find all pairs  $(n, k)$  such that every two distinct permutations have distinct  $k$ -mutations.\n\n*Remark*: For example, when  $(n, k) = (4, 2)$ , the  $2$ -mutation of  $(1, 2, 4, 3)$  is  $(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5)$ .\n\n*Proposed by Borna \u0160imi\u0107*","t":[{"b":4,"e":1.0,"k":"flat","v":0.82589,"x":1.0,"p":[[0,34,0.0,0.88393,0.12595,0.85714,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,12,0,14],[4,34,0.1176,0.82589,0.16263,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,9,0,11],[8,34,0.2353,0.83035,0.20652,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,4,0,0,10,0,13],[12,34,0.3529,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[16,34,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,34,0.9412,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[34,34,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":0.8616,"x":0.99107,"p":[[0,38,0.0,0.89285,0.13832,0.85714,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,11,0,16],[4,38,0.1053,0.8616,0.16164,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,13,0,13],[8,38,0.2105,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,38,0.3158,0.92409,0.10709,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[16,38,0.4211,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[20,38,0.5263,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[24,38,0.6316,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[28,38,0.7368,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[32,38,0.8421,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,38,0.9474,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[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]]}]},{"i":"702d5f013564f492","q":"Find all prime numbers  $p$  such that  $y^2 = x^3+4x$  has exactly  $p$  solutions in integers modulo  $p$ .\n\nIn other words, determine all prime numbers  $p$  with the following property:\nthere exist exactly  $p$  ordered pairs of integers  $(x,y)$  such that  $x,y \\in \\{0,1,\\dots,p-1\\}$ and \\[ p \\text{ divides } y^2 - x^3 - 4x. \\]","t":[{"b":2,"e":0.57143,"k":"flat","v":0.59374,"x":0.72321,"p":[[0,20,0.0,0.59819,0.16146,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,10,0,0,6,0,0,4,0,1],[4,20,0.2,0.64286,0.22304,0.42857,0.64286,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,0,0,0,7,0,0,7,0,0,6,0,0,7,0,3],[8,20,0.4,0.59374,0.1992,0.42857,0.57121,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,13,0,0,3,0,0,8,0,0,4,0,2],[12,20,0.6,0.69643,0.09942,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,22,0,0,4,0,0],[16,20,0.8,0.72321,0.07936,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,22,0,0,6,0,0],[20,20,1.0,0.7232,0.10064,0.71429,0.71429,0.75,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,19,0,0,8,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.62054,"x":0.74094,"p":[[0,25,0.0,0.62054,0.23854,0.42857,0.71429,0.75001,0.14286,1.0,0,4,0,0,0,2,0,0,1,0,0,10,0,0,2,0,0,9,0,0,4,0,4],[4,25,0.16,0.6875,0.15335,0.67857,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,17,0,0,6,0,1],[8,25,0.32,0.74094,0.05583,0.71429,0.71429,0.71429,0.71,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0],[12,25,0.48,0.72768,0.06546,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,0,5,0,0],[16,25,0.64,0.6964,0.09282,0.67857,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,21,0,0,2,0,1],[20,25,0.8,0.70982,0.09771,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,20,0,0,6,0,0],[24,25,0.96,0.73661,0.08828,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,0,9,0,0],[25,25,1.0,0.71429,0.07143,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,0,4,0,0]]}]},{"i":"da3eb6d011bcfb38","q":"Let  $ P(x) \\equal{} x^3 \\plus{} mx \\plus{} n$  be an integer polynomial satisfying that if  $ P(x) \\minus{} P(y)$  is divisible by 107, then  $ x \\minus{} y$  is divisible by 107 as well, where  $ x$  and  $ y$  are integers. Prove that 107 divides  $ m$ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.37054,"x":0.54464,"p":[[0,45,0.0,0.41295,0.21109,0.2857,0.42857,0.42858,0.14286,1.0,0,2,0,0,0,5,0,1,6,0,0,14,0,0,2,0,0,2,0,0,0,0,2],[4,45,0.0889,0.37054,0.20159,0.25,0.42857,0.42857,0.0,1.0,1,1,0,1,0,7,0,0,5,0,0,15,0,0,2,0,0,0,0,0,1,0,1],[8,45,0.1778,0.45536,0.29545,0.24999,0.35714,0.60714,0.14286,1.0,0,4,0,0,0,8,0,0,8,0,0,5,0,0,3,0,0,1,0,0,3,0,4],[12,45,0.2667,0.54464,0.26107,0.28571,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,4,0,0,5,0,0,4,0,0,8,0,0,4,0,2],[16,45,0.3556,0.43301,0.24348,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,4,0,0,6,0,0,9,0,0,6,0,0,1,0,0,3,0,1],[20,45,0.4444,0.45085,0.22617,0.2857,0.42859,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,7,0,0,8,0,0,6,0,0,4,0,0,0,0,2],[24,45,0.5333,0.44196,0.24577,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,7,0,0,7,0,0,6,0,0,5,0,0,3,0,0,3,0,1],[28,45,0.6222,0.46878,0.23209,0.25,0.5005,0.71429,0.14286,0.85714,0,0,0,0,0,8,0,0,2,0,0,6,0,0,7,0,0,7,0,0,2,0,0],[32,45,0.7111,0.52228,0.25154,0.28571,0.4998,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,4,0,0,7,0,0,4,0,0,6,0,0,5,0,1],[36,45,0.8,0.41964,0.22851,0.28571,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,4,0,0,7,0,0,8,0,0,6,0,0,2,0,0,3,0,0],[40,45,0.8889,0.46426,0.25753,0.25,0.571,0.57143,0.0,1.0,1,1,0,1,0,7,0,0,4,0,0,3,0,0,10,0,0,3,0,0,3,0,1],[44,45,0.9778,0.45978,0.24151,0.28571,0.42857,0.60714,0.0,1.0,1,1,0,1,0,4,0,0,8,0,0,5,0,0,6,0,0,5,0,0,2,0,1],[45,45,1.0,0.45094,0.22047,0.28571,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,4,0,0,7,0,0,7,0,0,4,0,0,8,0,0,1,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.34366,"x":0.56691,"p":[[0,25,0.0,0.34366,0.15521,0.25002,0.42857,0.42857,0.14,0.85714,0,0,0,0,0,8,0,0,7,0,0,15,0,0,1,0,0,0,0,0,1,0,0],[4,25,0.16,0.38838,0.12991,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,1,0,0,21,0,0,4,0,0,0,0,0,0,0,0],[8,25,0.32,0.56691,0.20972,0.42857,0.571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,11,0,0,4,0,0,10,0,0,0,0,3],[12,25,0.48,0.50445,0.15145,0.42857,0.42857,0.57111,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,22,0,0,3,0,0,4,0,0,1,0,1],[16,25,0.64,0.43749,0.16727,0.42857,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,2,0,0,15,0,0,6,0,0,4,0,0,0,0,0],[20,25,0.8,0.4375,0.13333,0.42857,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,21,0,0,4,0,0,1,0,0,1,0,0],[24,25,0.96,0.39732,0.13709,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,4,0,0,6,0,0,16,0,0,5,0,0,1,0,0,0,0,0],[25,25,1.0,0.44638,0.13711,0.42857,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,2,0,0,5,0,0,14,0,0,9,0,0,2,0,0,0,0,0]]}]},{"i":"eaaac964be03cbfd","q":"Let $\\mathbb{Q}$ be the set of rational numbers. Given a rational number $a \\neq 0$, find, with proof, all functions $f: \\mathbb{Q} \\rightarrow \\mathbb{Q}$ satisfying the equation\n\n$$\nf(f(x)+a y)=a f(y)+x\n$$\n\nfor all $x, y \\in \\mathbb{Q}$.","t":[{"b":1,"e":1.0,"k":"flat","v":0.90179,"x":0.99107,"p":[[0,19,0.0,0.92857,0.11845,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,22],[4,19,0.2105,0.90179,0.16146,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,2,0,22],[8,19,0.4211,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[12,19,0.6316,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,19,0.8421,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[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":5,"e":0.71429,"k":"flat","v":0.82142,"x":0.95982,"p":[[0,11,0.0,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[4,11,0.3636,0.9241,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[8,11,0.7273,0.9375,0.18877,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,27],[11,11,1.0,0.82142,0.16753,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,6,0,0,13,0,9]]}]},{"i":"e2c8084fb4f0a9bb","q":"Let  $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$  have rational coefficients and have roots  $r_{1}$ ,  $r_{2}$ , and  $r_{3}$ . If  $r_{1}-r_{2}$  is rational, must  $r_{1}$ ,  $r_{2}$ , and  $r_{3}$  be rational?","t":[{"b":1,"e":0.071,"k":"falling","v":0.35267,"x":0.90179,"p":[[0,26,0.0,0.62498,0.31893,0.28571,0.64286,0.85714,0.0,1.0,2,7,0,2,0,1,0,0,6,0,0,3,0,0,4,0,0,1,0,0,8,0,7],[4,26,0.1538,0.59372,0.37476,0.2857,0.57143,1.0,0.0,1.0,5,10,0,5,0,1,0,0,5,0,0,2,0,0,4,0,0,0,0,0,5,0,10],[8,26,0.3077,0.84822,0.19541,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,12,0,13],[12,26,0.4615,0.90179,0.16536,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,9,0,19],[16,26,0.6154,0.57823,0.3177,0.42857,0.571,0.85714,0.0,1.0,3,7,0,3,0,2,0,0,1,0,0,9,0,0,4,0,0,2,1,0,3,0,7],[20,26,0.7692,0.53347,0.2812,0.28571,0.4286,0.74996,0.14286,1.0,0,4,0,0,0,5,0,1,3,0,0,8,0,0,4,0,0,3,0,0,4,0,4],[24,26,0.9231,0.38384,0.24865,0.14286,0.42857,0.42857,0.0,1.0,1,2,0,1,0,9,0,0,5,0,0,11,0,0,1,0,0,2,0,0,1,0,2],[26,26,1.0,0.35267,0.22582,0.14286,0.42857,0.42857,0.0,1.0,2,2,0,2,0,8,0,0,5,0,0,13,0,0,2,0,0,0,0,0,0,0,2]]},{"b":7,"e":0.571,"k":"falling","v":0.52008,"x":0.85042,"p":[[0,36,0.0,0.67408,0.26544,0.4286,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,5,0,0,6,0,0,4,0,0,5,0,8],[4,36,0.1111,0.61607,0.33586,0.42857,0.57143,1.0,0.0,1.0,3,10,0,3,0,1,0,0,3,0,0,7,0,0,3,0,0,2,0,0,3,0,10],[8,36,0.2222,0.62054,0.38234,0.2857,0.71429,1.0,0.0,1.0,3,14,0,3,0,4,0,0,3,0,0,5,0,0,0,0,0,2,0,0,1,0,14],[12,36,0.3333,0.83033,0.22145,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0,8,0,15],[16,36,0.4444,0.77678,0.26229,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,2,0,0,8,0,13],[20,36,0.5556,0.84374,0.20318,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,5,0,17],[24,36,0.6667,0.85042,0.17893,0.71429,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,1,0,3,0,0,6,0,16],[28,36,0.7778,0.58034,0.27185,0.28571,0.57143,0.71429,0.0,1.0,1,6,0,1,0,0,0,0,8,0,0,4,0,0,6,0,0,6,0,0,1,0,6],[32,36,0.8889,0.55802,0.29744,0.39286,0.57121,0.75,0.0,1.0,2,6,0,2,0,2,0,0,4,0,0,7,0,0,5,0,0,4,0,0,2,0,6],[36,36,1.0,0.52008,0.31855,0.2857,0.42859,0.85714,0.0,1.0,2,5,0,2,0,2,0,0,11,0,0,2,0,1,2,0,0,2,0,0,5,0,5]]}]},{"i":"d867a3fc32ad9cbe","q":"Let  $n$  be a positive integer.\nFind the greatest common divisor of the numbers  $\\binom{2n}{1},\\binom{2n}{3},\\binom{2n}{5},...,\\binom{2n}{2n-1}$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,21,0.0,0.94194,0.12304,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[4,21,0.1905,0.93304,0.12869,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,6,0,23],[8,21,0.381,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,21,0.5714,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[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],[20,21,0.9524,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"flat","v":0.88839,"x":0.99554,"p":[[0,32,0.0,0.88839,0.17029,0.82143,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,3,0,21],[4,32,0.125,0.93304,0.14719,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26],[8,32,0.25,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[12,32,0.375,0.96429,0.10714,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[16,32,0.5,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[20,32,0.625,0.92857,0.15567,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,26],[24,32,0.75,0.96428,0.0945,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[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],[32,32,1.0,0.92411,0.11837,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,22]]}]},{"i":"493f40f9ee05507f","q":"Let  $ABC$  a triangle  with  $AB<AC$  and let  $I$  it\u00b4s incenter. Let  $\\Gamma$  the circumcircle of  $\\triangle  BIC$ .  $AI$  intersect  $\\Gamma$  again in  $P$ . Let  $Q$  a point in side  $AC$  such that  $AB=AQ$  and let  $R$  a point in  $AB$  with  $B$  between  $A$  and  $R$  such that  $AR=AC$ . Prove that  $IQPR$  is cyclic.","t":[{"b":3,"e":0.42857,"k":"flat","v":0.24108,"x":0.43304,"p":[[0,52,0.0,0.38379,0.30827,0.14286,0.42857,0.46525,0.0,1.0,4,3,1,4,0,10,0,0,1,0,0,9,0,0,1,0,0,2,0,0,2,0,3],[4,52,0.0769,0.36161,0.23415,0.14286,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,13,0,0,1,0,0,14,0,0,0,0,0,2,0,0,0,0,2],[8,52,0.1538,0.43304,0.11564,0.42857,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,26,0,0,0,0,0,3,0,0,0,0,0],[12,52,0.2308,0.33929,0.1915,0.14286,0.42857,0.42857,0.0,0.71429,2,0,0,2,0,10,0,0,0,0,0,17,0,0,0,0,0,3,0,0,0,0,0],[16,52,0.3077,0.33929,0.15047,0.14286,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,0,0,0,20,0,0,0,0,0,1,0,0,0,0,0],[20,52,0.3846,0.36161,0.19227,0.14286,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,10,0,0,0,0,0,17,0,0,0,0,0,4,0,0,0,0,0],[24,52,0.4615,0.36607,0.18189,0.14286,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,9,0,0,2,0,0,19,0,0,0,0,0,1,0,0,0,0,1],[28,52,0.5385,0.39286,0.2369,0.14286,0.42857,0.42858,0.0,1.0,1,2,0,1,0,9,0,0,0,0,0,18,0,0,0,0,0,1,0,0,1,0,2],[32,52,0.6154,0.36607,0.15947,0.14289,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,9,0,0,1,0,0,19,0,0,1,0,0,2,0,0,0,0,0],[36,52,0.6923,0.33929,0.14617,0.14289,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,7,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.29009,0.1451,0.14286,0.35714,0.42857,0.0,0.4286,1,0,0,1,0,13,0,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.24108,0.16146,0.14286,0.14286,0.42857,0.0,0.4286,4,0,0,4,0,15,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.41965,0.08703,0.42857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,29,0,0,0,0,0,1,0,0,0,0,0],[52,52,1.0,0.41518,0.04165,0.42857,0.42857,0.42858,0.2857,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.17402,"x":0.42411,"p":[[0,29,0.0,0.42411,0.3668,0.14286,0.1429,0.71429,0.0,1.0,3,7,0,3,0,14,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,7],[4,29,0.1379,0.20965,0.14727,0.14286,0.14286,0.1429,0.0,0.71429,1,0,0,1,0,24,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[8,29,0.2759,0.22768,0.21387,0.14286,0.14286,0.1429,0.0,0.85714,2,0,0,2,0,24,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,0],[12,29,0.4138,0.2142,0.15976,0.14286,0.14286,0.14286,0.14,0.71429,0,0,0,0,0,26,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[16,29,0.5517,0.34367,0.31316,0.14286,0.14286,0.5,0.14,1.0,0,4,0,0,0,21,0,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,4],[20,29,0.6897,0.17411,0.12234,0.14286,0.14286,0.14286,0.0,0.4286,4,0,0,4,0,22,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.17402,0.09936,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,27,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.17411,0.10555,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,25,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.19643,0.18814,0.14286,0.14286,0.14287,0.0,1.0,4,1,0,4,0,22,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"24829b46ccca49d0","q":"In the acute-angled triangle  $ ABC$ ,  $ D$  is the intersection of the altitude passing through  $ A$  with  $ BC$  and  $ I_a$  is the excenter of the triangle with respect to  $ A$ .  $ K$  is a point on the extension of  $ AB$  from  $ B$ , for which  $ \\angle AKI_a\\equal{}90^\\circ\\plus{}\\frac 34\\angle C$ .  $ I_aK$  intersects the extension of  $ AD$  at  $ L$ . Prove that  $ DI_a$  bisects the angle  $ \\angle AI_aB$  iff  $ AL\\equal{}2R$ . ( $ R$  is the circumradius of  $ ABC$ )","t":[{"b":3,"e":0.14286,"k":"flat","v":0.09366,"x":0.22321,"p":[[0,62,0.0,0.14286,0.09449,0.14286,0.14286,0.1429,0.0,0.28571,7,0,1,7,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,62,0.0645,0.21875,0.09438,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,12,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,62,0.129,0.22321,0.10062,0.14286,0.2857,0.28571,0.0,0.42857,3,0,0,3,0,9,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,62,0.1935,0.2008,0.10018,0.14286,0.14286,0.2857,0.0,0.4286,2,0,0,2,0,17,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,62,0.2581,0.18295,0.10252,0.14286,0.14286,0.2857,0.0,0.42857,4,0,0,4,0,16,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,62,0.3226,0.10714,0.10101,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,62,0.3871,0.15152,0.07939,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,22,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,62,0.4516,0.15179,0.08702,0.14286,0.14286,0.1429,0.0,0.28571,5,0,0,5,0,20,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,62,0.5161,0.13839,0.10999,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,16,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,62,0.5806,0.1517,0.07088,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.1517,0.10063,0.14214,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.11152,0.06898,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,62,0.7742,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,62,0.8387,0.12938,0.07455,0.14286,0.14286,0.1429,0.0,0.28571,6,0,0,6,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,62,0.9032,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.11152,0.07768,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[62,62,1.0,0.09376,0.08459,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.1429,"k":"flat","v":0.08036,"x":0.20537,"p":[[0,91,0.0,0.13393,0.10062,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,19,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,91,0.044,0.20537,0.10062,0.14286,0.2857,0.28571,0.0,0.286,4,0,0,4,0,10,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,91,0.0879,0.20536,0.11258,0.14286,0.2857,0.28571,0.0,0.42857,5,0,0,5,0,9,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,91,0.1319,0.18304,0.08917,0.14286,0.14286,0.2857,0.0,0.28571,3,0,0,3,0,17,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,91,0.1758,0.15179,0.10677,0.10714,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,14,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,91,0.2198,0.16063,0.08566,0.14286,0.14286,0.1786,0.0,0.28571,4,0,0,4,0,20,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,91,0.2637,0.16063,0.08566,0.14286,0.14286,0.1786,0.0,0.28571,4,0,0,4,0,20,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,91,0.3077,0.17411,0.09268,0.14286,0.14286,0.2857,0.0,0.42857,3,0,0,3,0,20,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,91,0.3516,0.14277,0.10714,0.105,0.14286,0.1429,0.0,0.42857,8,0,0,8,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,91,0.3956,0.17857,0.09449,0.14286,0.14286,0.2857,0.0,0.42857,3,0,0,3,0,19,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,91,0.4396,0.12501,0.08564,0.10714,0.14286,0.1429,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,91,0.4835,0.12491,0.10563,0.0,0.14286,0.14287,0.0,0.42857,10,0,0,10,0,17,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,91,0.5275,0.12482,0.11149,0.0,0.14286,0.1786,0.0,0.28571,12,0,0,12,0,12,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,91,0.5714,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,91,0.6154,0.10259,0.08913,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,91,0.6593,0.11152,0.07768,0.0,0.14286,0.14286,0.0,0.2857,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,91,0.7033,0.10259,0.08167,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,91,0.7473,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,91,0.7912,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,91,0.8352,0.11143,0.09927,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,91,0.8791,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,91,0.9231,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,91,0.967,0.09823,0.08329,0.0,0.14286,0.1429,0.0,0.28571,12,0,0,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[91,91,1.0,0.09813,0.06616,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7395ef9a48b61e17","q":"Let  $ABC$  be a triangle with  $AB = 3$ ,  $AC = 8$ ,  $BC = 7$  and let  $M$  and  $N$  be the midpoints of  $\\overline{AB}$  and  $\\overline{AC}$ , respectively. Point  $T$  is selected on side  $BC$  so that  $AT = TC$ . The circumcircles of triangles  $BAT$ ,  $MAN$  intersect at  $D$ . Compute  $DC$ 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 $ABCD$  be a cyclic quadrilateral. Let  $P$  be the intersection of  $BC$  and  $AD$ . Line  $AC$  intersects the circumcircle of triangle  $BDP$  in points  $S$  and  $T$ , with  $S$  between  $A$  and  $C$ . Line  $BD$  intersects the circumcircle of triangle  $ACP$  in points  $U$  and  $V$ , with  $U$  between  $B$  and  $D$ . Prove that  $PS$  =  $PT$  =  $PU$  =  $PV$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.07134,"x":0.67847,"p":[[0,38,0.0,0.59364,0.39636,0.25,0.714,1.0,0.0,1.0,3,13,0,3,0,5,0,0,6,0,0,1,0,0,1,0,0,0,0,0,3,0,13],[4,38,0.1053,0.62045,0.46521,0.0,1.0,1.0,0.0,1.0,9,19,0,9,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19],[8,38,0.2105,0.59366,0.42434,0.14286,0.78571,1.0,0.0,1.0,6,14,0,6,0,5,0,0,1,0,0,2,0,0,0,0,0,2,0,0,2,0,14],[12,38,0.3158,0.67847,0.41508,0.2857,1.0,1.0,0.0,1.0,6,19,0,6,0,1,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,19],[16,38,0.4211,0.59821,0.45237,0.14286,1.0,1.0,0.0,1.0,7,17,0,7,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,17],[20,38,0.5263,0.35266,0.42705,0.0,0.14286,1.0,0.0,1.0,15,9,0,15,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,9],[24,38,0.6316,0.5714,0.38795,0.25,0.4998,1.0,0.0,1.0,4,12,0,4,0,4,0,0,4,0,0,4,0,0,2,0,0,0,0,0,2,0,12],[28,38,0.7368,0.62491,0.43126,0.14286,1.0,1.0,0.0,1.0,6,17,0,6,0,5,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,17],[32,38,0.8421,0.25437,0.28958,0.0,0.1429,0.2857,0.0,1.0,10,3,0,10,0,8,0,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,3],[36,38,0.9474,0.19633,0.31084,0.0,0.0,0.2857,0.0,1.0,18,3,0,18,0,4,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[38,38,1.0,0.07134,0.10708,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.11152,"x":0.74545,"p":[[0,46,0.0,0.45981,0.42368,0.0,0.35714,1.0,0.0,1.0,11,10,1,11,0,2,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0,0,10],[4,46,0.087,0.74545,0.38766,0.53571,1.0,1.0,0.0,1.0,4,21,0,4,0,3,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,21],[8,46,0.1739,0.67409,0.42142,0.35714,1.0,1.0,0.0,1.0,7,19,0,7,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,19],[12,46,0.2609,0.64283,0.42559,0.14289,1.0,1.0,0.0,1.0,7,17,0,7,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,0,1,0,17],[16,46,0.3478,0.5625,0.41945,0.14286,0.71429,1.0,0.0,1.0,7,13,0,7,0,3,0,0,4,0,0,0,0,0,1,0,0,4,0,0,0,0,13],[20,46,0.4348,0.55804,0.45507,0.0,0.7143,1.0,0.0,1.0,9,16,0,9,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,16],[24,46,0.5217,0.45979,0.44281,0.0,0.28571,1.0,0.0,1.0,13,11,0,13,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,1,0,11],[28,46,0.6087,0.51339,0.45296,0.0,0.35716,1.0,0.0,1.0,10,14,0,10,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,14],[32,46,0.6957,0.39275,0.3712,0.14214,0.28571,0.6425,0.0,1.0,7,7,0,7,0,7,0,0,5,0,0,4,0,0,1,0,0,0,0,0,1,0,7],[36,46,0.7826,0.23206,0.3585,0.0,0.0,0.2857,0.0,1.0,18,4,0,18,0,5,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,4],[40,46,0.8696,0.22322,0.28333,0.0,0.14286,0.28571,0.0,1.0,10,3,0,10,0,12,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[44,46,0.9565,0.2365,0.26873,0.0,0.14286,0.2857,0.0,1.0,11,2,0,11,0,7,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[46,46,1.0,0.11152,0.19798,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,9,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0]]}]},{"i":"de7d5401324fdebb","q":"In a sports league, each team uses a set of at most  $t$  signature colors. A set  $S$  of teams is*color-identifiable* if one can assign each team in  $S$  one of their signature colors, such that no team in  $S$  is assigned any signature color of a different team in  $S$ .\n\nFor all positive integers  $n$  and  $t$ , determine the maximum integer  $g(n, t)$  such that: In any sports league with exactly  $n$  distinct colors present over all teams, one can always find a color-identifiable set of size at least  $g(n, t)$ .","t":[{"b":3,"e":0.57143,"k":"rising","v":0.38393,"x":0.64731,"p":[[0,24,0.0,0.38393,0.24598,0.14286,0.42857,0.57143,0.0,1.0,4,1,3,4,0,7,0,0,0,0,0,11,0,0,6,0,0,3,0,0,0,0,1],[4,24,0.1667,0.44195,0.29528,0.10714,0.57121,0.60714,0.0,0.85714,8,0,0,8,0,1,0,0,0,0,0,6,0,0,9,0,0,4,0,0,4,0,0],[8,24,0.3333,0.64731,0.23142,0.57143,0.71429,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,0,0,0,3,0,0,10,0,0,10,0,0,3,0,4],[12,24,0.5,0.63836,0.20199,0.57132,0.57143,0.85704,0.0,1.0,1,1,0,1,0,0,0,0,1,0,0,4,0,0,11,0,0,6,0,0,8,0,1],[16,24,0.6667,0.60712,0.22588,0.571,0.57143,0.75,0.0,1.0,2,2,0,2,0,0,0,0,0,0,0,5,0,0,14,0,0,3,0,0,6,0,2],[20,24,0.8333,0.61155,0.11427,0.57142,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,20,0,0,7,0,0,1,0,1],[24,24,1.0,0.63389,0.19213,0.57143,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,1,0,0,21,0,0,3,0,0,2,0,4]]},{"b":7,"e":0.85714,"k":"rising","v":0.08034,"x":0.68748,"p":[[0,46,0.0,0.48214,0.25692,0.42857,0.42857,0.57143,0.0,1.0,1,5,1,1,0,3,0,0,3,0,0,16,0,0,4,0,0,0,0,0,0,0,5],[4,46,0.087,0.44642,0.32488,0.0,0.57143,0.71429,0.0,1.0,9,1,0,9,0,1,0,0,1,0,0,3,0,0,8,0,0,5,0,0,4,0,1],[8,46,0.1739,0.50444,0.26722,0.42857,0.57143,0.71429,0.0,1.0,5,1,0,5,0,1,0,0,1,0,0,3,0,0,13,0,0,6,0,0,2,0,1],[12,46,0.2609,0.45981,0.31488,0.10714,0.57143,0.71429,0.0,1.0,8,1,0,8,0,1,0,0,1,0,0,4,0,0,8,0,0,5,0,0,4,0,1],[16,46,0.3478,0.31695,0.30873,0.0,0.35714,0.57143,0.0,0.85714,14,0,0,14,0,0,0,0,2,0,0,5,0,0,6,0,0,2,0,0,3,0,0],[20,46,0.4348,0.36606,0.34798,0.0,0.42857,0.60707,0.0,1.0,12,2,0,12,0,3,0,0,0,0,0,4,0,0,5,0,0,3,0,0,3,0,2],[24,46,0.5217,0.33929,0.29397,0.0,0.35714,0.57143,0.0,0.85714,11,0,0,11,0,1,0,0,4,0,0,6,0,0,3,0,0,5,0,0,2,0,0],[28,46,0.6087,0.08929,0.23623,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[32,46,0.6957,0.08034,0.19209,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[36,46,0.7826,0.14732,0.28678,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0],[40,46,0.8696,0.17411,0.28956,0.0,0.0,0.32143,0.0,1.0,22,1,0,22,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,1],[44,46,0.9565,0.54908,0.25281,0.42857,0.64286,0.71429,0.0,0.85714,3,0,0,3,0,1,0,0,3,0,0,4,0,0,5,0,0,12,0,0,4,0,0],[46,46,1.0,0.68748,0.16536,0.57143,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,13,0,0,10,0,0]]}]},{"i":"f8b477a139e89864","q":"A real number is written next to each vertex of a regular pentagon. All five numbers are different. A triple of vertices is called **successful** if they form an isosceles triangle for which the number written on the top vertex is either larger than both numbers written on the base vertices, or smaller than both. Find the maximum possible number of successful 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,0,0,0,0,0,2,0,0,6,0,0,5,0,0,5,0,14],[220,238,0.9244,0.87051,0.16892,0.82143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,7,0,17],[224,238,0.9412,0.88839,0.14167,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,2,0,19],[228,238,0.958,0.89731,0.11426,0.857,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,16],[232,238,0.9748,0.88392,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,18],[236,238,0.9916,0.90612,0.13195,0.82143,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,20],[238,238,1.0,0.92857,0.10102,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,20]]}]},{"i":"a9820afd2c1d9e80","q":"Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations\n\n$$\np x^{2}+2 q x+r=0, \\quad q x^{2}+2 r x+p=0, \\quad r x^{2}+2 p x+q=0\n$$\n\nhave a common root, say $\\alpha$. Prove that\n\n(a) $\\alpha$ is real and negative; and\n\n(b) the third equation has non-real roots.","t":[{"b":0,"e":1.0,"k":"flat","v":0.91071,"x":0.96428,"p":[[0,12,0.0,0.94195,0.07018,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[4,12,0.3333,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[8,12,0.6667,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[12,12,1.0,0.91964,0.09407,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,16]]},{"b":3,"e":0.85714,"k":"flat","v":0.91071,"x":0.96875,"p":[[0,9,0.0,0.91071,0.18123,0.85714,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,19],[4,9,0.4444,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[8,9,0.8889,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[9,9,1.0,0.94196,0.06786,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,2,18]]}]},{"i":"b4607c54c32f0e94","q":"7. (GBR) Let $A B C$ be a triangle such that $\\angle A C B=2 \\angle A B C$. Let $D$ be the point on the side $B C$ such that $C D=2 B D$. The segment $A D$ is extended to $E$ so that $A D=D E$. Prove that  $$ \\angle E C B+180^{\\circ}=2 \\angle E B C . $$","t":[{"b":0,"e":0.57143,"k":"rising","v":0.35265,"x":0.67857,"p":[[0,29,0.0,0.35265,0.24737,0.14286,0.2857,0.46418,0.0,1.0,1,2,0,1,0,11,0,0,8,0,0,4,0,0,4,0,0,2,0,0,0,0,2],[4,29,0.1379,0.37499,0.33834,0.14286,0.28571,0.57143,0.0,1.0,7,5,0,7,0,6,0,0,6,0,0,3,0,0,3,0,0,2,0,0,0,0,5],[8,29,0.2759,0.56249,0.31529,0.28571,0.57121,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,7,0,0,4,0,0,6,0,0,1,0,0,2,0,8],[12,29,0.4138,0.54892,0.28397,0.28571,0.57143,0.71429,0.0,1.0,1,5,0,1,0,3,0,0,5,0,0,6,0,0,5,0,0,5,0,0,2,0,5],[16,29,0.5517,0.56247,0.30915,0.2857,0.57143,0.857,0.14286,1.0,0,5,0,0,0,7,0,0,4,0,0,2,0,0,5,0,0,4,0,0,5,0,5],[20,29,0.6897,0.62041,0.30656,0.28571,0.57143,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,5,0,0,0,0,0,8,0,0,4,0,0,3,0,8],[24,29,0.8276,0.55357,0.28959,0.28571,0.64286,0.71429,0.0,1.0,2,4,0,2,0,2,0,0,6,0,0,4,0,0,2,0,0,10,0,0,2,0,4],[28,29,0.9655,0.60699,0.29662,0.28571,0.64071,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,7,0,0,4,0,0,3,0,0,7,0,0,1,0,8],[29,29,1.0,0.67857,0.25753,0.57142,0.71429,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,2,0,0,2,0,0,7,0,0,8,0,0,4,0,7]]},{"b":4,"e":0.14286,"k":"falling","v":0.12946,"x":0.37498,"p":[[0,11,0.0,0.37498,0.27374,0.14286,0.2857,0.57143,0.0,1.0,3,2,0,3,0,8,0,0,8,0,0,3,0,0,3,0,0,5,0,0,0,0,2],[4,11,0.3636,0.15179,0.11812,0.14286,0.14286,0.14286,0.0,0.4286,6,0,0,6,0,22,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.12946,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.16518,0.12931,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,23,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"537730ecdb6b5a23","q":"Find all natural numbers  $m$  such that\n\\[1! \\cdot 3! \\cdot 5! \\cdots (2m-1)! = \\biggl( \\frac{m(m+1)}{2}\\biggr) !.\\]","t":[{"b":6,"e":0.42857,"k":"flat","v":0.37946,"x":0.68303,"p":[[0,26,0.0,0.50446,0.2525,0.42857,0.42857,0.42857,0.0,1.0,1,6,0,1,0,0,0,0,4,0,0,21,0,0,0,0,0,0,0,0,0,0,6],[4,26,0.1538,0.37946,0.1411,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,11,0,0,18,0,0,0,0,0,0,0,0,0,0,1],[8,26,0.3077,0.40625,0.17536,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,9,0,0,19,0,0,0,0,0,0,0,0,0,0,2],[12,26,0.4615,0.60266,0.25688,0.42857,0.42859,0.85704,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,14,0,0,3,0,0,3,0,0,2,0,7],[16,26,0.6154,0.68303,0.26422,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,11,0,0,1,0,0,4,0,0,5,0,9],[20,26,0.7692,0.62499,0.26184,0.42857,0.42859,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,14,0,0,2,0,0,2,0,0,3,0,8],[24,26,0.9231,0.50445,0.17851,0.42857,0.42857,0.42858,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,24,0,0,2,0,0,2,0,0,0,0,3],[26,26,1.0,0.49108,0.23941,0.42857,0.42857,0.42857,0.14286,1.0,0,5,0,0,0,2,0,0,4,0,0,20,0,0,0,0,0,1,0,0,0,0,5]]},{"b":7,"e":0.42857,"k":"flat","v":0.33482,"x":0.54911,"p":[[0,21,0.0,0.54911,0.25532,0.42857,0.42857,0.53574,0.28571,1.0,0,7,0,0,0,0,0,0,4,0,0,20,0,0,0,0,0,0,0,0,1,0,7],[4,21,0.1905,0.44643,0.21651,0.42857,0.42857,0.42857,0.0,1.0,2,3,1,2,0,0,0,0,4,0,0,22,0,0,0,0,0,1,0,0,0,0,3],[8,21,0.381,0.50893,0.29653,0.28571,0.42857,0.57145,0.14286,1.0,0,8,0,0,0,3,0,0,8,0,0,13,0,0,0,0,0,0,0,0,0,0,8],[12,21,0.5714,0.5,0.22016,0.42857,0.42857,0.42857,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,23,0,0,0,0,0,0,0,0,0,0,5],[16,21,0.7619,0.33482,0.15815,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,13,0,0,12,0,0,0,0,0,0,0,0,0,0,1],[20,21,0.9524,0.37054,0.08645,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,9,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.40625,0.05187,0.42857,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5cf9d9a70697c87d","q":"Consider two points $A\\left(x_{1}, y_{1}\\right)$ and $B\\left(x_{2}, y_{2}\\right)$ on the graph of the function $y=\\frac{1}{x}$ such that $0<x_{1}<x_{2}$ and $|A B|=2 \\cdot|O A|(O$ is the reference point, i.e., $O(0,0))$. Let $C$ be the midpoint of the segment $A B$. Prove that the angle between the $x$-axis and the ray $O A$ is equal to three times the angle between $x$-axis and the ray $O C$.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.06697,"x":0.14509,"p":[[0,52,0.0,0.14063,0.18078,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,15,0,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,52,0.0769,0.12054,0.17536,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,52,0.1538,0.12054,0.17536,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,52,0.2308,0.09821,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,52,0.3077,0.14509,0.16797,0.14286,0.14286,0.14286,0.0,1.0,7,1,0,7,0,22,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,52,0.3846,0.09152,0.07625,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,1,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.09152,0.13151,0.0,0.10714,0.14286,0.0,0.71429,15,0,0,15,1,15,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,52,0.5385,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,52,0.6154,0.09822,0.09062,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.07366,0.07882,0.0,0.03571,0.14286,0.0,0.2857,16,0,0,16,1,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.10705,0.17495,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,52,0.8462,0.06697,0.07973,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.06697,0.07973,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.08036,0.08702,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.04241,"x":0.16071,"p":[[0,32,0.0,0.125,0.15465,0.0,0.14286,0.14286,0.0,0.85714,11,0,0,11,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,32,0.125,0.16071,0.23623,0.0,0.14286,0.14286,0.0,1.0,12,2,0,12,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,32,0.25,0.10937,0.0778,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,1,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.10705,0.07982,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.09822,0.07524,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.0759,0.07974,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.04241,0.0615,0.0,0.0,0.08929,0.0,0.1429,21,0,0,21,3,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6b92381bab8050a1","q":"Let  $ a,b,c\\in\\mathbb{R}_+^*, $  and  $ f:[0,a]\\longrightarrow [0,b] $  bijective and non-decreasing. Prove that:  $$  \\frac{1}{b}\\int_0^a f^2 (x)dx +\\frac{1}{a}\\int_0^b \\left( f^{-1} (x)\\right)^2dx\\le ab.  $$","t":[{"b":0,"e":1.0,"k":"rising","v":0.36607,"x":1.0,"p":[[0,36,0.0,0.36607,0.36235,0.0,0.28571,0.5,0.0,1.0,10,6,0,10,0,3,0,0,6,0,0,5,0,0,0,0,0,2,0,0,0,0,6],[4,36,0.1111,0.52232,0.39059,0.14286,0.28571,1.0,0.0,1.0,4,11,0,4,0,5,0,0,8,0,0,0,0,0,1,0,0,3,0,0,0,0,11],[8,36,0.2222,0.67411,0.38834,0.28571,1.0,1.0,0.0,1.0,4,17,0,4,0,2,0,0,3,0,0,3,0,0,0,0,0,3,0,0,0,0,17],[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],[36,36,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":0.0,"k":"falling","v":0.02232,"x":0.46427,"p":[[0,21,0.0,0.3883,0.35404,0.14286,0.2857,0.71429,0.0,1.0,6,6,0,6,0,8,0,0,5,0,0,4,0,0,0,0,0,3,0,0,0,0,6],[4,21,0.1905,0.25,0.30929,0.0,0.14286,0.32143,0.0,1.0,12,3,0,12,0,8,0,0,4,0,0,3,0,0,0,0,0,2,0,0,0,0,3],[8,21,0.381,0.36161,0.34439,0.10714,0.28571,0.50002,0.0,1.0,8,5,0,8,0,6,0,0,5,0,0,5,0,0,0,0,0,3,0,0,0,0,5],[12,21,0.5714,0.46427,0.30514,0.10717,0.71429,0.71429,0.0,0.71429,8,0,0,8,0,1,0,0,2,0,0,2,0,0,2,0,0,17,0,0,0,0,0],[16,21,0.7619,0.04911,0.1411,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,21,0.9524,0.04018,0.1439,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[21,21,1.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,1,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"a421c91421ce8f7b","q":"$\\$$ indy has $\\$ 100$ in pennies (worth $\\$ 0.01$ each), nickels (worth $\\$ 0.05$ each), dimes (worth $\\$ 0.10$ each), and quarters (worth $\\$ 0.25$ each). Prove that she can split her coins into two piles, each with total value exactly $\\$ 50$.","t":[{"b":0,"e":0.71429,"k":"rising","v":0.41963,"x":0.683,"p":[[0,5,0.0,0.41963,0.24983,0.14286,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,11,0,0,3,0,0,6,0,0,3,0,0,8,0,0,0,0,1],[4,5,0.8,0.44197,0.23517,0.14286,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,10,0,0,1,0,0,8,0,0,2,0,0,11,0,0,0,0,0],[5,5,1.0,0.683,0.1703,0.57132,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,15,0,0,2,0,4]]},{"b":4,"e":0.0,"k":"falling","v":0.18295,"x":0.39731,"p":[[0,26,0.0,0.39731,0.25933,0.14286,0.28571,0.71429,0.14286,1.0,0,1,0,0,0,12,0,0,5,0,0,5,0,0,1,0,0,7,0,0,1,0,1],[4,26,0.1538,0.3616,0.24479,0.14286,0.42857,0.4286,0.0,1.0,2,1,0,2,0,11,0,0,2,0,0,10,0,0,1,0,0,5,0,0,0,0,1],[8,26,0.3077,0.35715,0.22588,0.14286,0.42857,0.4286,0.0,0.71429,3,0,0,3,0,8,0,0,4,0,0,10,0,0,1,0,0,6,0,0,0,0,0],[12,26,0.4615,0.29462,0.19861,0.14286,0.2857,0.42857,0.0,0.71429,3,0,0,3,0,11,0,0,7,0,0,5,0,0,4,0,0,2,0,0,0,0,0],[16,26,0.6154,0.30348,0.2106,0.14286,0.2857,0.42857,0.0,1.0,1,1,0,1,0,14,0,0,6,0,0,6,0,0,3,0,0,1,0,0,0,0,1],[20,26,0.7692,0.18295,0.12996,0.14286,0.14286,0.2857,0.0,0.4286,5,0,0,5,0,18,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.19196,0.15403,0.14286,0.14286,0.28571,0.0,0.571,6,0,0,6,0,17,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[26,26,1.0,0.23185,0.2106,0.14214,0.14286,0.32143,0.0,0.85714,5,0,0,5,0,17,0,0,2,0,0,4,0,0,2,0,0,1,0,0,1,0,0]]}]},{"i":"19aeb5cdbbad433b","q":"2. (CZS 1) We are given $3 n$ points $A_{1}, A_{2}, \\ldots, A_{3 n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_{i}$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.10268,"x":0.34375,"p":[[0,18,0.0,0.34375,0.35689,0.0,0.42857,0.71429,0.0,1.0,15,2,0,15,0,0,0,0,0,0,0,7,0,0,1,0,0,4,0,0,3,0,2],[4,18,0.2222,0.10268,0.1525,0.0,0.0,0.2857,0.0,0.5714,21,0,0,21,0,1,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,18,0.4444,0.2366,0.10479,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.23214,0.08564,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,8,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.24107,0.09061,0.2857,0.2857,0.28571,0.0,0.28571,3,0,0,3,0,4,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.22767,0.11214,0.2857,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.4286,"k":"rising","v":0.24106,"x":0.49106,"p":[[0,19,0.0,0.24106,0.31223,0.0,0.0,0.42857,0.0,1.0,17,2,0,17,0,2,0,0,2,0,0,5,0,0,1,0,0,3,0,0,0,0,2],[4,19,0.2105,0.43295,0.20368,0.42857,0.42857,0.46431,0.0,0.85714,3,0,0,3,0,1,0,0,3,0,0,17,0,0,2,0,0,5,0,0,1,0,0],[8,19,0.4211,0.49106,0.19212,0.42857,0.42857,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,3,0,0,14,0,0,3,0,0,10,0,0,0,0,0],[12,19,0.6316,0.40625,0.23176,0.2857,0.42857,0.50002,0.0,0.71429,5,0,0,5,0,1,0,0,4,0,0,14,0,0,0,0,0,8,0,0,0,0,0],[16,19,0.8421,0.45084,0.13878,0.42857,0.42857,0.4642,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,19,0,0,4,0,0,4,0,0,0,0,0],[19,19,1.0,0.45984,0.15863,0.42857,0.42857,0.571,0.0,0.71429,1,0,0,1,0,0,0,0,5,0,0,17,0,0,3,0,0,6,0,0,0,0,0]]}]},{"i":"e7914bad25443406","q":"Let  $ ABC$  be a triangle, and let  $ X$ ,  $ Y$ ,  $ Z$  be three points on the segments  $ BC$ ,  $ CA$ ,  $ AB$ , respectively. Denote by  $ X^{\\prime}$ ,  $ Y^{\\prime}$ ,  $ Z^{\\prime}$  the reflections of these points  $ X$ ,  $ Y$ ,  $ Z$  in the midpoints of the segments  $ BC$ ,  $ CA$ ,  $ AB$ , respectively. Prove that  $ \\left\\vert XYZ\\right\\vert \\equal{}\\left\\vert X^{\\prime}Y^{\\prime}Z^{\\prime}\\right\\vert$ .","t":[{"b":0,"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],[4,42,0.0952,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,42,0.1905,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,42,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,42,0.4762,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,42,0.5714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,42,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]},{"b":4,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,19,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,19,0.2105,0.94643,0.18472,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,28],[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":"518ee836faae1538","q":"In triangle  $ABC$ ,  $AB = 28$ ,  $AC = 36$ , and  $BC = 32$ . Let  $D$  be the point on segment  $BC$  satisfying  $\\angle BAD = \\angle DAC$ , and let  $E$  be the unique point such that  $DE \\parallel AB$  and line  $AE$  is tangent to the circumcircle of  $ABC$ . Find the length of segment  $AE$ .\n\n*Ray Li*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.30339,"x":0.41518,"p":[[0,51,0.0,0.41064,0.18826,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,8,0,0,8,0,0,7,0,0,4,0,0,0,0,0],[4,51,0.0784,0.41518,0.17261,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,12,0,0,8,0,0,6,0,0,4,0,0,0,0,0],[8,51,0.1569,0.31696,0.17399,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,8,0,0,9,0,0,8,0,0,4,0,0,1,0,0,0,0,0],[12,51,0.2353,0.37055,0.17445,0.28571,0.28586,0.4286,0.0,0.71429,1,0,0,1,0,4,0,0,12,0,0,8,0,0,4,0,0,3,0,0,0,0,0],[16,51,0.3137,0.36607,0.19865,0.24999,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,6,0,0,9,0,0,4,0,0,9,0,0,2,0,0,0,0,0],[20,51,0.3922,0.30339,0.17422,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,10,0,0,8,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[24,51,0.4706,0.39284,0.1675,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,5,0,0,10,0,0,7,0,0,8,0,0,2,0,0,0,0,0],[28,51,0.549,0.41071,0.14617,0.28571,0.42857,0.57141,0.0,0.71429,1,0,0,1,0,1,0,0,9,0,0,12,0,0,8,0,0,1,0,0,0,0,0],[32,51,0.6275,0.375,0.12242,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,11,0,0,15,0,0,4,0,0,0,0,0,0,0,0],[36,51,0.7059,0.38393,0.15746,0.28571,0.42857,0.42858,0.0,0.71429,2,0,0,2,0,1,0,0,10,0,0,12,0,0,6,0,0,1,0,0,0,0,0],[40,51,0.7843,0.35714,0.12877,0.28571,0.35714,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,13,0,0,12,0,0,4,0,0,0,0,0,0,0,0],[44,51,0.8627,0.39286,0.11845,0.28571,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,3,0,0,7,0,0,17,0,0,5,0,0,0,0,0,0,0,0],[48,51,0.9412,0.40625,0.12931,0.28571,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,2,0,0,9,0,0,14,0,0,6,0,0,1,0,0,0,0,0],[51,51,1.0,0.37499,0.13241,0.28571,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,13,0,0,9,0,0,7,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.33928,"x":0.37052,"p":[[0,20,0.0,0.35268,0.17491,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,7,0,0,10,0,0,11,0,0,3,0,0,0,0,0,0,0,1],[4,20,0.2,0.35713,0.17126,0.25,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,8,0,0,9,0,0,8,0,0,5,0,0,2,0,0,0,0,0],[8,20,0.4,0.3616,0.12867,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,9,0,0,14,0,0,4,0,0,0,0,0,0,0,0],[12,20,0.6,0.36152,0.1238,0.28571,0.28571,0.42857,0.14,0.57143,0,0,0,0,0,3,0,0,14,0,0,10,0,0,5,0,0,0,0,0,0,0,0],[16,20,0.8,0.33928,0.13243,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,3,0,0,14,0,0,12,0,0,1,0,0,1,0,0,0,0,0],[20,20,1.0,0.37052,0.12805,0.2857,0.42857,0.4286,0.14286,0.57143,0,0,0,0,0,4,0,0,10,0,0,13,0,0,5,0,0,0,0,0,0,0,0]]}]},{"i":"f4b7b5d4bbb15efd","q":"Let $n$ be an odd natural number and $x$ and $y$ be two rational numbers such that\n\n$$\nx^{n}+2 y=y^{n}+2 x\n$$\n\nShow that $x=y$.","t":[{"b":4,"e":0.28571,"k":"falling","v":0.33928,"x":0.71874,"p":[[0,53,0.0,0.71429,0.30723,0.42857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,3,0,0,3,0,0,4,0,0,1,0,0,4,0,0,4,0,13],[4,53,0.0755,0.71429,0.23146,0.64286,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,5,0,0,0,0,0,14,0,0,1,0,9],[8,53,0.1509,0.67411,0.24546,0.42857,0.71429,0.78571,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,7,0,0,3,0,0,12,0,0,0,0,8],[12,53,0.2264,0.66518,0.22191,0.42859,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,7,0,0,3,0,0,13,0,0,1,0,6],[16,53,0.3019,0.62946,0.24448,0.42857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,9,0,0,0,0,0,9,0,0,4,0,5],[20,53,0.3774,0.61607,0.26351,0.42857,0.71429,0.75,0.0,1.0,1,6,0,1,0,0,0,0,4,0,0,9,0,0,1,0,0,9,0,0,2,0,6],[24,53,0.4528,0.60268,0.21939,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,10,0,0,4,0,0,10,0,0,1,0,4],[28,53,0.5283,0.71874,0.24611,0.42857,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,8,0,0,0,0,12],[32,53,0.6038,0.58036,0.27184,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,3,0,0,9,0,0,2,0,0,8,0,0,2,0,5],[36,53,0.6792,0.66071,0.27375,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,5,0,0,6,0,0,0,0,0,10,0,0,1,0,9],[40,53,0.7547,0.6875,0.23808,0.42857,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,1,0,0,8,0,0,0,0,0,13,0,0,1,0,8],[44,53,0.8302,0.58927,0.23077,0.39286,0.64286,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,8,0,0,4,0,0,4,0,0,12,0,0,0,0,4],[48,53,0.9057,0.48214,0.18123,0.39286,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,14,0,0,3,0,0,5,0,0,1,0,1],[52,53,0.9811,0.33928,0.15047,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,4,0,0,16,0,0,11,0,0,0,0,0,0,0,0,0,0,1],[53,53,1.0,0.37053,0.13296,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,17,0,0,10,0,0,3,0,0,0,0,0,1,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.15179,"x":0.82143,"p":[[0,47,0.0,0.82143,0.26,0.71429,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,5,0,0,2,0,19],[4,47,0.0851,0.79018,0.2474,0.71429,0.78571,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,11,0,0,1,0,15],[8,47,0.1702,0.65625,0.28316,0.42857,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,2,0,0,8,0,0,5,0,0,4,0,0,0,0,11],[12,47,0.2553,0.70089,0.27976,0.42857,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,4,0,0,4,0,0,1,0,0,10,0,0,1,0,11],[16,47,0.3404,0.75446,0.1931,0.67857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,12,0,0,3,0,9],[20,47,0.4255,0.76339,0.21902,0.71429,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,15,0,0,1,0,11],[24,47,0.5106,0.66963,0.21853,0.53539,0.71429,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,4,0,0,3,0,0,13,0,0,3,0,5],[28,47,0.5957,0.69643,0.26184,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,2,0,0,7,0,0,2,0,0,9,0,0,0,0,11],[32,47,0.6809,0.67857,0.23145,0.57143,0.71429,0.74996,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,4,0,0,3,0,0,14,0,0,2,0,6],[36,47,0.766,0.30804,0.23449,0.14286,0.14286,0.42857,0.14286,0.85714,0,0,0,0,0,20,0,0,0,0,0,6,0,0,0,0,0,5,0,0,1,0,0],[40,47,0.8511,0.25,0.19885,0.14286,0.14286,0.21432,0.14286,0.71429,0,0,0,0,0,24,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0],[44,47,0.9362,0.29911,0.29744,0.14286,0.14286,0.21429,0.0,1.0,1,3,0,1,0,23,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,3],[47,47,1.0,0.15179,0.15947,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"8782026a00169098","q":"At a certain orphanage, every pair of orphans are either friends or enemies. For every three of an orphan's friends, an even number of pairs of them are enemies. Prove that it's possible to assign each orphan two parents such that every pair of friends shares exactly one parent, but no pair of enemies does, and no three parents are in a love triangle (where each pair of them has a child).","t":[{"b":6,"e":1.0,"k":"flat","v":0.66963,"x":0.79467,"p":[[0,28,0.0,0.70536,0.28333,0.57143,0.71429,0.89286,0.0,1.0,1,8,0,1,0,3,0,0,1,0,0,0,0,0,5,0,0,7,0,0,7,0,8],[4,28,0.1429,0.79467,0.24979,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,1,0,0,3,0,0,3,0,0,10,0,12],[8,28,0.2857,0.74107,0.27764,0.57143,0.85714,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,0,0,0,2,0,0,4,0,0,3,0,0,12,0,8],[12,28,0.4286,0.77677,0.30918,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,0,5,0,17],[16,28,0.5714,0.66963,0.34707,0.53539,0.85714,0.89286,0.0,1.0,4,8,0,4,0,2,0,0,1,0,0,1,0,0,4,0,0,1,0,0,11,0,8],[20,28,0.7143,0.73212,0.26428,0.5713,0.85714,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,0,3,0,0,4,0,0,5,0,0,8,0,9],[24,28,0.8571,0.71869,0.27779,0.57132,0.78571,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,2,0,0,1,0,0,5,0,0,5,0,0,6,0,10],[28,28,1.0,0.73661,0.23719,0.57143,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,3,0,0,8,0,9]]},{"b":7,"e":0.85714,"k":"flat","v":0.60712,"x":0.80357,"p":[[0,27,0.0,0.80357,0.23077,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,6,0,0,9,0,12],[4,27,0.1481,0.74107,0.27301,0.57143,0.85707,1.0,0.0,1.0,2,10,0,2,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,7,0,10],[8,27,0.2963,0.68302,0.21939,0.5354,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,5,0,0,3,0,0,12,0,0,3,0,6],[12,27,0.4444,0.63393,0.2878,0.42857,0.71429,0.85714,0.0,1.0,2,4,0,2,0,2,0,0,1,0,0,5,0,0,4,0,0,5,0,0,9,0,4],[16,27,0.5926,0.64732,0.3499,0.39286,0.78564,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,3,0,0,1,0,0,3,0,0,4,0,0,8,0,8],[20,27,0.7407,0.60712,0.30723,0.49968,0.71429,0.85714,0.0,1.0,4,4,0,4,0,0,0,0,4,0,0,0,0,0,7,0,0,6,0,0,7,0,4],[24,27,0.8889,0.70075,0.19019,0.57143,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,2,0,0,7,0,0,7,0,0,13,0,1],[27,27,1.0,0.67853,0.2143,0.571,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,5,0,0,3,0,0,11,0,0,9,0,2]]}]},{"i":"d586d5d2a212f19b","q":"For any polynomial  $P(x)=a_0+a_1x+\\ldots+a_kx^k$  with integer coefficients, the number of odd coefficients is denoted by  $o(P)$ . For  $i-0,1,2,\\ldots$  let  $Q_i(x)=(1+x)^i$ . Prove that if  $i_1,i_2,\\ldots,i_n$  are integers satisfying  $0\\le i_1<i_2<\\ldots<i_n$ , then: \\[ o(Q_{i_1}+Q_{i_2}+\\ldots+Q_{i_n})\\ge o(Q_{i_1}).  \\]","t":[{"b":2,"e":0.28571,"k":"flat","v":0.20982,"x":0.28561,"p":[[0,43,0.0,0.22768,0.09354,0.14286,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,10,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.24553,0.22084,0.14286,0.14286,0.2857,0.0,1.0,1,1,0,1,0,21,0,0,6,0,0,0,0,0,1,0,0,1,0,0,1,0,1],[8,43,0.186,0.23214,0.15872,0.14286,0.21428,0.28571,0.0,1.0,1,1,0,1,0,15,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,43,0.2791,0.27232,0.19678,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,15,0,0,13,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[16,43,0.3721,0.28561,0.24747,0.14286,0.14286,0.28571,0.14,1.0,0,3,0,0,0,17,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[20,43,0.4651,0.23674,0.11639,0.14286,0.2857,0.28571,0.14286,0.71429,0,0,0,0,0,15,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,43,0.5581,0.23214,0.09942,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,12,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,43,0.6512,0.20982,0.10092,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,17,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,43,0.7442,0.25,0.07986,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,10,0,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.24999,0.09445,0.14286,0.28571,0.28571,0.14286,0.571,0,0,0,0,0,11,0,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,43,0.9302,0.23214,0.06916,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.21428,0.07143,0.14286,0.21428,0.28571,0.14286,0.28571,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.14732,"x":0.29464,"p":[[0,29,0.0,0.1875,0.0974,0.14286,0.14286,0.28571,0.0,0.28571,4,0,0,4,0,14,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.14732,0.11564,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,22,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,29,0.2759,0.20982,0.1636,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,21,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,29,0.4138,0.29464,0.20182,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,12,0,0,15,0,0,2,0,0,0,0,0,1,0,0,1,0,1],[16,29,0.5517,0.17857,0.06186,0.14286,0.14286,0.17857,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.21419,0.15976,0.14286,0.14286,0.2857,0.14,1.0,0,1,0,0,0,22,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[24,29,0.8276,0.24999,0.14722,0.14286,0.2857,0.28571,0.14286,0.85714,0,0,0,0,0,15,0,0,14,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[28,29,0.9655,0.19196,0.07668,0.14286,0.14286,0.2857,0.14286,0.4286,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.19634,0.06923,0.14286,0.14286,0.28571,0.14,0.28571,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fbe79503f2c10eb3","q":"Let  $BD$  be an altitude of  $\\triangle ABC$  with  $AB < BC$  and  $\\angle B > 90^\\circ$ . Let  $M$  be the midpoint of  $AC$ , and point  $K$  be symmetric to point  $D$  with respect to point  $M$ . A perpendicular drawn from point  $M$  to the line  $BC$  intersects line  $AB$  at point  $L$ . Prove that  $\\angle MBL = \\angle MKL$ .\n\n*Proposed by Oleksandra Yakovenko*","t":[{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.34817,"p":[[0,138,0.0,0.20078,0.2078,0.0,0.14286,0.42857,0.0,0.57143,13,0,2,13,0,6,0,0,4,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[4,138,0.029,0.29463,0.20182,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,9,0,0,8,0,0,5,0,0,4,0,0,2,0,0,0,0,0],[8,138,0.058,0.22768,0.18851,0.14286,0.14288,0.2857,0.0,0.71429,6,0,0,6,0,12,0,0,8,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[12,138,0.087,0.25,0.18557,0.14286,0.2143,0.32143,0.0,0.71429,5,0,0,5,0,11,0,0,8,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[16,138,0.1159,0.28114,0.19395,0.14286,0.28571,0.42857,0.0,0.57143,6,0,0,6,0,7,0,0,6,0,0,8,0,0,5,0,0,0,0,0,0,0,0],[20,138,0.1449,0.21875,0.1636,0.14286,0.14288,0.28571,0.0,0.57143,6,0,0,6,0,11,0,0,10,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[24,138,0.1739,0.28574,0.22868,0.14286,0.28571,0.4286,0.0,0.71429,7,0,0,7,0,8,0,0,4,0,0,7,0,0,3,0,0,3,0,0,0,0,0],[28,138,0.2029,0.29462,0.21702,0.14286,0.2857,0.571,0.0,0.57143,7,0,0,7,0,6,0,0,6,0,0,4,0,0,9,0,0,0,0,0,0,0,0],[32,138,0.2319,0.24543,0.20899,0.0,0.21428,0.42857,0.0,0.57143,9,0,0,9,0,7,0,0,6,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[36,138,0.2609,0.28571,0.17496,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,5,0,0,11,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[40,138,0.2899,0.19642,0.18469,0.0,0.14286,0.28571,0.0,0.71429,9,0,0,9,0,12,0,0,4,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[44,138,0.3188,0.30354,0.2194,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,6,0,0,7,0,0,7,0,0,3,0,0,3,0,0,0,0,0],[48,138,0.3478,0.2857,0.22867,0.14286,0.2857,0.46418,0.0,0.71429,7,0,0,7,0,8,0,0,5,0,0,4,0,0,6,0,0,2,0,0,0,0,0],[52,138,0.3768,0.34817,0.18531,0.2857,0.35714,0.42858,0.0,0.71429,3,0,0,3,0,4,0,0,9,0,0,10,0,0,4,0,0,2,0,0,0,0,0],[56,138,0.4058,0.20981,0.20818,0.0,0.14286,0.32143,0.0,0.71429,11,0,0,11,0,8,0,0,5,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[60,138,0.4348,0.27677,0.22568,0.0,0.28571,0.4642,0.0,0.57143,9,0,0,9,0,6,0,0,3,0,0,6,0,0,8,0,0,0,0,0,0,0,0],[64,138,0.4638,0.19195,0.16979,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,11,0,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[68,138,0.4928,0.19188,0.15818,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,9,0,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[72,138,0.5217,0.19633,0.18814,0.0,0.14286,0.32143,0.0,0.57143,10,0,0,10,0,11,0,0,3,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[76,138,0.5507,0.33478,0.18418,0.14289,0.28571,0.4286,0.0,0.71429,2,0,0,2,0,8,0,0,7,0,0,8,0,0,6,0,0,1,0,0,0,0,0],[80,138,0.5797,0.2767,0.1748,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,12,0,0,4,0,0,11,0,0,1,0,0,1,0,0,0,0,0],[84,138,0.6087,0.27231,0.19998,0.14286,0.2857,0.4286,0.0,0.57143,6,0,0,6,0,9,0,0,5,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[88,138,0.6377,0.28124,0.1767,0.14286,0.2857,0.42857,0.0,0.57143,4,0,0,4,0,9,0,0,7,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[92,138,0.6667,0.21427,0.17494,0.0,0.2857,0.32143,0.0,0.571,10,0,0,10,0,5,0,0,9,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[96,138,0.6957,0.23658,0.2039,0.10714,0.14286,0.4286,0.0,0.57143,8,0,0,8,0,10,0,0,5,0,0,3,0,0,6,0,0,0,0,0,0,0,0],[100,138,0.7246,0.2142,0.20828,0.0,0.14286,0.32143,0.0,0.71429,10,0,0,10,0,9,0,0,5,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[104,138,0.7536,0.28125,0.2004,0.14286,0.21429,0.42858,0.0,0.71429,4,0,0,4,0,12,0,0,3,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[108,138,0.7826,0.27676,0.20802,0.14286,0.2857,0.42857,0.0,0.71429,5,0,0,5,0,10,0,0,7,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[112,138,0.8116,0.20517,0.17108,0.14,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,13,0,0,6,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[116,138,0.8406,0.31241,0.16927,0.14286,0.28571,0.4286,0.0,0.57143,3,0,0,3,0,7,0,0,7,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[120,138,0.8696,0.32142,0.19883,0.14286,0.28571,0.4286,0.0,0.71429,4,0,0,4,0,7,0,0,6,0,0,8,0,0,6,0,0,1,0,0,0,0,0],[124,138,0.8986,0.18295,0.17584,0.0,0.14286,0.2857,0.0,0.71429,10,0,0,10,0,10,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[128,138,0.9275,0.2589,0.18008,0.14286,0.2857,0.32143,0.0,0.57143,5,0,0,5,0,9,0,0,10,0,0,3,0,0,5,0,0,0,0,0,0,0,0],[132,138,0.9565,0.20088,0.18159,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,11,0,0,5,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[136,138,0.9855,0.2232,0.18533,0.10714,0.1429,0.28571,0.0,0.57143,8,0,0,8,0,9,0,0,8,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[138,138,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.10714,"x":0.29014,"p":[[0,95,0.0,0.29014,0.23546,0.10714,0.21431,0.571,0.0,0.71429,8,0,1,8,0,8,0,0,1,0,0,6,0,0,8,0,0,1,0,0,0,0,0],[4,95,0.0421,0.28123,0.2244,0.14286,0.2857,0.32143,0.0,0.71429,5,0,0,5,0,10,0,0,9,0,0,1,0,0,3,0,0,4,0,0,0,0,0],[8,95,0.0842,0.24999,0.17855,0.14286,0.2857,0.42857,0.0,0.57143,6,0,0,6,0,9,0,0,7,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[12,95,0.1263,0.26785,0.20436,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,9,0,0,7,0,0,4,0,0,5,0,0,1,0,0,0,0,0],[16,95,0.1684,0.21872,0.19875,0.0,0.2143,0.2857,0.0,0.71429,10,0,0,10,0,6,0,0,10,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[20,95,0.2105,0.25,0.21724,0.10714,0.21428,0.42858,0.0,0.71429,8,0,0,8,0,8,0,0,7,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[24,95,0.2526,0.21874,0.21717,0.0,0.14286,0.32143,0.0,0.71429,12,0,0,12,0,5,0,0,7,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[28,95,0.2947,0.26331,0.2234,0.105,0.2857,0.42857,0.0,0.71429,8,0,0,8,0,7,0,0,7,0,0,5,0,0,2,0,0,3,0,0,0,0,0],[32,95,0.3368,0.16955,0.2096,0.0,0.14143,0.28571,0.0,0.71429,15,0,0,15,0,7,0,0,4,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[36,95,0.3789,0.17411,0.17399,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,7,0,0,9,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[40,95,0.4211,0.13392,0.16339,0.0,0.07143,0.2857,0.0,0.571,16,0,0,16,0,7,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[44,95,0.4632,0.16509,0.1825,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,9,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[48,95,0.5053,0.15179,0.18877,0.0,0.14286,0.2857,0.0,0.71429,15,0,0,15,0,8,0,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[52,95,0.5474,0.20087,0.20468,0.0,0.14286,0.32143,0.0,0.71429,12,0,0,12,0,7,0,0,5,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[56,95,0.5895,0.14714,0.13115,0.0,0.14286,0.1786,0.0,0.42857,10,0,0,10,0,14,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[60,95,0.6316,0.17411,0.20119,0.0,0.14286,0.2857,0.0,0.71429,15,0,0,15,0,5,0,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[64,95,0.6737,0.13392,0.15122,0.0,0.14286,0.17868,0.0,0.571,14,0,0,14,0,10,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[68,95,0.7158,0.11159,0.15036,0.0,0.0,0.1786,0.0,0.571,18,0,0,18,0,6,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[72,95,0.7579,0.10714,0.13363,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,10,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[76,95,0.8,0.1607,0.1812,0.0,0.14286,0.2857,0.0,0.57143,14,0,0,14,0,8,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[80,95,0.8421,0.16516,0.18246,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,9,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[84,95,0.8842,0.16954,0.18011,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,10,0,0,5,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[88,95,0.9263,0.14286,0.15567,0.0,0.14286,0.1429,0.0,0.57143,12,0,0,12,0,13,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[92,95,0.9684,0.15846,0.17097,0.0,0.14286,0.2857,0.0,0.571,13,0,0,13,0,9,0,0,5,0,1,2,0,0,2,0,0,0,0,0,0,0,0],[95,95,1.0,0.24107,0.13091,0.14289,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,10,0,0,14,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"e9f29123950e2e2d","q":"Let  $1000 \\leq n = \\text{ABCD}_{10} \\leq 9999$  be a positive integer whose digits  $\\text{ABCD}$  satisfy the divisibility condition:   $$ 1111 | (\\text{ABCD} + \\text{AB} \\times \\text{CD}). $$   Determine the smallest possible value of  $n$ .","t":[{"b":2,"e":0.57143,"k":"flat","v":0.53125,"x":0.71429,"p":[[0,115,0.0,0.60714,0.29233,0.39286,0.71429,0.75,0.0,1.0,1,6,0,1,0,4,0,0,3,0,0,1,0,0,6,0,0,9,0,0,2,0,6],[4,115,0.0348,0.68302,0.17763,0.57143,0.71429,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,10,0,0,4,0,4],[8,115,0.0696,0.6875,0.14032,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,24,0,0,0,0,2],[12,115,0.1043,0.6875,0.0974,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,27,0,0,1,0,0],[16,115,0.1391,0.67857,0.09449,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,25,0,0,1,0,0],[20,115,0.1739,0.70536,0.07087,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,27,0,0,2,0,0],[24,115,0.2087,0.69643,0.07784,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,28,0,0,1,0,0],[28,115,0.2435,0.70089,0.05486,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,27,0,0,1,0,0],[32,115,0.2783,0.70089,0.04164,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[36,115,0.313,0.6875,0.09062,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,29,0,0,0,0,0],[40,115,0.3478,0.68304,0.08552,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,25,0,0,1,0,0],[44,115,0.3826,0.6875,0.07523,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,25,0,0,1,0,0],[48,115,0.4174,0.70535,0.11811,0.71429,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,24,0,0,1,0,2],[52,115,0.4522,0.70536,0.11259,0.71429,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,26,0,0,0,0,2],[56,115,0.487,0.70536,0.09407,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,24,0,0,0,0,2],[60,115,0.5217,0.70076,0.07455,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,26,0,0,0,0,1],[64,115,0.5565,0.69196,0.06298,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[68,115,0.5913,0.70536,0.03458,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0],[72,115,0.6261,0.68749,0.05579,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[76,115,0.6609,0.70536,0.06121,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,29,0,0,1,0,0],[80,115,0.6957,0.71429,0.05051,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,2,0,0],[84,115,0.7304,0.61159,0.10853,0.57143,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,14,0,0,0,0,0],[88,115,0.7652,0.55804,0.15714,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,6,0,0,10,0,0,12,0,0,0,0,0],[92,115,0.8,0.56695,0.13115,0.53539,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,5,0,0,14,0,0,10,0,0,0,0,0],[96,115,0.8348,0.60714,0.13832,0.57143,0.64286,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,3,0,0,11,0,0,16,0,0,0,0,0],[100,115,0.8696,0.5803,0.13334,0.571,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,4,0,0,15,0,0,11,0,0,0,0,0],[104,115,0.9043,0.5714,0.15972,0.42859,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,5,0,0,9,0,0,14,0,0,0,0,0],[108,115,0.9391,0.61159,0.1439,0.57143,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,14,0,0,11,0,0,1,0,1],[112,115,0.9739,0.59822,0.13092,0.53572,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,12,0,0,11,0,0,0,0,1],[115,115,1.0,0.53125,0.1394,0.42857,0.57143,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,9,0,0,11,0,0,8,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.41964,"x":0.69641,"p":[[0,34,0.0,0.67841,0.27432,0.571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,2,0,0,2,0,0,4,0,0,9,0,0,5,0,7],[4,34,0.1176,0.62499,0.21944,0.5354,0.71429,0.71429,0.0,1.0,1,2,0,1,0,0,0,0,4,0,0,3,0,0,4,0,0,15,0,0,3,0,2],[8,34,0.2353,0.5982,0.27067,0.42857,0.57143,0.75,0.14286,1.0,0,6,0,0,0,3,0,0,4,0,0,5,0,0,6,0,0,6,0,0,2,0,6],[12,34,0.3529,0.69641,0.21355,0.57132,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,7,0,0,4,0,7],[16,34,0.4706,0.5357,0.23419,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,4,0,0,5,0,0,11,0,0,6,0,0,0,0,3],[20,34,0.5882,0.49999,0.25,0.28571,0.42857,0.57143,0.0,1.0,1,4,0,1,0,1,0,0,7,0,0,11,0,0,5,0,0,2,0,0,1,0,4],[24,34,0.7059,0.4866,0.19186,0.39286,0.42857,0.57143,0.0,1.0,1,1,0,1,0,0,0,0,7,0,0,9,0,0,10,0,0,3,0,0,1,0,1],[28,34,0.8235,0.41964,0.21706,0.28571,0.42857,0.4286,0.14286,1.0,0,3,0,0,0,3,0,0,11,0,0,12,0,0,3,0,0,0,0,0,0,0,3],[32,34,0.9412,0.47319,0.16144,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,3,0,0,15,0,0,9,0,0,2,0,0,0,0,1],[34,34,1.0,0.43749,0.16341,0.42857,0.42857,0.57111,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,16,0,0,8,0,0,0,0,0,0,0,1]]}]},{"i":"842881b85093e98c","q":"Find all prime numbers  $ p$  and  $ q$  such that  $ p^3 \\minus{} q^5 \\equal{} (p \\plus{} q)^2$ .","t":[{"b":3,"e":1.0,"k":"volatile","v":0.49554,"x":1.0,"p":[[0,12,0.0,0.49554,0.38297,0.14286,0.35714,1.0,0.0,1.0,3,11,0,3,0,7,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,11],[4,12,0.3333,0.63838,0.35889,0.28571,0.78564,1.0,0.0,1.0,1,13,0,1,0,4,0,0,6,0,0,3,0,0,1,0,0,1,0,0,3,0,13],[8,12,0.6667,0.82589,0.30875,0.89286,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,24],[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]]},{"b":4,"e":0.14286,"k":"falling","v":0.07143,"x":0.51338,"p":[[0,15,0.0,0.48437,0.35747,0.24999,0.35714,1.0,0.0,1.0,2,9,0,2,1,5,0,0,8,0,0,6,0,0,0,0,0,0,0,0,1,0,9],[4,15,0.2667,0.51338,0.40226,0.14286,0.28571,1.0,0.0,1.0,5,11,0,5,0,4,0,0,9,0,0,0,0,0,1,0,0,0,0,0,2,0,11],[8,15,0.5333,0.07589,0.07973,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.07143,0.08564,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,2,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b1f80b5dace05fc","q":"Determine all pairs of prime numbers  $(p, q)$  which satisfy the equation\n\\[\np^3+q^3+1=p^2q^2\n\\]","t":[{"b":1,"e":0.14286,"k":"flat","v":0.08482,"x":0.20086,"p":[[0,37,0.0,0.14732,0.14934,0.0,0.14286,0.28571,0.0,0.57143,12,0,0,12,0,11,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,37,0.1081,0.16964,0.15335,0.0,0.14286,0.2857,0.0,0.57143,10,0,0,10,0,11,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,37,0.2162,0.20071,0.16318,0.14286,0.14286,0.2857,0.0,0.71429,4,0,0,4,0,19,0,0,5,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[12,37,0.3243,0.20086,0.21081,0.0,0.14286,0.2857,0.0,0.71429,9,0,0,9,0,14,0,0,3,0,0,1,0,0,3,0,0,2,0,0,0,0,0],[16,37,0.4324,0.16964,0.1729,0.10714,0.14286,0.14287,0.0,0.71429,8,0,0,8,0,17,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[20,37,0.5405,0.08482,0.10012,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.14277,0.07986,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.14732,0.0977,0.14286,0.14286,0.1786,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.1517,0.09408,0.14286,0.14286,0.1786,0.0,0.28571,6,0,0,6,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.17411,0.06901,0.14286,0.14286,0.17857,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.18742,0.06629,0.14286,0.14286,0.2857,0.14,0.286,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.13838,"x":0.36151,"p":[[0,22,0.0,0.13838,0.1493,0.0,0.14286,0.17857,0.0,0.571,13,0,0,13,0,11,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,22,0.1818,0.22768,0.21086,0.14286,0.14286,0.32143,0.0,0.71429,7,0,0,7,0,14,0,0,3,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[8,22,0.3636,0.18295,0.18295,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,11,0,0,7,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[12,22,0.5455,0.24999,0.19883,0.14286,0.14286,0.32143,0.0,0.71429,5,0,0,5,0,13,0,0,6,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[16,22,0.7273,0.36151,0.19888,0.24999,0.35714,0.4286,0.0,0.71429,2,0,0,2,0,6,0,0,8,0,0,9,0,0,3,0,0,4,0,0,0,0,0],[20,22,0.9091,0.19178,0.14561,0.14286,0.14286,0.2857,0.0,0.71429,4,0,0,4,0,18,0,0,8,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[22,22,1.0,0.18303,0.07349,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"44860aba491ccb4d","q":"Find all functions  $f : R \\to R$  which satisfy   $f  \\left(\\frac{\\sqrt3}{3} x\\right) = \\sqrt3 f(x) - \\frac{2\\sqrt3}{3} x$  \n\nand  $f(x)f(y) = f(xy) + f  \\left(\\frac{x}{y} \\right) $    for all  $x, y \\in R$ , with  $y \\ne 0$","t":[{"b":0,"e":0.2857,"k":"flat","v":0.25446,"x":0.30804,"p":[[0,10,0.0,0.25893,0.14032,0.14286,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,6,0,0,15,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,10,0.4,0.30356,0.13241,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,7,0,0,18,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[8,10,0.8,0.30804,0.13415,0.25002,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,7,0,0,12,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[10,10,1.0,0.25446,0.15458,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,11,0,0,11,0,0,4,0,0,3,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.1741,"x":0.31249,"p":[[0,41,0.0,0.21429,0.14725,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,12,0,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,41,0.0976,0.1875,0.10971,0.14286,0.2143,0.28571,0.0,0.28571,6,0,0,6,0,10,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.25446,0.13236,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,10,0,0,15,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,41,0.2927,0.1741,0.12234,0.10714,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,10,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.21879,0.11845,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,0,15,0,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,41,0.4878,0.25,0.10714,0.14286,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,8,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.20982,0.10705,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,16,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.21875,0.10705,0.14286,0.2857,0.28571,0.0,0.42857,3,0,0,3,0,11,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.24107,0.09061,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,12,0,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,41,0.878,0.31249,0.15333,0.25,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,7,0,0,14,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[40,41,0.9756,0.27232,0.13054,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,4,0,0,20,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[41,41,1.0,0.24553,0.15251,0.14286,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,10,0,0,11,0,0,5,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"4ca4f6d86d10216b","q":"Let  $P, Q,R$  be three polynomials with real coefficients such that \\[P(Q(x)) + P(R(x))=\\text{constant}\\] for all  $x$ . Prove that  $P(x)=\\text{constant}$  or  $Q(x)+R(x)=\\text{constant}$  for all  $x$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.79237,"x":0.95088,"p":[[0,53,0.0,0.86159,0.19721,0.85711,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2,0,0,8,0,17],[4,53,0.0755,0.79237,0.24834,0.57132,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,1,3,0,0,4,0,0,1,0,0,6,0,15],[8,53,0.1509,0.83926,0.21944,0.82132,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,14],[12,53,0.2264,0.83034,0.246,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,4,0,0,4,0,18],[16,53,0.3019,0.87499,0.16656,0.82132,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,7,0,17],[20,53,0.3774,0.90177,0.18013,0.857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,21],[24,53,0.4528,0.90848,0.18055,0.91071,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,3,1,23],[28,53,0.5283,0.90175,0.14486,0.857,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,20],[32,53,0.6038,0.84371,0.2458,0.82132,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,0,5,0,19],[36,53,0.6792,0.91294,0.10823,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,11,0,17],[40,53,0.7547,0.88391,0.13573,0.82132,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,8,0,16],[44,53,0.8302,0.87496,0.18128,0.857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,3,0,0,10,0,16],[48,53,0.9057,0.88838,0.19145,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,7,0,20],[52,53,0.9811,0.92855,0.10103,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,19],[53,53,1.0,0.95088,0.11634,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25]]},{"b":7,"e":1.0,"k":"flat","v":0.8281,"x":0.97767,"p":[[0,15,0.0,0.87053,0.18681,0.85713,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,11,0,16],[4,15,0.2667,0.8281,0.21862,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,1,0,8,0,14],[8,15,0.5333,0.9241,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,9,0,20],[12,15,0.8,0.95089,0.11072,1.0,1.0,1.0,0.5714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[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]]}]},{"i":"afaacf70c461c78d","q":"Let  $x$  be a positive real number. Define\n\\[\n  A = \\sum_{k=0}^{\\infty} \\frac{x^{3k}}{(3k)!}, \\quad\n  B = \\sum_{k=0}^{\\infty} \\frac{x^{3k+1}}{(3k+1)!}, \\quad\\text{and}\\quad\n  C = \\sum_{k=0}^{\\infty} \\frac{x^{3k+2}}{(3k+2)!}.\n\\] Given that  $A^3+B^3+C^3 + 8ABC = 2014$ , compute  $ABC$ .\n\n*Proposed by Evan Chen*","t":[{"b":2,"e":0.71429,"k":"flat","v":0.67857,"x":0.84821,"p":[[0,49,0.0,0.67857,0.28347,0.57143,0.64286,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,2,0,0,2,0,0,9,0,0,4,0,0,1,0,11],[4,49,0.0816,0.84821,0.20183,0.67857,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,0,4,0,18],[8,49,0.1633,0.79018,0.27196,0.71429,0.85714,1.0,0.0,1.0,1,15,1,1,0,1,0,0,2,0,0,0,0,0,2,0,0,7,0,0,4,0,15],[12,49,0.2449,0.75,0.24744,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,0,0,0,10,0,0,4,0,0,2,0,13],[16,49,0.3265,0.80804,0.21011,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,14,0,0,1,0,14],[20,49,0.4082,0.71875,0.16554,0.57143,0.71429,0.74996,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,15,0,0,3,0,5],[24,49,0.4898,0.77232,0.11769,0.71429,0.71429,0.75,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,2,0,6],[28,49,0.5714,0.77232,0.15093,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,18,0,0,0,0,9],[32,49,0.6531,0.7232,0.15127,0.57143,0.71429,0.75,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,14,0,0,3,0,5],[36,49,0.7347,0.68302,0.12746,0.57143,0.71429,0.71429,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,14,0,0,1,0,3],[40,49,0.8163,0.77232,0.17807,0.57143,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,7,0,0,4,0,10],[44,49,0.898,0.78125,0.17122,0.57143,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,4,0,10],[48,49,0.9796,0.72321,0.14258,0.57143,0.71429,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,12,0,0,5,0,4],[49,49,1.0,0.7857,0.15154,0.71429,0.71429,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,0,3,0,9]]},{"b":5,"e":1.0,"k":"rising","v":0.79018,"x":1.0,"p":[[0,39,0.0,0.79018,0.25997,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,3,0,0,5,0,15],[4,39,0.1026,0.94196,0.13296,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[8,39,0.2051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[39,39,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2c8157dab94f5e2a","q":"For some integer  $m$ , the polynomial  $x^3-2011x+m$  has the three integer roots  $a$ ,  $b$ , and  $c$ . Find  $|a|+|b|+|c|$ .","t":[{"b":3,"e":0.85714,"k":"flat","v":0.97768,"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],[4,51,0.0784,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,51,0.1569,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,51,0.2353,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,51,0.3137,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,51,0.3922,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,51,0.4706,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,51,0.549,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,51,0.6275,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,51,0.7059,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,51,0.7843,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,51,0.8627,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[48,51,0.9412,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[51,51,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":7,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,35,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,35,0.1143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,35,0.2286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[16,35,0.4571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[28,35,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"8ceb381d10689547","q":"Let  $ n $  be a natural number. Find all integer numbers that can be written as  $$  \\frac{1}{a_1} +\\frac{2}{a_2} +\\cdots +\\frac{n}{a_n} ,  $$  where  $ a_1,a_2,...,a_n $  are natural numbers.","t":[{"b":2,"e":0.42857,"k":"volatile","v":0.40179,"x":0.85268,"p":[[0,13,0.0,0.65178,0.31122,0.28571,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,9,0,0,6,0,0,2,0,0,0,0,0,3,0,12],[4,13,0.3077,0.79464,0.24984,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,1,0,0,1,0,18],[8,13,0.6154,0.85268,0.26362,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,3,0,22],[12,13,0.9231,0.41964,0.11811,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,25,0,0,0,0,0,0,0,0,0,0,1],[13,13,1.0,0.40179,0.16917,0.28571,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,14,0,0,16,0,0,0,0,0,0,0,0,0,0,2]]},{"b":3,"e":1.0,"k":"rising","v":0.58481,"x":1.0,"p":[[0,22,0.0,0.70982,0.31437,0.39286,0.85714,1.0,0.0,1.0,1,12,0,1,0,1,0,0,6,0,0,1,0,0,2,0,0,3,0,0,6,0,12],[4,22,0.1818,0.83036,0.25111,0.67857,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,2,0,0,2,0,20],[8,22,0.3636,0.58481,0.29094,0.28571,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,10,0,0,8,0,0,1,0,0,4,0,0,0,0,9],[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":"6d0831b72cebe34f","q":"Let $a, b$, and $c$ be non-negative real numbers, no two of which are equal. Prove that\n\n$$\n\\frac{a^{2}}{(b-c)^{2}}+\\frac{b^{2}}{(c-a)^{2}}+\\frac{c^{2}}{(a-b)^{2}}>2\n$$","t":[{"b":1,"e":0.57143,"k":"rising","v":0.08036,"x":0.58034,"p":[[0,16,0.0,0.08036,0.24984,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,16,0.25,0.50446,0.48179,0.0,0.64286,1.0,0.0,1.0,15,14,0,15,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,14],[8,16,0.5,0.51786,0.15872,0.42857,0.57143,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,9,0,0,9,0,0,7,0,0,1,0,0],[12,16,0.75,0.58034,0.15126,0.42857,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,7,0,0,8,0,0,13,0,0,1,0,0],[16,16,1.0,0.47768,0.1411,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,12,0,0,8,0,0,5,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.01786,"x":0.23661,"p":[[0,7,0.0,0.20534,0.36931,0.0,0.0,0.10714,0.0,1.0,24,3,0,24,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,3],[4,7,0.5714,0.23661,0.39222,0.0,0.0,0.46429,0.0,1.0,23,4,0,23,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,4],[7,7,1.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b7bfa525a72f4c4","q":"$x$ ,  $y$  and  $z$  are real numbers where the sum of any two among them is not  $1$ . Show that, \\[ \\dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \\dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \\ge 2(x+y+z) - \\dfrac{3}{4}\\]Find all triples  $(x,y,z)$  of real numbers satisfying the equality case.","t":[{"b":4,"e":0.14286,"k":"falling","v":0.27009,"x":0.86161,"p":[[0,73,0.0,0.81696,0.31183,0.75,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,23],[4,73,0.0548,0.86161,0.2435,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,22],[8,73,0.1096,0.75446,0.35577,0.42857,1.0,1.0,0.0,1.0,1,21,0,1,0,4,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,21],[12,73,0.1644,0.77679,0.32721,0.64286,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,2,0,0,3,0,0,0,0,0,3,0,0,2,0,19],[16,73,0.2192,0.83929,0.3004,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,24],[20,73,0.274,0.58929,0.37923,0.14286,0.57143,1.0,0.0,1.0,3,13,0,3,0,6,0,0,1,0,0,3,0,0,6,0,0,0,0,0,0,0,13],[24,73,0.3288,0.75,0.33881,0.57143,1.0,1.0,0.0,1.0,1,17,0,1,0,5,0,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,17],[28,73,0.3836,0.78125,0.32924,0.57143,1.0,1.0,0.0,1.0,1,20,0,1,0,3,0,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,20],[32,73,0.4384,0.67411,0.35217,0.42857,0.85714,1.0,0.0,1.0,1,16,0,1,0,4,0,0,2,0,0,6,0,0,2,0,0,1,0,0,0,0,16],[36,73,0.4932,0.58482,0.36133,0.14286,0.64286,1.0,0.0,1.0,1,11,0,1,0,8,0,0,3,0,0,1,0,0,3,0,0,5,0,0,0,0,11],[40,73,0.5479,0.53572,0.35174,0.14289,0.57143,1.0,0.0,1.0,1,9,0,1,0,8,0,0,5,0,0,1,0,0,5,0,0,2,0,0,1,0,9],[44,73,0.6027,0.57143,0.38466,0.14286,0.57143,1.0,0.0,1.0,2,13,0,2,0,8,0,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,13],[48,73,0.6575,0.37499,0.2829,0.14286,0.2857,0.57143,0.0,1.0,1,3,0,1,0,12,0,0,7,0,0,2,0,0,5,0,0,1,0,0,1,0,3],[52,73,0.7123,0.30804,0.24251,0.14286,0.21428,0.42858,0.0,0.85714,4,0,0,4,0,12,0,0,4,0,0,5,0,0,2,0,0,4,0,0,1,0,0],[56,73,0.7671,0.375,0.25443,0.14286,0.35714,0.57143,0.0,1.0,2,1,0,2,0,10,0,0,4,0,0,6,0,0,6,0,0,1,0,0,2,0,1],[60,73,0.8219,0.30804,0.24772,0.14286,0.14286,0.42858,0.0,0.85714,3,0,0,3,0,14,0,0,4,0,0,4,0,0,2,0,0,3,0,0,2,0,0],[64,73,0.8767,0.3125,0.24337,0.14286,0.28571,0.46431,0.0,0.857,4,0,0,4,0,11,0,0,6,0,0,3,0,0,3,0,0,4,0,0,1,0,0],[68,73,0.9315,0.29464,0.28107,0.0,0.28571,0.46431,0.0,0.85714,10,0,0,10,0,5,0,0,6,0,0,3,0,0,2,0,0,4,0,0,2,0,0],[72,73,0.9863,0.27009,0.2285,0.14286,0.14288,0.42857,0.0,0.85714,5,0,0,5,1,11,0,0,6,0,0,3,0,0,3,0,0,2,0,0,1,0,0],[73,73,1.0,0.27232,0.24578,0.14286,0.14286,0.42858,0.0,0.85714,6,0,0,6,0,13,0,0,3,0,0,3,0,0,3,0,0,3,0,0,1,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.23661,"x":0.97321,"p":[[0,109,0.0,0.86161,0.25123,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,3,0,0,4,0,21],[4,109,0.0367,0.95089,0.13175,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[8,109,0.0734,0.90625,0.20395,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,1,0,25],[12,109,0.1101,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[16,109,0.1468,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[20,109,0.1835,0.85714,0.25505,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,4,0,0,0,0,23],[24,109,0.2202,0.83929,0.24936,0.67857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,0,1,0,21],[28,109,0.2569,0.80802,0.29367,0.57132,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,0,1,0,21],[32,109,0.2936,0.80804,0.30641,0.67857,1.0,1.0,0.0,1.0,1,21,0,1,0,2,0,0,1,0,0,2,0,0,2,0,0,2,0,0,1,0,21],[36,109,0.3303,0.84821,0.26711,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,23],[40,109,0.367,0.92857,0.22588,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[44,109,0.4037,0.8125,0.3163,0.57143,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,1,0,0,1,0,0,4,0,0,0,0,0,1,0,22],[48,109,0.4404,0.87054,0.27516,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,25],[52,109,0.4771,0.82589,0.30459,0.67857,1.0,1.0,0.0,1.0,2,23,0,2,0,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,23],[56,109,0.5138,0.54464,0.39357,0.14286,0.57143,1.0,0.0,1.0,5,11,0,5,0,6,0,0,2,0,0,0,0,0,5,0,0,3,0,0,0,0,11],[60,109,0.5505,0.31697,0.32681,0.14286,0.1429,0.42857,0.0,1.0,7,4,0,7,0,10,0,0,6,0,0,2,0,0,1,0,0,1,0,0,1,0,4],[64,109,0.5872,0.3125,0.32427,0.10714,0.14286,0.57143,0.0,1.0,8,3,0,8,0,10,0,0,4,0,0,1,0,0,4,0,0,0,0,0,2,0,3],[68,109,0.6239,0.50445,0.35532,0.14286,0.42859,0.85714,0.0,1.0,2,7,0,2,0,9,0,0,3,0,0,3,0,0,3,0,0,2,0,0,3,0,7],[72,109,0.6606,0.42856,0.39123,0.14286,0.28571,0.89286,0.0,1.0,7,8,0,7,0,8,0,0,2,0,0,3,0,0,2,0,0,1,0,0,1,0,8],[76,109,0.6972,0.39731,0.36198,0.14286,0.1429,0.60714,0.0,1.0,5,6,0,5,0,12,0,0,1,0,0,2,0,0,4,0,0,1,0,0,1,0,6],[80,109,0.7339,0.4375,0.35881,0.14286,0.28571,0.75,0.0,1.0,2,7,0,2,0,13,0,0,2,0,0,4,0,0,1,0,0,2,0,0,1,0,7],[84,109,0.7706,0.42411,0.3737,0.14286,0.28571,0.78571,0.0,1.0,5,8,0,5,0,9,0,0,4,0,0,3,0,0,2,0,0,1,0,0,0,0,8],[88,109,0.8073,0.23661,0.249,0.14286,0.14286,0.28571,0.0,1.0,7,2,0,7,0,12,0,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[92,109,0.844,0.375,0.27374,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,15,0,0,4,0,0,3,0,0,3,0,0,3,0,0,3,0,1],[96,109,0.8807,0.37946,0.31665,0.14286,0.28571,0.60714,0.0,1.0,6,3,0,6,0,6,0,0,7,0,0,2,0,0,3,0,0,4,0,0,1,0,3],[100,109,0.9174,0.36606,0.23672,0.14286,0.42857,0.57111,0.0,1.0,4,1,0,4,0,6,0,0,4,0,0,9,0,0,6,0,0,2,0,0,0,0,1],[104,109,0.9541,0.50893,0.34058,0.24999,0.57143,0.75,0.0,1.0,3,7,0,3,0,5,0,0,6,0,0,1,0,0,6,0,0,3,0,0,1,0,7],[108,109,0.9908,0.2991,0.24836,0.14286,0.2857,0.57143,0.0,0.85714,7,0,0,7,0,7,0,0,8,0,0,0,0,0,7,0,0,2,0,0,1,0,0],[109,109,1.0,0.29241,0.25402,0.14286,0.14286,0.46429,0.0,0.85714,5,0,0,5,1,12,0,0,3,0,0,3,0,0,3,0,0,4,0,0,1,0,0]]}]},{"i":"396f2b085a9d1e40","q":"Let  $(A,+, \\cdot)$  be a ring in which the set of invertible elements is finite. Prove that the following assertions are equivalent:\n[list=1]\n[*] For any non-invertible element  $a \\in A,$  there is a non-invertible element  $b \\in A$  such that  $ab=a+b.$ [*] Any non-invertible element of the ring  $A$  is nilpotent.\n[/list]\n\n*Mihai Opincariu*","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.84375,"p":[[0,102,0.0,0.62487,0.26179,0.42857,0.71429,0.75,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,7,0,0,1,0,0,11,0,0,3,0,5],[4,102,0.0392,0.84375,0.2,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,8,0,0,3,0,17],[8,102,0.0784,0.54017,0.23346,0.42857,0.4998,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,3,0,0,10,0,0,2,0,0,10,0,0,3,0,1],[12,102,0.1176,0.69194,0.25283,0.4286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,6,0,0,4,0,0,7,0,0,3,0,9],[16,102,0.1569,0.67857,0.23145,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,11,0,0,0,0,0,11,0,0,1,0,8],[20,102,0.1961,0.59371,0.22046,0.4286,0.57143,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,10,0,0,8,0,0,7,0,0,1,0,4],[24,102,0.2353,0.62053,0.23037,0.42857,0.64286,0.74996,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,10,0,0,3,0,0,8,0,0,4,0,4],[28,102,0.2745,0.55354,0.20438,0.42857,0.4286,0.71429,0.1429,1.0,0,3,0,0,0,1,0,0,2,0,0,14,0,0,4,0,0,8,0,0,0,0,3],[32,102,0.3137,0.62944,0.23921,0.42857,0.71429,0.75,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,11,0,0,2,0,0,9,0,0,3,0,5],[36,102,0.3529,0.65177,0.19212,0.4286,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,13,0,0,0,0,5],[40,102,0.3922,0.64731,0.22864,0.42857,0.71429,0.85704,0.1429,1.0,0,5,0,0,0,1,0,0,1,0,0,9,0,0,4,0,0,8,0,0,4,0,5],[44,102,0.4314,0.63379,0.19859,0.42857,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,13,0,0,3,0,3],[48,102,0.4706,0.45535,0.14031,0.42857,0.42859,0.4642,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,20,0,0,4,0,0,4,0,0,0,0,0],[52,102,0.5098,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,102,0.549,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,102,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,102,0.6275,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,102,0.6667,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,102,0.7059,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[76,102,0.7451,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[80,102,0.7843,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,102,0.8235,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,102,0.8627,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,102,0.902,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,102,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,102,0.9804,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[102,102,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.4955,"x":0.88838,"p":[[0,30,0.0,0.65175,0.23127,0.42857,0.71429,0.85704,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,4,0,0,2,0,0,8,0,0,10,0,2],[4,30,0.1333,0.4955,0.26721,0.2857,0.4286,0.71429,0.0,1.0,1,4,0,1,0,4,0,0,5,0,0,8,0,0,5,0,0,5,0,0,0,0,4],[8,30,0.2667,0.49998,0.23145,0.42857,0.4286,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,3,0,0,13,0,0,3,0,0,8,0,0,0,0,2],[12,30,0.4,0.54461,0.29329,0.28571,0.571,0.71429,0.0,1.0,3,4,0,3,0,1,0,0,5,0,0,6,0,0,2,0,0,9,0,0,2,0,4],[16,30,0.5333,0.6294,0.2283,0.4286,0.57143,0.71429,0.14286,1.0,0,5,0,0,0,2,0,0,0,0,0,8,0,0,7,0,0,8,0,0,2,0,5],[20,30,0.6667,0.7857,0.20517,0.67857,0.71429,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,9,0,0,2,0,13],[24,30,0.8,0.73213,0.23352,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,10,0,0,4,0,9],[28,30,0.9333,0.88838,0.1461,0.857,1.0,1.0,0.4286,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,8,0,17],[30,30,1.0,0.87499,0.14174,0.857,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,11,0,14]]}]},{"i":"f55b297022f3ca19","q":"A square  $ABCD$  is given. A point  $P$  is chosen inside the triangle  $ABC$  such that  $\\angle CAP = 15^\\circ = \\angle BCP$ . A point  $Q$  is chosen such that  $APCQ$  is an isosceles trapezoid:  $PC \\parallel AQ$ , and  $AP=CQ, AP\\nparallel CQ$ . Denote by  $N$  the midpoint of  $PQ$ . Find the angles of the triangle  $CAN$ .","t":[{"b":4,"e":0.42857,"k":"falling","v":0.44643,"x":0.60268,"p":[[0,153,0.0,0.60268,0.16262,0.42857,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,13,0,0,4,0,0],[4,153,0.0261,0.53125,0.1439,0.42857,0.42859,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,19,0,0,0,0,0,12,0,0,0,0,0],[8,153,0.0523,0.54464,0.1448,0.42857,0.42859,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,17,0,0,1,0,0,13,0,0,0,0,0],[12,153,0.0784,0.48661,0.12299,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,24,0,0,0,0,0,7,0,0,0,0,0],[16,153,0.1046,0.49107,0.15947,0.42857,0.42857,0.71429,0.0,0.7143,1,0,0,1,0,0,0,0,1,0,0,21,0,0,0,0,0,9,0,0,0,0,0],[20,153,0.1307,0.49107,0.1181,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0],[24,153,0.1569,0.52679,0.1357,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,11,0,0,0,0,0],[28,153,0.183,0.51339,0.13767,0.42857,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,21,0,0,0,0,0,10,0,0,0,0,0],[32,153,0.2092,0.50893,0.12846,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,9,0,0,0,0,0],[36,153,0.2353,0.54464,0.14032,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,0,13,0,0,0,0,0],[40,153,0.2614,0.52679,0.1357,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,11,0,0,0,0,0],[44,153,0.2876,0.47768,0.11633,0.42857,0.42857,0.42858,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,25,0,0,0,0,0,6,0,0,0,0,0],[48,153,0.3137,0.53572,0.13832,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,12,0,0,0,0,0],[52,153,0.3399,0.5625,0.14258,0.42857,0.4286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,15,0,0,0,0,0],[56,153,0.366,0.48214,0.11152,0.42857,0.42857,0.42858,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,6,0,0,0,0,0],[60,153,0.3922,0.47321,0.10372,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,5,0,0,0,0,0],[64,153,0.4183,0.53572,0.13832,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,12,0,0,0,0,0],[68,153,0.4444,0.50893,0.12846,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,9,0,0,0,0,0],[72,153,0.4706,0.49107,0.12339,0.42857,0.42857,0.46431,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,23,0,0,1,0,0,7,0,0,0,0,0],[76,153,0.4967,0.5,0.12372,0.42857,0.42857,0.50002,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,8,0,0,0,0,0],[80,153,0.5229,0.50893,0.1234,0.42857,0.42857,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,22,0,0,2,0,0,8,0,0,0,0,0],[84,153,0.549,0.51339,0.12807,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,22,0,0,1,0,0,9,0,0,0,0,0],[88,153,0.5752,0.5,0.12372,0.42857,0.42857,0.50002,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,8,0,0,0,0,0],[92,153,0.6013,0.54018,0.1461,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,18,0,0,0,0,0,13,0,0,0,0,0],[96,153,0.6275,0.54018,0.16262,0.42857,0.42859,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,16,0,0,0,0,0,14,0,0,0,0,0],[100,153,0.6536,0.54911,0.13882,0.42857,0.42859,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,13,0,0,0,0,0],[104,153,0.6797,0.52679,0.1357,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,11,0,0,0,0,0],[108,153,0.7059,0.49999,0.1237,0.42857,0.42857,0.49995,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,8,0,0,0,0,0],[112,153,0.732,0.51786,0.13243,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,10,0,0,0,0,0],[116,153,0.7582,0.49107,0.11811,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0],[120,153,0.7843,0.52679,0.14913,0.42857,0.42857,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,20,0,0,0,0,0,10,0,0,1,0,0],[124,153,0.8105,0.46875,0.10853,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,26,0,0,0,0,0,5,0,0,0,0,0],[128,153,0.8366,0.54911,0.14773,0.42857,0.42857,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,17,0,0,0,0,0,14,0,0,0,0,0],[132,153,0.8627,0.51786,0.13243,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,10,0,0,0,0,0],[136,153,0.8889,0.5,0.12372,0.42857,0.42857,0.50002,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,8,0,0,0,0,0],[140,153,0.915,0.54017,0.13709,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,0,12,0,0,0,0,0],[144,153,0.9412,0.57143,0.14285,0.42857,0.57144,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0],[148,153,0.9673,0.51786,0.13243,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,10,0,0,0,0,0],[152,153,0.9935,0.48219,0.1115,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,6,0,0,0,0,0],[153,153,1.0,0.44643,0.06916,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.44197,"x":0.54464,"p":[[0,175,0.0,0.5357,0.13832,0.42857,0.42859,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,17,0,0,3,0,0,11,0,0,0,0,0],[4,175,0.02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,0.14223,0.42857,0.42857,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,19,0,0,1,0,0,10,0,0,0,0,0],[175,175,1.0,0.46433,0.09447,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,4,0,0,0,0,0]]}]},{"i":"80c50a1d5121a8f7","q":"As  $a$ ,  $b$  and  $c$  range over *all* real numbers, let  $m$  be the smallest possible value of   $$ 2\\left(a+b+c\\right)^2+\\left(ab-4\\right)^2+\\left(bc-4\\right)^2+\\left(ca-4\\right)^2 $$   and  $n$  be the number of ordered triplets  $\\left(a,b,c\\right)$  such that the above quantity is minimized. Compute  $m+n$ .\n\n*2016 CCA Math Bonanza Team #8*","t":[{"b":5,"e":0.85714,"k":"flat","v":0.8125,"x":0.85714,"p":[[0,20,0.0,0.8125,0.1448,0.85714,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,21,0,4],[4,20,0.2,0.85714,0.15567,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,16,0,11],[8,20,0.4,0.83928,0.13243,0.85714,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,23,0,5],[12,20,0.6,0.8125,0.16146,0.85714,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,22,0,4],[16,20,0.8,0.85714,0.08748,0.85714,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,3],[20,20,1.0,0.85267,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.8125,"x":0.86607,"p":[[0,56,0.0,0.86606,0.11811,0.85714,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,21,0,8],[4,56,0.0714,0.8125,0.16536,0.85714,0.85714,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,23,0,4],[8,56,0.1429,0.8125,0.12595,0.85714,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,28,0,0],[12,56,0.2143,0.83482,0.06298,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,25,0,1],[16,56,0.2857,0.86161,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,2],[20,56,0.3571,0.85714,0.06186,0.85714,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,29,0,2],[24,56,0.4286,0.84821,0.04971,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0],[28,56,0.5,0.85267,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,1],[32,56,0.5714,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[36,56,0.6429,0.81696,0.12492,0.85714,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,29,0,0],[40,56,0.7143,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[44,56,0.7857,0.82143,0.12372,0.85714,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,29,0,0],[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],[56,56,1.0,0.86161,0.02486,0.85714,0.85714,0.85714,0.85714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1]]}]},{"i":"0651895a943d4402","q":"Given a sequence  $a_1,a_2,\\ldots $  of non-negative real numbers satisfying the conditions:\n\n1.  $a_n + a_{2n} \\geq 3n$ ;\n2.  $a_{n+1}+n \\leq 2\\sqrt{a_n \\left(n+1\\right)}$ for all  $n\\in\\mathbb N$  (where  $\\mathbb N=\\left\\{1,2,3,...\\right\\}$ ).\n\n(1) Prove that the inequality  $a_n \\geq n$  holds for every  $n \\in \\mathbb N$ .\n(2) Give an example of such a sequence.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.37053,"x":0.46429,"p":[[0,36,0.0,0.41964,0.17474,0.28571,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,12,0,0,17,0,0,0,0,0,1,0,0,0,0,2],[4,36,0.1111,0.4375,0.19541,0.28571,0.42857,0.42857,0.28571,1.0,0,3,0,0,0,0,0,0,11,0,0,17,0,0,1,0,0,0,0,0,0,0,3],[8,36,0.2222,0.46429,0.21429,0.28571,0.42857,0.42857,0.2857,1.0,0,4,0,0,0,0,0,0,9,0,0,18,0,0,1,0,0,0,0,0,0,0,4],[12,36,0.3333,0.375,0.06916,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.39286,0.06186,0.39286,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.41071,0.04725,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.38393,0.06622,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.37946,0.06785,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.37053,0.07873,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,14,0,0,17,0,0,1,0,0,0,0,0,0,0,0],[36,36,1.0,0.38393,0.06622,0.28571,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"rising","v":0.41518,"x":1.0,"p":[[0,30,0.0,0.43754,0.17835,0.28571,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,10,0,0,18,0,0,0,0,0,2,0,0,0,0,2],[4,30,0.1333,0.41518,0.13534,0.28571,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,9,0,0,21,0,0,0,0,0,1,0,0,0,0,1],[8,30,0.2667,0.91518,0.20782,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,27],[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],[30,30,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"5beee60b4c0d3fba","q":"For all $x, y, z>0$ satisfying $\\frac{x}{y z}+\\frac{y}{z x}+\\frac{z}{x y} \\leq x+y+z$, prove that\n\n$$\n\\frac{1}{x^{2}+y+z}+\\frac{1}{y^{2}+z+x}+\\frac{1}{z^{2}+x+y} \\leq 1\n$$","t":[{"b":5,"e":0.14286,"k":"flat","v":0.09821,"x":0.14286,"p":[[0,72,0.0,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,72,0.0556,0.125,0.05923,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,72,0.1111,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,72,0.1667,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,72,0.2222,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,72,0.2778,0.13375,0.07934,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,72,0.3333,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,72,0.3889,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,72,0.4444,0.12045,0.06295,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,72,0.5,0.12946,0.05486,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,72,0.5556,0.12947,0.09006,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,72,0.6111,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,72,0.6667,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,72,0.7222,0.11161,0.09932,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,72,0.7778,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,72,0.8333,0.11143,0.06894,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,72,0.8889,0.10714,0.07143,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,72,0.9444,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,72,1.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0625,"x":0.15179,"p":[[0,132,0.0,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,132,0.0303,0.12054,0.06298,0.14286,0.14286,0.14286,0.0,0.2857,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,132,0.0606,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,132,0.0909,0.15179,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,23,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,132,0.1212,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,132,0.1515,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,132,0.1818,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,132,0.2121,0.14286,0.10102,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,22,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,132,0.2424,0.12937,0.05484,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,132,0.2727,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,132,0.303,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,132,0.3333,0.10714,0.07986,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,132,0.3636,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,132,0.3939,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,132,0.4242,0.11161,0.09268,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,132,0.4545,0.12482,0.09939,0.105,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[64,132,0.4848,0.125,0.08564,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,132,0.5152,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,132,0.5455,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,132,0.5758,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,1,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,132,0.6061,0.10714,0.07143,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,132,0.6364,0.10268,0.08171,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,132,0.6667,0.11161,0.07771,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,132,0.697,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,132,0.7273,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,132,0.7576,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,132,0.7879,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,132,0.8182,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,132,0.8485,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[116,132,0.8788,0.125,0.08564,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[120,132,0.9091,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,132,0.9394,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,132,0.9697,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,132,1.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"39a3fa1c439912c2","q":"Let  $T$  a set with 2007 points on the plane, without any 3 collinear points.\r\nLet  $P$  any point which belongs to  $T$ . \r\nProve that the number of triangles that contains the point  $P$  inside and \r\nits vertices are from  $T$ , is even.","t":[{"b":1,"e":0.2857,"k":"rising","v":0.04464,"x":0.24097,"p":[[0,33,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,2,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.24097,0.21854,0.0,0.2857,0.42857,0.0,1.0,9,1,0,9,0,6,0,0,8,0,0,7,0,0,1,0,0,0,0,0,0,0,1],[8,33,0.2424,0.13829,0.18026,0.0,0.0,0.2857,0.0,0.57143,18,0,0,18,0,3,0,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[12,33,0.3636,0.09375,0.14555,0.0,0.0,0.17857,0.0,0.57143,21,0,0,21,0,3,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,33,0.4848,0.15179,0.18536,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,1,0,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[20,33,0.6061,0.08482,0.13767,0.0,0.0,0.2857,0.0,0.42857,23,0,0,23,0,0,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.10714,0.13832,0.0,0.0,0.1786,0.0,0.42857,18,0,0,18,0,6,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.11607,0.14913,0.0,0.0,0.2857,0.0,0.42857,18,0,0,18,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.16071,0.15872,0.0,0.2143,0.28571,0.0,0.42857,15,0,0,15,0,1,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.21429,0.16752,0.0,0.2143,0.42857,0.0,0.4286,9,0,0,9,0,7,0,0,7,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"rising","v":0.05357,"x":0.35704,"p":[[0,37,0.0,0.05357,0.18123,0.0,0.0,0.0,0.0,1.0,27,1,6,27,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,37,0.1081,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.12053,0.16409,0.0,0.0,0.17857,0.0,0.57143,18,0,0,18,0,6,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[12,37,0.3243,0.20534,0.21703,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,4,0,0,4,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[16,37,0.4324,0.2857,0.22302,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,4,0,0,8,0,0,7,0,0,2,0,0,3,0,0,0,0,0],[20,37,0.5405,0.26337,0.24768,0.0,0.2857,0.42857,0.0,0.71429,12,0,0,12,0,3,0,0,4,0,0,7,0,0,3,0,0,3,0,0,0,0,0],[24,37,0.6486,0.30801,0.18931,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,6,0,0,8,0,0,11,0,0,2,0,0,0,0,0,1,0,0],[28,37,0.7568,0.3214,0.26242,0.10714,0.28571,0.571,0.0,1.0,8,1,0,8,0,3,0,0,9,0,0,3,0,0,5,0,0,3,0,0,0,0,1],[32,37,0.8649,0.35704,0.17138,0.28571,0.35714,0.42858,0.0,0.71429,3,0,0,3,0,1,0,0,12,0,0,11,0,0,3,0,0,2,0,0,0,0,0],[36,37,0.973,0.27236,0.16892,0.14286,0.2857,0.42857,0.0,0.57143,7,0,0,7,0,2,0,0,11,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[37,37,1.0,0.28572,0.22868,0.14286,0.2857,0.42857,0.0,0.85714,6,0,0,6,0,8,0,0,6,0,0,9,0,0,0,0,0,1,0,0,2,0,0]]}]},{"i":"0d9d96f8c08138be","q":"Let  $S$  be the set of all positive integers of the form  $19a+85b$ , where  $a,b$  are arbitrary positive integers. On the real axis, the points of  $S$  are colored in red and the remaining integer numbers are colored in green. Find, with proof, whether or not there exists a point  $A$  on the real axis such that any two points with integer coordinates which are symmetrical with respect to  $A$  have necessarily distinct colors.","t":[{"b":2,"e":1.0,"k":"rising","v":0.51339,"x":0.99554,"p":[[0,83,0.0,0.60713,0.43301,0.14286,0.92857,1.0,0.0,1.0,5,16,0,5,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,16],[4,83,0.0482,0.73214,0.37585,0.28571,1.0,1.0,0.0,1.0,1,20,0,1,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,20],[8,83,0.0964,0.51339,0.44011,0.14286,0.28573,1.0,0.0,1.0,6,13,0,6,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,13],[12,83,0.1446,0.5892,0.39255,0.14286,0.64286,1.0,0.0,1.0,1,13,0,1,0,10,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,13],[16,83,0.1928,0.66072,0.37072,0.14289,0.85714,1.0,0.14286,1.0,0,16,0,0,0,9,0,0,0,0,0,2,0,0,4,0,0,1,0,0,0,0,16],[20,83,0.241,0.75884,0.35088,0.57143,1.0,1.0,0.0,1.0,2,19,0,2,0,4,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,19],[24,83,0.2892,0.5223,0.43391,0.14286,0.4998,1.0,0.0,1.0,6,13,0,6,0,9,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,13],[28,83,0.3373,0.54911,0.39627,0.14286,0.57143,1.0,0.0,1.0,4,11,0,4,0,8,0,0,1,0,0,1,0,0,3,0,0,3,0,0,1,0,11],[32,83,0.3855,0.59822,0.39357,0.14286,0.57143,1.0,0.0,1.0,3,14,0,3,0,6,0,0,2,0,0,4,0,0,2,0,0,0,0,0,1,0,14],[36,83,0.4337,0.79015,0.30303,0.57143,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,0,0,0,0,0,0,5,0,0,2,0,0,3,0,18],[40,83,0.4819,0.83927,0.31085,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,24],[44,83,0.5301,0.82589,0.34944,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,25],[48,83,0.5783,0.69195,0.39948,0.25,1.0,1.0,0.0,1.0,2,19,0,2,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,19],[52,83,0.6265,0.60713,0.41802,0.14286,0.85714,1.0,0.0,1.0,3,15,0,3,0,9,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,15],[56,83,0.6747,0.79018,0.31941,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,20],[60,83,0.7229,0.69641,0.37585,0.28571,1.0,1.0,0.0,1.0,1,18,0,1,0,6,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,18],[64,83,0.7711,0.82587,0.28063,0.67857,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,2,0,0,1,0,0,3,0,0,1,0,0,2,0,21],[68,83,0.8193,0.85714,0.26,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,23],[72,83,0.8675,0.93304,0.1636,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,26],[76,83,0.9157,0.93749,0.18192,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,28],[80,83,0.9639,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[83,83,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":0.0,"k":"falling","v":0.16053,"x":0.83482,"p":[[0,82,0.0,0.63839,0.42706,0.14286,1.0,1.0,0.0,1.0,4,18,1,4,0,6,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,18],[4,82,0.0488,0.83482,0.31966,0.96425,1.0,1.0,0.0,1.0,1,24,0,1,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,24],[8,82,0.0976,0.79462,0.30917,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,3,0,0,0,0,0,1,0,0,4,0,0,1,0,0,3,0,19],[12,82,0.1463,0.79017,0.32141,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,3,0,0,1,0,0,0,0,0,5,0,0,1,0,0,0,0,21],[16,82,0.1951,0.79902,0.34984,0.71429,1.0,1.0,0.0,1.0,2,23,0,2,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,23],[20,82,0.2439,0.77679,0.37276,0.75001,1.0,1.0,0.0,1.0,2,22,0,2,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,22],[24,82,0.2927,0.60259,0.41619,0.14286,0.78571,1.0,0.0,1.0,5,14,0,5,0,6,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,14],[28,82,0.3415,0.65624,0.38109,0.14286,0.85714,1.0,0.0,1.0,2,15,0,2,0,7,0,0,0,0,0,1,0,0,5,0,0,0,0,0,2,0,15],[32,82,0.3902,0.70536,0.40867,0.14289,1.0,1.0,0.0,1.0,3,20,0,3,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,20],[36,82,0.439,0.56247,0.3976,0.14286,0.57121,1.0,0.0,1.0,2,13,0,2,0,10,0,0,2,0,0,0,0,0,4,0,0,1,0,0,0,0,13],[40,82,0.4878,0.55357,0.39569,0.14286,0.42859,1.0,0.0,1.0,2,13,0,2,0,9,0,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,13],[44,82,0.5366,0.58036,0.42996,0.14286,0.78571,1.0,0.0,1.0,6,14,0,6,0,6,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,14],[48,82,0.5854,0.57141,0.43595,0.14286,0.78571,1.0,0.0,1.0,6,14,0,6,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,14],[52,82,0.6341,0.60268,0.40206,0.14286,0.78571,1.0,0.0,1.0,2,14,0,2,0,8,0,0,4,0,0,0,0,0,1,0,0,1,0,0,2,0,14],[56,82,0.6829,0.55795,0.44809,0.105,0.71429,1.0,0.0,1.0,8,15,0,8,0,5,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,15],[60,82,0.7317,0.60268,0.39405,0.14286,0.57143,1.0,0.0,1.0,2,14,0,2,0,9,0,0,0,0,0,2,0,0,4,0,0,0,0,0,1,0,14],[64,82,0.7805,0.65616,0.41177,0.14286,0.92857,1.0,0.0,1.0,3,16,0,3,0,8,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,16],[68,82,0.8293,0.50446,0.42556,0.14286,0.28571,1.0,0.0,1.0,6,11,0,6,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0,3,0,11],[72,82,0.878,0.61159,0.39324,0.14286,0.71429,1.0,0.0,1.0,3,14,0,3,0,7,0,0,1,0,0,1,0,0,3,0,0,3,0,0,0,0,14],[76,82,0.9268,0.63393,0.39113,0.14289,0.78571,1.0,0.0,1.0,2,15,0,2,0,7,0,0,2,0,0,2,0,0,1,0,0,2,0,0,1,0,15],[80,82,0.9756,0.42857,0.35892,0.14286,0.28571,0.64282,0.0,1.0,4,7,0,4,0,9,0,0,5,0,0,2,0,0,4,0,0,0,0,0,1,0,7],[82,82,1.0,0.16053,0.13242,0.14214,0.14286,0.1429,0.0,0.57143,6,0,0,6,0,20,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"cb385bf747d3c1b5","q":"Find the largest positive integer  $n$  for which the inequality \n\\[ \\frac{a+b+c}{abc+1}+\\sqrt[n]{abc} \\leq \\frac{5}{2}\\]\nholds true for all  $a, b, c \\in [0,1]$ . Here we make the convention  $\\sqrt[1]{abc}=abc$ .","t":[{"b":5,"e":0.71429,"k":"flat","v":0.80804,"x":0.83929,"p":[[0,26,0.0,0.82142,0.21428,0.71429,0.85707,1.0,0.0,1.0,1,15,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,2,0,15],[4,26,0.1538,0.8125,0.25111,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,3,0,15],[8,26,0.3077,0.8125,0.16536,0.71429,0.78571,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,4,0,12],[12,26,0.4615,0.83929,0.19804,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,0,0,18],[16,26,0.6154,0.83481,0.15201,0.71429,0.71429,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,1,0,14],[20,26,0.7692,0.80804,0.1411,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,3,0,10],[24,26,0.9231,0.82588,0.15461,0.71429,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,2,0,13],[26,26,1.0,0.8125,0.14914,0.71429,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,3,0,11]]},{"b":7,"e":0.71429,"k":"flat","v":0.79464,"x":0.84821,"p":[[0,5,0.0,0.84821,0.15947,0.71429,0.85714,1.0,0.5714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,4,0,15],[4,5,0.8,0.83482,0.14334,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,8,0,11],[5,5,1.0,0.79464,0.13803,0.71429,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,8,0,7]]}]},{"i":"c2d604c21c259657","q":"Let $a, b$ and $c$ be positive integers, all distinct, and suppose that $p=ab+bc+ca$ is a prime number.\na) Prove that $a^{2}, b^{2}$ and $c^{2}$ give different remainders when divided by $p$.\nb) Prove that $a^{3}, b^{3}$ and $c^{3}$ give different remainders when divided by $p$.","t":[{"b":1,"e":0.85714,"k":"rising","v":0.69196,"x":0.94196,"p":[[0,16,0.0,0.69196,0.2699,0.42857,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,1,0,0,13,0,0,3,0,0,0,0,0,3,0,12],[4,16,0.25,0.89731,0.17943,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,6,0,21],[8,16,0.5,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[12,16,0.75,0.89285,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[16,16,1.0,0.85267,0.06667,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,3]]},{"b":7,"e":0.42857,"k":"falling","v":0.42857,"x":0.75446,"p":[[0,27,0.0,0.69195,0.28147,0.42857,0.64264,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,13,0,0,1,0,0,1,0,0,2,0,13],[4,27,0.1481,0.73659,0.28147,0.42857,0.85707,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,9,0,0,2,0,0,2,0,0,2,0,15],[8,27,0.2963,0.75446,0.27254,0.42857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,11,0,0,0,0,0,2,0,0,2,0,16],[12,27,0.4444,0.53125,0.2237,0.42857,0.42857,0.46431,0.2857,1.0,0,4,0,0,0,0,0,0,3,0,0,21,0,0,1,0,0,0,0,0,3,0,4],[16,27,0.5926,0.45094,0.10169,0.42857,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,26,0,0,2,0,0,1,0,0,1,0,0],[20,27,0.7407,0.43304,0.05629,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,30,0,0,0,0,0,1,0,0,0,0,0],[24,27,0.8889,0.44648,0.09942,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,1],[27,27,1.0,0.42857,1e-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":"1a6982d671c1a5ea","q":"Prove or disprove the following statement. If  $g:(0,1) \\to (0,1)$  is an increasing function and satisfies $g(x) > x$  for all  $x \\in (0,1)$ , then there exists a continuous function  $f:(0,1) \\to \\mathbb{R}$  satisfying  $f(x) < f(g(x)) $  for all  $x \\in (0,1)$ , but  $f$  is not an increasing function.","t":[{"b":1,"e":0.42857,"k":"rising","v":0.29454,"x":0.625,"p":[[0,56,0.0,0.29454,0.33684,0.0,0.14286,0.57111,0.0,1.0,13,2,7,13,0,6,0,0,1,0,0,3,0,0,2,0,0,3,0,0,2,0,2],[4,56,0.0714,0.58036,0.29867,0.42857,0.71429,0.71429,0.0,1.0,4,4,0,4,0,2,0,0,0,0,0,5,0,0,0,0,0,17,0,0,0,0,4],[8,56,0.1429,0.55356,0.3004,0.42859,0.71429,0.71429,0.0,1.0,6,1,0,6,0,1,0,0,0,0,0,2,0,0,2,0,0,18,0,0,2,0,1],[12,56,0.2143,0.625,0.25692,0.42857,0.71429,0.71429,0.0,1.0,1,4,0,1,0,3,0,0,1,0,0,4,0,0,1,0,0,17,0,0,1,0,4],[16,56,0.2857,0.625,0.26426,0.71429,0.71429,0.71429,0.0,1.0,3,2,0,3,0,2,0,0,0,0,0,1,0,0,1,0,0,21,0,0,2,0,2],[20,56,0.3571,0.45536,0.35072,0.0,0.71429,0.71429,0.0,1.0,10,2,0,10,0,1,0,0,2,0,0,1,0,0,1,0,0,14,0,0,1,0,2],[24,56,0.4286,0.49997,0.36069,0.0,0.71429,0.71429,0.0,1.0,10,3,0,10,0,0,0,0,0,0,0,2,0,0,3,0,0,11,0,0,3,0,3],[28,56,0.5,0.47768,0.34369,0.0,0.71429,0.71429,0.0,1.0,9,2,0,9,0,1,0,0,1,0,0,3,0,0,1,0,0,13,0,0,2,0,2],[32,56,0.5714,0.52232,0.30849,0.35714,0.71429,0.71429,0.0,1.0,6,2,0,6,0,2,0,0,0,0,0,3,0,0,4,0,0,14,0,0,1,0,2],[36,56,0.6429,0.50444,0.24739,0.42857,0.57143,0.71429,0.0,0.71429,4,0,0,4,0,2,0,0,0,0,0,7,0,0,5,0,0,14,0,0,0,0,0],[40,56,0.7143,0.56694,0.25873,0.571,0.71429,0.71429,0.0,0.71429,4,0,0,4,0,2,0,0,0,0,0,1,0,0,3,0,0,22,0,0,0,0,0],[44,56,0.7857,0.53121,0.25811,0.42857,0.64286,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,1,0,0,6,0,0,4,0,0,15,0,0,0,0,1],[48,56,0.8571,0.62044,0.16216,0.571,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,0,0,0,1,0,0,10,0,0,19,0,0,0,0,0],[52,56,0.9286,0.50893,0.23673,0.42857,0.57143,0.71429,0.0,0.71429,3,0,0,3,0,2,0,0,1,0,0,9,0,0,2,0,0,15,0,0,0,0,0],[56,56,1.0,0.54448,0.1532,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,10,0,0,10,0,0,10,0,0,0,0,0]]},{"b":2,"e":0.571,"k":"flat","v":0.10714,"x":0.61159,"p":[[0,42,0.0,0.37043,0.37438,0.0,0.21429,0.71429,0.0,1.0,11,4,4,11,0,5,0,0,2,0,0,3,0,0,1,0,0,3,0,0,3,0,4],[4,42,0.0952,0.61159,0.26782,0.67857,0.71429,0.71429,0.0,1.0,4,1,0,4,0,1,0,0,0,0,0,1,0,0,2,0,0,20,0,0,3,0,1],[8,42,0.1905,0.55357,0.31288,0.25001,0.71429,0.71429,0.0,1.0,5,1,0,5,0,3,0,0,1,0,0,1,0,0,0,0,0,17,0,0,4,0,1],[12,42,0.2857,0.46874,0.36287,0.0,0.71429,0.71429,0.0,1.0,11,3,0,11,0,0,0,0,0,0,0,3,0,0,1,0,0,13,0,0,1,0,3],[16,42,0.381,0.4107,0.36899,0.0,0.42859,0.71429,0.0,1.0,12,4,0,12,0,1,0,0,1,0,0,3,0,0,4,0,0,6,0,0,1,0,4],[20,42,0.4762,0.33034,0.36146,0.0,0.0,0.71429,0.0,1.0,17,1,0,17,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,1,0,1],[24,42,0.5714,0.47322,0.35792,0.0,0.71429,0.71429,0.0,1.0,10,3,0,10,0,1,0,0,0,0,0,3,0,0,1,0,0,13,0,0,1,0,3],[28,42,0.6667,0.40177,0.36497,0.0,0.42857,0.71429,0.0,1.0,12,3,0,12,0,2,0,0,1,0,0,2,0,0,1,0,0,11,0,0,0,0,3],[32,42,0.7619,0.36159,0.33499,0.0,0.42857,0.71429,0.0,1.0,12,2,0,12,0,2,0,0,1,0,0,5,0,0,2,0,0,8,0,0,0,0,2],[36,42,0.8571,0.15624,0.2865,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0],[40,42,0.9524,0.10714,0.25,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0],[42,42,1.0,0.44643,0.22517,0.42857,0.42857,0.71429,0.0,0.71429,4,0,0,4,0,1,0,0,2,0,0,14,0,0,2,0,0,9,0,0,0,0,0]]}]},{"i":"ef914079d5bae7ab","q":"Given a sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of non-negative real numbers satisfying the conditions\n\n(1) $a_{n}+a_{2 n} \\geq 3 n$\n\n(2) $a_{n+1}+n \\leq 2 \\sqrt{a_{n} \\cdot(n+1)}$\n\nfor all indices $n=1,2 \\ldots$\n\n(a) Prove that the inequality $a_{n} \\geq n$ holds for every $n \\in \\mathbb{N}$.\n\n(b) Give an example of such a sequence.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.26786,"x":0.32589,"p":[[0,92,0.0,0.28126,0.05629,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,92,0.0435,0.27232,0.07457,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,92,0.087,0.29018,0.07563,0.28571,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,92,0.1304,0.30803,0.12428,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,92,0.1739,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,92,0.2174,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,92,0.2609,0.27678,0.07936,0.28571,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,92,0.3043,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,92,0.3478,0.2991,0.04164,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,92,0.3913,0.28571,0.06186,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,92,0.4348,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,92,0.4783,0.32589,0.12993,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[48,92,0.5217,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[52,92,0.5652,0.28571,1e-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],[56,92,0.6087,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,92,0.6522,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,92,0.6957,0.3125,0.10374,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[68,92,0.7391,0.28571,1e-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],[72,92,0.7826,0.28571,1e-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],[76,92,0.8261,0.28571,1e-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],[80,92,0.8696,0.28571,1e-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],[84,92,0.913,0.28571,1e-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],[88,92,0.9565,0.28571,1e-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],[92,92,1.0,0.28124,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.27678,"x":0.34822,"p":[[0,99,0.0,0.34375,0.16312,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,27,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[4,99,0.0404,0.33036,0.1729,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,99,0.0808,0.29464,0.11259,0.28571,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,99,0.1212,0.30803,0.1017,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,99,0.1616,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,99,0.202,0.28571,1e-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],[24,99,0.2424,0.27678,0.07936,0.28571,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,99,0.2828,0.29464,0.11259,0.28571,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[32,99,0.3232,0.3125,0.10374,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[36,99,0.3636,0.30803,0.1017,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,99,0.404,0.28125,0.05629,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,99,0.4444,0.34822,0.18877,0.28571,0.28571,0.28571,0.143,1.0,0,1,0,0,0,1,0,0,27,0,0,0,0,0,1,0,0,0,0,0,2,0,1],[48,99,0.4848,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,99,0.5253,0.3125,0.10374,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[56,99,0.5657,0.30804,0.1017,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[60,99,0.6061,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,99,0.6465,0.28571,1e-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],[68,99,0.6869,0.33482,0.13175,0.28571,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[72,99,0.7273,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,99,0.7677,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,99,0.8081,0.28572,5e-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],[84,99,0.8485,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,99,0.8889,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,99,0.9293,0.28571,1e-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],[96,99,0.9697,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[99,99,1.0,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dc02594f7a0375c3","q":"Let  $F_1=1, F_2=1,$  and  $F_{n+2}=F_{n+1}+F_n.$  Then,   $$ S = \\sum_{n=1}^{\\infty} \\arctan\\left(\\frac{1}{F_n}\\right)\\arctan\\left(\\frac{1}{F_{n+1}}\\right) $$   Find  $\\lfloor 80S \\rfloor.$ (Hint: it may be useful to note that  $\\arctan(\\tfrac{1}{1}) = \\arctan(\\tfrac{1}{2})+\\arctan(\\tfrac{1}{3}).$ )","t":[{"b":0,"e":1.0,"k":"rising","v":0.70536,"x":1.0,"p":[[0,30,0.0,0.70536,0.39276,0.28571,1.0,1.0,0.0,1.0,4,18,0,4,0,3,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,18],[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],[8,30,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,30,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[20,30,0.6667,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":4,"e":1.0,"k":"rising","v":0.54018,"x":1.0,"p":[[0,46,0.0,0.54018,0.41762,0.14286,0.64286,1.0,0.0,1.0,6,11,0,6,0,6,0,0,3,0,0,0,0,0,1,0,0,2,0,0,3,0,11],[4,46,0.087,0.76339,0.33045,0.64286,0.92857,1.0,0.0,1.0,1,16,0,1,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,0,7,0,16],[8,46,0.1739,0.92411,0.21424,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,26],[12,46,0.2609,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,46,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,46,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,46,0.7826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,46,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]}]},{"i":"e65ba3d1a1b28dd1","q":"A  $3\\times3$  grid is to be painted with three colors (red, green, and blue) such that\n[list=i]\n[*] no two squares that share an edge are the same color and\n[*] no two corner squares on the same edge of the grid have the same color.\n[/list]\nAs an example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? (Rotations and reflections are considered distinct colorings.)","t":[{"b":3,"e":0.0,"k":"falling","v":0.12501,"x":0.75,"p":[[0,34,0.0,0.63393,0.34615,0.39286,0.71429,1.0,0.0,1.0,1,12,0,1,0,6,0,0,1,0,0,5,0,0,0,0,0,7,0,0,0,0,12],[4,34,0.1176,0.75,0.32733,0.53571,1.0,1.0,0.14286,1.0,0,18,0,0,0,5,0,0,1,0,0,2,0,0,1,0,0,5,0,0,0,0,18],[8,34,0.2353,0.64286,0.30305,0.42857,0.71429,0.78571,0.14286,1.0,0,8,0,0,0,7,0,0,0,0,0,2,0,0,0,0,0,15,0,0,0,0,8],[12,34,0.3529,0.46866,0.36991,0.14286,0.35714,0.71429,0.0,1.0,3,7,0,3,0,12,0,0,1,0,0,1,0,0,1,0,0,7,0,0,0,0,7],[16,34,0.4706,0.25,0.26964,0.14286,0.14286,0.32143,0.0,1.0,7,1,0,7,0,16,0,0,1,0,0,2,0,0,0,0,0,5,0,0,0,0,1],[20,34,0.5882,0.17411,0.19475,0.10714,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,19,0,0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[24,34,0.7059,0.12501,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.17857,0.15972,0.14286,0.14286,0.14287,0.0,0.71429,6,0,0,6,0,20,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[32,34,0.9412,0.17838,0.17129,0.14214,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,21,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[34,34,1.0,0.13831,0.145,0.0,0.14286,0.14286,0.0,0.71429,9,0,0,9,0,20,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.61607,"x":0.82589,"p":[[0,10,0.0,0.61607,0.39357,0.14286,0.71429,1.0,0.0,1.0,2,14,0,2,0,8,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,14],[4,10,0.4,0.67857,0.35714,0.35714,0.71429,1.0,0.0,1.0,1,14,0,1,0,7,0,0,0,0,0,2,0,0,0,0,0,7,0,0,1,0,14],[8,10,0.8,0.82589,0.23619,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,16],[10,10,1.0,0.77678,0.17105,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,9]]}]},{"i":"215919579ef644ae","q":"Two circles $k_{1}$ and $k_{2}$ intersect a.t points $A$ and $B$. A circle $k_{3}$ centered at $A$ meet $k_{1}$ at $M$ and $P$ and $k_{2}$ at $N$ and $Q$, such that $N$ and $Q$ are on different sides of $M P$ and $A B>A M$.\n\nProve rhat the angles $\\angle M B Q$ and $\\angle N B P$ are equal.","t":[{"b":3,"e":0.28571,"k":"falling","v":0.05804,"x":0.62946,"p":[[0,66,0.0,0.62946,0.36045,0.2857,0.42857,1.0,0.0,1.0,2,15,1,2,0,0,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,15],[4,66,0.0606,0.32589,0.29502,0.10714,0.2857,0.42857,0.0,1.0,8,4,0,8,0,1,0,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,4],[8,66,0.1212,0.31696,0.20434,0.2857,0.28571,0.28571,0.0,1.0,2,2,0,2,0,4,0,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,2],[12,66,0.1818,0.30357,0.17405,0.2857,0.28571,0.42857,0.0,1.0,3,1,0,3,0,3,0,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[16,66,0.2424,0.23662,0.18766,0.14286,0.2857,0.28571,0.0,1.0,7,1,0,7,0,4,0,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[20,66,0.303,0.26786,0.27141,0.10714,0.2857,0.28571,0.0,1.0,8,3,0,8,0,6,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[24,66,0.3636,0.26339,0.27458,0.0,0.2857,0.28571,0.0,1.0,9,3,0,9,0,5,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[28,66,0.4242,0.34821,0.27418,0.2857,0.28571,0.42857,0.0,1.0,4,4,0,4,0,3,0,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,4],[32,66,0.4848,0.28125,0.19393,0.14289,0.28571,0.28571,0.0,1.0,4,1,0,4,0,5,0,0,17,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[36,66,0.5455,0.16964,0.12595,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,9,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,66,0.6061,0.27678,0.24206,0.14286,0.2857,0.28571,0.0,1.0,6,2,0,6,0,6,0,0,14,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[44,66,0.6667,0.24553,0.2402,0.0,0.2857,0.28571,0.0,1.0,9,2,0,9,0,4,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[48,66,0.7273,0.2366,0.22192,0.10714,0.2857,0.28571,0.0,1.0,8,1,0,8,0,6,0,0,14,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[52,66,0.7879,0.16517,0.14334,0.0,0.2857,0.2857,0.0,0.42857,13,0,0,13,0,2,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,66,0.8485,0.21428,0.19561,0.10714,0.2143,0.2857,0.0,1.0,8,1,0,8,0,8,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[60,66,0.9091,0.19643,0.1948,0.0,0.2143,0.28571,0.0,1.0,10,1,0,10,0,6,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[64,66,0.9697,0.1116,0.1461,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"falling","v":0.05358,"x":0.66518,"p":[[0,62,0.0,0.66518,0.3722,0.28571,1.0,1.0,0.0,1.0,3,17,2,3,0,0,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,17],[4,62,0.0645,0.20536,0.2141,0.0,0.21428,0.28571,0.0,1.0,12,1,0,12,0,4,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[8,62,0.129,0.25437,0.1915,0.14286,0.2857,0.28571,0.0,1.0,6,1,0,6,0,5,0,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[12,62,0.1935,0.23214,0.24936,0.0,0.2857,0.28571,0.0,1.0,12,2,0,12,0,1,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[16,62,0.2581,0.24991,0.27203,0.0,0.2857,0.28571,0.0,1.0,9,3,0,9,0,6,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[20,62,0.3226,0.20982,0.14279,0.10714,0.2857,0.28571,0.0,0.42857,8,0,0,8,0,5,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,62,0.3871,0.19643,0.14617,0.0,0.2857,0.28571,0.0,0.4286,10,0,0,10,0,3,0,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,62,0.4516,0.24107,0.22428,0.0,0.2857,0.28571,0.0,1.0,11,1,0,11,0,0,0,0,15,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[32,62,0.5161,0.27678,0.19865,0.24999,0.2857,0.42857,0.0,1.0,7,1,0,7,0,1,0,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[36,62,0.5806,0.18741,0.12081,0.105,0.2857,0.2857,0.0,0.28571,8,0,0,8,0,6,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.24554,0.20896,0.0,0.2857,0.32143,0.0,1.0,9,1,0,9,0,3,0,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[44,62,0.7097,0.24553,0.19638,0.10714,0.2857,0.28571,0.0,1.0,8,1,0,8,0,2,0,0,17,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[48,62,0.7742,0.19643,0.15047,0.0,0.2857,0.28571,0.0,0.4286,10,0,0,10,0,4,0,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[52,62,0.8387,0.15625,0.13054,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,8,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,62,0.9032,0.12501,0.14617,0.0,0.0,0.2857,0.0,0.42857,18,0,0,18,0,1,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.05358,0.09943,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[62,62,1.0,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"87541afa85fde78d","q":"In the triangle  $ABC$  its incircle with center  $I$  touches its sides  $BC, CA$  and  $AB$  in the points  $A_1, B_1, C_1$  respectively. Through  $I$  is drawn a line  $\\ell$ . The points  $A', B'$  and  $C'$  are reflections of  $A_1, B_1, C_1$  with respect to the line  $\\ell$ . Prove that the lines  $AA', BB'$  and  $CC'$  intersects at a common point.","t":[{"b":0,"e":0.0,"k":"falling","v":0.04464,"x":0.58482,"p":[[0,60,0.0,0.45088,0.26989,0.28571,0.42859,0.71429,0.0,1.0,6,1,1,6,0,0,0,0,4,0,0,7,0,0,5,0,0,9,0,0,0,0,1],[4,60,0.0667,0.58482,0.36832,0.42857,0.64286,1.0,0.0,1.0,7,10,0,7,0,0,0,0,0,0,0,5,0,0,4,0,0,6,0,0,0,0,10],[8,60,0.1333,0.45536,0.4,0.0,0.42857,0.89286,0.0,1.0,10,8,0,10,0,2,0,0,3,0,0,3,0,0,2,0,0,3,0,0,1,0,8],[12,60,0.2,0.39732,0.39727,0.0,0.42857,0.71429,0.0,1.0,14,5,0,14,0,1,0,0,0,0,0,4,0,0,1,0,0,5,0,0,2,0,5],[16,60,0.2667,0.52232,0.40344,0.0,0.4286,1.0,0.0,1.0,9,10,0,9,0,1,0,0,0,0,0,7,0,0,2,0,0,1,0,0,2,0,10],[20,60,0.3333,0.47767,0.425,0.0,0.49979,1.0,0.0,1.0,12,9,0,12,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,0,1,0,9],[24,60,0.4,0.45536,0.40475,0.0,0.42857,0.89286,0.0,1.0,11,8,0,11,0,1,0,0,2,0,0,5,0,0,0,0,0,4,0,0,1,0,8],[28,60,0.4667,0.48213,0.3549,0.14286,0.4286,0.75,0.0,1.0,7,5,0,7,0,3,0,0,1,0,0,6,0,0,3,0,0,4,0,0,3,0,5],[32,60,0.5333,0.37945,0.40502,0.0,0.35714,0.71429,0.0,1.0,15,7,0,15,0,0,0,0,1,0,0,5,0,0,1,0,0,3,0,0,0,0,7],[36,60,0.6,0.44642,0.40049,0.0,0.42857,0.89286,0.0,1.0,11,8,0,11,0,1,0,0,2,0,0,5,0,0,2,0,0,2,0,0,1,0,8],[40,60,0.6667,0.31247,0.31426,0.0,0.35714,0.571,0.0,1.0,14,2,0,14,0,0,0,0,2,0,0,7,0,0,4,0,0,3,0,0,0,0,2],[44,60,0.7333,0.3616,0.34806,0.0,0.42857,0.57143,0.0,1.0,12,4,0,12,0,2,0,0,1,0,0,6,0,0,4,0,0,3,0,0,0,0,4],[48,60,0.8,0.28124,0.34345,0.0,0.07143,0.57143,0.0,1.0,16,3,0,16,0,2,0,0,3,0,0,1,0,0,4,0,0,3,0,0,0,0,3],[52,60,0.8667,0.24107,0.33396,0.0,0.0,0.42858,0.0,1.0,18,3,0,18,0,2,0,0,2,0,0,3,0,0,2,0,0,2,0,0,0,0,3],[56,60,0.9333,0.10268,0.17215,0.0,0.0,0.17857,0.0,0.42857,23,0,0,23,0,1,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.04464,0.12078,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"falling","v":0.33929,"x":0.54016,"p":[[0,41,0.0,0.50446,0.2789,0.39285,0.57143,0.71429,0.0,1.0,5,1,1,5,0,1,0,0,2,0,0,6,0,0,3,0,0,13,0,0,1,0,1],[4,41,0.0976,0.49999,0.37115,0.0,0.57143,0.75,0.0,1.0,9,5,0,9,0,1,0,0,1,0,0,3,0,0,3,0,0,7,0,0,3,0,5],[8,41,0.1951,0.52232,0.39059,0.10714,0.50001,1.0,0.0,1.0,8,9,0,8,0,2,0,0,0,0,0,6,0,0,3,0,0,2,0,0,2,0,9],[12,41,0.2927,0.48661,0.38276,0.0,0.4286,0.75,0.0,1.0,9,7,0,9,0,2,0,0,0,0,0,6,0,0,1,0,0,6,0,0,1,0,7],[16,41,0.3902,0.54016,0.37582,0.14286,0.57143,1.0,0.0,1.0,6,9,0,6,0,3,0,0,2,0,0,3,0,0,4,0,0,4,0,0,1,0,9],[20,41,0.4878,0.47322,0.4141,0.0,0.42859,0.89286,0.0,1.0,11,8,0,11,0,1,0,0,2,0,0,4,0,0,0,0,0,3,0,0,3,0,8],[24,41,0.5854,0.42856,0.31541,0.14289,0.42857,0.60714,0.0,1.0,6,4,0,6,0,3,0,0,3,0,0,10,0,0,2,0,0,3,0,0,1,0,4],[28,41,0.6829,0.33929,0.29827,0.0,0.35714,0.46429,0.0,1.0,10,2,0,10,0,2,0,0,4,0,0,8,0,0,2,0,0,4,0,0,0,0,2],[32,41,0.7805,0.3973,0.3383,0.0,0.42857,0.57111,0.0,1.0,9,3,0,9,0,3,0,0,1,0,0,9,0,0,3,0,0,0,0,0,4,0,3],[36,41,0.878,0.41516,0.34875,0.0,0.42857,0.60682,0.0,1.0,9,5,0,9,0,1,0,0,3,0,0,10,0,0,1,0,0,1,0,0,2,0,5],[40,41,0.9756,0.34371,0.23648,0.14286,0.28571,0.42858,0.0,1.0,3,1,0,3,0,8,0,0,7,0,0,7,0,0,4,0,0,1,0,0,1,0,1],[41,41,1.0,0.35268,0.22299,0.24999,0.35714,0.46429,0.0,0.85714,5,0,0,5,0,3,0,0,8,0,0,8,0,0,5,0,0,2,0,0,1,0,0]]}]},{"i":"54cec8a3fed8c770","q":"There are  $10$  students in a class planning to hold several parties during the summer vacation. Each student can attend at most three parties, and any two students must meet at least once in some party. Find the largest positive integer \\( m \\) such that no matter how the parties are arranged under the above conditions, there must be at least one party with at least \\( m \\) participants.\n*Proposed by Wang Huixing*","t":[{"b":0,"e":0.28571,"k":"flat","v":0.125,"x":0.28571,"p":[[0,110,0.0,0.21409,0.15977,0.14286,0.14286,0.28571,0.0,0.71429,4,0,2,4,0,15,0,0,10,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[4,110,0.0364,0.28571,0.18211,0.24999,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,3,0,0,17,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[8,110,0.0727,0.25446,0.17029,0.2857,0.28571,0.28571,0.0,1.0,5,1,0,5,0,2,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,110,0.1091,0.2366,0.16982,0.14286,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,3,0,0,20,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[16,110,0.1455,0.17856,0.16748,0.0,0.2857,0.28571,0.0,0.57143,13,0,0,13,0,2,0,0,15,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[20,110,0.1818,0.20535,0.19541,0.0,0.2857,0.28571,0.0,1.0,11,1,0,11,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,110,0.2182,0.16955,0.13572,0.0,0.21428,0.28571,0.0,0.42857,11,0,0,11,0,5,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,110,0.2545,0.1875,0.13092,0.0,0.2857,0.28571,0.0,0.28571,10,0,0,10,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,110,0.2909,0.17857,0.13363,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,5,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,110,0.3273,0.21428,0.11845,0.14286,0.28571,0.28571,0.0,0.42857,6,0,0,6,0,5,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,110,0.3636,0.18304,0.16065,0.0,0.21435,0.28571,0.0,0.57143,11,0,0,11,0,5,0,0,14,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[44,110,0.4,0.1383,0.12619,0.0,0.14286,0.28571,0.0,0.28571,13,0,0,13,0,7,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,110,0.4364,0.17411,0.13236,0.0,0.2857,0.28571,0.0,0.28571,11,0,0,11,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,110,0.4727,0.14277,0.12372,0.0,0.14286,0.2857,0.0,0.28571,12,0,0,12,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,110,0.5091,0.18303,0.1197,0.14286,0.14286,0.28571,0.0,0.42857,7,0,0,7,0,10,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,110,0.5455,0.19196,0.14987,0.10714,0.2143,0.28571,0.0,0.71429,8,0,0,8,0,8,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[64,110,0.5818,0.16518,0.12931,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,8,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,110,0.6182,0.125,0.12752,0.0,0.14286,0.28571,0.0,0.28571,15,0,0,15,0,6,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,110,0.6545,0.1875,0.10374,0.14286,0.14286,0.28571,0.0,0.28571,5,0,0,5,0,12,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,110,0.6909,0.16509,0.11357,0.105,0.14286,0.28571,0.0,0.28571,8,0,0,8,0,11,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,110,0.7273,0.17411,0.09932,0.14286,0.14286,0.28571,0.0,0.28571,5,0,0,5,0,15,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,110,0.7636,0.16509,0.11357,0.105,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,11,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,110,0.8,0.16964,0.10374,0.14286,0.14286,0.28571,0.0,0.28571,6,0,0,6,0,14,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,110,0.8364,0.19643,0.10564,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,10,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,110,0.8727,0.17857,0.11845,0.14286,0.14286,0.28571,0.0,0.42857,7,0,0,7,0,11,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,110,0.9091,0.17857,0.10101,0.14286,0.14286,0.28571,0.0,0.28571,5,0,0,5,0,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,110,0.9455,0.17411,0.09268,0.14286,0.14286,0.2857,0.0,0.4286,3,0,0,3,0,20,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[108,110,0.9818,0.21411,0.08002,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,14,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[110,110,1.0,0.20089,0.08645,0.14286,0.1429,0.28571,0.0,0.28571,2,0,0,2,0,15,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.125,"x":0.28571,"p":[[0,133,0.0,0.1875,0.11538,0.14286,0.14286,0.2857,0.0,0.4286,5,0,2,5,0,14,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,133,0.0301,0.28571,0.22304,0.14286,0.28571,0.32143,0.0,1.0,7,1,0,7,0,2,0,0,15,0,0,5,0,0,0,0,0,2,0,0,0,0,1],[8,133,0.0602,0.27232,0.18681,0.24999,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,1,0,0,17,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[12,133,0.0902,0.20982,0.12869,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,2,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,133,0.1203,0.27232,0.19352,0.2857,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,0,0,0,21,0,0,0,0,0,1,0,0,3,0,0,0,0,0],[20,133,0.1504,0.26785,0.18814,0.2857,0.28571,0.28571,0.0,1.0,6,1,0,6,0,1,0,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[24,133,0.1805,0.19196,0.14555,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,3,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,133,0.2105,0.23214,0.12242,0.14286,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,5,0,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,133,0.2406,0.17411,0.13236,0.0,0.2857,0.28571,0.0,0.28571,11,0,0,11,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,133,0.2707,0.16071,0.1948,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,7,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,133,0.3008,0.16518,0.13415,0.0,0.14286,0.2857,0.0,0.42857,11,0,0,11,0,6,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,133,0.3308,0.20536,0.16342,0.0,0.2857,0.28571,0.0,0.71429,9,0,0,9,0,5,0,0,15,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[48,133,0.3609,0.20089,0.19186,0.0,0.2857,0.28571,0.0,0.85714,11,0,0,11,0,4,0,0,14,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[52,133,0.391,0.19641,0.16653,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,3,0,0,15,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[56,133,0.4211,0.18303,0.14826,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,9,0,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[60,133,0.4511,0.16518,0.14773,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,9,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[64,133,0.4812,0.17857,0.18558,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,8,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[68,133,0.5113,0.17411,0.11143,0.14286,0.14288,0.28571,0.0,0.28571,7,0,0,7,0,11,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,133,0.5414,0.20089,0.12807,0.14286,0.2857,0.28571,0.0,0.42857,7,0,0,7,0,7,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,133,0.5714,0.13839,0.13114,0.0,0.14286,0.2857,0.0,0.42857,13,0,0,13,0,8,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,133,0.6015,0.16518,0.16409,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,7,0,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[84,133,0.6316,0.125,0.13243,0.0,0.14286,0.2857,0.0,0.42857,15,0,0,15,0,7,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,133,0.6617,0.16053,0.13246,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,7,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,133,0.6917,0.13838,0.14049,0.0,0.14286,0.28571,0.0,0.571,13,0,0,13,0,9,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[96,133,0.7218,0.16509,0.14336,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,10,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[100,133,0.7519,0.21875,0.10705,0.14286,0.2857,0.28571,0.0,0.4286,3,0,0,3,0,11,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[104,133,0.782,0.19187,0.11075,0.14286,0.14286,0.28571,0.0,0.42857,5,0,0,5,0,12,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[108,133,0.812,0.18295,0.08175,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,19,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,133,0.8421,0.18304,0.10248,0.14286,0.14286,0.28571,0.0,0.28571,5,0,0,5,0,13,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,133,0.8722,0.17857,0.08748,0.14286,0.14286,0.2857,0.0,0.28571,3,0,0,3,0,18,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,133,0.9023,0.19643,0.10564,0.14286,0.1429,0.28571,0.0,0.42857,4,0,0,4,0,13,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[124,133,0.9323,0.22768,0.08645,0.14286,0.2857,0.28571,0.0,0.4286,1,0,0,1,0,12,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[128,133,0.9624,0.16964,0.09061,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,18,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,133,0.9925,0.22759,0.1467,0.14286,0.21428,0.28571,0.0,0.85714,2,0,0,2,0,14,0,0,14,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[133,133,1.0,0.19188,0.07673,0.14286,0.14286,0.2857,0.14,0.4286,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"02cf602a8144425f","q":"Find all natural numbers  $n$  such that the sum of the three largest divisors of  $n$  is  $1457$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,43,0.0,0.92857,0.10715,0.85714,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,20],[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],[8,43,0.186,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[12,43,0.2791,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,43,0.3721,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[20,43,0.4651,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[24,43,0.5581,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[28,43,0.6512,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[32,43,0.7442,0.93304,0.24218,1.0,1.0,1.0,0.0,1.0,2,29,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[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],[40,43,0.9302,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[43,43,1.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]},{"b":4,"e":0.85714,"k":"flat","v":0.83927,"x":0.97321,"p":[[0,51,0.0,0.91963,0.19215,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,26],[4,51,0.0784,0.85268,0.2461,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,1,0,0,2,0,22],[8,51,0.1569,0.83927,0.24421,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,1,0,0,1,0,21],[12,51,0.2353,0.9107,0.17408,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,6,0,22],[16,51,0.3137,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[20,51,0.3922,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[24,51,0.4706,0.91964,0.1234,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,0,21],[28,51,0.549,0.94196,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,23],[32,51,0.6275,0.91516,0.11773,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,19],[36,51,0.7059,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[40,51,0.7843,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[44,51,0.8627,0.93304,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[48,51,0.9412,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[51,51,1.0,0.9241,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,21]]}]},{"i":"4887618a817fec92","q":"In a class of  $ n\\geq 4$  some students are friends. In this class any  $ n \\minus{} 1$  students can be seated in a round table such that every student is sitting next to a friend of him in both sides, but  $ n$  students can not be seated in that way. Prove that the minimum value of  $ n$  is  $ 10$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.14286,"x":0.18741,"p":[[0,10,0.0,0.16027,0.05937,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.16929,0.09073,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,29,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,10,0.8,0.18741,0.10975,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,26,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[10,10,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.1425,"x":0.17411,"p":[[0,23,0.0,0.16473,0.08086,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,29,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,23,0.1739,0.17411,0.07771,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.1517,0.03461,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.14705,0.02492,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.14679,0.02497,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.1425,0.00095,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7c2ce1f0240f08d2","q":"Given reals  $x_i\\ne 0, i=1,\\cdots,n$  where  $\\sum\\limits_{j=1}^n x_i=0$ . Find the minimum value of   $$ \\left( \\sum_{i=1}^n x_i^2\\right) \\left( \\sum_{i=1}^n x_i^{-2} \\right) $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.91071,"x":0.98214,"p":[[0,23,0.0,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[4,23,0.1739,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[8,23,0.3478,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[12,23,0.5217,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[16,23,0.6957,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[20,23,0.8696,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[23,23,1.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29]]},{"b":1,"e":0.71429,"k":"flat","v":0.79464,"x":0.9375,"p":[[0,6,0.0,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[4,6,0.6667,0.87054,0.13997,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,1,0,17],[6,6,1.0,0.79464,0.12846,0.71429,0.71429,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,9]]}]},{"i":"86b33283d3fe1174","q":"Let  $ABC$  be an isosceles triangle with  $AB=4$ ,  $BC=CA=6$ . On the segment  $AB$  consecutively lie points  $X_{1},X_{2},X_{3},\\ldots$  such that the lengths of the segments  $AX_{1},X_{1}X_{2},X_{2}X_{3},\\ldots$  form an infinite geometric progression with starting value  $3$  and common ratio  $\\frac{1}{4}$ . On the segment  $CB$  consecutively lie points  $Y_{1},Y_{2},Y_{3},\\ldots$  such that the lengths of the segments  $CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\\ldots$  form an infinite geometric progression with starting value  $3$  and common ratio  $\\frac{1}{2}$ . On the segment  $AC$  consecutively lie points  $Z_{1},Z_{2},Z_{3},\\ldots$  such that the lengths of the segments  $AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\\ldots$  form an infinite geometric progression with starting value  $3$  and common ratio  $\\frac{1}{2}$ . Find all triplets of positive integers  $(a,b,c)$  such that the segments  $AY_{a}$ ,  $BZ_{b}$  and  $CX_{c}$  are concurrent.","t":[{"b":3,"e":0.71429,"k":"flat","v":0.82142,"x":0.88838,"p":[[0,8,0.0,0.88838,0.16265,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,1,0,21],[4,8,0.5,0.82142,0.1557,0.71429,0.71429,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,1,0,13],[8,8,1.0,0.8482,0.15128,0.71429,0.85707,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,2,0,15]]},{"b":4,"e":1.0,"k":"flat","v":0.89284,"x":1.0,"p":[[0,61,0.0,0.89284,0.17499,0.71429,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,0,0,23],[4,61,0.0656,0.92411,0.14719,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,25],[8,61,0.1311,0.94643,0.12753,1.0,1.0,1.0,0.5714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,27],[12,61,0.1967,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[16,61,0.2623,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,61,0.3279,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[28,61,0.459,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,61,0.5246,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,61,0.5902,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[44,61,0.7213,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,61,0.7869,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,61,0.8525,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,61,0.918,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,61,0.9836,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[61,61,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"d4047fa446dcd0c0","q":"An isosceles triangle  $ABC$  is inscribed in a circle with  $\\angle ACB = 90^o$  and  $EF$  is a chord of the circle such that neither E nor  $F$  coincide with  $C$ . Lines  $CE$  and  $CF$  meet  $AB$  at  $D$  and  $G$  respectively. Prove that  $|CE|\\cdot |DG| = |EF| \\cdot  |CG|$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.01777,"x":0.15624,"p":[[0,29,0.0,0.14285,0.21127,0.0,0.0,0.2857,0.0,0.857,19,0,0,19,0,4,0,0,2,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[4,29,0.1379,0.15624,0.19997,0.0,0.0,0.42857,0.0,0.571,18,0,0,18,0,4,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[8,29,0.2759,0.08036,0.15947,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,7,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[12,29,0.4138,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.03572,0.08749,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.04902,0.09173,0.0,0.0,0.14071,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.42857,"k":"flat","v":0.14737,"x":0.26784,"p":[[0,25,0.0,0.15624,0.22966,0.0,0.0,0.1786,0.0,1.0,17,1,0,17,0,7,0,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[4,25,0.16,0.17856,0.20201,0.0,0.14286,0.32143,0.0,0.71429,14,0,0,14,0,7,0,0,3,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[8,25,0.32,0.20982,0.26722,0.0,0.14286,0.32142,0.0,1.0,13,1,0,13,0,10,0,0,1,0,0,3,0,0,1,0,0,3,0,0,0,0,1],[12,25,0.48,0.26784,0.26182,0.0,0.14286,0.42857,0.0,1.0,11,1,0,11,0,6,0,0,0,0,0,11,0,0,1,0,0,2,0,0,0,0,1],[16,25,0.64,0.14737,0.1873,0.0,0.0,0.42857,0.0,0.43,18,0,0,18,0,4,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.19643,0.18123,0.0,0.14288,0.42857,0.0,0.4286,12,0,0,12,0,6,0,0,4,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.17857,0.17497,0.0,0.14286,0.42857,0.0,0.4286,12,0,0,12,0,9,0,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.21876,0.2082,0.0,0.14286,0.42857,0.0,0.71429,12,0,0,12,0,6,0,0,1,0,0,12,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"020ec49cb08fa36e","q":"In an acute traingle  $ABC$  with   $AB< BC$  let  $BH_b$  be its altitude, and let  $O$  be the circumcenter. A line through  $H_b$  parallel to  $CO$  meets  $BO$  at  $X$ . Prove that  $X$  and the midpoints of  $AB$  and  $AC$  are collinear.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02677,"p":[[0,15,0.0,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,1,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.05356,"p":[[0,38,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.05356,0.13712,0.0,0.0,0.0,0.0,0.571,27,0,1,27,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,38,0.4211,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,38,0.8421,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[38,38,1.0,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"294d4871f1ac36ee","q":"Let  $S_1$  be the area of the regular pentagon  $ABCDE$ . And let  $S_2$  be the area of the regular pentagon whose sides lie on the lines  $AC, CE, EB, BD, DA$ . What is values of  $\\frac{S_1}{S_2}$  ? $\\textbf{(A)}\\ \\frac{41}{6} \\qquad\\textbf{(B)}\\ \\frac{3+5\\sqrt5}{2} \\qquad\\textbf{(C)}\\ 4+\\sqrt5 \\qquad\\textbf{(D)}\\ \\frac{7+3\\sqrt5}2 \\qquad\\textbf{(E)}\\ \\text{None}$","t":[{"b":1,"e":0.71429,"k":"flat","v":0.77232,"x":0.91964,"p":[[0,40,0.0,0.80357,0.23891,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,7,0,0,4,0,15],[4,40,0.1,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[8,40,0.2,0.83482,0.20858,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,9,0,0,2,0,17],[12,40,0.3,0.87054,0.22968,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,3,0,0,2,0,22],[16,40,0.4,0.91071,0.14174,0.85711,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,21],[20,40,0.5,0.85267,0.15767,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,2,0,16],[24,40,0.6,0.86161,0.145,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,5,0,15],[28,40,0.7,0.88393,0.19704,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,2,0,21],[32,40,0.8,0.77232,0.23107,0.71429,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,14,0,0,0,0,13],[36,40,0.9,0.80357,0.19805,0.71429,0.78571,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,11,0,0,3,0,13],[40,40,1.0,0.83482,0.18595,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,11,0,0,1,0,16]]},{"b":6,"e":1.0,"k":"flat","v":0.79464,"x":0.87946,"p":[[0,5,0.0,0.87946,0.20238,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,1,0,22],[4,5,0.8,0.79464,0.18877,0.71429,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,16,0,0,1,0,12],[5,5,1.0,0.80804,0.20395,0.71429,0.78571,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,12,0,0,2,0,14]]}]},{"i":"d556df67b500024e","q":"Let  $ S$  be the set of all real numbers strictly greater than \u22121. Find all functions  $ f: S \\to S$  satisfying the two conditions:\r\n\r\n(a)  $ f(x \\plus{} f(y) \\plus{} xf(y)) \\equal{} y \\plus{} f(x) \\plus{} yf(x)$  for all  $ x, y$  in  $ S$ ;\r\n\r\n(b)  $ \\frac {f(x)}{x}$  is strictly increasing on each of the two intervals  $ \\minus{} 1 < x < 0$  and  $ 0 < x$ .","t":[{"b":0,"e":0.42857,"k":"rising","v":0.29017,"x":0.46872,"p":[[0,19,0.0,0.29017,0.22722,0.14286,0.21431,0.42857,0.0,0.85714,6,0,0,6,0,10,0,0,1,0,0,10,0,0,3,0,0,1,0,0,1,0,0],[4,19,0.2105,0.34375,0.12807,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,6,0,0,4,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.34821,0.17474,0.24999,0.42857,0.42857,0.0,0.71429,3,0,0,3,0,5,0,0,4,0,0,16,0,0,3,0,0,1,0,0,0,0,0],[12,19,0.6316,0.40625,0.17169,0.28571,0.42857,0.4643,0.0,0.85714,1,0,0,1,0,3,0,0,7,0,0,13,0,0,6,0,0,1,0,0,1,0,0],[16,19,0.8421,0.41067,0.23074,0.14286,0.42857,0.57111,0.0,1.0,1,1,0,1,0,8,0,0,3,0,0,9,0,0,7,0,0,2,0,0,1,0,1],[19,19,1.0,0.46872,0.21497,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,1,0,0,12,0,0,8,0,0,2,0,0,2,0,1]]},{"b":7,"e":0.57143,"k":"flat","v":0.21875,"x":0.37946,"p":[[0,16,0.0,0.2767,0.21416,0.0,0.35714,0.42857,0.0,0.57143,9,0,0,9,0,5,0,0,2,0,0,11,0,0,5,0,0,0,0,0,0,0,0],[4,16,0.25,0.21875,0.20198,0.0,0.14286,0.42857,0.0,0.71429,10,0,0,10,0,9,0,0,2,0,0,9,0,0,1,0,0,1,0,0,0,0,0],[8,16,0.5,0.26339,0.23176,0.0,0.21428,0.42857,0.0,1.0,9,1,0,9,0,7,0,0,1,0,0,13,0,0,1,0,0,0,0,0,0,0,1],[12,16,0.75,0.37946,0.21608,0.14286,0.42857,0.42858,0.0,0.85714,3,0,0,3,0,6,0,0,1,0,0,16,0,0,3,0,0,1,0,0,2,0,0],[16,16,1.0,0.32143,0.15972,0.14289,0.42857,0.42857,0.0,0.57143,3,0,0,3,0,6,0,0,5,0,0,16,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"f0d397ac90ff02d2","q":"Circles  $\\omega_1$  and  $\\omega_2$  intersect at points  $A$  and  $B$ , and  $M$  is the midpoint of  $AB$ . Points  $S_1$  and  $S_2$  lie on the line  $AB$  (but not between  $A$  and  $B$ ). The tangents drawn from  $S_1$  to  $\\omega_1$  touch it at  $X_1$  and  $Y_1$ , and the tangents drawn from  $S_2$  to  $\\omega_2$  touch it at  $X_2$  and  $Y_2$ . Prove that if the line  $X_1X_2$  passes through  $M$ , then line  $Y_1Y_2$  also passes through  $M$ .","t":[{"b":1,"e":0.28571,"k":"flat","v":0.3482,"x":0.47315,"p":[[0,35,0.0,0.40172,0.18701,0.28571,0.42859,0.57141,0.0,0.57143,3,0,0,3,0,2,0,0,7,0,0,6,0,0,14,0,0,0,0,0,0,0,0],[4,35,0.1143,0.3482,0.20182,0.2857,0.28571,0.57143,0.0,0.57143,5,0,1,5,0,2,0,0,10,0,0,4,0,0,11,0,0,0,0,0,0,0,0],[8,35,0.2286,0.47315,0.16532,0.42857,0.5712,0.57143,0.0,0.57143,2,0,0,2,0,1,0,0,3,0,0,5,0,0,21,0,0,0,0,0,0,0,0],[12,35,0.3429,0.44186,0.15294,0.28571,0.4998,0.57111,0.0,0.57143,1,0,0,1,0,1,0,0,8,0,0,6,0,0,16,0,0,0,0,0,0,0,0],[16,35,0.4571,0.4553,0.15331,0.42857,0.571,0.57143,0.0,0.57143,1,0,0,1,0,2,0,0,4,0,0,8,0,0,17,0,0,0,0,0,0,0,0],[20,35,0.5714,0.424,0.14042,0.28571,0.42857,0.571,0.14286,0.57143,0,0,0,0,0,2,0,0,10,0,0,7,0,0,13,0,0,0,0,0,0,0,0],[24,35,0.6857,0.44191,0.1611,0.28571,0.571,0.57143,0.0,0.57143,1,0,0,1,0,2,0,0,7,0,0,5,0,0,17,0,0,0,0,0,0,0,0],[28,35,0.8,0.44185,0.15318,0.28571,0.571,0.57143,0.14,0.57143,0,0,0,0,0,3,0,0,8,0,0,4,0,0,17,0,0,0,0,0,0,0,0],[32,35,0.9143,0.41958,0.15941,0.28571,0.4286,0.57143,0.0,0.57143,1,0,0,1,0,1,0,0,12,0,0,3,0,0,15,0,0,0,0,0,0,0,0],[35,35,1.0,0.45974,0.1416,0.39286,0.571,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,7,0,0,7,0,0,17,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.2857,"x":0.43296,"p":[[0,11,0.0,0.35257,0.19883,0.25002,0.35714,0.571,0.0,0.57143,4,0,0,4,0,4,0,0,8,0,0,5,0,0,11,0,0,0,0,0,0,0,0],[4,11,0.3636,0.2857,0.21722,0.0,0.28571,0.4642,0.0,0.57143,9,0,1,9,0,2,0,0,9,0,0,4,0,0,8,0,0,0,0,0,0,0,0],[8,11,0.7273,0.43292,0.14949,0.28571,0.4286,0.57143,0.14,0.57143,0,0,0,0,0,3,0,0,8,0,0,6,0,0,15,0,0,0,0,0,0,0,0],[11,11,1.0,0.43296,0.16157,0.28571,0.4998,0.57143,0.0,0.57143,1,0,0,1,0,2,0,0,8,0,0,5,0,0,16,0,0,0,0,0,0,0,0]]}]},{"i":"50ba2683c510ab4d","q":"Let  $\\mathbb{Z^+}$  denote the set of positive integers. Find all functions  $f:\\mathbb{Z^+} \\to \\mathbb{Z^+}$  satisfying the condition  $$  f(a) + f(b) \\mid (a + b)^2 $$  for all  $a,b \\in \\mathbb{Z^+}$","t":[{"b":1,"e":0.85714,"k":"flat","v":0.60266,"x":0.70978,"p":[[0,9,0.0,0.66517,0.17717,0.4286,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,15,0,0,3,0,3],[4,9,0.4444,0.70978,0.18381,0.571,0.71429,0.85704,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,12,0,0,7,0,4],[8,9,0.8889,0.60266,0.17398,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,10,0,0,6,0,0,10,0,0,5,0,0],[9,9,1.0,0.62048,0.14557,0.42859,0.64286,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,7,0,0,12,0,0,4,0,0]]},{"b":6,"e":0.571,"k":"flat","v":0.44641,"x":0.66517,"p":[[0,45,0.0,0.61155,0.16841,0.42857,0.57121,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,5,0,0,10,0,0,4,0,1],[4,45,0.0889,0.6428,0.20518,0.42857,0.64286,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,8,0,0,6,0,0,7,0,0,6,0,3],[8,45,0.1778,0.66517,0.20079,0.42857,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,8,0,0,1,0,0,11,0,0,8,0,2],[12,45,0.2667,0.63392,0.2141,0.42857,0.64286,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,12,0,0,3,0,0,9,0,0,2,0,5],[16,45,0.3556,0.55356,0.16268,0.42857,0.4286,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,17,0,0,2,0,0,9,0,0,3,0,0],[20,45,0.4444,0.58925,0.18813,0.42857,0.571,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,13,0,0,2,0,0,10,0,0,4,0,1],[24,45,0.5333,0.56247,0.1554,0.42857,0.571,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,14,0,0,6,0,0,8,0,0,3,0,0],[28,45,0.6222,0.54458,0.15743,0.42857,0.571,0.60707,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,13,0,0,9,0,0,6,0,0,1,0,1],[32,45,0.7111,0.48651,0.14243,0.42857,0.42857,0.4642,0.14,0.85714,0,0,0,0,0,1,0,0,0,0,0,23,0,0,3,0,0,3,0,0,2,0,0],[36,45,0.8,0.51784,0.14615,0.42857,0.42859,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,19,0,0,6,0,0,3,0,0,3,0,0],[40,45,0.8889,0.48212,0.13714,0.42857,0.42857,0.4642,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,22,0,0,4,0,0,3,0,0,0,0,1],[44,45,0.9778,0.44641,0.09275,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,24,0,0,3,0,0,2,0,0,0,0,0],[45,45,1.0,0.47762,0.13646,0.42857,0.42857,0.571,0.1429,0.857,0,0,0,0,0,1,0,0,2,0,0,19,0,0,6,0,0,3,0,0,1,0,0]]}]},{"i":"6c237e72c9f905a7","q":"Write the natural numbers from left to right in ascending order. Every minute, we perform an operation. After  $m$  minutes, we divide the entire available series into consecutive blocks of  $m$  numbers. We leave the first block unchanged and in each of the other blocks we move all the numbers except the first one one place to the left, and move the first one to the end of the block. Prove that throughout the process, each natural number will only move a finite number of times.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.13391,"x":0.37052,"p":[[0,49,0.0,0.13391,0.14694,0.0,0.14286,0.14286,0.0,0.57143,11,0,6,11,0,17,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,49,0.0816,0.37052,0.25965,0.14286,0.35714,0.57143,0.0,1.0,1,1,0,1,0,14,0,0,1,0,0,6,0,0,4,0,0,4,0,0,1,0,1],[8,49,0.1633,0.31695,0.26421,0.14286,0.21429,0.42858,0.0,0.85714,5,0,0,5,0,11,0,0,3,0,0,6,0,0,2,0,0,2,0,0,3,0,0],[12,49,0.2449,0.33918,0.24685,0.14286,0.2857,0.57111,0.0,1.0,1,1,0,1,0,14,0,0,5,0,0,3,0,0,5,0,0,2,0,0,1,0,1],[16,49,0.3265,0.34372,0.28981,0.14286,0.14286,0.57143,0.0,0.85714,5,0,0,5,0,13,0,0,0,0,0,1,0,0,7,0,0,3,0,0,3,0,0],[20,49,0.4082,0.35713,0.27198,0.14286,0.35714,0.57143,0.0,1.0,4,2,0,4,0,10,0,0,2,0,0,6,0,0,6,0,0,2,0,0,0,0,2],[24,49,0.4898,0.3124,0.22433,0.14286,0.14286,0.42857,0.14,0.85714,0,0,0,0,0,18,0,0,2,0,0,5,0,0,4,0,0,1,0,0,2,0,0],[28,49,0.5714,0.33473,0.25413,0.14286,0.14288,0.42858,0.0,1.0,1,1,0,1,0,16,0,0,1,0,0,7,0,0,3,0,0,1,0,0,2,0,1],[32,49,0.6531,0.31695,0.20742,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,14,0,0,8,0,0,3,0,0,5,0,0,0,0,0,2,0,0],[36,49,0.7347,0.3214,0.223,0.14286,0.28571,0.57143,0.0,0.71429,4,0,0,4,0,11,0,0,2,0,0,4,0,0,10,0,0,1,0,0,0,0,0],[40,49,0.8163,0.34372,0.25216,0.14286,0.14286,0.57111,0.14286,1.0,0,1,0,0,0,17,0,0,2,0,0,3,0,0,7,0,0,0,0,0,2,0,1],[44,49,0.898,0.32589,0.21498,0.14286,0.21428,0.57143,0.0,0.85714,1,0,0,1,0,15,0,0,2,0,0,4,0,0,9,0,0,0,0,0,1,0,0],[48,49,0.9796,0.33478,0.19755,0.14286,0.35714,0.42858,0.14286,1.0,0,1,0,0,0,13,0,0,3,0,0,11,0,0,4,0,0,0,0,0,0,0,1],[49,49,1.0,0.28125,0.16554,0.14286,0.2143,0.42857,0.0,0.57143,1,0,0,1,0,15,0,0,4,0,0,8,0,0,4,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.0892,"x":0.51783,"p":[[0,49,0.0,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.2857,13,0,5,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.47766,0.27573,0.14286,0.4286,0.71429,0.0,1.0,1,1,0,1,0,8,0,0,2,0,0,6,0,0,4,0,0,6,0,0,4,0,1],[8,49,0.1633,0.41955,0.27426,0.14286,0.42857,0.57143,0.0,1.0,2,3,0,2,0,9,0,0,0,0,0,10,0,0,7,0,0,0,0,0,1,0,3],[12,49,0.2449,0.42406,0.29336,0.14286,0.42857,0.57143,0.0,1.0,1,2,0,1,0,12,0,0,2,0,0,3,0,0,8,0,0,0,0,0,4,0,2],[16,49,0.3265,0.41963,0.30079,0.14286,0.42857,0.57143,0.0,1.0,4,3,0,4,0,7,0,0,3,0,0,5,0,0,6,0,0,3,0,0,1,0,3],[20,49,0.4082,0.37499,0.2829,0.14286,0.21428,0.57143,0.0,1.0,2,1,0,2,0,14,0,0,1,0,0,3,0,0,5,0,0,4,0,0,2,0,1],[24,49,0.4898,0.51783,0.25938,0.39286,0.57143,0.60714,0.0,1.0,1,2,0,1,0,5,0,0,2,0,0,5,0,0,11,0,0,2,0,0,4,0,2],[28,49,0.5714,0.25874,0.20966,0.14286,0.14286,0.42857,0.0,0.85714,2,0,0,2,0,20,0,0,1,0,0,2,0,0,6,0,0,0,0,0,1,0,0],[32,49,0.6531,0.36604,0.25487,0.14286,0.28571,0.57111,0.0,0.85714,2,0,0,2,0,11,0,0,4,0,0,5,0,0,6,0,0,0,0,0,4,0,0],[36,49,0.7347,0.39282,0.25251,0.14286,0.42857,0.571,0.14286,1.0,0,1,0,0,0,12,0,0,3,0,0,8,0,0,4,0,0,1,0,0,3,0,1],[40,49,0.8163,0.24106,0.17653,0.14286,0.14286,0.42857,0.0,0.57143,2,0,0,2,0,20,0,0,1,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[44,49,0.898,0.24107,0.16917,0.14286,0.14286,0.32143,0.0,0.71429,2,0,0,2,0,18,0,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[48,49,0.9796,0.28105,0.17317,0.14286,0.14288,0.42857,0.0,0.57143,1,0,0,1,0,16,0,0,3,0,0,7,0,0,5,0,0,0,0,0,0,0,0],[49,49,1.0,0.27678,0.17105,0.14286,0.21428,0.42857,0.0,0.57143,2,0,0,2,0,14,0,0,4,0,0,8,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"b6ca7c3454c8e2f4","q":"A warehouse contains  $175$  boots of size  $8$ ,  $175$  boots of size  $9$  and  $200$  boots of size  $10$ . Of these  $550$  boots,  $250$  are for the left foot and  $300$  for the right foot. Let  $n$  denote the total number of usable pairs of boots in the warehouse. (A usable pair consists of a left and a right boot of the same size.)\n\n(a) Is  $n=50$  possible?\n\n(b) Is  $n=51$  possible?","t":[{"b":4,"e":0.42857,"k":"flat","v":0.37945,"x":0.5089,"p":[[0,90,0.0,0.37945,0.1268,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,19,0,0,6,0,0,6,0,0,1,0,0,0,0,0],[4,90,0.0444,0.47766,0.18422,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,6,0,0,7,0,0,5,0,0,2,0,0],[8,90,0.0889,0.42409,0.16934,0.28571,0.42857,0.57111,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,8,0,0,6,0,0,2,0,0,0,0,1],[12,90,0.1333,0.41516,0.16505,0.28571,0.28571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,18,0,0,3,0,0,8,0,0,2,0,0,1,0,0],[16,90,0.1778,0.46426,0.17126,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,6,0,0,10,0,0,2,0,0,2,0,0],[20,90,0.2222,0.47765,0.18072,0.28571,0.42859,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,7,0,0,8,0,0,5,0,0,0,0,1],[24,90,0.2667,0.45087,0.16791,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,14,0,0,4,0,0,10,0,0,3,0,0,1,0,0],[28,90,0.3111,0.46426,0.1515,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,6,0,0,11,0,0,4,0,0,0,0,0],[32,90,0.3556,0.43748,0.15946,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,14,0,0,6,0,0,9,0,0,2,0,0,1,0,0],[36,90,0.4,0.41067,0.14169,0.28571,0.28571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,3,0,0,11,0,0,1,0,0,0,0,0],[40,90,0.4444,0.42854,0.15565,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,6,0,0,7,0,0,4,0,0,0,0,0],[44,90,0.4889,0.3839,0.12592,0.28571,0.28571,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,18,0,0,7,0,0,6,0,0,1,0,0,0,0,0],[48,90,0.5333,0.46871,0.12989,0.42857,0.42857,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,7,0,0,12,0,0,10,0,0,3,0,0,0,0,0],[52,90,0.5778,0.45532,0.1801,0.28571,0.42859,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,14,0,0,4,0,0,10,0,0,3,0,0,0,0,1],[56,90,0.6222,0.44192,0.1571,0.28571,0.42857,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,13,0,0,7,0,0,9,0,0,2,0,0,1,0,0],[60,90,0.6667,0.50445,0.14718,0.42857,0.4998,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,10,0,0,9,0,0,7,0,0,0,0,0],[64,90,0.7111,0.39284,0.13362,0.28571,0.28571,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,18,0,0,5,0,0,8,0,0,1,0,0,0,0,0],[68,90,0.7556,0.46875,0.20897,0.2857,0.42857,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,15,0,0,3,0,0,8,0,0,4,0,0,0,0,2],[72,90,0.8,0.43301,0.16162,0.28571,0.35714,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,3,0,0,9,0,0,4,0,0,0,0,0],[76,90,0.8444,0.40625,0.14773,0.28571,0.35714,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,9,0,0,3,0,0,4,0,0,0,0,0],[80,90,0.8889,0.5089,0.20805,0.28571,0.57121,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,2,0,0,11,0,0,3,0,0,3,0,1],[84,90,0.9333,0.42409,0.15354,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,7,0,0,6,0,0,4,0,0,0,0,0],[88,90,0.9778,0.44643,0.17405,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,11,0,0,8,0,0,9,0,0,2,0,0,0,0,1],[90,90,1.0,0.46872,0.16455,0.28571,0.42857,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,11,0,0,7,0,0,9,0,0,4,0,0,1,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.37946,"x":0.50893,"p":[[0,74,0.0,0.37946,0.11633,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,18,0,0,7,0,0,7,0,0,0,0,0,0,0,0],[4,74,0.0541,0.42857,0.14726,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,7,0,0,8,0,0,3,0,0,0,0,0],[8,74,0.1081,0.44641,0.14616,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,4,0,0,13,0,0,2,0,0,0,0,0],[12,74,0.1622,0.50893,0.14698,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,6,0,0,13,0,0,6,0,0,0,0,0],[16,74,0.2162,0.46429,0.15972,0.28571,0.4286,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,11,0,0,7,0,0,10,0,0,3,0,0,1,0,0],[20,74,0.2703,0.46426,0.14724,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,8,0,0,10,0,0,4,0,0,0,0,0],[24,74,0.3243,0.46425,0.15969,0.28571,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,5,0,0,10,0,0,5,0,0,0,0,0],[28,74,0.3784,0.45532,0.15332,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,6,0,0,10,0,0,4,0,0,0,0,0],[32,74,0.4324,0.43302,0.14499,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,8,0,0,8,0,0,3,0,0,0,0,0],[36,74,0.4865,0.47319,0.14912,0.39286,0.42859,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,9,0,0,14,0,0,0,0,0,0,0,1],[40,74,0.5405,0.43304,0.14054,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,9,0,0,10,0,0,0,0,0,1,0,0],[44,74,0.5946,0.48658,0.19839,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,5,0,0,9,0,0,3,0,0,2,0,1],[48,74,0.6486,0.45088,0.16792,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,5,0,0,7,0,0,6,0,0,0,0,0],[52,74,0.7027,0.46869,0.17937,0.28571,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,6,0,0,9,0,0,4,0,0,0,0,1],[56,74,0.7568,0.43298,0.13587,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,6,0,0,12,0,0,1,0,0,0,0,0],[60,74,0.8108,0.46427,0.14285,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,7,0,0,12,0,0,3,0,0,0,0,0],[64,74,0.8649,0.39731,0.12232,0.28571,0.35714,0.57111,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,7,0,0,9,0,0,0,0,0,0,0,0],[68,74,0.9189,0.43302,0.14052,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,7,0,0,10,0,0,2,0,0,0,0,0],[72,74,0.973,0.44639,0.1417,0.28571,0.42859,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,6,0,0,12,0,0,2,0,0,0,0,0],[74,74,1.0,0.42856,0.13362,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,12,0,0,10,0,0,8,0,0,2,0,0,0,0,0]]}]},{"i":"116feb014e6cdb27","q":"Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \\neq 1$. For a positive integer $n$, define $\\operatorname{divs}(n)$ to be the set of positive divisors of $n$.\n\nA positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which\n\n$$\nf[\\operatorname{divs}(m)]=\\operatorname{divs}(n)\n$$\n\nProve that for any such $f$, there are finitely many $f$-cool integers.\n(The notation $f[S]$ for some set $S$ denotes the set $\\{f(s): s \\in S\\}$.)\nRemark 1. The original problem statement was \"For a fixed non-constant polynomial $f(x) \\neq$ $x$, prove that there are finitely many composite $f$-cool integers.\" Note that this allows $f(1)=1$. Try this problem for an added challenge!","t":[{"b":2,"e":0.14286,"k":"flat","v":0.125,"x":0.16965,"p":[[0,34,0.0,0.16965,0.0974,0.14286,0.14286,0.2857,0.0,0.42857,4,0,0,4,0,19,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.165,0.06304,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.15607,0.09009,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,22,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.14732,0.08364,0.14286,0.14286,0.14286,0.0,0.4286,4,0,0,4,0,24,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.15179,0.11259,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,23,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,34,0.5882,0.16964,0.08328,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,26,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.13839,0.09771,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,20,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.125,0.05922,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.14723,0.0563,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.15607,0.0417,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.04018,"x":0.23205,"p":[[0,58,0.0,0.15607,0.0746,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.23205,0.18818,0.14286,0.14286,0.28571,0.0,0.85714,2,0,0,2,0,19,0,0,7,0,0,0,0,0,2,0,0,1,0,0,1,0,0],[8,58,0.1379,0.14732,0.16554,0.10714,0.14286,0.14286,0.0,0.85714,8,0,0,8,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[12,58,0.2069,0.15179,0.12339,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,19,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,58,0.2759,0.14286,0.18558,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,58,0.3448,0.11161,0.09268,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.16071,0.14617,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,20,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[28,58,0.4828,0.12045,0.08071,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.10714,0.11294,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.09821,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2bfae4ede8932829","q":"Let $n$ be a natural number greater than 1. Prove that there exists a natural number $m$ greater than $n^{n}$ such that\n\n$$\n\\frac{n^{m}-m^{n}}{n+m}\n$$\n\nis a natural number.\n\n(Nikola Petrovi\u0107)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.11607,"x":0.5223,"p":[[0,42,0.0,0.11607,0.26592,0.0,0.0,0.14286,0.0,1.0,23,2,19,23,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[4,42,0.0952,0.5223,0.31054,0.14286,0.57143,0.71429,0.0,1.0,2,4,0,2,0,8,0,0,0,0,0,2,0,0,6,0,0,9,0,0,1,0,4],[8,42,0.1905,0.4732,0.2889,0.24999,0.35714,0.71429,0.14286,1.0,0,3,0,0,0,8,0,0,8,0,0,1,0,0,4,0,0,6,0,0,2,0,3],[12,42,0.2857,0.43298,0.25623,0.2857,0.35714,0.60714,0.0,1.0,2,1,0,2,0,5,0,0,9,0,0,1,0,0,7,0,0,6,0,0,1,0,1],[16,42,0.381,0.50442,0.26721,0.28571,0.571,0.71429,0.14286,1.0,0,3,0,0,0,7,0,0,4,0,0,3,0,0,8,0,0,6,0,0,1,0,3],[20,42,0.4762,0.51338,0.31715,0.14286,0.57143,0.71429,0.0,1.0,4,3,0,4,0,5,0,0,2,0,0,1,0,0,5,0,0,10,0,0,2,0,3],[24,42,0.5714,0.34812,0.23949,0.14286,0.2857,0.60714,0.0,0.71429,1,0,0,1,0,12,0,0,9,0,0,0,0,0,2,0,0,8,0,0,0,0,0],[28,42,0.6667,0.35268,0.24218,0.14286,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,12,0,0,4,0,0,4,0,0,3,0,0,7,0,0,0,0,0],[32,42,0.7619,0.33482,0.23854,0.14286,0.28571,0.57143,0.0,0.71429,4,0,0,4,0,8,0,0,8,0,0,3,0,0,3,0,0,6,0,0,0,0,0],[36,42,0.8571,0.32134,0.24229,0.14286,0.2857,0.57143,0.0,0.71429,3,0,0,3,0,11,0,0,9,0,0,0,0,0,2,0,0,7,0,0,0,0,0],[40,42,0.9524,0.30343,0.20748,0.14286,0.2857,0.46418,0.0,0.71429,3,0,0,3,0,11,0,0,7,0,0,3,0,0,6,0,0,2,0,0,0,0,0],[42,42,1.0,0.25891,0.21848,0.14286,0.14288,0.42858,0.0,0.85714,4,0,0,4,0,16,0,0,3,0,0,3,0,0,4,0,0,1,0,0,1,0,0]]},{"b":5,"e":0.57143,"k":"rising","v":0.16518,"x":0.70982,"p":[[0,78,0.0,0.16518,0.29038,0.0,0.0,0.14286,0.0,1.0,18,2,16,18,0,8,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[4,78,0.0513,0.70982,0.26119,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,0,0,0,3,0,0,5,0,0,9,0,0,2,0,10],[8,78,0.1026,0.54462,0.33395,0.25,0.57143,0.71429,0.0,1.0,3,7,0,3,0,5,0,0,3,0,0,1,0,0,6,0,0,7,0,0,0,0,7],[12,78,0.1538,0.54911,0.3409,0.14286,0.71429,0.71429,0.0,1.0,3,6,0,3,0,7,0,0,0,0,0,3,0,0,1,0,0,11,0,0,1,0,6],[16,78,0.2051,0.5848,0.28428,0.39286,0.64286,0.71429,0.14286,1.0,0,5,0,0,0,6,0,0,2,0,0,3,0,0,5,0,0,9,0,0,2,0,5],[20,78,0.2564,0.48213,0.30878,0.24999,0.57121,0.71429,0.0,1.0,4,3,0,4,0,4,0,0,4,0,0,3,0,0,7,0,0,4,0,0,3,0,3],[24,78,0.3077,0.41964,0.26229,0.14286,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,9,0,0,8,0,0,5,0,0,0,0,0,8,0,0,0,0,2],[28,78,0.359,0.48658,0.27861,0.14286,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,10,0,0,3,0,0,1,0,0,5,0,0,9,0,0,3,0,1],[32,78,0.4103,0.49988,0.27905,0.2857,0.571,0.71429,0.0,1.0,1,3,0,1,0,5,0,0,7,0,0,2,0,0,5,0,0,8,0,0,1,0,3],[36,78,0.4615,0.44639,0.30039,0.14286,0.35714,0.71429,0.0,1.0,2,4,0,2,0,7,0,0,7,0,0,2,0,0,5,0,0,5,0,0,0,0,4],[40,78,0.5128,0.52231,0.26392,0.28571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,3,0,0,1,0,0,7,0,0,12,0,0,0,0,2],[44,78,0.5641,0.3705,0.30482,0.14286,0.2143,0.57143,0.0,1.0,4,2,0,4,0,12,0,0,2,0,0,1,0,0,7,0,0,2,0,0,2,0,2],[48,78,0.6154,0.4107,0.2519,0.14286,0.42859,0.60714,0.0,1.0,2,1,0,2,0,7,0,0,6,0,0,5,0,0,4,0,0,7,0,0,0,0,1],[52,78,0.6667,0.33033,0.27532,0.14286,0.14286,0.57111,0.0,1.0,4,1,0,4,0,13,0,0,3,0,0,2,0,0,3,0,0,6,0,0,0,0,1],[56,78,0.7179,0.33916,0.28518,0.14286,0.14286,0.571,0.0,1.0,2,2,0,2,0,16,0,0,3,0,0,1,0,0,3,0,0,5,0,0,0,0,2],[60,78,0.7692,0.46424,0.32926,0.14286,0.571,0.71429,0.0,1.0,5,4,0,5,0,5,0,0,4,0,0,1,0,0,6,0,0,6,0,0,1,0,4],[64,78,0.8205,0.33032,0.26588,0.14286,0.14288,0.571,0.0,1.0,3,1,0,3,0,14,0,0,3,0,0,2,0,0,4,0,0,5,0,0,0,0,1],[68,78,0.8718,0.38837,0.28622,0.14286,0.35714,0.57143,0.0,1.0,2,2,0,2,0,13,0,0,1,0,0,3,0,0,7,0,0,3,0,0,1,0,2],[72,78,0.9231,0.43747,0.28999,0.14286,0.28571,0.60714,0.0,1.0,1,3,0,1,0,9,0,0,7,0,0,0,0,0,7,0,0,4,0,0,1,0,3],[76,78,0.9744,0.30344,0.2044,0.14286,0.28571,0.4642,0.0,0.71429,2,0,0,2,0,13,0,0,6,0,0,3,0,0,6,0,0,2,0,0,0,0,0],[78,78,1.0,0.35256,0.25755,0.14286,0.2857,0.57111,0.0,1.0,2,1,0,2,0,13,0,0,3,0,0,4,0,0,4,0,0,5,0,0,0,0,1]]}]},{"i":"fdb47d8eb9bdbac0","q":"If  $n,p,q \\in \\mathbb{N}, p<q $  then   $$ {{(p+q)n}\\choose{n}} \\sum \\limits_{k=0}^n  (-1)^k {{n}\\choose{k}} {{(p+q-1)n}\\choose{pn-k}}= {{(p+q)n}\\choose{pn}} \\sum \\limits_{k=0}^{\\left [\\frac{n}{2} \\right ]}  (-1)^k {{pn}\\choose{k}} {{(q-p)n}\\choose{n-2k}}  $$","t":[{"b":0,"e":0.28571,"k":"falling","v":0.42409,"x":0.82143,"p":[[0,51,0.0,0.62946,0.27861,0.42857,0.57143,1.0,0.0,1.0,1,9,0,1,0,0,0,0,3,0,0,9,0,0,7,0,0,1,0,0,2,0,9],[4,51,0.0784,0.74554,0.26901,0.71429,0.85714,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,2,0,0,1,0,0,2,0,0,5,0,0,13,0,7],[8,51,0.1569,0.78125,0.1988,0.71429,0.85714,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,20,0,3],[12,51,0.2353,0.7232,0.22286,0.57143,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,3,0,0,2,0,0,4,0,0,16,0,3],[16,51,0.3137,0.82143,0.16751,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,15,0,8],[20,51,0.3922,0.80357,0.13716,0.82143,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,22,0,2],[24,51,0.4706,0.74105,0.21261,0.57143,0.85714,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,3,0,0,17,0,3],[28,51,0.549,0.78125,0.19557,0.71429,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,5,0,0,17,0,5],[32,51,0.6275,0.6339,0.20807,0.571,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,0,0,0,6,0,0,10,0,0,9,0,0],[36,51,0.7059,0.60257,0.24691,0.42857,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,1,0,0,4,0,0,6,0,0,3,0,0,6,0,0,11,0,0],[40,51,0.7843,0.51339,0.26692,0.28571,0.57143,0.71429,0.0,1.0,3,1,0,3,0,1,0,0,6,0,0,4,0,0,6,0,0,7,0,0,4,0,1],[44,51,0.8627,0.5491,0.23176,0.28571,0.64286,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,11,0,0,1,0,0,3,0,0,11,0,0,5,0,0],[48,51,0.9412,0.54462,0.19045,0.42857,0.571,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,9,0,0,7,0,0,5,0,0,5,0,0],[51,51,1.0,0.42409,0.21865,0.2857,0.35714,0.57143,0.14286,0.85714,0,0,0,0,0,6,0,0,10,0,0,3,0,0,7,0,0,4,0,0,2,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.45528,"x":0.78125,"p":[[0,53,0.0,0.5491,0.27225,0.42857,0.4998,0.64286,0.0,1.0,2,5,0,2,0,1,0,0,2,0,0,11,0,0,8,0,0,0,0,0,3,0,5],[4,53,0.0755,0.78125,0.15561,0.71429,0.85714,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,10,0,0,14,0,4],[8,53,0.1509,0.62946,0.25218,0.42857,0.71429,0.85714,0.0,1.0,2,1,0,2,0,0,0,0,2,0,0,6,0,0,4,0,0,6,0,0,11,0,1],[12,53,0.2264,0.64284,0.22868,0.4286,0.71429,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,3,0,0,5,0,0,4,0,0,8,0,0,10,0,1],[16,53,0.3019,0.61607,0.24856,0.42857,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,2,0,0,5,0,0,8,0,0,9,0,1],[20,53,0.3774,0.65179,0.28558,0.53571,0.71429,0.85714,0.0,1.0,3,2,0,3,0,0,0,0,3,0,0,2,0,0,3,0,0,6,0,0,13,0,2],[24,53,0.4528,0.68302,0.24153,0.57132,0.71429,0.85714,0.0,1.0,1,2,0,1,0,1,0,0,2,0,0,3,0,0,2,0,0,9,0,0,12,0,2],[28,53,0.5283,0.66517,0.23853,0.53571,0.71429,0.85714,0.1429,1.0,0,3,0,0,0,1,0,0,5,0,0,2,0,0,3,0,0,9,0,0,9,0,3],[32,53,0.6038,0.60268,0.26663,0.28571,0.71429,0.85714,0.0,0.85714,2,0,0,2,0,0,0,0,7,0,0,1,0,0,3,0,0,8,0,0,11,0,0],[36,53,0.6792,0.61607,0.2299,0.42857,0.71429,0.75,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,5,0,0,2,0,0,12,0,0,8,0,0],[40,53,0.7547,0.61606,0.24074,0.53539,0.71429,0.75,0.0,1.0,1,1,0,1,0,1,0,0,5,0,0,1,0,0,5,0,0,11,0,0,7,0,1],[44,53,0.8302,0.49997,0.23417,0.28571,0.57143,0.60714,0.0,0.85714,2,0,0,2,0,2,0,0,5,0,0,4,0,0,11,0,0,4,0,0,4,0,0],[48,53,0.9057,0.51785,0.20748,0.28571,0.57143,0.60714,0.0,0.85714,1,0,0,1,0,0,0,0,8,0,0,4,0,0,11,0,0,4,0,0,4,0,0],[52,53,0.9811,0.45979,0.16261,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,7,0,0,9,0,0,5,0,0,0,0,0],[53,53,1.0,0.45528,0.16141,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,13,0,0,5,0,0,9,0,0,5,0,0,0,0,0]]}]},{"i":"553fa4523ba1b8e5","q":"12. (POL 6) Two congruent equilateral triangles $A B C$ and $A^{\\prime} B^{\\prime} C^{\\prime}$ in the plane are given. Show that the midpoints of the segments $A A^{\\prime}, B B^{\\prime}, C C^{\\prime}$ either are collinear or form an equilateral triangle.","t":[{"b":0,"e":1.0,"k":"flat","v":0.97768,"x":0.99107,"p":[[0,8,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,8,0.5,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,8,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.9107,"x":1.0,"p":[[0,16,0.0,0.9107,0.15467,0.85714,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,22],[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],[16,16,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"70bd1dd8d56528dd","q":"Determine the digits  $0\\leqslant c\\leqslant 9$  such that for any positive integer  $k{}$  there exists a positive integer  $n$  such that the last  $k{}$  digits of  $n^9$  are equal to  $c{}.$","t":[{"b":1,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,73,0.0,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[4,73,0.0548,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,73,0.1096,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,73,0.1644,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,73,0.2192,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[20,73,0.274,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,73,0.3288,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,73,0.3836,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,73,0.4384,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,73,0.4932,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,73,0.5479,0.94643,0.15047,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,27],[44,73,0.6027,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[48,73,0.6575,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[52,73,0.7123,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,73,0.7671,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[60,73,0.8219,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,73,0.8767,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[68,73,0.9315,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[72,73,0.9863,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[73,73,1.0,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":5,"e":1.0,"k":"flat","v":0.95536,"x":0.99554,"p":[[0,43,0.0,0.95536,0.14914,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[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],[16,43,0.3721,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[20,43,0.4651,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[24,43,0.5581,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,43,0.6512,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,43,0.7442,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,43,0.8372,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[40,43,0.9302,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[43,43,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"83c355dcd161af83","q":"The convex quadrilateral  $ABCD$  is has angle  $A$  equal to  $60^o$  , angle bisector of  $A$  the diagonal  $AC$  and  $\\angle ACD= 40^o$  and   $\\angle ACB = 120^o$ . Inside the quadrilateral the point  $P$  lies  such that  $\\angle PDA = 40^o$  and  $\\angle PBA = 10^o$ ;\na) Find the angle  $\\angle DPB$ ?\nb) Prove that  $P$  lies on the diagonal  $AC$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.82586,"x":0.89282,"p":[[0,12,0.0,0.87498,0.19482,0.85711,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,18],[4,12,0.3333,0.88615,0.16547,0.857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,6,1,18],[8,12,0.6667,0.89282,0.14292,0.857,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,7,0,18],[12,12,1.0,0.82586,0.19148,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,6,0,0,9,0,12]]},{"b":5,"e":1.0,"k":"falling","v":0.57807,"x":0.82139,"p":[[0,24,0.0,0.82139,0.22308,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,4,0,0,6,0,15],[4,24,0.1667,0.81246,0.2434,0.78561,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,2,0,0,1,0,0,4,0,0,0,0,0,10,0,14],[8,24,0.3333,0.58696,0.24337,0.42859,0.571,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,2,0,1,4,0,0,11,0,0,4,0,0,3,0,4],[12,24,0.5,0.57807,0.24377,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,3,0,0,10,1,0,4,0,0,4,0,3],[16,24,0.6667,0.64728,0.25251,0.53539,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,2,0,0,4,0,0,10,0,0,4,0,0,3,0,7],[20,24,0.8333,0.60039,0.22437,0.42859,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,1,5,0,0,9,0,0,5,0,0,3,0,4],[24,24,1.0,0.58255,0.2576,0.42857,0.571,0.85704,0.1429,1.0,0,5,0,0,0,1,0,0,6,0,0,7,0,0,8,0,0,1,0,0,3,1,5]]}]},{"i":"bc28ff7430bdc695","q":"25. N6 (FRA) ${ }^{\\text {IMO6 }}$ Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^{p}-p$ is not divisible by $q$.","t":[{"b":1,"e":0.0,"k":"falling","v":0.0,"x":0.55357,"p":[[0,43,0.0,0.55357,0.47749,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[4,43,0.093,0.48214,0.4514,0.14286,0.14286,1.0,0.0,1.0,7,13,0,7,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,13],[8,43,0.186,0.21875,0.34807,0.0,0.14286,0.14286,0.0,1.0,15,5,0,15,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[12,43,0.2791,0.49991,0.47252,0.0,0.14286,1.0,0.0,1.0,10,15,0,10,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[16,43,0.3721,0.42856,0.43448,0.0,0.14286,1.0,0.0,1.0,10,10,0,10,0,8,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,10],[20,43,0.4651,0.09821,0.24074,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[24,43,0.5581,0.1383,0.28456,0.0,0.0,0.14286,0.0,1.0,19,3,0,19,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[28,43,0.6512,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"falling","v":0.0,"x":0.68304,"p":[[0,21,0.0,0.65179,0.44311,0.14286,1.0,1.0,0.0,1.0,5,19,0,5,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,19],[4,21,0.1905,0.63393,0.44311,0.14286,1.0,1.0,0.0,1.0,6,18,0,6,0,6,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,18],[8,21,0.381,0.68304,0.41147,0.14286,1.0,1.0,0.0,1.0,4,17,0,4,0,6,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,17],[12,21,0.5714,0.67857,0.42708,0.14286,1.0,1.0,0.0,1.0,6,19,0,6,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,19],[16,21,0.7619,0.59811,0.44104,0.105,0.92857,1.0,0.0,1.0,8,16,0,8,0,3,0,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,16],[20,21,0.9524,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a6d2b76026b5bc33","q":"Let  $p$  and  $q$  be prime numbers. The sequence  $(x_n)$  is defined by  $x_1 = 1$ ,  $x_2 = p$  and  $x_{n+1} = px_n - qx_{n-1}$  for all  $n \\geq 2$ .\nGiven that there is some  $k$  such that  $x_{3k} = -3$ , find  $p$  and  $q$ .","t":[{"b":2,"e":0.14286,"k":"falling","v":0.26784,"x":0.56695,"p":[[0,38,0.0,0.47768,0.28928,0.2857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,7,0,0,7,0,0,6,0,0,2,0,0,4,0,0,2,0,4],[4,38,0.1053,0.53571,0.34069,0.14286,0.50001,0.85714,0.14286,1.0,0,6,0,0,0,10,0,0,3,0,0,3,0,0,3,0,0,1,0,0,6,0,6],[8,38,0.2105,0.56695,0.33404,0.2857,0.57121,0.85714,0.14286,1.0,0,7,0,0,0,6,0,0,8,0,0,1,0,0,3,0,0,1,0,0,6,0,7],[12,38,0.3158,0.40177,0.24855,0.2857,0.28571,0.4642,0.0,1.0,1,1,0,1,0,5,0,0,13,0,0,5,0,0,1,0,0,3,0,0,3,0,1],[16,38,0.4211,0.33928,0.22797,0.14286,0.28571,0.42858,0.0,1.0,1,2,0,1,0,8,0,0,14,0,0,3,0,0,3,0,0,1,0,0,0,0,2],[20,38,0.5263,0.28125,0.17308,0.14286,0.2857,0.42857,0.0,0.71429,2,0,0,2,0,12,0,0,8,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[24,38,0.6316,0.31701,0.19146,0.14286,0.2857,0.42857,0.14286,0.85714,0,0,0,0,0,12,0,0,9,0,0,7,0,0,2,0,0,0,0,0,2,0,0],[28,38,0.7368,0.33474,0.19112,0.2857,0.2857,0.42857,0.14,1.0,0,1,0,0,0,7,0,0,16,0,0,5,0,0,2,0,0,0,0,0,1,0,1],[32,38,0.8421,0.36151,0.25256,0.14286,0.28571,0.4642,0.14,1.0,0,2,0,0,0,14,0,0,4,0,0,6,0,0,3,0,0,3,0,0,0,0,2],[36,38,0.9474,0.33033,0.22708,0.14286,0.2857,0.42858,0.0,1.0,1,1,0,1,0,12,0,0,8,0,0,4,0,0,3,0,0,3,0,0,0,0,1],[38,38,1.0,0.26784,0.17401,0.14286,0.2143,0.28571,0.14286,0.857,0,0,0,0,0,16,0,0,10,0,0,3,0,0,1,0,0,1,0,0,1,0,0]]},{"b":6,"e":0.85714,"k":"rising","v":0.40615,"x":0.90625,"p":[[0,17,0.0,0.5133,0.33294,0.2857,0.42857,0.89286,0.14,1.0,0,8,0,0,0,7,0,0,8,0,0,4,0,0,3,0,0,0,0,0,2,0,8],[4,17,0.2353,0.4732,0.27993,0.2857,0.42857,0.57143,0.14286,1.0,0,5,0,0,0,5,0,0,10,0,0,4,0,0,7,0,0,0,0,0,1,0,5],[8,17,0.4706,0.40615,0.25036,0.1429,0.42857,0.57143,0.14,1.0,0,2,0,0,0,10,0,0,5,0,0,7,0,0,5,0,0,2,0,0,1,0,2],[12,17,0.7059,0.77679,0.31327,0.82143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,4,0,0,3,0,0,0,0,0,0,0,0,1,0,0,9,0,15],[16,17,0.9412,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[17,17,1.0,0.90178,0.08328,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,12]]}]},{"i":"a2009eda8be9d905","q":"Let\n\n$$\nS(n, r)=\\binom{n-1}{r-1}+\\binom{n-1}{r}+\\binom{n-1}{r+1}+\\cdots+\\binom{n-1}{n-1}\n$$\n\nfor all $n, r>0$, and in particular $S(n, r)=0$ if $r>n>0$. Prove that the number in row $n$ of the table, $r$ columns to the left of the 1 in the top row, is at most $S(n, r)$. (Hint: First prove that $S(n-1, r-1)+S(n-1, r)=S(n, r)$.)","t":[{"b":2,"e":0.57143,"k":"flat","v":0.54908,"x":0.5714,"p":[[0,8,0.0,0.56242,0.17834,0.39286,0.64271,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,2,0,0,6,0,0,16,0,0,0,0,0],[4,8,0.5,0.54908,0.18249,0.28571,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,9,0,0,2,0,0,6,0,0,15,0,0,0,0,0],[8,8,1.0,0.5714,0.17496,0.42857,0.64286,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,5,0,0,5,0,0,15,0,0,1,0,0]]},{"b":7,"e":0.2857,"k":"volatile","v":0.37053,"x":0.55357,"p":[[0,3,0.0,0.55357,0.17768,0.39286,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,8,0,0,3,0,0,6,0,0,15,0,0,0,0,0],[3,3,1.0,0.37053,0.17445,0.2857,0.28571,0.28571,0.1429,0.71429,0,0,0,0,0,1,0,0,24,0,0,0,0,0,1,0,0,6,0,0,0,0,0]]}]},{"i":"c793d6443655cb18","q":"Find all positive integers  $n$  such that there exists an infinite set  $A$  of positive integers with the following property: For all pairwise distinct numbers  $a_1, a_2, \\ldots , a_n \\in A$ , the numbers  $$ a_1 + a_2 + \\ldots + a_n \\text{ and } a_1\\cdot a_2\\cdot \\ldots\\cdot a_n $$  are coprime.","t":[{"b":0,"e":0.42857,"k":"falling","v":0.20536,"x":0.75446,"p":[[0,52,0.0,0.75446,0.35756,0.53571,1.0,1.0,0.0,1.0,3,19,3,3,0,2,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,19],[4,52,0.0769,0.33033,0.22426,0.14286,0.28571,0.4286,0.0,0.85714,5,0,0,5,0,5,0,0,8,0,0,7,0,0,4,0,0,2,0,0,1,0,0],[8,52,0.1538,0.36605,0.22283,0.14289,0.35714,0.57111,0.0,1.0,2,1,0,2,0,8,0,0,6,0,0,6,0,0,8,0,0,1,0,0,0,0,1],[12,52,0.2308,0.38384,0.21252,0.1429,0.42857,0.571,0.0,1.0,1,1,0,1,0,8,0,0,5,0,0,8,0,0,8,0,0,1,0,0,0,0,1],[16,52,0.3077,0.33472,0.22198,0.14286,0.28571,0.4286,0.0,0.85714,2,0,0,2,0,11,0,0,5,0,0,7,0,0,3,0,0,3,0,0,1,0,0],[20,52,0.3846,0.29454,0.20502,0.14286,0.2857,0.42858,0.0,0.71429,4,0,0,4,0,8,0,0,11,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[24,52,0.4615,0.34807,0.2473,0.14286,0.2857,0.571,0.0,1.0,3,1,0,3,0,9,0,0,7,0,0,3,0,0,7,0,0,1,0,0,1,0,1],[28,52,0.5385,0.30801,0.20855,0.14286,0.2857,0.4286,0.0,0.85714,2,0,0,2,0,12,0,0,7,0,0,4,0,0,5,0,0,1,0,0,1,0,0],[32,52,0.6154,0.20536,0.17835,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,13,0,0,7,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[36,52,0.6923,0.35703,0.2173,0.14286,0.28571,0.571,0.0,0.71429,2,0,0,2,0,8,0,0,8,0,0,5,0,0,4,0,0,5,0,0,0,0,0],[40,52,0.7692,0.34372,0.19837,0.2857,0.28571,0.42858,0.0,1.0,1,1,0,1,0,6,0,0,15,0,0,3,0,0,5,0,0,1,0,0,0,0,1],[44,52,0.8462,0.33471,0.19439,0.1429,0.28571,0.42858,0.0,0.71429,2,0,0,2,0,7,0,0,11,0,0,5,0,0,4,0,0,3,0,0,0,0,0],[48,52,0.9231,0.26774,0.17037,0.14286,0.2857,0.32143,0.0,0.71429,2,0,0,2,0,13,0,0,9,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[52,52,1.0,0.27673,0.19857,0.14286,0.2143,0.42858,0.0,0.71429,4,0,0,4,0,12,0,0,5,0,0,5,0,0,5,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.16965,"x":0.73658,"p":[[0,45,0.0,0.73658,0.32949,0.5354,1.0,1.0,0.0,1.0,1,18,1,1,0,3,0,0,1,0,0,3,0,0,5,0,0,1,0,0,0,0,18],[4,45,0.0889,0.24088,0.20347,0.105,0.14288,0.42857,0.0,0.71429,8,0,0,8,0,9,0,0,5,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[8,45,0.1778,0.375,0.26666,0.14286,0.28571,0.57143,0.0,1.0,4,1,0,4,0,6,0,0,8,0,0,4,0,0,3,0,0,5,0,0,1,0,1],[12,45,0.2667,0.30356,0.27835,0.14286,0.21428,0.5711,0.0,1.0,7,2,0,7,0,9,0,0,5,0,0,2,0,0,6,0,0,1,0,0,0,0,2],[16,45,0.3556,0.30332,0.26907,0.105,0.2143,0.571,0.0,1.0,8,1,0,8,0,8,0,0,3,0,0,3,0,0,7,0,0,2,0,0,0,0,1],[20,45,0.4444,0.3124,0.25867,0.14214,0.2857,0.4642,0.0,0.85714,7,0,0,7,0,7,0,0,5,0,0,5,0,0,4,0,0,2,0,0,2,0,0],[24,45,0.5333,0.29451,0.22259,0.14286,0.28571,0.4642,0.0,0.71429,6,0,0,6,0,8,0,0,6,0,0,4,0,0,6,0,0,2,0,0,0,0,0],[28,45,0.6222,0.37054,0.22261,0.14286,0.35714,0.57111,0.0,0.71429,1,0,0,1,0,11,0,0,4,0,0,5,0,0,6,0,0,5,0,0,0,0,0],[32,45,0.7111,0.28562,0.21433,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,9,0,0,5,0,0,10,0,0,2,0,0,0,0,0,0,0,1],[36,45,0.8,0.29464,0.2878,0.0,0.2143,0.46429,0.0,1.0,9,1,0,9,0,7,0,0,6,0,0,2,0,0,2,0,0,4,0,0,1,0,1],[40,45,0.8889,0.22765,0.21382,0.0,0.14286,0.42857,0.0,0.57143,11,0,0,11,0,6,0,0,6,0,0,3,0,0,6,0,0,0,0,0,0,0,0],[44,45,0.9778,0.16965,0.16536,0.0,0.14286,0.28571,0.0,0.57143,11,0,0,11,0,11,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[45,45,1.0,0.18304,0.17215,0.10714,0.14286,0.2857,0.0,0.71429,8,0,0,8,0,15,0,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"a7b5f30dd840e58c","q":"Does there exist a positive integer $k$ and a non-constant sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of positive integers such that $a_{n}=\\operatorname{gcd}\\left(a_{n+k}, a_{n+k+1}\\right)$ for all positive integers $n$?","t":[{"b":2,"e":0.71429,"k":"flat","v":0.21875,"x":0.3482,"p":[[0,45,0.0,0.24106,0.12075,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,13,0,0,15,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,45,0.0889,0.3482,0.24726,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,7,0,0,14,0,0,2,0,0,3,0,0,0,0,0,3,0,1],[8,45,0.1778,0.21875,0.15146,0.14286,0.14286,0.2857,0.0,0.71429,3,0,0,3,0,16,0,0,9,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[12,45,0.2667,0.28569,0.22865,0.14286,0.2143,0.4286,0.0,0.85714,5,0,0,5,0,11,0,0,5,0,0,5,0,0,3,0,0,2,0,0,1,0,0],[16,45,0.3556,0.2768,0.15125,0.14286,0.2857,0.32143,0.0,0.57143,3,0,0,3,0,7,0,0,14,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[20,45,0.4444,0.25893,0.16536,0.14286,0.2857,0.28571,0.0,0.85714,2,0,0,2,0,12,0,0,12,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[24,45,0.5333,0.29017,0.22441,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,9,0,0,7,0,0,6,0,0,3,0,0,0,0,0,2,0,0],[28,45,0.6222,0.31693,0.26897,0.14286,0.2143,0.42857,0.0,1.0,3,2,0,3,0,13,0,0,6,0,0,3,0,0,3,0,0,1,0,0,1,0,2],[32,45,0.7111,0.2589,0.19041,0.14286,0.2143,0.28571,0.0,0.71429,4,0,0,4,0,12,0,0,9,0,0,1,0,0,5,0,0,1,0,0,0,0,0],[36,45,0.8,0.26786,0.18472,0.14286,0.2857,0.28571,0.0,1.0,2,1,0,2,0,12,0,0,11,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[40,45,0.8889,0.29009,0.18037,0.14286,0.2857,0.28571,0.0,0.85714,1,0,0,1,0,11,0,0,13,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[44,45,0.9778,0.2857,0.24483,0.14286,0.2143,0.28571,0.0,1.0,3,2,0,3,0,13,0,0,9,0,0,2,0,0,2,0,0,1,0,0,0,0,2],[45,45,1.0,0.32587,0.2179,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,14,0,0,6,0,0,6,0,0,4,0,0,0,0,0,1,0,1]]},{"b":5,"e":0.2857,"k":"flat","v":0.16937,"x":0.34353,"p":[[0,62,0.0,0.16937,0.07533,0.14286,0.14286,0.17857,0.0,0.28571,2,0,1,2,0,22,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,62,0.0645,0.22757,0.167,0.14286,0.14286,0.28571,0.0,0.85714,3,0,0,3,0,15,0,0,10,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[8,62,0.129,0.24527,0.17956,0.14286,0.21428,0.28571,0.0,0.85714,4,0,0,4,0,12,0,0,9,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[12,62,0.1935,0.2499,0.175,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,9,0,0,11,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[16,62,0.2581,0.2364,0.14561,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,0,15,0,0,10,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[20,62,0.3226,0.22313,0.15952,0.14286,0.14286,0.28571,0.0,0.71429,3,0,0,3,0,17,0,0,6,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[24,62,0.3871,0.25446,0.19144,0.14286,0.2857,0.28571,0.0,1.0,3,1,0,3,0,11,0,0,14,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[28,62,0.4516,0.17411,0.10555,0.14286,0.14286,0.2857,0.0,0.4286,4,0,0,4,0,19,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,62,0.5161,0.25436,0.22515,0.14286,0.14288,0.2857,0.0,1.0,5,1,0,5,0,12,0,0,8,0,0,4,0,0,1,0,0,0,0,0,1,0,1],[36,62,0.5806,0.2275,0.1552,0.14286,0.14286,0.2857,0.0,0.57143,4,0,0,4,0,14,0,0,7,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[40,62,0.6452,0.23652,0.18427,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,19,0,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[44,62,0.7097,0.30356,0.24677,0.14286,0.2143,0.32143,0.0,1.0,1,2,0,1,0,15,0,0,8,0,0,3,0,0,2,0,0,0,0,0,1,0,2],[48,62,0.7742,0.34353,0.23664,0.14286,0.2857,0.42857,0.0,1.0,1,2,0,1,0,10,0,0,10,0,0,4,0,0,4,0,0,1,0,0,0,0,2],[52,62,0.8387,0.26767,0.19489,0.14286,0.2857,0.28571,0.0,0.85714,2,0,0,2,0,13,0,0,11,0,0,3,0,0,1,0,0,0,0,0,2,0,0],[56,62,0.9032,0.29464,0.25488,0.14286,0.2857,0.32143,0.0,1.0,5,2,0,5,0,8,0,0,11,0,0,4,0,0,1,0,0,0,0,0,1,0,2],[60,62,0.9677,0.33034,0.19376,0.24999,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,8,0,0,17,0,0,1,0,0,3,0,0,1,0,0,2,0,0],[62,62,1.0,0.30356,0.26903,0.14286,0.2143,0.28571,0.0,1.0,1,3,0,1,0,15,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,3]]}]},{"i":"c796fe0faa0297d7","q":"Consider the set\n\n$$\nA=\\left\\{1+\\frac{1}{k}: k=1,2,3, \\ldots\\right\\}\n$$\n\n(a) Prove that every integer $x \\geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different.\n(b) For every integer $x \\geq 2$, let $f(x)$ denote the minimum integer such that $x$ can be written as the product of $f(x)$ elements of $A$, which are not necessarily different.\nProve that there exist infinitely many pairs $(x, y)$ of integers with $x \\geq 2, y \\geq 2$, and\n\n$$\nf(x y)<f(x)+f(y) .\n$$\n\n(Pairs $\\left(x_{1}, y_{1}\\right)$ and $\\left(x_{2}, y_{2}\\right)$ are different if $x_{1} \\neq x_{2}$ or $\\left.y_{1} \\neq y_{2}\\right)$.\n(Mihail Baluna, Romania)","t":[{"b":2,"e":0.1429,"k":"flat","v":0.34374,"x":0.72766,"p":[[0,62,0.0,0.41065,0.22227,0.2857,0.42857,0.571,0.0,0.85714,4,0,4,4,0,0,0,0,10,0,0,6,0,0,7,0,0,4,0,0,1,0,0],[4,62,0.0645,0.63838,0.2923,0.42857,0.57141,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,4,0,0,8,0,0,3,0,0,2,0,0,4,0,9],[8,62,0.129,0.64279,0.27434,0.4286,0.64286,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,3,0,0,3,0,0,7,0,0,5,0,0,4,0,7],[12,62,0.1935,0.67855,0.24745,0.4286,0.64286,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,7,0,0,7,0,0,5,0,0,2,0,9],[16,62,0.2581,0.61601,0.27303,0.42857,0.571,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,6,0,0,5,0,0,8,0,0,2,0,0,2,0,8],[20,62,0.3226,0.66515,0.27805,0.4286,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,2,0,0,4,0,0,5,0,0,7,0,0,2,0,9],[24,62,0.3871,0.6562,0.19188,0.571,0.57143,0.75,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,10,0,0,7,0,0,4,0,4],[28,62,0.4516,0.66965,0.2753,0.42964,0.64286,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,3,0,0,4,0,0,7,0,0,3,0,0,4,0,9],[32,62,0.5161,0.62499,0.27372,0.42857,0.71429,0.85704,0.1429,1.0,0,5,0,0,0,2,0,0,5,0,0,6,0,0,2,0,0,5,0,0,7,0,5],[36,62,0.5806,0.72766,0.22407,0.57143,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,5,0,0,5,0,9],[40,62,0.6452,0.4464,0.16654,0.28571,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,13,0,0,7,0,0,1,0,0,2,0,0],[44,62,0.7097,0.4598,0.1776,0.39286,0.42857,0.571,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,15,0,0,6,0,0,0,0,0,2,0,1],[48,62,0.7742,0.50428,0.22022,0.42857,0.4286,0.571,0.14,1.0,0,2,0,0,0,3,0,0,4,0,0,11,0,0,7,0,0,3,0,0,2,0,2],[52,62,0.8387,0.45533,0.16916,0.39286,0.42857,0.4642,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,16,0,0,4,0,0,1,0,0,3,0,0],[56,62,0.9032,0.38375,0.12114,0.28571,0.42857,0.4286,0.14,0.57143,0,0,0,0,0,4,0,0,6,0,0,18,0,0,4,0,0,0,0,0,0,0,0],[60,62,0.9677,0.4151,0.14014,0.28571,0.42857,0.4286,0.14,0.71429,0,0,0,0,0,2,0,0,8,0,0,16,0,0,3,0,0,3,0,0,0,0,0],[62,62,1.0,0.34374,0.12805,0.2857,0.28571,0.4286,0.14286,0.71429,0,0,0,0,0,5,0,0,12,0,0,13,0,0,1,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.85714,"k":"rising","v":0.3302,"x":0.73211,"p":[[0,112,0.0,0.3302,0.21826,0.14286,0.28571,0.4286,0.0,0.85714,5,0,5,5,0,4,0,0,9,0,0,8,0,0,3,0,0,2,0,0,1,0,0],[4,112,0.0357,0.68297,0.24678,0.4286,0.64286,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,6,0,0,7,0,0,4,0,0,3,0,9],[8,112,0.0714,0.683,0.22229,0.571,0.71429,0.85714,0.1429,1.0,0,5,0,0,0,1,0,0,2,0,0,3,0,0,7,0,0,8,0,0,6,0,5],[12,112,0.1071,0.62498,0.25191,0.42857,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,10,0,0,7,0,0,2,0,0,3,0,7],[16,112,0.1429,0.6071,0.28347,0.28571,0.57143,0.89275,0.14286,1.0,0,8,0,0,0,2,0,0,7,0,0,2,0,0,8,0,0,4,0,0,1,0,8],[20,112,0.1786,0.65176,0.26712,0.42859,0.57143,1.0,0.1429,1.0,0,9,0,0,0,1,0,0,3,0,0,8,0,0,5,0,0,4,0,0,2,0,9],[24,112,0.2143,0.59818,0.30605,0.28571,0.64264,0.85714,0.14286,1.0,0,6,0,0,0,5,0,0,4,0,0,5,0,0,2,0,0,4,0,0,6,0,6],[28,112,0.25,0.72316,0.23131,0.571,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,3,0,0,10,0,0,3,0,0,4,0,10],[32,112,0.2857,0.62941,0.29852,0.42857,0.57141,1.0,0.0,1.0,1,9,0,1,0,3,0,0,1,0,0,6,0,0,7,0,0,3,0,0,2,0,9],[36,112,0.3214,0.6786,0.23686,0.42965,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,6,0,0,3,0,0,9,0,0,5,0,6],[40,112,0.3571,0.6785,0.27202,0.571,0.64286,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,1,0,0,3,0,0,9,0,0,3,0,0,4,0,9],[44,112,0.3929,0.66068,0.22232,0.5354,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,5,0,0,7,0,0,5,0,0,9,0,3],[48,112,0.4286,0.68746,0.20343,0.571,0.57143,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,2,0,0,8,0,5],[52,112,0.4643,0.61604,0.20024,0.4286,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,7,0,0,5,0,0,3,0,4],[56,112,0.5,0.66053,0.23644,0.571,0.64286,0.85704,0.14,1.0,0,5,0,0,0,2,0,0,1,0,0,4,0,0,9,0,0,5,0,0,6,0,5],[60,112,0.5357,0.73211,0.19481,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,5,0,0,9,0,6],[64,112,0.5714,0.65172,0.22853,0.4286,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,8,0,0,9,0,0,2,0,0,5,0,6],[68,112,0.6071,0.56247,0.24468,0.42857,0.571,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,5,0,0,8,0,0,6,0,0,2,0,0,7,0,2],[72,112,0.6429,0.66069,0.19805,0.4286,0.57143,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,9,0,0,2,0,0,9,0,3],[76,112,0.6786,0.57588,0.1838,0.42857,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,12,0,0,8,0,0,4,0,0,5,0,1],[80,112,0.7143,0.59817,0.2126,0.42859,0.57141,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,8,0,0,10,0,0,4,0,0,4,0,3],[84,112,0.75,0.57137,0.21128,0.42857,0.571,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,9,0,0,10,0,0,2,0,0,5,0,2],[88,112,0.7857,0.683,0.22513,0.42857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,2,0,0,10,0,5],[92,112,0.8214,0.52676,0.2034,0.42857,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,13,0,0,7,0,0,1,0,0,5,0,1],[96,112,0.8571,0.62948,0.18843,0.43,0.57141,0.85704,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,8,0,0,11,0,0,3,0,0,7,0,2],[100,112,0.8929,0.59371,0.20237,0.42857,0.571,0.85704,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,13,0,0,5,0,0,3,0,0,8,0,1],[104,112,0.9286,0.58481,0.19674,0.4286,0.571,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,14,0,0,8,0,0,1,0,0,6,0,2],[108,112,0.9643,0.51337,0.18161,0.42857,0.42859,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,17,0,0,6,0,0,2,0,0,3,0,1],[112,112,1.0,0.60266,0.21938,0.42857,0.57143,0.85704,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,11,0,0,6,0,0,3,0,0,6,0,3]]}]},{"i":"11870a135d507ccc","q":"Let $A B C D$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that  $$ \\frac{1}{3}\\left|A B^{3}-A D^{3}\\right| \\leq\\left|B C^{3}-C D^{3}\\right| \\leq 3\\left|A B^{3}-A D^{3}\\right| $$  When does equality hold?","t":[{"b":0,"e":0.4286,"k":"rising","v":0.42857,"x":0.61605,"p":[[0,109,0.0,0.42857,0.13363,0.42857,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,22,0,0,1,0,0,1,0,0,0,0,1],[4,109,0.0367,0.52231,0.18073,0.42857,0.42857,0.60714,0.28571,0.85714,0,0,0,0,0,0,0,0,4,0,0,16,0,0,4,0,0,3,0,0,5,0,0],[8,109,0.0734,0.54017,0.22227,0.42857,0.4286,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,6,0,0,11,0,0,9,0,0,0,0,0,2,0,4],[12,109,0.1101,0.49105,0.17834,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,16,0,0,6,0,0,2,0,0,2,0,1],[16,109,0.1468,0.4999,0.18915,0.42857,0.42859,0.57143,0.14,1.0,0,2,0,0,0,2,0,0,3,0,0,13,0,0,9,0,0,3,0,0,0,0,2],[20,109,0.1835,0.50443,0.2051,0.42857,0.42859,0.60714,0.0,1.0,1,1,0,1,0,0,0,0,6,0,0,11,0,0,6,0,0,5,0,0,2,0,1],[24,109,0.2202,0.51337,0.2392,0.28571,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,1,0,0,9,0,0,9,0,0,5,0,0,2,0,0,3,0,3],[28,109,0.2569,0.43302,0.18028,0.28571,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,1,0,0,12,0,0,10,0,0,6,0,0,1,0,0,1,0,1],[32,109,0.2936,0.49552,0.21124,0.28571,0.42857,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,9,0,0,11,0,0,7,0,0,1,0,0,1,0,3],[36,109,0.3303,0.433,0.15761,0.28571,0.42857,0.42858,0.14286,0.857,0,0,0,0,0,2,0,0,7,0,0,16,0,0,3,0,0,3,0,0,1,0,0],[40,109,0.367,0.52677,0.17655,0.42857,0.42857,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,17,0,0,4,0,0,3,0,0,5,0,0],[44,109,0.4037,0.49999,0.20203,0.42857,0.42857,0.60714,0.0,1.0,1,1,0,1,0,0,0,0,5,0,0,14,0,0,4,0,0,5,0,0,2,0,1],[48,109,0.4404,0.49999,0.22868,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,7,0,0,12,0,0,2,0,0,3,0,0,5,0,1],[52,109,0.4771,0.46875,0.20277,0.39286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,5,0,0,14,0,0,4,0,0,3,0,0,2,0,1],[56,109,0.5138,0.50438,0.237,0.28571,0.42857,0.60714,0.14,1.0,0,3,0,0,0,1,0,0,8,0,0,13,0,0,2,0,0,2,0,0,3,0,3],[60,109,0.5505,0.50892,0.18536,0.42857,0.42859,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,4,0,0,16,0,0,8,0,0,1,0,0,0,0,3],[64,109,0.5872,0.47765,0.14985,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,13,0,0,9,0,0,3,0,0,1,0,0],[68,109,0.6239,0.5,0.22587,0.42857,0.42857,0.57143,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,12,0,0,7,0,0,1,0,0,3,0,2],[72,109,0.6606,0.47766,0.21609,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,8,0,0,9,0,0,6,0,0,4,0,0,2,0,1],[76,109,0.6972,0.54461,0.22428,0.42857,0.42857,0.57143,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,13,0,0,7,0,0,2,0,0,0,0,5],[80,109,0.7339,0.52676,0.19703,0.42857,0.42859,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,11,0,0,8,0,0,3,0,0,2,0,2],[84,109,0.7706,0.5223,0.23584,0.39286,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,7,0,0,12,0,0,3,0,0,3,0,0,3,0,3],[88,109,0.8073,0.55793,0.20952,0.42857,0.57143,0.71429,0.14,1.0,0,2,0,0,0,2,0,0,3,0,0,8,0,0,7,0,0,9,0,0,1,0,2],[92,109,0.844,0.52677,0.20958,0.42857,0.4998,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,10,0,0,7,0,0,6,0,0,1,0,2],[96,109,0.8807,0.55802,0.18681,0.42857,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,11,0,0,11,0,0,2,0,0,3,0,2],[100,109,0.9174,0.558,0.19019,0.42857,0.571,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,12,0,0,6,0,0,7,0,0,3,0,1],[104,109,0.9541,0.58928,0.21943,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,12,0,0,8,0,0,3,0,0,3,0,4],[108,109,0.9908,0.61605,0.2034,0.42857,0.57143,0.857,0.14286,1.0,0,1,0,0,0,1,0,0,1,0,0,9,0,0,7,0,0,5,0,0,8,0,1],[109,109,1.0,0.58034,0.21109,0.42857,0.57143,0.60714,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,11,0,0,11,0,0,2,0,0,2,0,4]]},{"b":1,"e":0.28571,"k":"flat","v":0.40625,"x":0.61159,"p":[[0,69,0.0,0.40625,0.06298,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,25,0,0,1,0,0,0,0,0,0,0,0],[4,69,0.058,0.50892,0.20805,0.39286,0.42857,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,8,0,0,12,0,0,4,0,0,4,0,0,2,0,2],[8,69,0.1159,0.56695,0.26603,0.42857,0.42859,0.75,0.14286,1.0,0,7,0,0,0,1,0,0,6,0,0,10,0,0,6,0,0,1,0,0,1,0,7],[12,69,0.1739,0.50444,0.20197,0.42857,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,4,0,0,16,0,0,5,0,0,3,0,0,0,0,3],[16,69,0.2319,0.50893,0.18877,0.42857,0.42857,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,5,0,0,16,0,0,4,0,0,4,0,0,1,0,2],[20,69,0.2899,0.48665,0.15508,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,5,0,0,16,0,0,6,0,0,4,0,0,0,0,1],[24,69,0.3478,0.61159,0.22934,0.42857,0.4998,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,14,0,0,2,0,0,6,0,0,3,0,5],[28,69,0.4058,0.58482,0.27049,0.42857,0.5,0.75,0.0,1.0,1,6,0,1,0,1,0,0,3,0,0,11,0,0,3,0,0,5,0,0,2,0,6],[32,69,0.4638,0.54909,0.23176,0.42857,0.42857,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,3,0,0,14,0,0,4,0,0,4,0,0,3,0,3],[36,69,0.5217,0.59821,0.24856,0.42857,0.42857,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,16,0,0,0,0,0,4,0,0,3,0,6],[40,69,0.5797,0.57142,0.21428,0.42857,0.4286,0.60714,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,15,0,0,7,0,0,1,0,0,3,0,4],[44,69,0.6377,0.49554,0.22299,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,12,0,0,4,0,0,3,0,0,1,0,3],[48,69,0.6957,0.54017,0.21349,0.42857,0.5712,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,9,0,0,6,0,0,8,0,0,1,0,2],[52,69,0.7536,0.5,0.18211,0.42857,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,16,0,0,1,0,0,7,0,0,1,0,1],[56,69,0.8116,0.57142,0.19884,0.42857,0.5,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,14,0,0,7,0,0,2,0,0,5,0,2],[60,69,0.8696,0.51339,0.20473,0.42857,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,17,0,0,5,0,0,2,0,0,1,0,3],[64,69,0.9275,0.46426,0.19231,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,8,0,0,8,0,0,9,0,0,3,0,0,2,0,0],[68,69,0.9855,0.51337,0.21386,0.28571,0.4998,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,4,0,0,6,0,0,5,0,0,5,0,0],[69,69,1.0,0.46417,0.20216,0.28571,0.42857,0.57143,0.14,0.85714,0,0,0,0,0,1,0,0,12,0,0,7,0,0,6,0,0,2,0,0,4,0,0]]}]},{"i":"934bd73a741da0ff","q":"Let  $ A \\equal{} \\{1,2,\\ldots,2008\\}$ \r\na) Find the number of subset of  $ A$  which satisfy : the product of its elements is divisible by 7\r\nb) Let  $ N(i)$  denotes the number of subset of  $ A$  which sum of its elements remains  $ i$  when divided by 7. Prove that  $ N(0) \\minus{} N(1) \\plus{} N(2) \\minus{} N(3) \\plus{} N(4) \\minus{} N(5) \\plus{} N(6)\\minus{}N(7) \\equal{} 0$ \r\n\r\nEDITED : thx for cosinator.. BTW, your statement and my correction give 80% hint of the solution :D","t":[{"b":1,"e":0.5714,"k":"falling","v":0.49548,"x":0.80356,"p":[[0,26,0.0,0.77232,0.21683,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,3,0,0,0,0,0,4,0,0,10,0,0,4,0,11],[4,26,0.1538,0.80356,0.22233,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,8,0,0,4,0,14],[8,26,0.3077,0.64732,0.26241,0.28571,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,9,0,0,0,0,0,4,0,0,10,0,0,2,0,7],[12,26,0.4615,0.62052,0.22191,0.53571,0.71414,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,7,0,0,1,0,0,7,0,0,12,0,0,1,0,4],[16,26,0.6154,0.49998,0.24999,0.28571,0.49979,0.71429,0.0,1.0,1,3,1,1,0,0,0,0,13,0,0,2,0,0,7,0,0,5,0,0,1,0,3],[20,26,0.7692,0.49548,0.2051,0.28571,0.571,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,13,0,0,2,0,0,10,0,0,4,0,0,2,0,1],[24,26,0.9231,0.56696,0.22442,0.28571,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,9,0,0,3,0,0,7,0,0,9,0,0,1,0,3],[26,26,1.0,0.51782,0.21052,0.28571,0.5712,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,12,0,0,1,0,0,10,0,0,7,0,0,0,0,2]]},{"b":5,"e":0.85714,"k":"flat","v":0.84375,"x":0.98214,"p":[[0,21,0.0,0.84375,0.19352,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,2,0,17],[4,21,0.1905,0.89732,0.14827,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,9,0,18],[8,21,0.381,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[12,21,0.5714,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,21,0.7619,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,21,0.9524,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[21,21,1.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]}]},{"i":"c789cba2d271d52c","q":"Find all the pairs  $(a,b)$  of integer numbers such that: $\\triangleright$   $a-b-1|a^2+b^2$  $\\triangleright$     $\\frac{a^2+b^2}{2ab-1}=\\frac{20}{19}$","t":[{"b":3,"e":1.0,"k":"flat","v":0.86161,"x":1.0,"p":[[0,38,0.0,0.86161,0.21275,0.85714,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,11,0,16],[4,38,0.1053,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,38,0.2105,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,38,0.3158,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[16,38,0.4211,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[20,38,0.5263,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,38,0.6316,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,38,0.7368,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[38,38,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":4,"e":0.14286,"k":"falling","v":0.16062,"x":0.84375,"p":[[0,34,0.0,0.84375,0.26812,0.85714,1.0,1.0,0.0,1.0,1,18,1,1,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,8,0,18],[4,34,0.1176,0.84375,0.26088,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,5,0,20],[8,34,0.2353,0.68741,0.31646,0.28571,0.85714,1.0,0.14,1.0,0,10,0,0,0,2,0,0,8,0,0,1,0,0,1,0,0,1,0,0,9,0,10],[12,34,0.3529,0.76339,0.29582,0.75,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,12,0,12],[16,34,0.4706,0.67411,0.33165,0.28571,0.85714,0.85714,0.0,1.0,1,7,0,1,0,5,0,0,3,0,0,0,0,0,2,0,0,1,0,0,13,0,7],[20,34,0.5882,0.68304,0.30875,0.39286,0.85714,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,5,0,0,2,0,0,3,0,0,2,0,0,7,0,10],[24,34,0.7059,0.44196,0.33571,0.14286,0.2857,0.85714,0.14286,1.0,0,3,0,0,0,14,0,0,5,0,0,1,0,0,1,0,0,1,0,0,7,0,3],[28,34,0.8235,0.20987,0.16366,0.14286,0.14286,0.14287,0.0,0.85714,1,0,0,1,0,24,0,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[32,34,0.9412,0.19643,0.21651,0.14286,0.14286,0.14286,0.0,1.0,3,2,0,3,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[34,34,1.0,0.16062,0.14175,0.14286,0.14286,0.14286,0.0,0.85714,3,0,0,3,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"a3dfe8f911ea3e33","q":"Let  $G$  be a graph, not containing  $K_4$  as a subgraph and  $|V(G)|=3k$  (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in  $G$ ?","t":[{"b":1,"e":1.0,"k":"flat","v":0.84822,"x":1.0,"p":[[0,39,0.0,0.91072,0.1948,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,26],[4,39,0.1026,0.84822,0.2257,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,5,0,0,0,0,21],[8,39,0.2051,0.87054,0.23517,0.92857,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,2,0,0,0,0,24],[12,39,0.3077,0.8616,0.22155,0.82132,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,3,0,21],[16,39,0.4103,0.88393,0.21261,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,3,0,0,0,0,24],[20,39,0.5128,0.94643,0.15047,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,28],[24,39,0.6154,0.89286,0.21129,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,25],[28,39,0.7179,0.95536,0.17655,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30],[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],[39,39,1.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]},{"b":4,"e":0.42857,"k":"flat","v":0.9375,"x":1.0,"p":[[0,31,0.0,0.9375,0.17104,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,28],[4,31,0.129,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[8,31,0.2581,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[12,31,0.3871,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[16,31,0.5161,0.9464,0.14624,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,28],[20,31,0.6452,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[24,31,0.7742,0.9375,0.18536,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,28],[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":"4d601d2899b050d1","q":"A point  $ D$  is taken on the side  $ AB$  of a triangle  $ ABC$ . Two circles passing through  $ D$  and touching  $ AC$  and  $ BC$  at  $ A$  and  $ B$  respectively intersect again at point  $ E$ . Let  $ F$  be the point symmetric to  $ C$  with respect to the perpendicular bisector of  $ AB$ . Prove that the points  $ D,E,F$  lie on a line.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.33036,"x":0.39732,"p":[[0,47,0.0,0.39732,0.1665,0.28571,0.35714,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,16,0,0,13,0,0,0,0,0,1,0,0,1,0,1],[4,47,0.0851,0.375,0.16269,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,17,0,0,12,0,0,0,0,0,0,0,0,1,0,1],[8,47,0.1702,0.36161,0.07129,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.33482,0.06786,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,21,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.34376,0.07016,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.33482,0.06785,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,21,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.33929,0.07783,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,21,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[28,47,0.5957,0.35714,0.07143,0.28571,0.35714,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.34821,0.07087,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,18,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.33928,0.09279,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,17,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.33036,0.06622,0.28571,0.28571,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.34821,0.10062,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,16,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[47,47,1.0,0.375,0.06916,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.2857,"k":"flat","v":0.32143,"x":0.43308,"p":[[0,36,0.0,0.43308,0.18724,0.28571,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,12,0,0,16,0,0,0,0,0,0,0,0,3,0,1],[4,36,0.1111,0.39285,0.19561,0.28571,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,22,0,0,5,0,0,0,0,0,1,0,0,4,0,0],[8,36,0.2222,0.36605,0.17471,0.28571,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,0,0,0,16,0,0,10,0,0,2,0,0,0,0,0,2,0,0],[12,36,0.3333,0.40623,0.19919,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,18,0,0,7,0,0,1,0,0,1,0,0,4,0,0],[16,36,0.4444,0.37052,0.19838,0.28571,0.28571,0.42857,0.0,1.0,2,1,0,2,0,0,0,0,18,0,0,7,0,0,2,0,0,1,0,0,1,0,1],[20,36,0.5556,0.36161,0.07974,0.28571,0.35714,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,16,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[24,36,0.6667,0.36607,0.07936,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[28,36,0.7778,0.33482,0.06785,0.28571,0.28571,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,21,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.32589,0.08918,0.28571,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,20,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.32143,0.06186,0.2857,0.28571,0.32143,0.2857,0.4286,0,0,0,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9688018b4d98ceda","q":"Let $q=\\frac{3 p-5}{2}$ where $p$ is an odd prime, and let  $$ S_{q}=\\frac{1}{2 \\cdot 3 \\cdot 4}+\\frac{1}{5 \\cdot 6 \\cdot 7}+\\cdots+\\frac{1}{q(q+1)(q+2)} $$  Prove that if $\\frac{1}{p}-2 S_{q}=\\frac{m}{n}$ for integers $m$ and $n$, then $m-n$ is divisible by $p$.","t":[{"b":2,"e":1.0,"k":"flat","v":0.76338,"x":1.0,"p":[[0,19,0.0,0.86161,0.25874,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,0,1,0,24],[4,19,0.2105,0.76338,0.3126,0.53539,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,3,0,0,2,0,0,2,0,0,1,0,0,4,0,17],[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]]},{"b":4,"e":1.0,"k":"flat","v":0.84821,"x":0.98214,"p":[[0,9,0.0,0.84821,0.24468,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,3,0,20],[4,9,0.4444,0.84821,0.23941,0.71429,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,0,4,0,19],[8,9,0.8889,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[9,9,1.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]}]},{"i":"8cf1cb3370041166","q":"$10000$  nonzero digits are written in a  $100$ -by- $100$  table, one digit per cell.  From left to right, each row forms a  $100$ -digit integer.  From top to bottom, each column forms a  $100$ -digit integer.  So the rows and columns form  $200$  integers (each with  $100$  digits), not necessarily distinct.  Prove that if at least  $199$  of these  $200$  numbers are divisible by  $2013$ , then all of them are divisible by  $2013$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.75446,"x":1.0,"p":[[0,52,0.0,0.78571,0.30514,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,0,0,0,3,0,19],[4,52,0.0769,0.75446,0.32189,0.53571,1.0,1.0,0.0,1.0,1,18,1,1,0,2,0,0,2,0,0,3,0,0,4,0,0,0,0,0,2,0,18],[8,52,0.1538,0.95089,0.15407,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[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],[32,52,0.6154,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[40,52,0.7692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":7,"e":1.0,"k":"flat","v":0.86159,"x":1.0,"p":[[0,59,0.0,0.87946,0.22899,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,0,0,1,0,24],[4,59,0.0678,0.95534,0.12599,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,28],[8,59,0.1356,0.86159,0.2487,0.89286,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,24],[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],[16,59,0.2712,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,59,0.339,0.94197,0.17075,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,28],[24,59,0.4068,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[28,59,0.4746,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[32,59,0.5424,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[36,59,0.6102,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[40,59,0.678,0.96427,0.10719,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[44,59,0.7458,0.94643,0.14174,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,27],[48,59,0.8136,0.95089,0.12682,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[52,59,0.8814,0.96427,0.10719,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,28],[56,59,0.9492,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[59,59,1.0,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29]]}]},{"i":"a038577cd5ac5084","q":"Let  $n\\ge3$  be an integer. A regular  $n$ -gon  $P$  is given. We randomly select three distinct vertices of  $P$ . The probability that these three vertices form an isosceles triangle is  $1/m$ , where  $m$  is an integer. How many such integers  $n\\le 2024$  are there?\n\\[\\rm a. ~674\\qquad \\mathrm b. ~675\\qquad \\mathrm c. ~682 \\qquad\\mathrm d. ~684\\qquad\\mathrm e. ~685\\]","t":[{"b":0,"e":0.85714,"k":"flat","v":0.79464,"x":0.83036,"p":[[0,5,0.0,0.83036,0.18013,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,0,13,0,10],[4,5,0.8,0.79911,0.15916,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,12,0,7],[5,5,1.0,0.79464,0.12846,0.71429,0.85714,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,18,0,2]]},{"b":2,"e":0.71429,"k":"flat","v":0.77679,"x":0.875,"p":[[0,57,0.0,0.82142,0.10717,0.71429,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,15,0,5],[4,57,0.0702,0.78572,0.14285,0.71429,0.85714,0.85714,0.4286,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,12,0,5],[8,57,0.1404,0.84375,0.15303,0.82132,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,14,0,10],[12,57,0.2105,0.84375,0.16115,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,10,0,12],[16,57,0.2807,0.84375,0.12556,0.71429,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,13,0,9],[20,57,0.3509,0.87054,0.12556,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,13,0,12],[24,57,0.4211,0.87054,0.16115,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,10,0,15],[28,57,0.4912,0.87054,0.15303,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,9,0,15],[32,57,0.5614,0.82589,0.17399,0.82143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,14,0,10],[36,57,0.6316,0.87054,0.16506,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,16],[40,57,0.7018,0.875,0.15046,0.85714,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,10,0,15],[44,57,0.7719,0.81696,0.18638,0.71429,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,6,0,0,11,0,10],[48,57,0.8421,0.83036,0.19377,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,3,0,0,8,0,14],[52,57,0.9123,0.77679,0.15947,0.67857,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,7,0,0,11,0,6],[56,57,0.9825,0.84821,0.15947,0.82143,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,12,0,12],[57,57,1.0,0.79018,0.17491,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,7,0,0,10,0,8]]}]},{"i":"43b0050af874ce0c","q":"Let  $S_n$  denote the set of permutations of the sequence  $(1,2,\\dots, n)$ . For every permutation  $\\pi=(\\pi_1, \\dots, \\pi_n)\\in S_n$ , let  $\\mathrm{inv}(\\pi)$  be the number of pairs  $1\\le i < j \\le n$  with  $\\pi_i>\\pi_j$ ; i. e. the number of inversions in  $\\pi$ . Denote by  $f(n)$  the number of permutations  $\\pi\\in S_n$  for which  $\\mathrm{inv}(\\pi)$  is divisible by  $n+1$ .\nProve that there exist infinitely many primes  $p$  such that  $f(p-1)>\\frac{(p-1)!}{p}$ , and infinitely many primes  $p$  such that  $f(p-1)<\\frac{(p-1)!}{p}$ .\n\n(Proposed by Fedor Petrov, St. Petersburg State University)","t":[{"b":3,"e":1.0,"k":"flat","v":0.96429,"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],[4,31,0.129,0.96429,0.09449,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[8,31,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,31,0.3871,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[28,31,0.9032,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]},{"b":7,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,11,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,11,0.3636,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[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],[11,11,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"8539746f0a62de06","q":"Let $a_{0}, a_{1}, \\ldots, a_{n}$ be numbers from the interval $(0, \\pi / 2)$ such that $\\tan \\left(a_{0}-\\frac{\\pi}{4}\\right)+$ $\\tan \\left(a_{1}-\\frac{\\pi}{4}\\right)+\\cdots+\\tan \\left(a_{n}-\\frac{\\pi}{4}\\right) \\geq n-1$. Prove that  $$ \\tan a_{0} \\tan a_{1} \\cdots \\tan a_{n} \\geq n^{n+1} $$","t":[{"b":5,"e":1.0,"k":"rising","v":0.79463,"x":0.96429,"p":[[0,8,0.0,0.79463,0.24985,0.53539,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,4,0,0,2,0,17],[4,8,0.5,0.81696,0.25812,0.42859,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0,4,0,19],[8,8,1.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25]]},{"b":6,"e":1.0,"k":"flat","v":0.8125,"x":0.97322,"p":[[0,24,0.0,0.8125,0.26351,0.42857,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,21],[4,24,0.1667,0.91518,0.17807,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,1,0,25],[8,24,0.3333,0.86607,0.2141,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,0,3,0,21],[12,24,0.5,0.96428,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[16,24,0.6667,0.94643,0.11709,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,24],[20,24,0.8333,0.97322,0.10374,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[24,24,1.0,0.95982,0.11425,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,27]]}]},{"i":"d1ff8ff905ed7fce","q":"Decide whether the integers  $1,2,\\ldots,100$  can be arranged in the cells  $C(i, j)$  of a  $10\\times10$  matrix (where  $1\\le i,j\\le 10$ ), such that the following conditions are fullfiled:\ni) In every row, the entries add up to the same sum  $S$ .\nii) In every column, the entries also add up to this sum  $S$ .\niii) For every  $k = 1, 2, \\ldots, 10$  the ten entries  $C(i, j)$  with  $i-j\\equiv k\\bmod{10}$  add up to  $S$ .\n*(Proposed by Gerhard Woeginger, Austria)*","t":[{"b":1,"e":0.28571,"k":"flat","v":0.14732,"x":0.79911,"p":[[0,56,0.0,0.14732,0.15355,0.0,0.14286,0.17857,0.0,0.57143,12,0,0,12,0,12,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,56,0.0714,0.41518,0.41243,0.0,0.2857,1.0,0.0,1.0,9,9,0,9,0,6,0,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,9],[8,56,0.1429,0.42857,0.40248,0.14286,0.2857,0.89286,0.0,1.0,7,8,1,7,0,8,0,0,5,0,0,0,0,0,0,0,0,2,0,0,2,0,8],[12,56,0.2143,0.32803,0.31792,0.14286,0.21428,0.35714,0.0,1.0,5,3,0,5,1,10,0,0,8,0,0,0,0,0,1,0,0,2,0,0,2,0,3],[16,56,0.2857,0.29464,0.33681,0.14286,0.14286,0.2857,0.0,1.0,7,4,0,7,0,14,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2,0,4],[20,56,0.3571,0.41517,0.36484,0.14286,0.28571,0.85714,0.0,1.0,4,7,0,4,0,8,0,0,10,0,0,0,0,0,1,0,0,0,0,0,2,0,7],[24,56,0.4286,0.68293,0.35859,0.28571,0.85714,1.0,0.0,1.0,1,14,0,1,0,5,0,0,4,0,0,0,0,0,2,0,0,2,0,0,4,0,14],[28,56,0.5,0.64062,0.36748,0.14286,0.78571,1.0,0.0,1.0,1,11,0,1,1,7,0,0,1,0,0,1,0,0,1,0,0,4,0,0,5,0,11],[32,56,0.5714,0.57589,0.41108,0.14286,0.64286,1.0,0.0,1.0,4,13,0,4,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0,3,0,13],[36,56,0.6429,0.52677,0.41869,0.14286,0.49979,1.0,0.0,1.0,5,11,0,5,0,9,0,0,1,0,0,1,0,0,1,0,0,1,0,0,3,0,11],[40,56,0.7143,0.79911,0.30275,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,1,0,0,1,0,0,1,0,0,1,0,0,5,0,0,3,0,18],[44,56,0.7857,0.66518,0.34369,0.39286,0.71429,1.0,0.0,1.0,2,11,0,2,0,4,0,0,2,0,0,2,0,0,1,0,0,6,0,0,4,0,11],[48,56,0.8571,0.44196,0.40305,0.14286,0.14288,0.89286,0.0,1.0,4,8,0,4,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,8],[52,56,0.9286,0.43304,0.3416,0.14286,0.35714,0.75,0.0,1.0,6,3,0,6,0,5,0,0,5,0,0,3,0,0,3,0,0,2,0,0,5,0,3],[56,56,1.0,0.22768,0.18509,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,14,0,0,7,0,0,3,0,0,1,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.03125,"x":0.2009,"p":[[0,35,0.0,0.12946,0.19351,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,35,0.1143,0.16964,0.3597,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,35,0.2286,0.2009,0.35689,0.0,0.0,0.14287,0.0,1.0,20,4,0,20,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,4],[12,35,0.3429,0.1875,0.35792,0.0,0.0,0.14286,0.0,1.0,22,5,0,22,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[16,35,0.4571,0.10268,0.24284,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,35,0.5714,0.16955,0.31225,0.0,0.0,0.14286,0.0,1.0,19,3,0,19,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[24,35,0.6857,0.08482,0.24186,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,35,0.8,0.09375,0.23038,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[32,35,0.9143,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"70a7759b99dc301c","q":"Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that the following inequality holds:\n\n$$\n2\\left(a^{2}+b^{2}+c^{2}\\right)\\left(\\frac{1}{a^{2}}+\\frac{1}{b^{2}}+\\frac{1}{c^{2}}\\right) \\geqslant 3(a+b+c+a b+b c+c a)\n$$\n\n(Romania)","t":[{"b":2,"e":0.0,"k":"flat","v":0.02679,"x":0.14732,"p":[[0,42,0.0,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.10714,0.15972,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,8,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,42,0.1905,0.13393,0.14698,0.0,0.14286,0.2857,0.0,0.42857,15,0,0,15,0,7,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.0982,0.13087,0.0,0.0,0.14286,0.0,0.571,17,0,0,17,0,10,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,42,0.381,0.14732,0.16164,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,9,0,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,42,0.4762,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.12053,0.17169,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,8,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[28,42,0.6667,0.10267,0.1347,0.0,0.0,0.14286,0.0,0.571,17,0,0,17,0,9,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,42,0.7619,0.08036,0.14258,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,42,0.8571,0.06696,0.16746,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[40,42,0.9524,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.07143,0.21429,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1]]},{"b":6,"e":0.14286,"k":"rising","v":0.05804,"x":0.32143,"p":[[0,89,0.0,0.09366,0.14984,0.0,0.0,0.14286,0.0,0.71429,18,0,1,18,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,89,0.0449,0.11607,0.1357,0.0,0.14286,0.14287,0.0,0.57143,15,0,0,15,0,10,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,89,0.0899,0.12936,0.22116,0.0,0.0,0.17857,0.0,1.0,20,1,0,20,0,4,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[12,89,0.1348,0.10714,0.10102,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,89,0.1798,0.125,0.15872,0.0,0.07143,0.2857,0.0,0.71429,16,0,0,16,0,7,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,89,0.2247,0.08037,0.12847,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,89,0.2697,0.12054,0.14334,0.0,0.14286,0.14286,0.0,0.71429,13,0,0,13,0,14,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,89,0.3146,0.11607,0.11538,0.0,0.14286,0.17857,0.0,0.28571,14,0,0,14,0,10,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,89,0.3596,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,89,0.4045,0.08928,0.17033,0.0,0.0,0.14286,0.0,0.857,20,0,0,20,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,89,0.4494,0.11607,0.15746,0.0,0.0,0.17857,0.0,0.57143,18,0,0,18,0,6,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,89,0.4944,0.12946,0.18336,0.0,0.0,0.2857,0.0,0.85714,17,0,0,17,0,6,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[48,89,0.5393,0.10714,0.12372,0.0,0.07143,0.14287,0.0,0.42857,16,0,0,16,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,89,0.5843,0.09821,0.18013,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[56,89,0.6292,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,89,0.6742,0.11143,0.1223,0.0,0.14,0.14286,0.0,0.42857,15,0,0,15,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,89,0.7191,0.12947,0.12556,0.0,0.14286,0.2857,0.0,0.42857,13,0,0,13,0,10,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,89,0.764,0.08929,0.1171,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,89,0.809,0.13839,0.19061,0.0,0.07143,0.17857,0.0,0.71429,16,0,0,16,0,8,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[76,89,0.8539,0.13393,0.10062,0.0,0.14286,0.1429,0.0,0.28571,9,0,0,9,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.14277,0.12877,0.0,0.14286,0.2857,0.0,0.42857,12,0,0,12,0,9,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.18304,0.20589,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,11,0,0,6,0,0,1,0,0,0,0,0,3,0,0,0,0,0],[88,89,0.9888,0.23213,0.22796,0.0,0.1429,0.28571,0.0,0.85714,9,0,0,9,0,8,0,0,9,0,0,2,0,0,2,0,0,0,0,0,2,0,0],[89,89,1.0,0.32143,0.29233,0.10714,0.28571,0.46431,0.0,1.0,8,1,0,8,0,4,0,0,11,0,0,1,0,0,2,0,0,2,0,0,3,0,1]]}]},{"i":"8b838f8e2bbb5add","q":"Let  $a,b,c\\in \\mathbb N$  be such that  $a,b\\neq c$ . Prove that there are infinitely many prime numbers  $p$  for which there exists  $n\\in\\mathbb N$  that  $p|a^n+b^n-c^n$ .","t":[{"b":0,"e":0.14286,"k":"rising","v":0.10268,"x":0.32143,"p":[[0,17,0.0,0.10268,0.14826,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,9,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.11152,0.14165,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,14,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,17,0.4706,0.29464,0.11258,0.14289,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,9,0,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.30795,0.11369,0.24999,0.28571,0.42857,0.14,0.4286,0,0,0,0,0,8,0,0,11,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.31697,0.11701,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,12,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[17,17,1.0,0.32143,0.11294,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,13,0,0,12,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.08927,"p":[[0,8,0.0,0.08927,0.14169,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,8,0.5,0.06687,0.11831,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,9,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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]]}]},{"i":"70949b69dc0e90a5","q":"Determine all unordered triples  $(x,y,z)$  of integers for which the number  $\\sqrt{\\frac{2005}{x+y}}+\\sqrt{\\frac{2005}{y+z}}+\\sqrt{\\frac{2005}{z+x}}$  is an integer.","t":[{"b":4,"e":1.0,"k":"flat","v":0.91964,"x":0.95536,"p":[[0,5,0.0,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[4,5,0.8,0.91964,0.16342,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,21],[5,5,1.0,0.9241,0.09439,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,17]]},{"b":7,"e":1.0,"k":"flat","v":0.89286,"x":0.99554,"p":[[0,52,0.0,0.89286,0.12877,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,10,0,16],[4,52,0.0769,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[8,52,0.1538,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,52,0.2308,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,52,0.3077,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[20,52,0.3846,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[24,52,0.4615,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[28,52,0.5385,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[32,52,0.6154,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,52,0.6923,0.96429,0.08748,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[40,52,0.7692,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[44,52,0.8462,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,52,0.9231,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[52,52,1.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"9de442f90c4b76fd","q":"Find all triples  $(a, b, c)$  of positive integers such that  $a \\leq b$  and \\[a!+b!=c^4+2024\\]\n\n*Proposed by Otgonbayar Uuye.*","t":[{"b":4,"e":0.85714,"k":"flat","v":0.88393,"x":0.97321,"p":[[0,27,0.0,0.91964,0.17474,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,23],[4,27,0.1481,0.88393,0.2299,0.82143,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,1,0,23],[8,27,0.2963,0.93747,0.13341,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[12,27,0.4444,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[16,27,0.5926,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[20,27,0.7407,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,27,0.8889,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[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]]},{"b":5,"e":1.0,"k":"flat","v":0.90624,"x":0.99107,"p":[[0,71,0.0,0.90624,0.13654,0.85714,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,8,0,19],[4,71,0.0563,0.93304,0.12869,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,23],[8,71,0.1127,0.94643,0.09942,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[12,71,0.169,0.97321,0.08328,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[16,71,0.2254,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[20,71,0.2817,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[24,71,0.338,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,71,0.3944,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,71,0.4507,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,71,0.507,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[40,71,0.5634,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[44,71,0.6197,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[48,71,0.6761,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[52,71,0.7324,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[56,71,0.7887,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[60,71,0.8451,0.94196,0.10013,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,22],[64,71,0.9014,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[68,71,0.9577,0.94196,0.09354,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,21],[71,71,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"947d5c112f459381","q":"Find all functions  $f$  from the set of reals to itself so that for all reals  $x,y,$    $$ f(x)f(f(x)+y) = f(x^2) + f(xy). $$  *Proposed by Culver Kwan*","t":[{"b":2,"e":0.42857,"k":"flat","v":0.42856,"x":0.48214,"p":[[0,37,0.0,0.42857,0.17857,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,7,0,0,9,0,0,11,0,0,0,0,0,0,0,1],[4,37,0.1081,0.48214,0.15463,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,11,0,0,11,0,0,2,0,0,0,0,1],[8,37,0.2162,0.48214,0.13716,0.42857,0.50001,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,7,0,0,9,0,0,14,0,0,1,0,0,1,0,0],[12,37,0.3243,0.42856,0.10712,0.42857,0.42857,0.46418,0.14286,0.57143,0,0,0,0,0,1,0,0,6,0,0,17,0,0,8,0,0,0,0,0,0,0,0],[16,37,0.4324,0.44643,0.12242,0.42857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,2,0,0,4,0,0,14,0,0,12,0,0,0,0,0,0,0,0],[20,37,0.5405,0.47319,0.12076,0.42857,0.4998,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,7,0,0,9,0,0,15,0,0,1,0,0,0,0,0],[24,37,0.6486,0.4375,0.18877,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,5,0,0,14,0,0,6,0,0,1,0,0,1,0,1],[28,37,0.7568,0.45982,0.11701,0.42857,0.42857,0.42857,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,0,0,2,0,0],[32,37,0.8649,0.44195,0.06543,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,28,0,0,2,0,0,1,0,0,0,0,0],[36,37,0.973,0.45536,0.10374,0.42857,0.42857,0.42857,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,2,0,0,1,0,0],[37,37,1.0,0.45536,0.08328,0.42857,0.42857,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,26,0,0,3,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.41964,"x":0.54017,"p":[[0,25,0.0,0.46875,0.18638,0.39286,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,12,0,0,9,0,0,0,0,0,2,0,1],[4,25,0.16,0.41964,0.15542,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,11,0,0,8,0,0,10,0,0,0,0,0,1,0,0],[8,25,0.32,0.4375,0.14698,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,10,0,0,8,0,0,3,0,0,0,0,0],[12,25,0.48,0.44195,0.14444,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,9,0,0,10,0,0,11,0,0,0,0,0,1,0,0],[16,25,0.64,0.48659,0.10629,0.42857,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,5,0,0,9,0,0,18,0,0,0,0,0,0,0,0],[20,25,0.8,0.54017,0.11142,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,7,0,0,20,0,0,2,0,0,1,0,0],[24,25,0.96,0.50891,0.11811,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,3,0,0,22,0,0,1,0,0,0,0,0],[25,25,1.0,0.53121,0.11969,0.53539,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,5,0,0,21,0,0,3,0,0,0,0,0]]}]},{"i":"afbcc6da6f0bd8c5","q":"Let  $(x_n)$  define by  $x_1\\in \\left(0;\\dfrac{1}{2}\\right)$  and  $x_{n+1}=3x_n^2-2nx_n^3$  for all  $n\\ge 1$ .\na) Prove that  $(x_n)$  convergence to  $0$ .\n\nb) For each  $n\\ge 1$ , let  $y_n=x_1+2x_2+\\cdots+n x_n$ . Prove that  $(y_n)$  has a limit.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.51782,"x":0.80357,"p":[[0,102,0.0,0.51782,0.17766,0.42857,0.4286,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,13,0,0,4,0,0,8,0,0,2,0,0],[4,102,0.0392,0.73214,0.1915,0.71429,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,15,0,0,3,0,7],[8,102,0.0784,0.80357,0.19149,0.71429,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,12,0,0,2,0,13],[12,102,0.1176,0.71427,0.22304,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,8,0,0,1,0,0,10,0,0,4,0,8],[16,102,0.1569,0.732,0.21355,0.71321,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,15,0,0,1,0,9],[20,102,0.1961,0.74109,0.1904,0.67857,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,13,0,0,3,0,8],[24,102,0.2353,0.66962,0.21262,0.42857,0.71429,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,11,0,0,2,0,6],[28,102,0.2745,0.75892,0.19378,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,15,0,0,1,0,10],[32,102,0.3137,0.70981,0.22442,0.42859,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,10,0,0,0,0,10],[36,102,0.3529,0.73659,0.21757,0.5354,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,8,0,0,1,0,0,11,0,0,2,0,10],[40,102,0.3922,0.74107,0.23808,0.71429,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,13,0,0,1,0,11],[44,102,0.4314,0.68304,0.21939,0.42857,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,9,0,0,2,0,0,11,0,0,2,0,7],[48,102,0.4706,0.71427,0.21725,0.57132,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,13,0,0,2,0,8],[52,102,0.5098,0.66517,0.25155,0.42857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,10,0,0,2,0,0,9,0,0,0,0,9],[56,102,0.549,0.73659,0.21462,0.57143,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,12,0,0,1,0,10],[60,102,0.5882,0.61607,0.23808,0.42857,0.64286,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,8,0,0,3,0,0,10,0,0,0,0,6],[64,102,0.6275,0.61607,0.19377,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,11,0,0,1,0,0,13,0,0,3,0,2],[68,102,0.6667,0.68304,0.24415,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,9,0,0,3,0,8],[72,102,0.7059,0.74106,0.23809,0.67846,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,4,0,0,1,0,0,11,0,0,2,0,11],[76,102,0.7451,0.70089,0.21829,0.4286,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,8,0,0,1,0,0,13,0,0,1,0,8],[80,102,0.7843,0.62944,0.1851,0.42857,0.71429,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,13,0,0,3,0,2],[84,102,0.8235,0.62052,0.249,0.42857,0.71429,0.71429,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,8,0,0,2,0,0,10,0,0,1,0,6],[88,102,0.8627,0.65621,0.23382,0.42857,0.71429,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,3,0,0,8,0,0,3,0,0,10,0,0,1,0,7],[92,102,0.902,0.72321,0.23941,0.42857,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,10,0,0,1,0,11],[96,102,0.9412,0.73214,0.20124,0.67857,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,7,0,0,1,0,0,13,0,0,3,0,8],[100,102,0.9804,0.70979,0.23005,0.571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,5,0,0,2,0,0,14,0,0,0,0,9],[102,102,1.0,0.66517,0.1976,0.53539,0.71429,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,13,0,0,3,0,4]]},{"b":5,"e":0.71429,"k":"flat","v":0.46428,"x":0.74554,"p":[[0,54,0.0,0.56692,0.19392,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,8,0,0,9,0,0,8,0,0,1,0,2],[4,54,0.0741,0.73659,0.22336,0.53539,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,10,0,0,1,0,11],[8,54,0.1481,0.68304,0.23618,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,9,0,0,1,0,9],[12,54,0.2222,0.66963,0.22429,0.42857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,10,0,0,3,0,0,9,0,0,2,0,7],[16,54,0.2963,0.74554,0.23072,0.4286,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,10,0,0,1,0,12],[20,54,0.3704,0.65177,0.19865,0.42859,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,9,0,0,4,0,0,11,0,0,3,0,4],[24,54,0.4444,0.69195,0.22048,0.5354,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,8,0,0,2,0,8],[28,54,0.5185,0.61157,0.18293,0.42857,0.57143,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,10,0,0,7,0,0,10,0,0,1,0,3],[32,54,0.5926,0.65624,0.25719,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,10,0,0,2,0,0,9,0,0,1,0,8],[36,54,0.6667,0.63393,0.20806,0.42857,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,11,0,0,1,0,5],[40,54,0.7407,0.56695,0.2004,0.42857,0.42857,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,16,0,0,2,0,0,8,0,0,1,0,3],[44,54,0.8148,0.68304,0.19799,0.42857,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,16,0,0,2,0,5],[48,54,0.8889,0.62054,0.22477,0.42857,0.71429,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,9,0,0,2,0,0,10,0,0,3,0,4],[52,54,0.963,0.46428,0.19885,0.2857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,8,0,0,12,0,0,3,0,0,5,0,0,1,0,1],[54,54,1.0,0.55802,0.1868,0.42857,0.42859,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,14,0,0,3,0,0,8,0,0,3,0,1]]}]},{"i":"e398cead60af18dc","q":"(a) Prove that for every real number $t$ such that $0<t<\\frac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \\geq 0$ ), such that\n\n$$\n|x-m y| \\leq t y\n$$\n\n(b) Determine whether for every real number $t$ such that $0<t<\\frac{1}{2}$ there exists an infinite set $S$ of positive integers such that\n\n$$\n|x-m y|>t y\n$$\n\nfor every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m>0$ ).\n(Merlijn Staps, The Netherlands)","t":[{"b":0,"e":0.42857,"k":"falling","v":0.24107,"x":0.50004,"p":[[0,61,0.0,0.48212,0.3549,0.14286,0.42857,0.85714,0.0,1.0,7,3,5,7,0,3,0,0,2,0,0,5,0,0,2,0,0,3,0,0,7,0,3],[4,61,0.0656,0.50004,0.28792,0.28571,0.50071,0.71429,0.0,1.0,3,2,0,3,0,2,0,0,7,0,0,4,0,0,4,0,0,6,0,0,4,0,2],[8,61,0.1311,0.46878,0.32188,0.14286,0.4293,0.71429,0.0,1.0,5,4,0,5,0,4,0,0,3,0,0,5,0,0,4,0,0,6,0,0,1,0,4],[12,61,0.1967,0.43301,0.30194,0.14286,0.42857,0.71429,0.0,1.0,5,2,0,5,0,4,0,0,4,0,0,7,0,0,3,0,0,4,0,0,3,0,2],[16,61,0.2623,0.39285,0.2945,0.14286,0.35714,0.71429,0.0,0.85714,6,0,0,6,0,5,0,0,5,0,0,4,0,0,3,0,0,5,0,0,4,0,0],[20,61,0.3279,0.35267,0.255,0.14286,0.28571,0.57111,0.0,0.85714,6,0,0,6,0,4,0,0,7,0,0,6,0,0,4,0,0,3,0,0,2,0,0],[24,61,0.3934,0.46415,0.28355,0.14286,0.4998,0.71429,0.0,1.0,3,1,0,3,0,6,0,0,2,0,0,5,0,0,7,0,0,4,0,0,4,0,1],[28,61,0.459,0.3749,0.25699,0.14286,0.35714,0.57111,0.0,0.85714,3,0,0,3,0,8,0,0,5,0,0,7,0,0,4,0,0,1,0,0,4,0,0],[32,61,0.5246,0.33027,0.22436,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,7,0,0,7,0,0,8,0,0,2,0,0,3,0,0,1,0,0],[36,61,0.5902,0.37498,0.29177,0.14286,0.35714,0.57111,0.0,1.0,7,1,0,7,0,3,0,0,6,0,0,7,0,0,2,0,0,3,0,0,3,0,1],[40,61,0.6557,0.43303,0.29121,0.2857,0.42857,0.60714,0.0,1.0,4,2,0,4,0,3,0,0,7,0,0,8,0,0,2,0,0,2,0,0,4,0,2],[44,61,0.7213,0.34354,0.20487,0.25,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,6,0,0,11,0,0,8,0,0,2,0,0,1,0,0,2,0,0],[48,61,0.7869,0.36607,0.17105,0.2857,0.28571,0.4286,0.14286,0.85714,0,0,0,0,0,5,0,0,13,0,0,9,0,0,2,0,0,2,0,0,1,0,0],[52,61,0.8525,0.24107,0.16146,0.10714,0.2857,0.28571,0.0,0.57143,8,0,0,8,0,2,0,0,15,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[56,61,0.918,0.40615,0.21172,0.28571,0.42857,0.4286,0.0,1.0,2,1,0,2,0,1,0,0,11,0,0,11,0,0,4,0,0,0,0,0,2,0,1],[60,61,0.9836,0.37499,0.21942,0.2857,0.42857,0.4286,0.0,1.0,2,1,0,2,0,5,0,0,7,0,0,14,0,0,1,0,0,0,0,0,2,0,1],[61,61,1.0,0.27215,0.20012,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,9,0,0,7,0,0,9,0,0,0,0,0,1,0,0,1,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.18741,"x":0.76784,"p":[[0,70,0.0,0.54446,0.37551,0.14214,0.50001,0.85714,0.0,1.0,6,7,5,6,0,3,0,0,1,0,0,6,0,0,1,0,0,2,0,0,6,0,7],[4,70,0.0571,0.76784,0.21054,0.71429,0.85714,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,13,0,6],[8,70,0.1143,0.59378,0.31963,0.42859,0.71429,0.85714,0.0,1.0,4,5,0,4,0,2,0,0,1,0,0,3,0,0,5,0,0,7,0,0,5,0,5],[12,70,0.1714,0.58034,0.27185,0.42857,0.57143,0.85714,0.0,1.0,1,2,0,1,0,4,0,0,1,0,0,6,0,0,6,0,0,4,0,0,8,0,2],[16,70,0.2286,0.5982,0.3489,0.28571,0.71429,0.85714,0.0,1.0,5,6,0,5,0,1,0,0,3,0,0,3,0,0,1,0,0,6,0,0,7,0,6],[20,70,0.2857,0.50444,0.32532,0.1429,0.57143,0.74996,0.0,1.0,4,2,0,4,0,5,0,0,4,0,0,0,0,0,5,0,0,6,0,0,6,0,2],[24,70,0.3429,0.51784,0.28514,0.28571,0.57143,0.74996,0.0,1.0,3,1,0,3,0,3,0,0,3,0,0,5,0,0,7,0,0,3,0,0,7,0,1],[28,70,0.4,0.43304,0.32436,0.14286,0.42857,0.71429,0.0,1.0,6,3,0,6,0,4,0,0,4,0,0,5,0,0,4,0,0,3,0,0,3,0,3],[32,70,0.4571,0.28552,0.22598,0.14,0.28571,0.42858,0.0,0.85714,7,0,0,7,0,6,0,0,8,0,0,5,0,0,4,0,0,1,0,0,1,0,0],[36,70,0.5143,0.34821,0.28105,0.14286,0.28571,0.57111,0.0,0.85714,5,0,0,5,0,9,0,0,4,0,0,5,0,0,4,0,0,0,0,0,5,0,0],[40,70,0.5714,0.28107,0.31036,0.0,0.14286,0.42858,0.0,0.85714,11,0,0,11,0,7,0,0,5,0,0,2,0,0,0,0,0,2,0,0,5,0,0],[44,70,0.6286,0.39286,0.30723,0.14286,0.35714,0.71429,0.0,1.0,7,1,0,7,0,3,0,0,6,0,0,6,0,0,1,0,0,4,0,0,4,0,1],[48,70,0.6857,0.52676,0.29544,0.28571,0.57121,0.857,0.0,1.0,3,1,0,3,0,2,0,0,6,0,0,4,0,0,3,0,0,5,0,0,8,0,1],[52,70,0.7429,0.33035,0.2889,0.14286,0.2857,0.42858,0.0,1.0,7,2,0,7,0,5,0,0,8,0,0,6,0,0,1,0,0,1,0,0,2,0,2],[56,70,0.8,0.32588,0.263,0.10714,0.28571,0.42858,0.0,1.0,8,1,0,8,0,3,0,0,6,0,0,9,0,0,2,0,0,2,0,0,1,0,1],[60,70,0.8571,0.28125,0.2382,0.10714,0.2857,0.42857,0.0,1.0,8,1,0,8,0,5,0,0,6,0,0,11,0,0,0,0,0,0,0,0,1,0,1],[64,70,0.9143,0.24534,0.2481,0.0,0.14288,0.42857,0.0,1.0,10,1,0,10,0,7,0,0,6,0,0,5,0,0,1,0,0,2,0,0,0,0,1],[68,70,0.9714,0.21429,0.26486,0.0,0.14286,0.28571,0.0,1.0,12,1,0,12,0,9,0,0,5,0,0,2,0,0,0,0,0,2,0,0,1,0,1],[70,70,1.0,0.18741,0.1871,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,7,0,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"cbb294088bcd1943","q":"Four points $P, Q, R$, and $S$ lie in this order on a circle, such that $\\angle P S R=90^{\\circ}$. Let $H$ and $K$ be the feet of the perpendiculars from $Q$ to $P R$ and $P S$, respectively. Let $T$ be the intersection of $H K$ and $Q S$. Prove that $|S T|=|T Q|$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.05357,"x":0.77677,"p":[[0,82,0.0,0.14286,0.32927,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,82,0.0488,0.69643,0.42069,0.28571,1.0,1.0,0.0,1.0,7,20,0,7,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,20],[8,82,0.0976,0.625,0.43704,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,17],[12,82,0.1463,0.63392,0.44741,0.0,1.0,1.0,0.0,1.0,9,17,0,9,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,17],[16,82,0.1951,0.77677,0.33682,0.67857,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,3,0,0,0,0,0,2,0,0,4,0,0,0,0,20],[20,82,0.2439,0.55801,0.43207,0.0,0.57121,1.0,0.0,1.0,9,14,0,9,0,1,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,14],[24,82,0.2927,0.56694,0.43225,0.0,0.64286,1.0,0.0,1.0,9,14,0,9,0,1,0,0,3,0,0,0,0,0,3,0,0,2,0,0,0,0,14],[28,82,0.3415,0.54018,0.44138,0.0,0.57144,1.0,0.0,1.0,10,13,0,10,0,0,0,0,5,0,0,1,0,0,0,0,0,1,0,0,2,0,13],[32,82,0.3902,0.51783,0.4514,0.0,0.57121,1.0,0.0,1.0,11,13,0,11,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,13],[36,82,0.439,0.33036,0.4141,0.0,0.0,0.71429,0.0,1.0,17,7,0,17,0,1,0,0,3,0,0,1,0,0,0,0,0,3,0,0,0,0,7],[40,82,0.4878,0.51785,0.47076,0.0,0.64286,1.0,0.0,1.0,13,14,0,13,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,14],[44,82,0.5366,0.51786,0.43117,0.0,0.42857,1.0,0.0,1.0,9,12,0,9,0,2,0,0,5,0,0,0,0,0,1,0,0,2,0,0,1,0,12],[48,82,0.5854,0.28125,0.39364,0.0,0.0,0.5,0.0,1.0,18,6,0,18,0,2,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,6],[52,82,0.6341,0.49106,0.46694,0.0,0.35714,1.0,0.0,1.0,13,14,0,13,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,14],[56,82,0.6829,0.55802,0.45647,0.0,0.71421,1.0,0.0,1.0,11,15,0,11,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,15],[60,82,0.7317,0.42409,0.46495,0.0,0.07143,1.0,0.0,1.0,16,11,0,16,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,11],[64,82,0.7805,0.64723,0.41351,0.24999,0.92857,1.0,0.0,1.0,6,16,0,6,0,2,0,0,3,0,0,0,0,0,1,0,0,3,0,0,1,0,16],[68,82,0.8293,0.375,0.42968,0.0,0.07143,0.85714,0.0,1.0,16,7,0,16,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,0,3,0,7],[72,82,0.878,0.08928,0.21352,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[76,82,0.9268,0.07141,0.18893,0.0,0.0,0.0,0.0,0.857,26,0,0,26,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[80,82,0.9756,0.08036,0.21998,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[82,82,1.0,0.05357,0.12753,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.08482,"x":0.70982,"p":[[0,52,0.0,0.16964,0.30396,0.0,0.0,0.2857,0.0,1.0,21,3,0,21,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[4,52,0.0769,0.57587,0.39526,0.2857,0.57121,1.0,0.0,1.0,6,12,0,6,0,1,0,0,5,0,0,3,0,0,2,0,0,1,0,0,2,0,12],[8,52,0.1538,0.51786,0.47882,0.0,0.57143,1.0,0.0,1.0,13,15,0,13,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,15],[12,52,0.2308,0.61152,0.4393,0.0,0.92857,1.0,0.0,1.0,9,16,0,9,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,16],[16,52,0.3077,0.65178,0.45025,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,18],[20,52,0.3846,0.66517,0.42948,0.10714,1.0,1.0,0.0,1.0,8,17,0,8,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,17],[24,52,0.4615,0.62946,0.41933,0.10714,0.71429,1.0,0.0,1.0,8,15,0,8,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,0,0,0,15],[28,52,0.5385,0.65622,0.45438,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,19],[32,52,0.6154,0.70982,0.41417,0.39287,1.0,1.0,0.0,1.0,7,20,0,7,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,20],[36,52,0.6923,0.5625,0.46144,0.0,0.78571,1.0,0.0,1.0,11,15,0,11,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,15],[40,52,0.7692,0.57589,0.44533,0.0,0.71429,1.0,0.0,1.0,10,14,0,10,0,1,0,0,2,0,0,0,0,0,0,0,0,4,0,0,1,0,14],[44,52,0.8462,0.5625,0.45308,0.0,0.7143,1.0,0.0,1.0,10,16,0,10,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,16],[48,52,0.9231,0.63392,0.45588,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,18],[52,52,1.0,0.08482,0.22829,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,1]]}]},{"i":"1cd15688307590e7","q":"Let $m$ be a positive integer such that $m \\equiv 2(\\bmod 4)$. Show that there exists at most one factorization $m=a b$ where $a$ and $b$ are positive integers satisfying $0<a-b<\\sqrt{5+4 \\sqrt{4 m+1}}$.","t":[{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.61607,"p":[[0,43,0.0,0.59375,0.36441,0.25,0.64286,1.0,0.0,1.0,2,10,0,2,0,6,0,0,3,0,0,3,0,0,2,0,0,2,0,0,4,0,10],[4,43,0.093,0.61607,0.33964,0.28571,0.71429,0.85714,0.0,1.0,3,6,0,3,0,4,0,0,2,0,0,1,0,0,3,0,0,5,0,0,8,0,6],[8,43,0.186,0.61607,0.33012,0.28571,0.71429,0.85714,0.0,1.0,2,6,0,2,0,4,0,0,4,0,0,1,0,0,1,0,0,7,0,0,7,0,6],[12,43,0.2791,0.34374,0.34415,0.0,0.14286,0.71429,0.0,1.0,11,1,0,11,0,7,0,0,0,0,0,1,0,0,3,0,0,6,0,0,3,0,1],[16,43,0.3721,0.45536,0.35614,0.14286,0.42859,0.85714,0.0,1.0,7,3,0,7,0,4,0,0,4,0,0,2,0,0,3,0,0,3,0,0,6,0,3],[20,43,0.4651,0.29908,0.25592,0.10714,0.28571,0.57111,0.0,0.85714,8,0,0,8,0,7,0,0,5,0,0,2,0,0,7,0,0,2,0,0,1,0,0],[24,43,0.5581,0.25447,0.31285,0.0,0.14286,0.42857,0.0,1.0,13,2,0,13,0,7,0,0,3,0,0,3,0,0,0,0,0,3,0,0,1,0,2],[28,43,0.6512,0.24999,0.31133,0.0,0.14286,0.46418,0.0,1.0,15,1,0,15,0,5,0,0,2,0,0,2,0,0,3,0,0,2,0,0,2,0,1],[32,43,0.7442,0.15625,0.23787,0.0,0.0,0.17857,0.0,0.85714,18,0,0,18,0,6,0,0,2,0,0,3,0,0,0,0,0,2,0,0,1,0,0],[36,43,0.8372,0.20536,0.29867,0.0,0.14286,0.1786,0.0,1.0,15,2,0,15,0,9,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,2],[40,43,0.9302,0.12946,0.23787,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,4,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[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":7,"e":0.42857,"k":"flat","v":0.40178,"x":0.69197,"p":[[0,44,0.0,0.40178,0.36845,0.0,0.28571,0.85704,0.0,1.0,9,3,1,9,0,5,0,0,3,0,0,4,0,0,0,0,0,2,0,0,6,0,3],[4,44,0.0909,0.58035,0.35524,0.14286,0.71429,0.85714,0.0,1.0,3,6,0,3,0,6,0,0,1,0,0,3,0,0,2,0,0,3,0,0,8,0,6],[8,44,0.1818,0.69196,0.27689,0.42857,0.85707,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,3,0,0,4,0,0,2,0,0,4,0,0,11,0,6],[12,44,0.2727,0.64284,0.29881,0.42857,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,4,0,0,4,0,0,3,0,0,6,0,0,4,0,8],[16,44,0.3636,0.64284,0.28794,0.42857,0.71429,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,4,0,0,4,0,0,4,0,0,7,0,0,4,0,7],[20,44,0.4545,0.69197,0.28146,0.53572,0.78571,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,4,0,0,2,0,0,3,0,0,5,0,0,9,0,7],[24,44,0.5455,0.65624,0.24186,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,6,0,0,4,0,0,7,0,0,6,0,5],[28,44,0.6364,0.65624,0.2572,0.42857,0.64286,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,1,0,0,6,0,0,7,0,0,3,0,0,8,0,5],[32,44,0.7273,0.66517,0.28485,0.57143,0.71429,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,2,0,0,1,0,0,8,0,0,7,0,0,3,0,8],[36,44,0.8182,0.65172,0.2524,0.571,0.64286,0.85714,0.0,1.0,1,4,0,1,0,2,0,0,0,0,0,4,0,0,9,0,0,4,0,0,8,0,4],[40,44,0.9091,0.67408,0.20278,0.57132,0.71429,0.85711,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,8,0,0,4,0,5],[44,44,1.0,0.42857,0.14725,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,11,0,0,10,0,0,7,0,0,3,0,0,0,0,0]]}]},{"i":"be13bfa5f9b96ef5","q":"In a square  $ABCD$ , let  $P$  and  $Q$  be points on the sides  $BC$  and  $CD$  respectively, different from its endpoints, such that  $BP=CQ$ . Consider points  $X$  and  $Y$  such that  $X\\neq Y$ , in the segments  $AP$  and  $AQ$  respectively. Show that, for every  $X$  and  $Y$  chosen, there exists a triangle whose sides have lengths  $BX$ ,  $XY$  and  $DY$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.07581,"x":0.15625,"p":[[0,106,0.0,0.07581,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,106,0.0377,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,106,0.0755,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,106,0.1132,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,106,0.1509,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,106,0.1887,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,106,0.2264,0.11599,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,1,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,106,0.2642,0.14724,0.16164,0.14286,0.14286,0.14286,0.0,1.0,5,1,0,5,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,106,0.3019,0.11143,0.05897,0.14,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,106,0.3396,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,106,0.3774,0.13393,0.16728,0.10714,0.14286,0.14286,0.0,1.0,8,1,0,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,106,0.4151,0.11572,0.0556,0.14,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,106,0.4528,0.12947,0.16888,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,106,0.4906,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,106,0.5283,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,106,0.566,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,106,0.6038,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,106,0.6415,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,106,0.6792,0.12045,0.05184,0.14286,0.14286,0.14286,0.0,0.143,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,106,0.717,0.10706,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,106,0.7547,0.12501,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,106,0.7925,0.08911,0.06903,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,106,0.8302,0.13375,0.16727,0.105,0.14286,0.14286,0.0,1.0,8,1,0,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[92,106,0.8679,0.15625,0.15714,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[96,106,0.9057,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,106,0.9434,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,106,0.9811,0.12501,0.04725,0.14286,0.14286,0.14286,0.0,0.143,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[106,106,1.0,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.07134,"x":0.20536,"p":[[0,54,0.0,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.15625,0.20316,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,20,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,54,0.1481,0.14732,0.16164,0.14286,0.14286,0.14286,0.0,1.0,5,1,0,5,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,54,0.2222,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.125,0.07784,0.14286,0.14286,0.14286,0.0,0.4286,6,0,0,6,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.11143,0.05897,0.14,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.1384,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.14277,0.11294,0.14286,0.14286,0.14286,0.0,0.71429,4,0,0,4,0,27,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[32,54,0.5926,0.20536,0.26711,0.14286,0.14286,0.14286,0.0,1.0,6,3,0,6,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[36,54,0.6667,0.15179,0.22851,0.0,0.14286,0.14286,0.0,1.0,10,2,0,10,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[40,54,0.7407,0.1383,0.16554,0.14214,0.14286,0.14286,0.0,1.0,7,1,0,7,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,54,0.8148,0.13393,0.16728,0.10714,0.14286,0.14286,0.0,1.0,8,1,0,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,54,0.8889,0.07581,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.12501,0.17035,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"7c8d64699038d539","q":"How many 4-digit numbers  $ABCD$  are there with the property that  $|A-B|= |B-C|= |C-D|$ ?\n\nNote that the first digit  $A$  of a four-digit number cannot be zero.","t":[{"b":0,"e":1.0,"k":"flat","v":0.66964,"x":0.87946,"p":[[0,90,0.0,0.84821,0.23673,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,6,0,0,3,0,19],[4,90,0.0444,0.87946,0.16409,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,1,0,20],[8,90,0.0889,0.75893,0.29761,0.57143,0.85714,1.0,0.0,1.0,1,15,0,1,0,2,0,0,2,0,0,0,0,0,4,0,0,5,0,0,3,0,15],[12,90,0.1333,0.66964,0.29544,0.53571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,1,0,0,2,0,0,4,0,0,8,0,0,3,0,9],[16,90,0.1778,0.67857,0.31339,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,1,0,0,0,0,0,5,0,0,7,0,0,2,0,11],[20,90,0.2222,0.75892,0.26107,0.71429,0.78564,1.0,0.0,1.0,1,11,0,1,0,2,0,0,0,0,0,0,0,0,4,0,0,9,0,0,5,0,11],[24,90,0.2667,0.83035,0.17291,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,2,0,15],[28,90,0.3111,0.81249,0.15336,0.71429,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,4,0,11],[32,90,0.3556,0.82143,0.17496,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,3,0,14],[36,90,0.4,0.84375,0.16115,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,6,0,14],[40,90,0.4444,0.83928,0.17768,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,3,0,16],[44,90,0.4889,0.82142,0.16751,0.71429,0.85707,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,4,0,13],[48,90,0.5333,0.84821,0.17474,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,5,0,16],[52,90,0.5778,0.83036,0.15746,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,7,0,12],[56,90,0.6222,0.83033,0.18015,0.67857,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,4,0,15],[60,90,0.6667,0.76784,0.17036,0.57143,0.71429,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,9,0,0,4,0,9],[64,90,0.7111,0.82589,0.19144,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,2,0,16],[68,90,0.7556,0.79909,0.19517,0.57143,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,3,0,0,3,0,14],[72,90,0.8,0.79909,0.15918,0.71429,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,5,0,10],[76,90,0.8444,0.79462,0.17838,0.57143,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,0,3,0,12],[80,90,0.8889,0.74552,0.1461,0.57143,0.71429,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,12,0,0,6,0,5],[84,90,0.9333,0.83482,0.17169,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,3,0,15],[88,90,0.9778,0.7723,0.1851,0.57143,0.71429,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,6,0,0,3,0,11],[90,90,1.0,0.72321,0.17105,0.57143,0.71429,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,7,0,0,3,0,7]]},{"b":4,"e":1.0,"k":"flat","v":0.38838,"x":0.87945,"p":[[0,127,0.0,0.80355,0.27838,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,5,0,0,2,0,18],[4,127,0.0315,0.875,0.21354,0.82143,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,2,0,0,2,0,22],[8,127,0.063,0.63839,0.24219,0.57143,0.57143,0.71429,0.14286,1.0,0,6,0,0,0,3,0,0,1,0,0,2,0,0,11,0,0,8,0,0,1,0,6],[12,127,0.0945,0.64732,0.32924,0.35714,0.71429,1.0,0.14286,1.0,0,9,0,0,0,8,0,0,0,0,0,1,0,0,4,0,0,5,0,0,5,0,9],[16,127,0.126,0.82589,0.19144,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,0,2,0,16],[20,127,0.1575,0.73214,0.26426,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,3,0,0,1,0,0,0,0,0,6,0,0,8,0,0,3,0,11],[24,127,0.189,0.70536,0.3071,0.57143,0.78571,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,1,0,0,0,0,0,7,0,0,3,0,0,4,0,12],[28,127,0.2205,0.38838,0.29283,0.14286,0.28571,0.57111,0.14286,1.0,0,3,0,0,0,14,0,0,5,0,0,4,0,0,2,0,0,2,0,0,2,0,3],[32,127,0.252,0.45973,0.3384,0.14286,0.35714,0.71429,0.14,1.0,0,5,0,0,0,15,0,0,1,0,0,2,0,0,2,0,0,5,0,0,2,0,5],[36,127,0.2835,0.50445,0.32337,0.14286,0.49979,0.85714,0.14286,1.0,0,5,0,0,0,10,0,0,4,0,0,2,0,0,5,0,0,2,0,0,4,0,5],[40,127,0.315,0.5357,0.33693,0.28571,0.42857,1.0,0.14286,1.0,0,9,0,0,0,6,0,0,9,0,0,3,0,0,2,0,0,2,0,0,1,0,9],[44,127,0.3465,0.85266,0.16556,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,4,0,16],[48,127,0.378,0.81249,0.15748,0.71429,0.78571,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,5,0,11],[52,127,0.4094,0.8125,0.18363,0.67857,0.71429,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,0,0,0,15],[56,127,0.4409,0.80802,0.15817,0.71429,0.85707,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,7,0,10],[60,127,0.4724,0.83033,0.16921,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,4,0,14],[64,127,0.5039,0.84821,0.15542,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,6,0,14],[68,127,0.5354,0.87945,0.17171,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,3,0,20],[72,127,0.5669,0.79464,0.17105,0.67857,0.71429,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,0,4,0,11],[76,127,0.5984,0.74999,0.18559,0.57143,0.71429,1.0,0.571,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,6,0,0,2,0,10],[80,127,0.6299,0.82142,0.16368,0.71429,0.78571,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,3,0,13],[84,127,0.6614,0.79016,0.18207,0.57143,0.71429,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,7,0,0,3,0,12],[88,127,0.6929,0.83928,0.16656,0.71429,0.85714,1.0,0.5714,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,6,0,14],[92,127,0.7244,0.83034,0.16919,0.71429,0.78571,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,1,0,15],[96,127,0.7559,0.79016,0.17853,0.57143,0.78564,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,5,0,11],[100,127,0.7874,0.81246,0.17294,0.71421,0.78571,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,0,3,0,13],[104,127,0.8189,0.74104,0.17657,0.57143,0.71429,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,6,0,0,4,0,8],[108,127,0.8504,0.81696,0.17215,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,4,0,13],[112,127,0.8819,0.79464,0.17105,0.71429,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,12,0,0,1,0,12],[116,127,0.9134,0.76786,0.1948,0.57143,0.71429,1.0,0.5714,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,4,0,0,2,0,12],[120,127,0.9449,0.85713,0.1557,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,6,0,15],[124,127,0.9764,0.76785,0.17405,0.57143,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,10,0,0,2,0,10],[127,127,1.0,0.74999,0.18212,0.57143,0.71429,1.0,0.4286,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,8,0,0,3,0,9]]}]},{"i":"89c302f061f4ffed","q":"In acute-angled triangle  $ABC$ ,  $BH$  is the altitude of the vertex  $B$ . The points  $D$  and  $E$  are midpoints of  $AB$  and  $AC$  respectively. Suppose that  $F$  be the reflection of  $H$  with respect to  $ED$ . Prove that the line  $BF$  passes through circumcenter of  $ABC$ .\n\n by Davood Vakili","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.16964,"p":[[0,35,0.0,0.16964,0.23808,0.0,0.0,0.32143,0.0,1.0,18,1,0,18,0,3,0,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[4,35,0.1143,0.08036,0.15126,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.04911,0.11633,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,35,0.8,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.11161,"p":[[0,23,0.0,0.11152,0.17398,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,4,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.0892,0.15044,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,4,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.11161,0.17762,0.0,0.0,0.17857,0.0,0.42857,22,0,0,22,0,2,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,1,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c65dc2a268c8dce2","q":"Let  $a$  and  $b$  be two square-free, distinct natural numbers. Show that there exist  $c>0$  such that \n\\[\n\\left | \\{n\\sqrt{a}\\}-\\{n\\sqrt{b}\\} \\right |>\\frac{c}{n^3}\\]\nfor every positive integer  $n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,26,0.0,0.94196,0.18851,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[4,26,0.1538,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[12,26,0.4615,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,26,0.9231,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[26,26,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":0.85714,"k":"flat","v":0.95982,"x":0.99554,"p":[[0,8,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,8,0.5,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[8,8,1.0,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]}]},{"i":"66dbe5c04ece343b","q":"Given a triangle $ABC$  with  $AB<BC<AC$  inscribed in circle  $(c)$ . The circle  $c(A,AB)$  (with center  $A$  and radius  $AB$ ) interects the line  $BC$  at point  $D$  and the circle  $(c)$  at point  $H$ . The circle  $c(A,AC)$   (with center  $A$  and radius  $AC$ )  interects the line  $BC$  at point  $Z$  and the circle  $(c)$  at point  $E$ . Lines  $ZH$  and  $ED$  intersect at point  $T$ . Prove that the circumscribed circles of triangles  $TDZ$  and  $TEH$  are equal.","t":[{"b":1,"e":0.0,"k":"falling","v":0.03125,"x":0.65178,"p":[[0,56,0.0,0.65178,0.42699,0.2857,1.0,1.0,0.0,1.0,7,18,0,7,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,18],[4,56,0.0714,0.5491,0.41666,0.10714,0.57143,1.0,0.0,1.0,8,12,0,8,0,2,0,0,3,0,0,1,0,0,3,0,0,2,0,0,1,0,12],[8,56,0.1429,0.60714,0.40406,0.14286,0.71429,1.0,0.0,1.0,5,14,0,5,0,4,0,0,3,0,0,0,0,0,2,0,0,4,0,0,0,0,14],[12,56,0.2143,0.32589,0.41224,0.0,0.14286,0.67857,0.0,1.0,15,8,0,15,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,8],[16,56,0.2857,0.40179,0.40945,0.0,0.14288,1.0,0.0,1.0,9,9,0,9,0,8,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,9],[20,56,0.3571,0.40624,0.42274,0.0,0.28571,1.0,0.0,1.0,13,9,0,13,0,2,0,0,3,0,0,2,0,0,1,0,0,2,0,0,0,0,9],[24,56,0.4286,0.18304,0.32778,0.0,0.0,0.1429,0.0,1.0,20,3,0,20,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,3],[28,56,0.5,0.2991,0.39666,0.0,0.14286,0.60682,0.0,1.0,15,6,0,15,0,7,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,6],[32,56,0.5714,0.18304,0.29284,0.0,0.0,0.2857,0.0,1.0,17,3,0,17,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[36,56,0.6429,0.11161,0.24932,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[40,56,0.7143,0.2009,0.274,0.0,0.14286,0.1786,0.0,1.0,13,2,0,13,0,11,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[44,56,0.7857,0.23213,0.28514,0.0,0.14286,0.28571,0.0,1.0,13,2,0,13,0,6,0,0,6,0,0,2,0,0,1,0,0,2,0,0,0,0,2],[48,56,0.8571,0.19643,0.29827,0.0,0.14286,0.17857,0.0,1.0,15,3,0,15,0,9,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,3],[52,56,0.9286,0.08036,0.08703,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.11152,"x":0.75,"p":[[0,60,0.0,0.75,0.36246,0.28571,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,21],[4,60,0.0667,0.5,0.42708,0.0,0.42859,1.0,0.0,1.0,10,9,0,10,0,2,0,0,3,0,0,2,0,0,0,0,0,1,0,0,5,0,9],[8,60,0.1333,0.44642,0.43116,0.0,0.35695,1.0,0.0,1.0,12,9,0,12,0,4,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,9],[12,60,0.2,0.42857,0.41496,0.0,0.28571,1.0,0.0,1.0,9,10,0,9,0,5,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,10],[16,60,0.2667,0.36607,0.39599,0.0,0.21428,0.75,0.0,1.0,13,6,0,13,0,3,0,0,3,0,0,2,0,0,2,0,0,1,0,0,2,0,6],[20,60,0.3333,0.38839,0.4392,0.0,0.14288,1.0,0.0,1.0,14,10,0,14,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,10],[24,60,0.4,0.41503,0.39977,0.0,0.21428,0.75,0.0,1.0,10,6,0,10,0,6,0,0,2,0,0,0,0,0,1,0,0,5,0,0,2,0,6],[28,60,0.4667,0.58036,0.37105,0.2857,0.71429,1.0,0.0,1.0,5,9,0,5,0,2,0,0,4,0,0,3,0,0,0,0,0,6,0,0,3,0,9],[32,60,0.5333,0.41955,0.42854,0.0,0.14286,0.85714,0.0,1.0,13,7,0,13,0,4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,3,0,7],[36,60,0.6,0.50893,0.40554,0.10714,0.57143,1.0,0.0,1.0,8,9,0,8,0,3,0,0,4,0,0,0,0,0,2,0,0,4,0,0,2,0,9],[40,60,0.6667,0.34811,0.40557,0.0,0.14286,0.74996,0.0,1.0,14,6,0,14,0,4,0,0,3,0,0,0,0,0,1,0,0,2,0,0,2,0,6],[44,60,0.7333,0.25893,0.33964,0.0,0.14286,0.28571,0.0,1.0,14,3,0,14,0,6,0,0,5,0,0,0,0,0,1,0,0,1,0,0,2,0,3],[48,60,0.8,0.11152,0.21047,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,4,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[52,60,0.8667,0.33928,0.30461,0.14286,0.21431,0.4642,0.0,1.0,5,2,0,5,0,11,0,0,4,0,0,4,0,0,2,0,0,1,0,0,3,0,2],[56,60,0.9333,0.20088,0.14219,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,17,0,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[60,60,1.0,0.14267,0.1184,0.14,0.14286,0.14286,0.0,0.571,7,0,0,7,0,21,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"96636c17b9285a7f","q":"Find all triples of natural numbers  $(a, b, c)$  for which the number   $$ 2^a + 2^b + 2^c + 3 $$   is the square of an integer.","t":[{"b":3,"e":1.0,"k":"rising","v":0.52901,"x":0.99107,"p":[[0,257,0.0,0.52901,0.1897,0.42857,0.42857,0.51785,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,23,0,1,2,0,0,1,0,0,2,0,3],[4,257,0.0156,0.77677,0.21999,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,1,0,0,3,0,14],[8,257,0.0311,0.88839,0.18466,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,7,0,20],[12,257,0.0467,0.88838,0.20119,0.857,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,2,0,0,2,0,23],[16,257,0.0623,0.87054,0.20628,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,2,0,0,3,0,21],[20,257,0.0778,0.79015,0.23144,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,2,0,0,2,0,16],[24,257,0.0934,0.83482,0.21756,0.57143,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,1,0,0,4,0,18],[28,257,0.1089,0.83927,0.20126,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,1,0,0,5,0,17],[32,257,0.1245,0.7723,0.22264,0.57143,0.85707,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,2,0,0,4,0,13],[36,257,0.1401,0.78125,0.2474,0.53572,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,7,0,0,2,0,0,3,0,0,4,0,15],[40,257,0.1556,0.81695,0.24805,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,3,0,0,3,0,18],[44,257,0.1712,0.80357,0.21943,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,4,0,15],[48,257,0.1868,0.90625,0.15815,0.85711,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,22],[52,257,0.2023,0.8482,0.17837,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,2,0,0,12,0,13],[56,257,0.2179,0.84152,0.20651,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,2,0,1,2,0,0,3,0,0,7,0,16],[60,257,0.2335,0.77677,0.21999,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,4,0,0,4,0,13],[64,257,0.249,0.81248,0.2156,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,4,0,16],[68,257,0.2646,0.83482,0.23987,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,2,0,0,5,0,18],[72,257,0.2802,0.82588,0.23888,0.57143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,0,0,0,1,0,20],[76,257,0.2957,0.81696,0.20277,0.67857,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,4,0,15],[80,257,0.3113,0.82143,0.20516,0.57143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,6,0,15],[84,257,0.3268,0.81249,0.22144,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,3,0,0,4,0,16],[88,257,0.3424,0.74997,0.24225,0.57132,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,2,0,0,4,0,0,7,0,0,2,0,0,5,0,12],[92,257,0.358,0.80356,0.21054,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,1,0,0,6,0,14],[96,257,0.3735,0.72768,0.28873,0.42859,0.78571,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,7,0,0,4,0,0,3,0,0,2,0,14],[100,257,0.3891,0.68302,0.25936,0.42857,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,9,0,0,7,0,0,1,0,0,2,0,11],[104,257,0.4047,0.79909,0.23382,0.57143,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,3,0,0,5,0,15],[108,257,0.4202,0.81027,0.22275,0.57143,0.96429,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,2,1,16],[112,257,0.4358,0.83482,0.24251,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,1,0,0,2,0,20],[116,257,0.4514,0.83036,0.19704,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,0,6,0,15],[120,257,0.4669,0.8125,0.22428,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,2,0,17],[124,257,0.4825,0.875,0.1948,0.82143,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,3,0,21],[128,257,0.4981,0.78125,0.22299,0.57143,0.85707,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,4,0,0,3,0,14],[132,257,0.5136,0.75893,0.27534,0.42857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,9,0,0,3,0,0,0,0,0,3,0,16],[136,257,0.5292,0.87053,0.19019,0.857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,0,6,0,19],[140,257,0.5447,0.86159,0.22157,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,0,1,0,22],[144,257,0.5603,0.76786,0.22232,0.57143,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,1,0,0,8,0,11],[148,257,0.5759,0.81248,0.20342,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,6,0,14],[152,257,0.5914,0.82142,0.21726,0.67857,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,3,0,0,5,0,16],[156,257,0.607,0.86607,0.2111,0.85711,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,0,5,0,20],[160,257,0.6226,0.84374,0.24838,0.82143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,5,0,19],[164,257,0.6381,0.84822,0.22286,0.67857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,1,0,0,3,0,20],[168,257,0.6537,0.85265,0.20043,0.67857,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,3,0,19],[172,257,0.6693,0.83035,0.2096,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,4,0,17],[176,257,0.6848,0.87053,0.19678,0.85711,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,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Let $x_{1} \\geqslant x_{2}>0$ and $y_{1} \\geqslant y_{2}>0$ be real numbers such that\n\n$$\nx_{1} \\geqslant y_{1} \\text { and } x_{1} x_{2} \\geqslant y_{1} y_{2}\n$$\n\nProve that\n\n$$\nx_{1}+x_{2} \\geqslant y_{1}+y_{2}\n$$\n\nb) Let $x_{1} \\geqslant x_{2} \\geqslant \\ldots \\geqslant x_{n}>0$ and $y_{1} \\geqslant y_{2} \\geqslant \\ldots \\geqslant y_{n}>0$ be real numbers such that\n\n$$\nx_{1} x_{2} \\cdots x_{i} \\geqslant y_{1} y_{2} \\cdots y_{i} \\text { for } i=1, \\ldots, n\n$$\n\nProve that\n\n$$\nx_{1}+x_{2}+\\cdots+x_{n} \\geqslant y_{1}+y_{2}+\\cdots+y_{n}\n$$","t":[{"b":1,"e":1.0,"k":"flat","v":0.88393,"x":0.97321,"p":[[0,42,0.0,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[4,42,0.0952,0.88393,0.22142,0.92857,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,4,0,0,0,0,24],[8,42,0.1905,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[12,42,0.2857,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[16,42,0.381,0.89731,0.16068,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,0,0,22],[20,42,0.4762,0.94642,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[24,42,0.5714,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[28,42,0.6667,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[32,42,0.7619,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[36,42,0.8571,0.95088,0.1048,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[40,42,0.9524,0.92857,0.13832,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,3,0,24],[42,42,1.0,0.89284,0.13835,0.71429,1.0,1.0,0.571,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,3,0,19]]},{"b":5,"e":1.0,"k":"flat","v":0.88839,"x":0.99107,"p":[[0,30,0.0,0.88839,0.22228,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,1,0,24],[4,30,0.1333,0.93304,0.18893,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[8,30,0.2667,0.93304,0.17491,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,27],[12,30,0.4,0.91964,0.17474,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,25],[16,30,0.5333,0.91964,0.14258,0.85711,1.0,1.0,0.4286,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,23],[20,30,0.6667,0.92408,0.14725,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,3,0,24],[24,30,0.8,0.92411,0.14719,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,25],[28,30,0.9333,0.91963,0.13807,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,23],[30,30,1.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"f582a26c6199ad50","q":"5. (GDR 2) Prove the following assertion: The four altitudes of a tetrahedron $A B C D$ intersect in a point if and only if  $$ A B^{2}+C D^{2}=B C^{2}+A D^{2}=C A^{2}+B D^{2} $$","t":[{"b":0,"e":0.42857,"k":"falling","v":0.44197,"x":0.79017,"p":[[0,46,0.0,0.79017,0.23416,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,3,0,0,2,0,0,9,0,0,4,0,13],[4,46,0.087,0.77677,0.22852,0.57132,0.78564,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,7,0,0,2,0,14],[8,46,0.1739,0.61606,0.24074,0.42857,0.57121,0.75,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,14,0,0,2,0,0,7,0,0,3,0,5],[12,46,0.2609,0.72765,0.23246,0.571,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,4,0,0,1,0,12],[16,46,0.3478,0.59374,0.26989,0.42857,0.4286,0.89275,0.0,1.0,1,8,0,1,0,0,0,0,1,0,0,17,0,0,2,0,0,2,0,0,1,0,8],[20,46,0.4348,0.66072,0.27606,0.42857,0.64286,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,14,0,0,1,0,0,4,0,0,2,0,10],[24,46,0.5217,0.67848,0.27216,0.42857,0.71429,1.0,0.14,1.0,0,9,0,0,0,2,0,0,1,0,0,9,0,0,1,0,0,6,0,0,4,0,9],[28,46,0.6087,0.58929,0.25692,0.42857,0.42859,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,15,0,0,1,0,0,4,0,0,4,0,5],[32,46,0.6957,0.64284,0.28121,0.42857,0.71429,0.89275,0.0,1.0,2,8,0,2,0,0,0,0,1,0,0,9,0,0,3,0,0,7,0,0,2,0,8],[36,46,0.7826,0.72768,0.26088,0.42859,0.71429,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,8,0,0,3,0,0,4,0,0,2,0,13],[40,46,0.8696,0.625,0.26183,0.42857,0.50001,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,13,0,0,4,0,0,2,0,0,1,0,9],[44,46,0.9565,0.48661,0.2854,0.42857,0.42857,0.4286,0.0,1.0,3,6,0,3,0,1,0,0,2,0,0,19,0,0,0,0,0,1,0,0,0,0,6],[46,46,1.0,0.44197,0.31412,0.39285,0.42857,0.42858,0.0,1.0,6,6,0,6,0,1,0,0,1,0,0,18,0,0,0,0,0,0,0,0,0,0,6]]},{"b":5,"e":0.71429,"k":"flat","v":0.80355,"x":0.95088,"p":[[0,45,0.0,0.80355,0.19481,0.71429,0.85707,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,10,0,0,5,0,12],[4,45,0.0889,0.88379,0.19389,0.82132,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,4,0,20],[8,45,0.1778,0.95088,0.11076,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[12,45,0.2667,0.87946,0.18249,0.82143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,5,0,19],[16,45,0.3556,0.80804,0.2412,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,0,4,0,15],[20,45,0.4444,0.90177,0.18015,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,0,0,24],[24,45,0.5333,0.89284,0.1786,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,21],[28,45,0.6222,0.86605,0.20809,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,4,0,20],[32,45,0.7111,0.93303,0.14719,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,4,0,24],[36,45,0.8,0.92855,0.12881,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,23],[40,45,0.8889,0.87945,0.17898,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,3,0,20],[44,45,0.9778,0.89729,0.1143,0.85711,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,15],[45,45,1.0,0.92857,0.14286,0.85714,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,22]]}]},{"i":"6923372355f322c3","q":"15. (LUX 2) Angles of a given triangle $A B C$ are all smaller than $120^{\\circ}$. Equilateral triangles $A F B, B D C$ and $C E A$ are constructed in the exterior of $\\triangle A B C$. (a) Prove that the lines $A D, B E$, and $C F$ pass through one point $S$. (b) Prove that $S D+S E+S F=2(S A+S B+S C)$.","t":[{"b":3,"e":0.71429,"k":"falling","v":0.29909,"x":0.73658,"p":[[0,44,0.0,0.5982,0.28444,0.42859,0.71429,0.85704,0.0,1.0,3,2,0,3,0,2,0,0,0,0,0,5,0,0,5,0,0,7,0,0,8,0,2],[4,44,0.0909,0.71875,0.2461,0.71429,0.78571,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,3,0,0,1,0,0,1,0,0,9,0,0,11,0,5],[8,44,0.1818,0.72319,0.24984,0.71429,0.71429,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,0,0,0,1,0,0,1,0,0,12,0,0,11,0,4],[12,44,0.2727,0.67855,0.25754,0.5354,0.85707,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,4,0,0,2,0,0,1,0,0,6,0,0,16,0,1],[16,44,0.3636,0.73658,0.18596,0.67857,0.78564,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,8,0,0,13,0,3],[20,44,0.4545,0.60714,0.27199,0.39286,0.71429,0.85714,0.0,1.0,1,1,0,1,0,4,0,0,3,0,0,1,0,0,2,0,0,12,0,0,8,0,1],[24,44,0.5455,0.69193,0.25533,0.57132,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,0,0,0,2,0,0,4,0,0,7,0,0,11,0,4],[28,44,0.6364,0.63388,0.26472,0.571,0.71429,0.85704,0.0,1.0,1,3,1,1,0,3,0,0,2,0,0,1,0,0,6,0,0,9,0,0,7,0,3],[32,44,0.7273,0.683,0.23619,0.57143,0.71429,0.85714,0.14286,1.0,0,4,0,0,0,3,0,0,1,0,0,1,0,0,5,0,0,11,0,0,7,0,4],[36,44,0.8182,0.70978,0.23552,0.57132,0.857,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,6,0,0,14,0,3],[40,44,0.9091,0.66951,0.20365,0.57132,0.71429,0.85714,0.14,1.0,0,1,0,0,0,2,0,0,1,0,0,2,0,0,6,0,0,11,0,0,9,0,1],[44,44,1.0,0.29909,0.19676,0.14286,0.2857,0.42857,0.0,0.71429,3,0,0,3,0,9,0,0,10,0,0,5,0,0,2,0,0,3,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"falling","v":0.24999,"x":0.69195,"p":[[0,37,0.0,0.60266,0.27833,0.4286,0.57143,0.85714,0.0,1.0,3,2,2,3,0,1,0,0,1,0,0,4,0,0,8,0,0,4,0,0,9,0,2],[4,37,0.1081,0.66516,0.24382,0.67846,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,1,0,0,1,0,0,13,0,0,9,0,2],[8,37,0.2162,0.58929,0.28738,0.39286,0.71429,0.85714,0.0,1.0,2,2,0,2,0,3,0,0,3,0,0,3,0,0,2,0,0,10,0,0,7,0,2],[12,37,0.3243,0.69195,0.229,0.57143,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,3,0,0,2,0,0,3,0,0,9,0,0,12,0,2],[16,37,0.4324,0.59374,0.33902,0.2857,0.71429,0.85714,0.0,1.0,4,4,0,4,0,2,0,0,4,0,0,2,0,0,1,0,0,5,0,0,10,0,4],[20,37,0.5405,0.5624,0.30929,0.28571,0.71429,0.85714,0.0,1.0,4,2,0,4,0,2,0,0,3,0,0,2,0,0,4,0,0,8,0,0,7,0,2],[24,37,0.6486,0.54016,0.26421,0.39286,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,3,0,0,3,0,0,5,0,0,5,0,0,7,0,0,7,0,0],[28,37,0.7568,0.59819,0.27535,0.42857,0.64286,0.85714,0.0,1.0,2,2,0,2,0,1,0,0,4,0,0,4,0,0,5,0,0,5,0,0,9,0,2],[32,37,0.8649,0.6071,0.28572,0.42857,0.71429,0.85714,0.0,1.0,3,2,0,3,0,1,0,0,3,0,0,2,0,0,4,0,0,9,0,0,8,0,2],[36,37,0.973,0.37945,0.26632,0.21427,0.42857,0.57143,0.0,0.85714,8,0,0,8,0,0,0,0,5,0,0,9,0,0,4,0,0,4,0,0,2,0,0],[37,37,1.0,0.24999,0.24998,0.0,0.2143,0.42857,0.0,0.85714,11,0,0,11,0,5,0,0,7,0,0,4,0,0,1,0,0,3,0,0,1,0,0]]}]},{"i":"d150e5e9db5d8570","q":"7. (FRA 5) Let real numbers $x_{1}, x_{2}, \\ldots, x_{n}$ satisfy $0<x_{1}<x_{2}<\\cdots<$ $x_{n}<1$ and set $x_{0}=0, x_{n+1}=1$. Suppose that these numbers satisfy the following system of equations:  $$ \\sum_{j=0, j \\neq i}^{n+1} \\frac{1}{x_{i}-x_{j}}=0 \\quad \\text { where } i=1,2, \\ldots, n $$  Prove that $x_{n+1-i}=1-x_{i}$ for $i=1,2, \\ldots, n$.","t":[{"b":5,"e":1.0,"k":"flat","v":0.72322,"x":0.98661,"p":[[0,37,0.0,0.7946,0.21114,0.71429,0.78571,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,9,0,0,3,0,13],[4,37,0.1081,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,37,0.2162,0.92857,0.22868,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[12,37,0.3243,0.9375,0.13803,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,25],[16,37,0.4324,0.92411,0.19556,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,27],[20,37,0.5405,0.91964,0.20806,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,25],[24,37,0.6486,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[28,37,0.7568,0.92857,0.14725,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,25],[32,37,0.8649,0.89719,0.17229,0.82143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,0,2,0,22],[36,37,0.973,0.72322,0.17105,0.71429,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,20,0,0,2,0,5],[37,37,1.0,0.72764,0.1834,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,14,0,0,2,0,7]]},{"b":6,"e":1.0,"k":"flat","v":0.85268,"x":0.99107,"p":[[0,35,0.0,0.85268,0.2382,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,3,0,20],[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],[8,35,0.2286,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[12,35,0.3429,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,35,0.4571,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,35,0.5714,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,35,0.6857,0.9375,0.11811,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,24],[28,35,0.8,0.87946,0.19597,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,17],[32,35,0.9143,0.94642,0.09947,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[35,35,1.0,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23]]}]},{"i":"4769992b060dc6db","q":"Let  $x\\in\\mathbb{R}$ . What is the maximum value of the following expression:  $\\sqrt{x-2018}  + \\sqrt{2020-x}$  ?","t":[{"b":3,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,45,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,45,0.0889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,45,0.1778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[28,45,0.6222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":4,"e":1.0,"k":"flat","v":1.0,"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],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"e03681ce25923b05","q":"Find the point in the closed unit disc  $D=\\{ (x,y) | x^2+y^2\\le 1 \\}$  at which the function  $f(x,y)=x+y$  attains its maximum .","t":[{"b":2,"e":1.0,"k":"flat","v":0.9933,"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],[4,41,0.0976,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[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],[20,41,0.4878,0.99777,0.01242,1.0,1.0,1.0,0.9286,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31],[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],[36,41,0.878,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,41,0.9756,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":1.0,"k":"flat","v":0.98661,"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],[4,42,0.0952,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,42,0.1905,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[16,42,0.381,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,42,0.4762,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,42,0.5714,0.99777,0.01242,1.0,1.0,1.0,0.9286,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31],[28,42,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,42,0.7619,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[36,42,0.8571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,42,0.9524,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[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":"2555a57795394f26","q":"We say that a positive real number  $d$  is  $good$  if there exists an infinite squence  $a_1,a_2,a_3,...\\in (0,d)$  such that for each  $n$ , the points  $a_1,a_2,...,a_n$  partition the interval  $[0,d]$  into segments of length at most  $\\frac{1}{n}$  each . Find  $\\text{sup}\\{d| d  \\text{is  good}\\}$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,36,0.0,0.05804,0.18509,0.0,0.0,0.0,0.0,0.71429,29,0,17,29,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[4,36,0.1111,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.01562,0.06616,0.0,0.0,0.0,0.0,0.357,30,0,0,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,18,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,9,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"df1544e23011e6e4","q":"Find all quadruples of real numbers  $(a,b,c,d)$  such that the equalities \n\\[X^2 + a X + b = (X-a)(X-c) \\text{ and } X^2 + c X + d = (X-b)(X-d)\\]\nhold for all real numbers  $X$ .\n\n*Morteza Saghafian, Iran*","t":[{"b":0,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,49,0.0,0.92411,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21],[4,49,0.0816,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,49,0.1633,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[12,49,0.2449,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[16,49,0.3265,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[20,49,0.4082,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,49,0.4898,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,49,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,49,0.6531,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[44,49,0.898,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,49,0.9796,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[49,49,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":3,"e":0.71429,"k":"flat","v":0.91964,"x":0.96875,"p":[[0,21,0.0,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[4,21,0.1905,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[8,21,0.381,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[12,21,0.5714,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[16,21,0.7619,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[20,21,0.9524,0.91964,0.11811,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,21],[21,21,1.0,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23]]}]},{"i":"14208b25c50b8266","q":"$$ \\int_0^\\frac{\\pi}{2}\\frac{1}{4-3\\cos^2(x)}\\,dx $$  *Proposed by Connor Gordon*","t":[{"b":0,"e":1.0,"k":"flat","v":1.0,"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],[4,42,0.0952,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,42,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":3,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,52,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,52,0.0769,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,52,0.1538,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,52,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"5e4dd6994ec9620f","q":"The number obtained by taking the last two digits of  $5^{2024}$  in the same order is:","t":[{"b":1,"e":0.71429,"k":"flat","v":0.79464,"x":0.90179,"p":[[0,27,0.0,0.88393,0.16146,0.71429,1.0,1.0,0.57143,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,2,0,20],[4,27,0.1481,0.79464,0.2111,0.67857,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,9,0,0,1,0,14],[8,27,0.2963,0.90179,0.22142,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,24],[12,27,0.4444,0.87947,0.22898,0.71429,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,23],[16,27,0.5926,0.83036,0.26592,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,5,0,0,1,0,20],[20,27,0.7407,0.87054,0.22968,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,22],[24,27,0.8889,0.88393,0.20025,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,5,0,0,1,0,22],[27,27,1.0,0.80804,0.23313,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,5,0,0,6,0,14]]},{"b":5,"e":0.14286,"k":"flat","v":0.72321,"x":0.84821,"p":[[0,35,0.0,0.84375,0.26812,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,22],[4,35,0.1143,0.73659,0.25781,0.57143,0.71429,1.0,0.14286,1.0,0,12,0,0,0,3,0,0,0,0,0,0,0,0,8,0,0,8,0,0,1,0,12],[8,35,0.2286,0.82589,0.20119,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,9,0,0,3,0,15],[12,35,0.3429,0.82143,0.26486,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,6,0,0,1,0,19],[16,35,0.4571,0.84821,0.2141,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,4,0,0,2,0,19],[20,35,0.5714,0.77679,0.27184,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,0,0,0,1,0,0,5,0,0,6,0,0,1,0,16],[24,35,0.6857,0.7499,0.26508,0.57143,0.71429,1.0,0.14,1.0,0,14,0,0,0,3,0,0,0,0,0,0,0,0,8,0,0,7,0,0,0,0,14],[28,35,0.8,0.72321,0.22286,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,0,0,0,0,0,0,9,0,0,11,0,0,1,0,9],[32,35,0.9143,0.80804,0.2412,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,8,0,0,0,0,17],[35,35,1.0,0.77232,0.2854,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,7,0,0,1,0,16]]}]},{"i":"a71e2513d464af4c","q":"Let  $ABC$  be a triangle and  $w$  its incircle.   $w$  touches  $BC,CA$  at  $A_1,B_1$  respectively.  The second intersection of  $AA_1$  and  $w$  is  $A_2$ , similarly define  $B_2$ .  Then  $AB,A_1B_1,A_2B_2$  concur at a point  $C_3$ .","t":[{"b":1,"e":0.0,"k":"falling","v":0.0,"x":0.29464,"p":[[0,43,0.0,0.24107,0.14914,0.21427,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,0,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.29464,0.22286,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,2,0,0,13,0,0,6,0,0,2,0,0,1,0,0,0,0,1],[8,43,0.186,0.23204,0.24158,0.0,0.2857,0.28571,0.0,1.0,12,1,0,12,0,1,0,0,14,0,0,2,0,0,1,0,0,0,0,0,1,0,1],[12,43,0.2791,0.16509,0.1825,0.0,0.07,0.28571,0.0,0.57143,16,0,0,16,0,2,0,0,8,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[16,43,0.3721,0.12946,0.20628,0.0,0.0,0.17857,0.0,1.0,18,1,0,18,0,6,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[20,43,0.4651,0.06687,0.07965,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.06696,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.08928,0.11709,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.10714,0.13832,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,3,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"falling","v":0.01339,"x":0.2232,"p":[[0,49,0.0,0.20536,0.18536,0.0,0.2857,0.28571,0.0,0.85714,11,0,0,11,0,2,0,0,16,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[4,49,0.0816,0.21428,0.17496,0.0,0.28571,0.28571,0.0,0.71429,11,0,0,11,0,0,0,0,17,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[8,49,0.1633,0.2232,0.21108,0.0,0.2857,0.28571,0.0,0.85714,11,0,0,11,0,4,0,0,10,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[12,49,0.2449,0.19642,0.17402,0.0,0.14286,0.28571,0.0,0.571,11,0,0,11,0,6,0,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[16,49,0.3265,0.12946,0.1488,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,3,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.16518,0.21162,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,2,0,0,8,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[24,49,0.4898,0.12937,0.19678,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,3,0,0,4,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[28,49,0.5714,0.10714,0.19885,0.0,0.0,0.17857,0.0,1.0,21,1,0,21,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,49,0.6531,0.08929,0.13243,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"32bfe5945942e31f","q":"A high school has an odd number of classes, and in each class, there is an odd number of students. One student is chosen from each class to be part of the student committee. If the number of classes with more boys than girls is odd, show that the number of ways to form a student committee containing an odd number of boys exceeds the number of ways to form a student committee containing an odd number of girls.","t":[{"b":1,"e":1.0,"k":"flat","v":0.97768,"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],[4,21,0.1905,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,21,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[21,21,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":1.0,"k":"flat","v":0.99107,"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],[8,36,0.2222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,36,0.3333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,36,0.5556,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,36,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"f4164489e36285b2","q":"Let  $n$  be an given positive integer, the set  $S=\\{1,2,\\cdots,n\\}$ .For any nonempty set  $A$  and  $B$ , find the minimum of  $|A\\Delta S|+|B\\Delta S|+|C\\Delta S|,$  where  $C=\\{a+b|a\\in A,b\\in B\\}, X\\Delta Y=X\\cup Y-X\\cap Y.$","t":[{"b":3,"e":0.85714,"k":"falling","v":0.47764,"x":0.90178,"p":[[0,68,0.0,0.75,0.21428,0.67857,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,6,0,0,12,0,6],[4,68,0.0588,0.90178,0.18364,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,6,0,21],[8,68,0.1176,0.85714,0.17857,0.82143,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,9,0,15],[12,68,0.1765,0.8616,0.1838,0.85714,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,12,0,14],[16,68,0.2353,0.7857,0.23959,0.67846,0.85714,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,4,0,0,7,0,13],[20,68,0.2941,0.79017,0.21719,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,5,0,0,12,0,9],[24,68,0.3529,0.70087,0.22121,0.571,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,6,0,0,7,0,0,3,0,0,10,0,5],[28,68,0.4118,0.73214,0.21354,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,8,0,0,7,0,7],[32,68,0.4706,0.73214,0.19804,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,8,0,0,8,0,6],[36,68,0.5294,0.73213,0.22233,0.57132,0.85714,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,6,0,0,4,0,0,2,0,0,14,0,5],[40,68,0.5882,0.73213,0.21054,0.57132,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,6,0,0,2,0,0,9,0,0,7,0,7],[44,68,0.6471,0.70534,0.20184,0.57132,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,7,0,0,4,0,0,9,0,5],[48,68,0.7059,0.69195,0.2345,0.5354,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,5,0,0,6,0,7],[52,68,0.7647,0.71427,0.23146,0.5354,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,6,0,0,6,0,8],[56,68,0.8235,0.62052,0.23585,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,1,0,0,10,0,0,5,0,0,8,0,0,2,0,5],[60,68,0.8824,0.60713,0.22016,0.42857,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,8,0,0,2,0,0,11,0,0,3,0,3],[64,68,0.9412,0.49107,0.19541,0.39286,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,6,0,0,11,0,0,4,0,0,8,0,0,0,0,1],[68,68,1.0,0.47764,0.14553,0.42857,0.42859,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,12,0,0,9,0,0,3,0,0,1,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.74102,"x":0.89283,"p":[[0,49,0.0,0.76784,0.22233,0.67857,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,4,0,0,12,0,8],[4,49,0.0816,0.86607,0.24468,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,4,0,21],[8,49,0.1633,0.82143,0.22016,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,9,0,14],[12,49,0.2449,0.77677,0.26472,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,2,0,0,1,0,0,2,0,0,5,0,0,2,0,0,6,0,14],[16,49,0.3265,0.77679,0.25238,0.67857,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,5,0,0,6,0,13],[20,49,0.4082,0.77676,0.21708,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,7,0,0,8,0,10],[24,49,0.4898,0.76339,0.21609,0.67857,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,8,0,9],[28,49,0.5714,0.74102,0.25368,0.571,0.78571,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,3,0,0,4,0,12],[32,49,0.6531,0.83036,0.20958,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,2,0,0,8,0,15],[36,49,0.7347,0.85267,0.19719,0.85714,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,12,0,14],[40,49,0.8163,0.86606,0.21998,0.82132,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,5,0,19],[44,49,0.898,0.89283,0.13366,0.85714,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,13,0,15],[48,49,0.9796,0.84375,0.1488,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,11,0,11],[49,49,1.0,0.87054,0.13054,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,13,0,12]]}]},{"i":"619a1c5b290cb428","q":"25. (NZL) Let $U=\\{1,2, \\ldots, n\\}$, where $n \\geq 3$. A subset $S$ of $U$ is said to be split by an arrangement of the elements of $U$ if an element not in $S$ occurs in the arrangement somewhere between two elements of $S$. For example, 13542 splits $\\{1,2,3\\}$ but not $\\{3,4,5\\}$. Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an arrangement of the elements of $U$ that splits all of them.","t":[{"b":3,"e":0.71429,"k":"rising","v":0.0625,"x":0.6473,"p":[[0,38,0.0,0.0625,0.15126,0.0,0.0,0.0,0.0,0.71429,25,0,3,25,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,38,0.1053,0.32141,0.37964,0.0,0.14286,0.71429,0.0,1.0,15,4,0,15,0,3,0,0,2,0,0,2,0,0,1,0,0,3,0,0,2,0,4],[8,38,0.2105,0.33916,0.35313,0.0,0.14286,0.71407,0.0,1.0,12,2,0,12,0,5,0,0,2,0,0,1,0,0,3,0,0,4,0,0,3,0,2],[12,38,0.3158,0.25438,0.31693,0.0,0.14286,0.42858,0.0,1.0,14,2,0,14,0,5,0,0,4,0,0,3,0,0,1,0,0,1,0,0,2,0,2],[16,38,0.4211,0.45982,0.38421,0.0,0.42857,0.85714,0.0,1.0,11,3,0,11,0,0,0,0,2,0,0,5,0,0,0,0,0,3,0,0,8,0,3],[20,38,0.5263,0.37499,0.33834,0.0,0.28571,0.71429,0.0,1.0,9,2,0,9,0,4,0,0,5,0,0,3,0,0,2,0,0,3,0,0,4,0,2],[24,38,0.6316,0.38839,0.35217,0.0,0.35714,0.71429,0.0,1.0,10,4,0,10,0,3,0,0,3,0,0,4,0,0,3,0,0,4,0,0,1,0,4],[28,38,0.7368,0.56696,0.35978,0.25,0.71429,0.85714,0.0,1.0,6,6,0,6,0,2,0,0,1,0,0,4,0,0,1,0,0,7,0,0,5,0,6],[32,38,0.8421,0.45066,0.34459,0.14286,0.42857,0.71429,0.0,1.0,6,4,0,6,0,5,0,0,3,0,0,4,0,0,3,0,0,4,0,0,3,0,4],[36,38,0.9474,0.6473,0.3214,0.42859,0.71429,1.0,0.0,1.0,2,9,0,2,0,3,0,0,1,0,0,4,0,0,5,0,0,3,0,0,5,0,9],[38,38,1.0,0.50445,0.3194,0.24999,0.42859,0.75,0.0,1.0,2,5,0,2,0,6,0,0,4,0,0,5,0,0,4,0,0,3,0,0,3,0,5]]},{"b":5,"e":0.85714,"k":"rising","v":0.05348,"x":0.66964,"p":[[0,52,0.0,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,1,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.24998,0.32535,0.0,0.0,0.42858,0.0,1.0,17,2,0,17,0,3,0,0,0,0,0,5,0,0,2,0,0,2,0,0,1,0,2],[8,52,0.1538,0.17857,0.31541,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,1,0,2],[12,52,0.2308,0.25892,0.28219,0.0,0.14286,0.42857,0.0,0.85714,13,0,0,13,0,4,0,0,4,0,0,5,0,0,2,0,0,1,0,0,3,0,0],[16,52,0.3077,0.17857,0.25754,0.0,0.0,0.42857,0.0,0.85714,19,0,0,19,0,3,0,0,1,0,0,5,0,0,1,0,0,2,0,0,1,0,0],[20,52,0.3846,0.20089,0.30063,0.0,0.0,0.28571,0.0,1.0,19,1,0,19,0,2,0,0,4,0,0,1,0,0,2,0,0,1,0,0,2,0,1],[24,52,0.4615,0.1607,0.28958,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,3,0,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,0],[28,52,0.5385,0.19642,0.31083,0.0,0.0,0.32143,0.0,0.85714,21,0,0,21,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,4,0,0],[32,52,0.6154,0.22321,0.333,0.0,0.0,0.32143,0.0,1.0,19,2,0,19,0,2,0,0,3,0,0,2,0,0,0,0,0,2,0,0,2,0,2],[36,52,0.6923,0.11161,0.25439,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[40,52,0.7692,0.56697,0.3164,0.39286,0.71429,0.75,0.0,1.0,4,3,0,4,0,3,0,0,1,0,0,4,0,0,1,0,0,11,0,0,5,0,3],[44,52,0.8462,0.66964,0.25111,0.57143,0.71429,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,1,0,0,3,0,0,3,0,0,10,0,0,9,0,3],[48,52,0.9231,0.60266,0.25688,0.42859,0.71429,0.85704,0.0,0.85714,3,0,0,3,0,0,0,0,2,0,0,4,0,0,5,0,0,9,0,0,9,0,0],[52,52,1.0,0.64729,0.21424,0.571,0.71429,0.857,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,9,0,0,7,0,2]]}]},{"i":"a0af40a87b048365","q":"A number is written in each cell of the  $N \\times N$  square. Let's call cell  $C$  *good* if in one of the cells adjacent to  $C$  on the side, there is a number  $1$  more than in  $C$ , and in some other of the cells adjacent to  $C$  on the side, there is a number  $3$  more than in  $C$ . What is the largest possible number of good cells?\n*Proposed by A. Chebotarev*","t":[{"b":3,"e":0.85714,"k":"rising","v":0.42856,"x":0.91518,"p":[[0,92,0.0,0.42856,0.38959,0.0,0.49979,0.85714,0.0,0.85714,13,0,2,13,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,0,11,0,0],[4,92,0.0435,0.79911,0.31715,0.82143,0.85714,1.0,0.0,1.0,4,15,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,15],[8,92,0.087,0.75,0.27894,0.71429,0.85714,1.0,0.0,1.0,2,10,0,2,0,1,0,0,0,0,0,2,0,0,2,0,0,7,0,0,8,0,10],[12,92,0.1304,0.85714,0.19233,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,9,0,14],[16,92,0.1739,0.85714,0.15152,0.85714,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,14,0,11],[20,92,0.2174,0.84821,0.20497,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,9,0,14],[24,92,0.2609,0.83036,0.19377,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,10,0,11],[28,92,0.3043,0.85714,0.15152,0.71429,0.85714,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,11,0,12],[32,92,0.3478,0.85714,0.11845,0.71429,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,10,0,11],[36,92,0.3913,0.86607,0.10677,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,17,0,9],[40,92,0.4348,0.84821,0.1234,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,11,0,10],[44,92,0.4783,0.74999,0.29451,0.71429,0.85714,1.0,0.0,1.0,3,10,0,3,0,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,10,0,10],[48,92,0.5217,0.89286,0.10714,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,14],[52,92,0.5652,0.84375,0.19019,0.82143,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,13,0,11],[56,92,0.6087,0.81696,0.18293,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,12,0,8],[60,92,0.6522,0.86607,0.12846,0.71429,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,9,0,13],[64,92,0.6957,0.78571,0.2369,0.71429,0.85714,0.89286,0.0,1.0,2,8,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,12,0,8],[68,92,0.7391,0.88392,0.10972,0.85711,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,12,0,13],[72,92,0.7826,0.81696,0.23483,0.71429,0.85714,1.0,0.0,1.0,2,10,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,13,0,10],[76,92,0.8261,0.87052,0.12559,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,10,0,13],[80,92,0.8696,0.83929,0.27141,0.85714,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,16],[84,92,0.913,0.83037,0.14472,0.71429,0.85714,1.0,0.43,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,12,0,9],[88,92,0.9565,0.91518,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[92,92,1.0,0.87946,0.11904,0.71429,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,9,0,14]]},{"b":6,"e":0.0,"k":"falling","v":0.04018,"x":0.86607,"p":[[0,67,0.0,0.39732,0.39404,0.0,0.28571,0.85714,0.0,1.0,14,1,7,14,0,1,0,0,2,0,0,1,0,0,0,0,0,4,0,0,9,0,1],[4,67,0.0597,0.70535,0.33869,0.67857,0.85714,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,0,13,0,8],[8,67,0.1194,0.63839,0.34623,0.53571,0.71429,0.85714,0.0,1.0,6,6,0,6,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,0,8,0,6],[12,67,0.1791,0.73661,0.26027,0.71429,0.85714,0.85714,0.0,1.0,2,6,0,2,0,1,0,0,0,0,0,1,0,0,2,0,0,9,0,0,11,0,6],[16,67,0.2388,0.66518,0.36177,0.39286,0.78571,1.0,0.0,1.0,5,10,0,5,0,1,0,0,2,0,0,1,0,0,0,0,0,7,0,0,6,0,10],[20,67,0.2985,0.68303,0.33831,0.57143,0.78564,1.0,0.0,1.0,4,9,0,4,0,2,0,0,0,0,0,1,0,0,2,0,0,7,0,0,7,0,9],[24,67,0.3582,0.69197,0.32948,0.67857,0.78571,0.89286,0.0,1.0,5,8,0,5,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,8,0,8],[28,67,0.4179,0.36161,0.40718,0.0,0.07143,0.75,0.0,1.0,16,4,0,16,0,2,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,4],[32,67,0.4776,0.69643,0.3549,0.67857,0.85714,1.0,0.0,1.0,6,9,0,6,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,10,0,9],[36,67,0.5373,0.86607,0.13333,0.82132,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,12,0,12],[40,67,0.597,0.74105,0.30606,0.71429,0.85707,1.0,0.0,1.0,4,10,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,7,0,10],[44,67,0.6567,0.77678,0.27184,0.71429,0.85714,0.89286,0.0,1.0,3,8,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,14,0,8],[48,67,0.7164,0.76786,0.23351,0.71429,0.85714,0.85714,0.0,1.0,2,7,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,10,0,7],[52,67,0.7761,0.79018,0.20511,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,11,0,0,10,0,8],[56,67,0.8358,0.75446,0.31183,0.71429,0.85714,1.0,0.0,1.0,4,11,0,4,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,9,0,11],[60,67,0.8955,0.81249,0.12081,0.71429,0.78571,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,9,0,7],[64,67,0.9552,0.50446,0.37284,0.0,0.71429,0.75,0.0,1.0,11,1,0,11,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,7,0,1],[67,67,1.0,0.04018,0.1394,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"3a1ce1fc421f4784","q":"Let five points on a circle be labelled $A, B, C, D$, and $E$ in clockwise order. Assume $A E=D E$ and let $P$ be the intersection of $A C$ and $B D$. Let $Q$ be the point on the line through $A$ and $B$ such that $A$ is between $B$ and $Q$ and $A Q=D P$. Similarly, let $R$ be the point on the line through $C$ and $D$ such that $D$ is between $C$ and $R$ and $D R=A P$. Prove that $P E$ is perpendicular to $Q R$.","t":[{"b":4,"e":0.0,"k":"falling","v":0.02679,"x":0.31686,"p":[[0,109,0.0,0.27232,0.29312,0.0,0.35714,0.42857,0.0,1.0,15,2,5,15,0,0,0,0,1,0,0,12,0,0,1,0,0,1,0,0,0,0,2],[4,109,0.0367,0.31686,0.34581,0.0,0.14286,0.4642,0.0,1.0,12,3,0,12,0,6,0,0,0,0,0,6,0,0,1,0,0,2,0,0,2,0,3],[8,109,0.0734,0.25893,0.35792,0.0,0.14286,0.21432,0.0,1.0,13,5,0,13,0,11,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,5],[12,109,0.1101,0.19642,0.27373,0.0,0.14286,0.1786,0.0,1.0,14,2,0,14,0,10,0,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,2],[16,109,0.1468,0.29017,0.33019,0.0,0.14286,0.4642,0.0,1.0,10,3,0,10,0,11,0,0,0,0,0,3,0,0,2,0,0,2,0,0,1,0,3],[20,109,0.1835,0.12491,0.26183,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[24,109,0.2202,0.16517,0.26988,0.0,0.0,0.14286,0.0,1.0,18,2,0,18,0,7,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,2],[28,109,0.2569,0.14286,0.25505,0.0,0.0,0.14286,0.0,1.0,18,2,0,18,0,9,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[32,109,0.2936,0.2232,0.29866,0.0,0.14286,0.42857,0.0,1.0,13,3,0,13,0,10,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[36,109,0.3303,0.12947,0.2212,0.0,0.0,0.14287,0.0,1.0,17,1,0,17,0,11,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[40,109,0.367,0.12054,0.20551,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,9,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[44,109,0.4037,0.18749,0.27763,0.0,0.14286,0.1429,0.0,1.0,14,2,0,14,0,12,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,2],[48,109,0.4404,0.09375,0.15815,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,12,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[52,109,0.4771,0.125,0.20124,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,14,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[56,109,0.5138,0.13393,0.21706,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,12,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[60,109,0.5505,0.10714,0.23958,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[64,109,0.5872,0.12945,0.25841,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,8,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[68,109,0.6239,0.09822,0.19704,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[72,109,0.6606,0.10268,0.17941,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[76,109,0.6972,0.07571,0.18199,0.0,0.0,0.14071,0.0,1.0,22,1,0,22,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[80,109,0.7339,0.09339,0.14098,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,11,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[84,109,0.7706,0.05357,0.09943,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,109,0.8073,0.07134,0.09443,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,109,0.844,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,109,0.8807,0.10714,0.15972,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,11,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[100,109,0.9174,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,109,0.9541,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,109,0.9908,0.06249,0.13328,0.0,0.0,0.03571,0.0,0.571,24,0,0,24,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[109,109,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.1429,"k":"falling","v":0.04018,"x":0.29911,"p":[[0,66,0.0,0.29911,0.26089,0.0,0.42857,0.42857,0.0,1.0,12,1,2,12,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,1],[4,66,0.0606,0.16517,0.25026,0.0,0.0,0.14287,0.0,1.0,17,1,0,17,0,8,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,1],[8,66,0.1212,0.11161,0.18466,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,7,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[12,66,0.1818,0.12052,0.17166,0.0,0.0,0.14286,0.0,0.571,18,0,0,18,0,8,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[16,66,0.2424,0.12045,0.20549,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,11,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[20,66,0.303,0.10705,0.24742,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[24,66,0.3636,0.08028,0.09402,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,66,0.4242,0.1741,0.25185,0.0,0.0,0.21432,0.0,1.0,17,1,0,17,0,7,0,0,0,0,0,3,0,0,4,0,0,0,0,0,0,0,1],[32,66,0.4848,0.05804,0.18851,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,66,0.5455,0.13392,0.18874,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,9,0,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[40,66,0.6061,0.10268,0.2402,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,66,0.6667,0.04911,0.16602,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[48,66,0.7273,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,66,0.7879,0.07143,0.15152,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[56,66,0.8485,0.06242,0.07927,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,66,0.9091,0.04911,0.06786,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.0625,0.11812,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[66,66,1.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b9d8ee20ff5b055a","q":"There are more than  $n^2$  stones on the table. Peter and Vasya play a game, Peter starts. Each turn, a player can take any prime number less than  $n$  stones, or any multiple of  $n$  stones, or  $1$  stone. Prove that Peter always can take the last stone (regardless of Vasya's strategy).\n\n*S Berlov*","t":[{"b":2,"e":0.0,"k":"falling","v":0.01777,"x":0.19196,"p":[[0,57,0.0,0.19196,0.13175,0.14286,0.14286,0.2857,0.0,0.57143,4,0,1,4,0,18,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,57,0.0702,0.17857,0.29881,0.0,0.0,0.14286,0.0,1.0,19,2,7,19,0,6,0,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,2],[8,57,0.1404,0.08929,0.18472,0.0,0.0,0.14286,0.0,1.0,20,1,4,20,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,57,0.2105,0.03125,0.08552,0.0,0.0,0.0,0.0,0.42857,27,0,1,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.10268,0.1439,0.0,0.0,0.14286,0.0,0.57143,17,0,2,17,0,11,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,57,0.3509,0.11589,0.1801,0.0,0.07,0.14286,0.0,0.71429,16,0,2,16,0,13,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[24,57,0.4211,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,1,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.04018,0.06424,0.0,0.0,0.14286,0.0,0.143,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.08034,0.16338,0.0,0.0,0.14286,0.0,0.71429,22,0,1,22,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[36,57,0.6316,0.05795,0.14217,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,57,0.7018,0.15161,0.21111,0.0,0.14143,0.14286,0.0,1.0,14,1,0,14,0,11,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[44,57,0.7719,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.05786,0.11201,0.0,0.0,0.14071,0.0,0.57143,22,0,0,22,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,57,0.9825,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[57,57,1.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01786,"x":0.26784,"p":[[0,76,0.0,0.12018,0.08064,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,76,0.0526,0.10268,0.25313,0.0,0.0,0.0,0.0,1.0,25,1,14,25,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[8,76,0.1053,0.10714,0.16366,0.0,0.0,0.14286,0.0,0.57143,19,0,8,19,0,7,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,76,0.1579,0.10713,0.17854,0.0,0.0,0.14286,0.0,0.57143,20,0,5,20,0,7,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[16,76,0.2105,0.11143,0.18463,0.0,0.0,0.14286,0.0,0.71429,18,0,3,18,0,10,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[20,76,0.2632,0.12482,0.17403,0.0,0.0,0.14286,0.0,0.57143,17,0,5,17,0,9,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[24,76,0.3158,0.16518,0.1992,0.0,0.14286,0.14286,0.0,0.85714,12,0,1,12,0,13,0,0,1,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[28,76,0.3684,0.18749,0.2659,0.0,0.0,0.32143,0.0,1.0,17,1,3,17,0,6,0,0,1,0,0,2,0,0,4,0,0,1,0,0,0,0,1],[32,76,0.4211,0.15625,0.26088,0.0,0.07143,0.14286,0.0,1.0,16,2,2,16,0,11,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[36,76,0.4737,0.15625,0.19678,0.0,0.07143,0.2857,0.0,0.57143,16,0,3,16,0,7,0,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[40,76,0.5263,0.20973,0.23956,0.0,0.14286,0.28571,0.0,1.0,10,1,2,10,0,11,0,0,5,0,0,3,0,0,1,0,0,0,0,0,1,0,1],[44,76,0.5789,0.19196,0.23313,0.0,0.14286,0.17857,0.0,1.0,10,1,0,10,0,14,0,0,3,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[48,76,0.6316,0.26784,0.29394,0.0,0.14286,0.42857,0.0,1.0,10,2,1,10,0,9,0,0,3,0,0,4,0,0,2,0,0,1,0,0,1,0,2],[52,76,0.6842,0.13366,0.18876,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,19,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[56,76,0.7368,0.19187,0.19761,0.0,0.14286,0.17857,0.0,0.57143,9,0,1,9,0,15,0,0,2,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[60,76,0.7895,0.15625,0.17262,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,13,0,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[64,76,0.8421,0.17393,0.19478,0.0,0.14286,0.17857,0.0,0.71429,11,0,0,11,0,13,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[68,76,0.8947,0.13375,0.16341,0.0,0.14,0.17857,0.0,0.57143,15,0,0,15,0,9,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[72,76,0.9474,0.07134,0.15968,0.0,0.0,0.14071,0.0,0.71429,23,0,0,23,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[76,76,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"db2bc00c066bce0b","q":"Let  $ABC$  be an acute triangle,  $H$  its orthocentre,  $D$  a point on the side  $[BC]$ , and  $P$  a point such that  $ADPH$  is a parallelogram.\nShow that  $\\angle BPC > \\angle BAC$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,86,0.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,86,0.0465,0.04464,0.10972,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,86,0.093,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,86,0.1395,0.03562,0.09439,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,86,0.186,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,86,0.2326,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,86,0.2791,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,86,0.3256,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,86,0.3721,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,86,0.4186,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,86,0.4651,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,86,0.5116,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,86,0.5581,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,86,0.6047,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,86,0.6512,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,86,0.6977,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,86,0.7442,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,86,0.7907,0.03571,0.11294,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[72,86,0.8372,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,86,0.8837,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,86,0.9302,0.03125,0.08553,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,86,0.9767,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[86,86,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,100,0.0,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,1,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,100,0.04,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,100,0.08,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,100,0.12,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,100,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,100,0.2,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,100,0.24,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,100,0.28,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,100,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,100,0.36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,100,0.4,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,100,0.44,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,100,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,100,0.52,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,100,0.56,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,100,0.6,0.03125,0.12745,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[64,100,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,100,0.68,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[72,100,0.72,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,100,0.76,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,100,0.8,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,100,0.84,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,100,0.88,0.02679,0.09061,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,100,0.92,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,100,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,100,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a08369a7fff87abc","q":"Find all real triples  $(a,b,c)$  satisfying\n\\[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\\]","t":[{"b":1,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"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],[4,25,0.16,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,25,0.32,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,25,0.48,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[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],[20,25,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[25,25,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":0.85714,"k":"flat","v":0.98661,"x":1.0,"p":[[0,34,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,34,0.1176,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,34,0.2353,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,34,0.3529,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,34,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,34,0.5882,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,34,0.7059,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[32,34,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"1975e5bb01a9bc6e","q":"The square polynomial  $x^2+ax+b+1$  has natural roots. Prove that  $(a^2+b^2)$  is a composite number.","t":[{"b":3,"e":1.0,"k":"flat","v":1.0,"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],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,29,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"7b45b37cd7d3c557","q":"If  $ a,b,c>0, $  then  $ \\sum_{\\text{cyc}} \\frac{a}{2a+b+c}\\le 3/4. $","t":[{"b":2,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,9,0.0,0.82143,0.32537,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,24],[4,9,0.4444,0.9241,0.20198,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,27],[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]]},{"b":3,"e":1.0,"k":"flat","v":0.72768,"x":1.0,"p":[[0,28,0.0,0.90625,0.25407,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,28],[4,28,0.1429,0.85268,0.26841,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,4,0,0,1,0,0,0,0,0,2,0,23],[8,28,0.2857,0.90625,0.20705,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,26],[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],[20,28,0.7143,0.84375,0.27516,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,0,0,0,5,0,0,0,0,0,1,0,0,1,0,23],[24,28,0.8571,0.72768,0.33761,0.42857,1.0,1.0,0.14286,1.0,0,19,0,0,0,3,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,19],[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":"574a646a389b7787","q":"Two circles of different radii are cut out of cardboard. Each circle is subdivided into  $200$  equal sectors. On each circle  $100$  sectors are painted white and the other  $100$  are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least  $100$  sectors on the small circle lie over sectors of the same color on the big circle.","t":[{"b":6,"e":1.0,"k":"flat","v":0.99107,"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],[4,18,0.2222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":7,"e":1.0,"k":"flat","v":1.0,"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],[8,35,0.2286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,35,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,35,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"64b470f0b20a22c3","q":"Let  $F$  be the set of all  $n-tuples$   $(A_1,A_2,\u2026,A_n)$  such that each  $A_i$  is a subset of  ${1,2,\u2026,2019}$ . Let  $\\mid{A}\\mid$  denote the number of elements o the set  $A$  . Find  $\\sum_{(A_1,\u2026,A_n)\\in{F}}^{}\\mid{A_1\\cup{A_2}\\cup...\\cup{A_n}}\\mid$","t":[{"b":2,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,55,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,55,0.0727,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[8,55,0.1455,0.92857,0.11294,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,21],[12,55,0.2182,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[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],[44,55,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,55,0.8727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":0.97768,"x":1.0,"p":[[0,105,0.0,0.98201,0.06968,1.0,1.0,1.0,0.71,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,105,0.0381,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,105,0.0762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,105,0.1143,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,105,0.1524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,105,0.1905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,105,0.2286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,105,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,105,0.3048,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,105,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,105,0.381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,105,0.419,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,105,0.4571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,105,0.4952,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,105,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,105,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,105,0.6095,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,105,0.6476,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,105,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,105,0.7238,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,105,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,105,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,105,0.8381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,105,0.8762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,105,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,105,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,105,0.9905,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[105,105,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"80780839070557a7","q":"Prove that, for all real $a \\geqslant 0$, we have\n\n$$\na^{3}+2 \\geqslant a^{2}+2 \\sqrt{a}\n$$","t":[{"b":1,"e":1.0,"k":"flat","v":0.88393,"x":0.95536,"p":[[0,25,0.0,0.91964,0.14258,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,0,0,24],[4,25,0.16,0.88393,0.16917,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,1,0,20],[8,25,0.32,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[12,25,0.48,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[16,25,0.64,0.91518,0.14223,0.82143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,1,0,23],[20,25,0.8,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[24,25,0.96,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[25,25,1.0,0.89285,0.15568,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,1,0,21]]},{"b":4,"e":1.0,"k":"flat","v":0.89272,"x":0.96875,"p":[[0,38,0.0,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[4,38,0.1053,0.95536,0.12078,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,28],[8,38,0.2105,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[12,38,0.3158,0.89272,0.16766,0.71429,1.0,1.0,0.4286,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,0,0,22],[16,38,0.4211,0.90179,0.16536,0.71429,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,0,0,23],[20,38,0.5263,0.9375,0.19541,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,28],[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],[28,38,0.7368,0.95536,0.10972,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,27],[32,38,0.8421,0.92411,0.13356,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,24],[36,38,0.9474,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[38,38,1.0,0.91518,0.13296,0.82143,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,22]]}]},{"i":"7c9661271c2e43c9","q":"15. (GDR 3) Suppose $\\tan \\alpha=p / q$, where $p$ and $q$ are integers and $q \\neq 0$. Prove that the number $\\tan \\beta$ for which $\\tan 2 \\beta=\\tan 3 \\alpha$ is rational only when $p^{2}+q^{2}$ is the square of an integer.","t":[{"b":2,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,41,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,41,0.0976,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,41,0.1951,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":1.0,"k":"flat","v":0.9866,"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],[4,30,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,30,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,30,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,30,0.6667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,30,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,30,0.9333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[30,30,1.0,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]}]},{"i":"6c73409f2fca772c","q":"In each cell of  $5 \\times 5$  table there is one number from  $1$  to  $5$  such that every number occurs exactly once in every row and in every column. Number in one column is *good positioned* if following holds:\n - In every row, every number which is left from *good positoned* number is smaller than him, and every number which is right to him is greater than him, or vice versa.\n -  In every column, every number which is above from *good positoned* number is smaller than him, and every number which is below to him is greater than him, or vice versa.\nWhat is maximal number of good positioned numbers that can occur in this table?","t":[{"b":0,"e":0.57143,"k":"flat","v":0.20537,"x":0.45092,"p":[[0,127,0.0,0.28573,0.30929,0.0,0.21431,0.57143,0.0,0.85714,15,0,13,15,0,1,0,0,3,0,0,4,0,0,2,0,0,5,0,0,2,0,0],[4,127,0.0315,0.29911,0.22689,0.24999,0.28571,0.32143,0.0,0.85714,7,0,2,7,0,1,0,0,16,0,0,3,0,0,2,0,0,1,0,0,2,0,0],[8,127,0.063,0.34373,0.24963,0.2857,0.28571,0.32143,0.0,1.0,4,2,0,4,0,2,0,0,18,0,0,2,0,0,2,0,0,1,0,0,1,0,2],[12,127,0.0945,0.38393,0.22428,0.2857,0.35714,0.57143,0.0,0.71429,5,0,0,5,0,0,0,0,11,0,0,5,0,0,6,0,0,5,0,0,0,0,0],[16,127,0.126,0.20537,0.18189,0.0,0.2143,0.28571,0.0,0.71429,10,0,0,10,0,6,0,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[20,127,0.1575,0.41964,0.22286,0.28571,0.28571,0.46431,0.0,1.0,1,1,0,1,0,1,0,0,15,0,0,7,0,0,3,0,0,1,0,0,3,0,1],[24,127,0.189,0.36159,0.15144,0.2857,0.28571,0.4286,0.0,0.71429,1,0,0,1,0,2,0,0,16,0,0,6,0,0,6,0,0,1,0,0,0,0,0],[28,127,0.2205,0.35268,0.13825,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,19,0,0,9,0,0,2,0,0,0,0,0,1,0,0],[32,127,0.252,0.41515,0.22687,0.28571,0.28571,0.571,0.0,1.0,1,1,0,1,0,2,0,0,15,0,0,5,0,0,3,0,0,3,0,0,2,0,1],[36,127,0.2835,0.29018,0.16164,0.2857,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,1,0,0,17,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[40,127,0.315,0.33482,0.16602,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,1,0,0,16,0,0,8,0,0,2,0,0,2,0,0,0,0,0],[44,127,0.3465,0.33035,0.09062,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,19,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[48,127,0.378,0.39732,0.18809,0.2857,0.28571,0.46429,0.14286,0.85714,0,0,0,0,0,4,0,0,13,0,0,7,0,0,3,0,0,4,0,0,1,0,0],[52,127,0.4094,0.37946,0.13174,0.28571,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,0,0,0,14,0,0,12,0,0,4,0,0,1,0,0,0,0,0],[56,127,0.4409,0.37052,0.12297,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,17,0,0,9,0,0,4,0,0,1,0,0,0,0,0],[60,127,0.4724,0.37052,0.13294,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,16,0,0,10,0,0,4,0,0,1,0,0,0,0,0],[64,127,0.5039,0.37945,0.12167,0.28571,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,15,0,0,11,0,0,4,0,0,1,0,0,0,0,0],[68,127,0.5354,0.42857,0.18211,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,1,0,0,10,0,0,10,0,0,7,0,0,1,0,0,2,0,0],[72,127,0.5669,0.37052,0.13765,0.28571,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,0,0,0,12,0,0,13,0,0,5,0,0,0,0,0,0,0,0],[76,127,0.5984,0.40179,0.1448,0.28571,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,1,0,0,13,0,0,13,0,0,1,0,0,4,0,0,0,0,0],[80,127,0.6299,0.38391,0.13568,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,11,0,0,14,0,0,4,0,0,1,0,0,0,0,0],[84,127,0.6614,0.40627,0.14332,0.28571,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,1,0,0,8,0,0,16,0,0,4,0,0,2,0,0,0,0,0],[88,127,0.6929,0.41518,0.08268,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,24,0,0,1,0,0,1,0,0,0,0,0],[92,127,0.7244,0.42856,0.14724,0.28571,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,17,0,0,2,0,0,1,0,0,2,0,0],[96,127,0.7559,0.45092,0.12929,0.42857,0.42857,0.42895,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,20,0,0,3,0,0,4,0,0,0,0,0],[100,127,0.7874,0.41072,0.09936,0.42857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,6,0,0,22,0,0,2,0,0,1,0,0,0,0,0],[104,127,0.8189,0.3929,0.14287,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,10,0,0,15,0,0,3,0,0,2,0,0,0,0,0],[108,127,0.8504,0.44197,0.09689,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,23,0,0,3,0,0,2,0,0,0,0,0],[112,127,0.8819,0.4509,0.1017,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,24,0,0,2,0,0,3,0,0,0,0,0],[116,127,0.9134,0.44643,0.07784,0.42857,0.42857,0.4286,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0],[120,127,0.9449,0.44196,0.13054,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,22,0,0,2,0,0,2,0,0,1,0,0],[124,127,0.9764,0.42857,0.08748,0.42857,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0,1,0,0],[127,127,1.0,0.42857,0.07143,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,27,0,0,1,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.01339,"x":0.33928,"p":[[0,95,0.0,0.26786,0.27141,0.0,0.28571,0.46431,0.0,0.85714,14,0,14,14,0,1,0,0,4,0,0,5,0,0,5,0,0,2,0,0,1,0,0],[4,95,0.0421,0.33928,0.24157,0.24999,0.28571,0.42857,0.0,1.0,4,1,0,4,0,4,0,0,13,0,0,6,0,0,1,0,0,1,0,0,2,0,1],[8,95,0.0842,0.29911,0.23517,0.0,0.28571,0.4286,0.0,1.0,9,1,0,9,0,0,0,0,11,0,0,6,0,0,5,0,0,0,0,0,0,0,1],[12,95,0.1263,0.23214,0.17767,0.10714,0.2857,0.28571,0.0,0.71429,8,0,0,8,0,5,0,0,13,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[16,95,0.1684,0.20536,0.20806,0.0,0.28571,0.28571,0.0,1.0,12,1,0,12,0,2,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[20,95,0.2105,0.25889,0.22985,0.0,0.28571,0.32143,0.0,0.85714,9,0,0,9,0,5,0,0,10,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[24,95,0.2526,0.20982,0.12869,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,2,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,95,0.2947,0.15625,0.17261,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,2,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[32,95,0.3368,0.16964,0.2126,0.0,0.07143,0.28571,0.0,1.0,16,1,0,16,0,1,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[36,95,0.3789,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,95,0.4211,0.11606,0.1836,0.0,0.0,0.2857,0.0,0.57143,22,0,0,22,0,0,0,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[44,95,0.4632,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,95,0.5053,0.06696,0.17122,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[52,95,0.5474,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,95,0.5895,0.11159,0.15036,0.0,0.0,0.28571,0.0,0.571,19,0,0,19,0,3,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,95,0.6316,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,95,0.6737,0.09821,0.16146,0.0,0.0,0.17857,0.0,0.57143,22,0,0,22,0,2,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[68,95,0.7158,0.12058,0.17904,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,2,0,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[72,95,0.7579,0.08464,0.13288,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[76,95,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,95,0.8421,0.05357,0.13243,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[84,95,0.8842,0.05348,0.11703,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[88,95,0.9263,0.07143,0.15567,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[92,95,0.9684,0.03098,0.05855,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[95,95,1.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f9b68862eb71cacb","q":"Find all natural integers  $n$  such that  $(n^3 + 39n - 2)n! + 17\\cdot 21^n + 5$  is a square.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.23205,"x":0.76339,"p":[[0,89,0.0,0.76339,0.34369,0.28571,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,21],[4,89,0.0449,0.63839,0.36244,0.28571,0.64286,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[8,89,0.0899,0.36161,0.28118,0.24999,0.28571,0.28571,0.14286,1.0,0,5,0,0,0,8,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[12,89,0.1348,0.57143,0.35535,0.28571,0.28571,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[16,89,0.1798,0.56241,0.36421,0.28571,0.28571,1.0,0.14,1.0,0,13,0,0,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[20,89,0.2247,0.35706,0.27206,0.24999,0.28571,0.28571,0.14,1.0,0,4,0,0,0,8,0,0,19,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[24,89,0.2697,0.27678,0.14258,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,89,0.3146,0.3125,0.18363,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[32,89,0.3596,0.28571,0.13832,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,89,0.4045,0.27678,0.14258,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,89,0.4494,0.29464,0.19212,0.24999,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,8,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,89,0.4944,0.26339,0.05187,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,89,0.5393,0.29018,0.19393,0.14286,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,9,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[52,89,0.5843,0.27232,0.09689,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,6,0,0,25,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[56,89,0.6292,0.25893,0.14914,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,11,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,89,0.6742,0.27232,0.14445,0.25002,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[64,89,0.7191,0.25,0.06186,0.24999,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,89,0.764,0.25893,0.05575,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,89,0.809,0.25446,0.05906,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,89,0.8539,0.25884,0.05595,0.2857,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,89,0.8989,0.23205,0.06928,0.14286,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,89,0.9438,0.24107,0.06621,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,89,0.9888,0.24991,0.06201,0.24999,0.2857,0.28571,0.14,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[89,89,1.0,0.25446,0.05906,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.25,"x":0.97768,"p":[[0,66,0.0,0.7142,0.37129,0.28571,1.0,1.0,0.14,1.0,0,20,0,0,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[4,66,0.0606,0.69196,0.37476,0.28571,1.0,1.0,0.14286,1.0,0,19,0,0,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[8,66,0.1212,0.65178,0.37276,0.28571,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[12,66,0.1818,0.89732,0.26058,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[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],[20,66,0.303,0.86161,0.27776,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[24,66,0.3636,0.88393,0.27067,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[28,66,0.4242,0.95089,0.17354,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[32,66,0.4848,0.93304,0.2082,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[36,66,0.5455,0.86161,0.28901,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[40,66,0.6061,0.34812,0.25245,0.28571,0.28571,0.28571,0.14,1.0,0,4,0,0,0,6,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[44,66,0.6667,0.30803,0.16793,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[48,66,0.7273,0.30357,0.18814,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,6,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[52,66,0.7879,0.25,0.06186,0.24999,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,66,0.8485,0.28571,0.13832,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[60,66,0.9091,0.25446,0.05906,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.27669,0.14266,0.2857,0.28571,0.28571,0.14,1.0,0,1,0,0,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[66,66,1.0,0.2991,0.13534,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,3,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"926e374348f996d2","q":"Consider 5 positive integers. By adding them two at a time in all possible ways, 10 integers are generated. Show that these 10 integers cannot be 10 consecutive integers.","t":[{"b":1,"e":1.0,"k":"rising","v":0.54018,"x":1.0,"p":[[0,20,0.0,0.54018,0.4556,0.0,0.57143,1.0,0.0,1.0,11,15,0,11,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,15],[4,20,0.2,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,20,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":6,"e":1.0,"k":"rising","v":0.64286,"x":1.0,"p":[[0,51,0.0,0.64732,0.44174,0.21427,1.0,1.0,0.0,1.0,8,19,0,8,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,19],[4,51,0.0784,0.66509,0.44562,0.105,1.0,1.0,0.0,1.0,8,20,0,8,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,20],[8,51,0.1569,0.64286,0.46566,0.0,1.0,1.0,0.0,1.0,10,20,0,10,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[12,51,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,51,0.3137,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,51,0.3922,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,51,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,51,0.549,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,51,0.6275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[40,51,0.7843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,51,0.8627,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,51,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[51,51,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a3f60e2e39ac62f1","q":"Find all ordered pairs of positive integers  $(r, s)$  for which there are exactly  $35$  ordered pairs of positive integers  $(a, b)$  such that the least common multiple of  $a$  and  $b$  is  $2^r \\cdot 3^s$ .","t":[{"b":2,"e":0.28571,"k":"flat","v":0.10259,"x":0.17857,"p":[[0,68,0.0,0.14732,0.11564,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,11,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,68,0.0588,0.16071,0.1171,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,10,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,68,0.1176,0.17857,0.12372,0.0,0.2857,0.28571,0.0,0.28571,9,0,0,9,0,6,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,68,0.1765,0.15625,0.12037,0.0,0.14286,0.28571,0.0,0.28571,10,0,0,10,0,9,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,68,0.2353,0.15625,0.12555,0.0,0.14286,0.28571,0.0,0.28571,11,0,0,11,0,7,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,68,0.2941,0.14286,0.12372,0.0,0.14286,0.2857,0.0,0.28571,12,0,0,12,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,68,0.3529,0.16518,0.10779,0.14286,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,13,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,68,0.4118,0.125,0.12242,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,8,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,68,0.4706,0.12946,0.12555,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,7,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,68,0.5294,0.17857,0.12372,0.0,0.2857,0.28571,0.0,0.28571,9,0,0,9,0,6,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,68,0.5882,0.16518,0.11904,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,9,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,68,0.6471,0.16964,0.11538,0.10714,0.14286,0.28571,0.0,0.28571,8,0,0,8,0,10,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,68,0.7059,0.10259,0.11422,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,68,0.7647,0.16518,0.11904,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,9,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.17857,0.11294,0.14286,0.14286,0.28571,0.0,0.28571,7,0,0,7,0,10,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,68,0.8824,0.16071,0.11152,0.10714,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,12,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.15625,0.11495,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,11,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,68,1.0,0.1741,0.12234,0.0,0.21428,0.2857,0.0,0.28571,9,0,0,9,0,7,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.12054,"x":0.16964,"p":[[0,56,0.0,0.12946,0.10926,0.0,0.14286,0.17857,0.0,0.28571,11,0,0,11,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.16518,0.11904,0.0,0.14286,0.28571,0.0,0.28571,9,0,0,9,0,9,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,56,0.1429,0.15616,0.12038,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,9,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,56,0.2143,0.15625,0.12037,0.0,0.14286,0.28571,0.0,0.28571,10,0,0,10,0,9,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.14732,0.11564,0.0,0.14286,0.2857,0.0,0.28571,10,0,0,10,0,11,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.12054,0.11355,0.0,0.14286,0.17857,0.0,0.28571,13,0,0,13,0,11,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.16964,0.11538,0.10714,0.14286,0.28571,0.0,0.28571,8,0,0,8,0,10,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.12946,0.12037,0.0,0.14286,0.2857,0.0,0.28571,13,0,0,13,0,9,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.125,0.12242,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,8,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,56,0.6429,0.16071,0.13716,0.0,0.2857,0.28571,0.0,0.28571,13,0,0,13,0,2,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.13393,0.12846,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.15178,0.12339,0.0,0.14286,0.2857,0.0,0.28571,11,0,0,11,0,8,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.13393,0.11258,0.0,0.14286,0.2857,0.0,0.28571,11,0,0,11,0,12,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.14286,0.10714,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.15625,0.12555,0.0,0.14286,0.28571,0.0,0.28571,11,0,0,11,0,7,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"244e703ed7e08c68","q":"Each square of an  $n \\times n$  grid is coloured either blue or red, where  $n$  is a positive integer. There are  $k$  blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form  $S_B$ . He then performs the same calculation on the red cells to compute  $S_R$ .\nIf  $S_B-  S_R = 50$ , determine (with proof) all possible values of  $k$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,50,0.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,50,0.08,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,50,0.16,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,50,0.24,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,50,0.32,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,50,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,50,0.64,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,50,0.72,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,50,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,50,0.88,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,50,0.96,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[50,50,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.9375,"x":0.99107,"p":[[0,93,0.0,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[4,93,0.043,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[8,93,0.086,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,93,0.129,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[16,93,0.172,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,93,0.2151,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[24,93,0.2581,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[28,93,0.3011,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[32,93,0.3441,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,93,0.3871,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[40,93,0.4301,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[44,93,0.4731,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[48,93,0.5161,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[52,93,0.5591,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[56,93,0.6022,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[60,93,0.6452,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[64,93,0.6882,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[68,93,0.7312,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[72,93,0.7742,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[76,93,0.8172,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[80,93,0.8602,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[84,93,0.9032,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[88,93,0.9462,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[92,93,0.9892,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[93,93,1.0,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27]]}]},{"i":"c5978edaf342a400","q":"Let  $a_0, a_1,\\dots, a_{19} \\in \\mathbb{R}$  and   $$ P(x) = x^{20} + \\sum_{i=0}^{19}a_ix^i, x \\in \\mathbb{R}. $$   If  $P(x)=P(-x)$  for all  $x \\in \\mathbb{R}$ , and   $$ P(k)=k^2, $$   for  $k=0, 1, 2, \\dots, 9$  then find   $$ \\lim_{x\\rightarrow 0} \\frac{P(x)}{\\sin^2x}. $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.86161,"x":0.94643,"p":[[0,47,0.0,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[4,47,0.0851,0.89286,0.06186,0.85714,0.85714,0.89286,0.85714,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[8,47,0.1702,0.87946,0.05187,0.85714,0.85714,0.85714,0.85714,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,5],[12,47,0.2553,0.87946,0.0724,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,7],[16,47,0.3404,0.86161,0.04351,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,29,0,2],[20,47,0.4255,0.86607,0.08702,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,23,0,6],[24,47,0.5106,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[28,47,0.5957,0.87054,0.05486,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,4],[32,47,0.6809,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[36,47,0.766,0.875,0.07784,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,25,0,6],[40,47,0.8511,0.88393,0.06622,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,7],[44,47,0.9362,0.875,0.05923,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,5],[47,47,1.0,0.87946,0.0724,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,7]]},{"b":5,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,58,0.0,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[4,58,0.069,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[8,58,0.1379,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[12,58,0.2069,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[16,58,0.2759,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[20,58,0.3448,0.93304,0.07973,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[24,58,0.4138,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[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],[32,58,0.5517,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,58,0.6207,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,58,0.6897,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[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],[48,58,0.8276,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,58,0.8966,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[56,58,0.9655,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[58,58,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"66701168e4986646","q":"Let  $ABC$  be a triangle with  $AB = AC$ . Let  $H$  be the orthocenter of  $ABC$ . Point $E$  is the midpoint of  $AC$  and point  $D$  lies on the side  $BC$  such that  $3CD = BC$ . Prove that $BE \\perp HD$ .\n\n*Proposed by Tran Quang Hung - Vietnam*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,22,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,22,0.7273,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,22,0.9091,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[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":"flat","v":0.0,"x":0.0625,"p":[[0,31,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"1d86ef6d967b33fd","q":"A class of  $10$  students took a math test. Each problem was solved by exactly  $7$  of the students. If the first nine students each solved  $4$  problems, how many problems did the tenth student solve?","t":[{"b":2,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,16,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,16,0.25,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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]]},{"b":7,"e":1.0,"k":"flat","v":0.98661,"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],[4,44,0.0909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"3d42916df2d8351d","q":"MKCD\n\nLet $x, y, z$ be positive real numbers that satisfy the equality $x^{2}+y^{2}+z^{2}=3$. Prove that\n\n$$\n\\frac{x^{2}+y z}{x^{2}+y z+1}+\\frac{y^{2}+z x}{y^{2}+z x+1}+\\frac{z^{2}+x y}{z^{2}+x y+1} \\leq 2\n$$","t":[{"b":1,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,44,0.0,0.93304,0.21424,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[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]]},{"b":5,"e":0.0,"k":"falling","v":0.40177,"x":0.95982,"p":[[0,24,0.0,0.92411,0.2172,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[4,24,0.1667,0.95982,0.1394,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,29],[8,24,0.3333,0.48214,0.43558,0.0,0.57141,1.0,0.0,1.0,12,11,0,12,0,1,0,0,2,0,0,0,0,0,4,0,0,2,0,0,0,0,11],[12,24,0.5,0.5982,0.40945,0.21429,0.71429,1.0,0.0,1.0,8,13,0,8,0,0,0,0,3,0,0,1,0,0,1,0,0,6,0,0,0,0,13],[16,24,0.6667,0.59375,0.41205,0.14286,0.71429,1.0,0.0,1.0,7,13,0,7,0,3,0,0,1,0,0,1,0,0,1,0,0,6,0,0,0,0,13],[20,24,0.8333,0.40177,0.39839,0.0,0.28571,0.71429,0.0,1.0,13,6,0,13,0,1,0,0,3,0,0,2,0,0,1,0,0,5,0,0,1,0,6],[24,24,1.0,0.43304,0.4439,0.0,0.28573,1.0,0.0,1.0,15,9,0,15,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,9]]}]},{"i":"153a8b68ded68260","q":"Let  $a,b$  be the smaller sides of a right triangle. Let  $c$  be the hypothenuse and  $h$  be the altitude from the right angle. Fint the maximal value of  $\\frac{c+h}{a+b}$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.98214,"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],[8,43,0.186,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,43,0.2791,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,43,0.3721,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,43,0.4651,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[28,43,0.6512,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,43,0.7442,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,43,0.8372,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[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":7,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,25,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,25,0.16,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,25,0.32,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,25,0.48,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[16,25,0.64,0.94643,0.17035,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,29],[20,25,0.8,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[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":"c9e305a68b24e642","q":"What is the sum of all primes  $p$  such that  $7^p - 6^p + 2$  is divisible by 43?","t":[{"b":6,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,59,0.0,0.95536,0.14914,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[4,59,0.0678,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,59,0.1356,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,59,0.2712,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,59,0.339,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,59,0.4068,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,59,0.4746,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,59,0.5424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,59,0.6102,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,59,0.678,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,59,0.7458,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[52,59,0.8814,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,59,0.9492,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[59,59,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"flat","v":0.93304,"x":1.0,"p":[[0,63,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,63,0.0635,0.95536,0.14032,1.0,1.0,1.0,0.4286,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,29],[8,63,0.127,0.9375,0.20183,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[12,63,0.1905,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[16,63,0.254,0.94196,0.18851,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,27],[20,63,0.3175,0.94196,0.18509,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,27],[24,63,0.381,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27],[28,63,0.4444,0.93304,0.2082,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[32,63,0.5079,0.9732,0.10379,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[36,63,0.5714,0.95982,0.1439,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[40,63,0.6349,0.95536,0.14914,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,29],[44,63,0.6984,0.94643,0.18123,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[48,63,0.7619,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[52,63,0.8254,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[56,63,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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":"493b34cbd83c4e4a","q":"Compute the largest integer not exceeding   $$ \\frac{2549^3}{2547\\cdot 2548} -\\frac{2547^3}{2548\\cdot 2549} $$","t":[{"b":4,"e":1.0,"k":"flat","v":0.99554,"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],[4,70,0.0571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,70,0.1143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,70,0.1714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,70,0.2286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,70,0.3429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,70,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[36,70,0.5143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,70,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,70,0.6286,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[56,70,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,70,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,70,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,70,0.9714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[70,70,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,21,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,21,0.1905,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[20,21,0.9524,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[21,21,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]}]},{"i":"34a1ba2a328898b7","q":"Consider all  $6$ -digit numbers of the form  $abccba$  where  $b$  is odd. Determine the number of all such  $6$ -digit numbers that are divisible by  $7$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,32,0.0,0.9375,0.14698,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,27],[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],[8,32,0.25,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,32,0.375,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[16,32,0.5,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,32,0.625,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,32,0.75,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[28,32,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":0.57143,"k":"flat","v":0.8482,"x":0.98214,"p":[[0,42,0.0,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[4,42,0.0952,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,42,0.1905,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,42,0.2857,0.8482,0.18538,0.67857,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,2,0,18],[16,42,0.381,0.91518,0.15093,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,23],[20,42,0.4762,0.91964,0.15947,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,25],[24,42,0.5714,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[28,42,0.6667,0.875,0.18814,0.67857,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,0,0,22],[32,42,0.7619,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[36,42,0.8571,0.94642,0.13247,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,27],[40,42,0.9524,0.875,0.18123,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,1,0,21],[42,42,1.0,0.92411,0.15966,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,26]]}]},{"i":"777ca5ed7ff35b49","q":"Two positive integers  $m$  and  $n$  are both less than  $500$  and  $\\text{lcm}(m,n) = (m-n)^2$ . What is the maximum possible value of  $m+n$ ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.92857,"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],[12,56,0.2143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,56,0.2857,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,56,0.3571,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,56,0.4286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,56,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[52,56,0.9286,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[56,56,1.0,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19]]},{"b":7,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,60,0.0,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[4,60,0.0667,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,60,0.1333,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,60,0.2,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,60,0.5333,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,60,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"d997a8b3c2fdecfb","q":"Let  $ABC$  be a triangle.  Let  $D$  be the midpoint of  $\\overline{BC}$ , let  $E$  be the midpoint of  $\\overline{AD}$ , and let  $F$  be the midpoint of  $\\overline{BE}$ .  Let  $G$  be the point where the lines  $AB$  and  $CF$  intersect.  What is the value of  $\\frac{AG}{AB}$ ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.96875,"x":0.99554,"p":[[0,37,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,37,0.1081,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[8,37,0.2162,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,37,0.3243,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,37,0.4324,0.97768,0.05187,1.0,1.0,1.0,0.8571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,37,0.5405,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,37,0.6486,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,37,0.7568,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,37,0.8649,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,37,0.973,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[37,37,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":4,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,22,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,22,0.1818,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[12,22,0.5455,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"613299ff6bd5e853","q":"The sequence $f(1), f(2), f(3), \\ldots$ is defined by  $$ f(n)=\\frac{1}{n}\\left(\\left\\lfloor\\frac{n}{1}\\right\\rfloor+\\left\\lfloor\\frac{n}{2}\\right\\rfloor+\\cdots+\\left\\lfloor\\frac{n}{n}\\right\\rfloor\\right), $$  where $\\lfloor x\\rfloor$ denotes the integer part of $x$.  (a) Prove that $f(n+1)>f(n)$ infinitely often.  (b) Prove that $f(n+1)<f(n)$ infinitely often.  (South Africa)","t":[{"b":3,"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],[4,30,0.1333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,30,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,30,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":6,"e":0.71429,"k":"falling","v":0.73214,"x":0.99107,"p":[[0,36,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,36,0.1111,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,36,0.2222,0.89732,0.13474,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,1,0,20],[12,36,0.3333,0.89285,0.12372,0.71429,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,6,0,17],[16,36,0.4444,0.82142,0.12878,0.71429,0.71429,1.0,0.714,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,4,0,10],[20,36,0.5556,0.84374,0.12556,0.71429,0.85707,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,7,0,11],[24,36,0.6667,0.83929,0.1171,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,10,0,9],[28,36,0.7778,0.77232,0.09353,0.71429,0.71429,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,7,0,3],[32,36,0.8889,0.79911,0.10012,0.71429,0.71429,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,11,0,4],[36,36,1.0,0.73214,0.05923,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,2,0,1]]}]},{"i":"c975b4849045e0f4","q":"Let $n \\geq 3$ be an integer such that $4 n+1$ is a prime number. Prove that $4 n+1$ divides $n^{2 n}-1$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.88391,"x":1.0,"p":[[0,105,0.0,0.96875,0.09268,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,28],[4,105,0.0381,0.96428,0.0945,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,27],[8,105,0.0762,0.92857,0.12372,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[12,105,0.1143,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[16,105,0.1524,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[20,105,0.1905,0.94196,0.123,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[24,105,0.2286,0.90178,0.13092,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,18],[28,105,0.2667,0.92411,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,11,0,19],[32,105,0.3048,0.88391,0.14917,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,11,0,16],[36,105,0.3429,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21],[40,105,0.381,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[44,105,0.419,0.91071,0.12242,0.85714,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,18],[48,105,0.4571,0.88839,0.14166,0.85714,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,10,0,16],[52,105,0.4952,0.90625,0.11633,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,17],[56,105,0.5333,0.88838,0.12238,0.85714,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,13,0,14],[60,105,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,105,0.6095,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,105,0.6476,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,105,0.6857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[76,105,0.7238,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[80,105,0.7619,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[84,105,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[88,105,0.8381,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[92,105,0.8762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[96,105,0.9143,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[100,105,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[104,105,0.9905,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[105,105,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":1,"e":0.57143,"k":"flat","v":0.8616,"x":0.98214,"p":[[0,39,0.0,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[4,39,0.1026,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,39,0.2051,0.90179,0.14032,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,19],[12,39,0.3077,0.86607,0.16728,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,6,0,17],[16,39,0.4103,0.90179,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,13,0,15],[20,39,0.5128,0.90179,0.1357,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,19],[24,39,0.6154,0.8616,0.14054,0.82132,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,12,0,12],[28,39,0.7179,0.88839,0.14167,0.71429,1.0,1.0,0.5714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,18],[32,39,0.8205,0.89286,0.13363,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,8,0,17],[36,39,0.9231,0.88393,0.1448,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,10,0,16],[39,39,1.0,0.91517,0.11769,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,8,0,19]]}]},{"i":"f442486bfb2c1e32","q":"What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,57,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,57,0.0702,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,57,0.1404,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,57,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[40,57,0.7018,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.0,"p":[[0,30,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ac83eab02a875be6","q":"Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$","t":[{"b":2,"e":1.0,"k":"flat","v":1.0,"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],[4,88,0.0455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,88,0.0909,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,88,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,88,0.4091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,88,0.4545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[60,88,0.6818,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[72,88,0.8182,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[80,88,0.9091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[84,88,0.9545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":4,"e":0.14286,"k":"falling","v":0.16518,"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],[4,30,0.1333,0.66964,0.42773,0.14286,1.0,1.0,0.0,1.0,2,20,0,2,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[8,30,0.2667,0.5892,0.43859,0.14286,1.0,1.0,0.0,1.0,2,17,0,2,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[12,30,0.4,0.56696,0.43373,0.14286,0.57143,1.0,0.0,1.0,1,16,0,1,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[16,30,0.5333,0.48214,0.42969,0.14286,0.14286,1.0,0.0,1.0,2,13,0,2,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[20,30,0.6667,0.40616,0.40112,0.14286,0.14286,1.0,0.0,1.0,1,10,0,1,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[24,30,0.8,0.3258,0.35761,0.14286,0.14286,0.14286,0.0,1.0,1,7,0,1,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[28,30,0.9333,0.21875,0.2525,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[30,30,1.0,0.16518,0.15198,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"c6cdc7d35bab5b4f","q":"Let  $\\left( A,+, \\cdot \\right)$  be a ring that verifies the following properties:\r\n(i) it has a unit,  $1$ , and its order is  $p$ , a prime number;\r\n(ii) there is  $B \\subset A, \\, |B| = p$ , such that: for all  $x,y \\in A$ , there is  $b \\in B$  such that  $xy = byx$ .\r\n\r\nProve that  $A$  is commutative.\r\n\r\n*Ion Savu*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.08473,"x":0.18081,"p":[[0,38,0.0,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.11598,0.17654,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,38,0.2105,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.10259,0.18291,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,38,0.4211,0.10045,0.06404,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,1,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.12947,0.16888,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,38,0.6316,0.18081,0.22159,0.14286,0.14286,0.14286,0.0,1.0,5,2,0,5,1,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,38,0.7368,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.08706,"x":0.13393,"p":[[0,20,0.0,0.08706,0.06855,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,1,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.13393,0.16728,0.10714,0.14286,0.14286,0.0,1.0,8,1,0,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,20,0.4,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"44871fc9ba7431c9","q":"Suppose that  $ a, b, c, d$  are real numbers satisfying  $ a \\geq b \\geq c \\geq d \\geq 0$ ,  $ a^2 \\plus{} d^2 \\equal{} 1$ ,  $ b^2 \\plus{} c^2 \\equal{} 1$ , and  $ ac \\plus{} bd \\equal{} 1/3$ . Find the value of  $ ab \\minus{} cd$ .","t":[{"b":0,"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":[[0,33,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[8,33,0.2424,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"28537c326a846aad","q":"Among the  $900$  residents of Aimeville, there are  $195$  who own a diamond ring,  $367$  who own a set of golf clubs, and  $562$  who own a garden spade. In addition, each of the  $900$  residents owns a bag of candy hearts. There are  $437$  residents who own exactly two of these things, and  $234$  residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.","t":[{"b":1,"e":0.28571,"k":"flat","v":0.92857,"x":0.98214,"p":[[0,55,0.0,0.97321,0.07524,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[4,55,0.0727,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,55,0.1455,0.92857,0.15568,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,1,0,25],[12,55,0.2182,0.92857,0.16752,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,25],[16,55,0.2909,0.92857,0.19233,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,26],[20,55,0.3636,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[24,55,0.4364,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[28,55,0.5091,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,55,0.5818,0.94196,0.21086,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[36,55,0.6545,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,55,0.7273,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[44,55,0.8,0.93304,0.18893,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[48,55,0.8727,0.95089,0.13651,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,28],[52,55,0.9455,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[55,55,1.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":2,"e":0.71429,"k":"flat","v":0.79909,"x":0.91518,"p":[[0,24,0.0,0.82143,0.31339,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,1,0,22],[4,24,0.1667,0.91518,0.15916,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,2,0,23],[8,24,0.3333,0.88839,0.22228,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,3,0,22],[12,24,0.5,0.875,0.22232,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,3,0,0,1,0,23],[16,24,0.6667,0.84375,0.28652,0.82143,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,3,0,21],[20,24,0.8333,0.83034,0.27766,0.71429,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,1,0,20],[24,24,1.0,0.79909,0.32706,0.67846,1.0,1.0,0.0,1.0,2,21,0,2,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,21]]}]},{"i":"06ff370b80ca2b57","q":"Prove: if  $2^{2^n-1}-1$  is a prime, then  $n$  is a prime.","t":[{"b":0,"e":1.0,"k":"flat","v":0.9375,"x":1.0,"p":[[0,38,0.0,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[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],[8,38,0.2105,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,38,0.3158,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[28,38,0.7368,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,38,0.8421,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,38,0.9474,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":1,"e":1.0,"k":"flat","v":0.94195,"x":1.0,"p":[[0,32,0.0,0.96428,0.08749,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,26],[4,32,0.125,0.94195,0.12811,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[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],[12,32,0.375,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[16,32,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[28,32,0.875,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,32,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"85a36b638dddbe21","q":"A computer is programmed to randomly generate a string of six symbols using only the letters  $A,B,C$ . What is the probability that the string will not contain three consecutive  $A$ 's?","t":[{"b":4,"e":1.0,"k":"flat","v":0.92857,"x":0.99554,"p":[[0,47,0.0,0.97321,0.10374,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,29],[4,47,0.0851,0.95982,0.1394,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,29],[8,47,0.1702,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[12,47,0.2553,0.95089,0.11633,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,25],[16,47,0.3404,0.92857,0.18211,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,3,0,26],[20,47,0.4255,0.93304,0.16746,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,26],[24,47,0.5106,0.93304,0.18205,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,27],[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],[36,47,0.766,0.96875,0.12745,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[40,47,0.8511,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[44,47,0.9362,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[47,47,1.0,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.93304,"x":0.99554,"p":[[0,18,0.0,0.95089,0.15815,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,28],[4,18,0.2222,0.93304,0.16746,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,26],[8,18,0.4444,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,18,0.6667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]}]},{"i":"868c69ec31d26abd","q":"Prove that whenever the player must choose the side of a coin, the optimal strategy is to choose heads if more coins have been determined tails than heads and to choose tails if more coins have been determined heads than tails.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,10,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,26,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.00893,0.0346,0.0,0.0,0.0,0.0,0.143,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"0330d175bb789ab2","q":"Which point on the circle  $ (x \\minus{} 11)^2 \\plus{} (y \\minus{} 13)^2 \\equal{} 116$  is farthest from the point  $ (41, 25)$ ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.99107,"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],[16,27,0.5926,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,27,0.7407,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,27,0.8889,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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":0.98661,"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],[4,37,0.1081,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[12,37,0.3243,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,37,0.4324,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,37,0.5405,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[37,37,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a556e688369a2ecb","q":"Determine the number of arrangements  $ a_1,a_2,...,a_{10}$  of the numbers  $ 1,2,...,10$  such that  $ a_i>a_{2i}$  for  $ 1 \\le i \\le 5$  and  $ a_i>a_{2i\\plus{}1}$  for  $ 1 \\le i \\le 4$ .","t":[{"b":1,"e":0.2857,"k":"falling","v":0.1384,"x":0.39286,"p":[[0,18,0.0,0.39286,0.26245,0.24999,0.28571,0.42857,0.14286,1.0,0,4,0,0,0,8,0,0,10,0,0,9,0,0,0,0,0,1,0,0,0,0,4],[4,18,0.2222,0.20527,0.11264,0.14286,0.14286,0.28571,0.0,0.4286,1,0,0,1,0,21,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.26786,0.25692,0.14286,0.14286,0.28571,0.0,1.0,4,2,0,4,0,15,0,0,6,0,0,3,0,0,1,0,0,0,0,0,1,0,2],[12,18,0.6667,0.32126,0.33321,0.14286,0.14286,0.28571,0.0,1.0,1,6,0,1,0,21,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[16,18,0.8889,0.33482,0.31665,0.14286,0.21428,0.32144,0.0,1.0,3,5,0,3,0,13,0,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,5],[18,18,1.0,0.1384,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.28571,"x":0.55357,"p":[[0,37,0.0,0.50447,0.37113,0.14286,0.35714,1.0,0.14286,1.0,0,10,0,0,0,12,0,0,4,0,0,4,0,0,0,0,0,1,0,0,1,0,10],[4,37,0.1081,0.38839,0.27718,0.14289,0.28571,0.42858,0.14286,1.0,0,4,0,0,0,9,0,0,12,0,0,4,0,0,2,0,0,0,0,0,1,0,4],[8,37,0.2162,0.28571,0.14286,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,11,0,0,9,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[12,37,0.3243,0.34822,0.24206,0.14289,0.28571,0.42857,0.0,1.0,1,3,0,1,0,8,0,0,12,0,0,7,0,0,1,0,0,0,0,0,0,0,3],[16,37,0.4324,0.38393,0.25111,0.14286,0.35714,0.46431,0.0,1.0,2,2,0,2,0,7,0,0,7,0,0,8,0,0,4,0,0,1,0,0,1,0,2],[20,37,0.5405,0.50893,0.30291,0.28571,0.42857,0.67857,0.0,1.0,1,8,0,1,0,2,0,0,8,0,0,12,0,0,1,0,0,0,0,0,0,0,8],[24,37,0.6486,0.41965,0.26471,0.28571,0.42857,0.46431,0.0,1.0,2,4,0,2,0,4,0,0,8,0,0,10,0,0,4,0,0,0,0,0,0,0,4],[28,37,0.7568,0.40178,0.20025,0.2857,0.42857,0.42858,0.0,1.0,1,2,0,1,0,2,0,0,11,0,0,12,0,0,4,0,0,0,0,0,0,0,2],[32,37,0.8649,0.52679,0.30813,0.28571,0.42857,0.78571,0.0,1.0,2,8,0,2,0,1,0,0,7,0,0,10,0,0,3,0,0,1,0,0,0,0,8],[36,37,0.973,0.49108,0.29,0.28571,0.42857,0.57143,0.14286,1.0,0,7,0,0,0,4,0,0,8,0,0,11,0,0,2,0,0,0,0,0,0,0,7],[37,37,1.0,0.55357,0.30462,0.28571,0.42857,1.0,0.0,1.0,1,9,0,1,0,1,0,0,7,0,0,11,0,0,2,0,0,1,0,0,0,0,9]]}]},{"i":"f2db84e4a1af1f5b","q":"Let  $n$  be a positive integer. For any  $k$ , denote by  $a_k$  the number of permutations of  $\\{1,2,\\dots,n\\}$  with exactly  $k$  disjoint cycles. (For example, if  $n=3$  then  $a_2=3$  since  $(1)(23)$ ,  $(2)(31)$ ,  $(3)(12)$  are the only such permutations.) Evaluate\n\\[ a_n n^n + a_{n-1} n^{n-1} + \\dots + a_1 n. \\]*Proposed by Sammy Luo*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,63,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,63,0.0635,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,63,0.127,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,63,0.1905,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,63,0.254,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,63,0.3175,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,63,0.381,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,63,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,63,0.5079,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,63,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,63,0.6349,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,63,0.6984,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,63,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,63,0.8254,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,63,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,63,0.9524,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[63,63,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"03a7c16d834be374","q":"The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6 \\leq P(x) \\leq 4$ or $6 \\leq P(x) \\leq 16$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,27,0.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,27,0.1481,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,27,0.2963,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,27,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":0.97321,"x":1.0,"p":[[0,12,0.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[4,12,0.3333,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,12,0.6667,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[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":"6a2beb335207bba1","q":"Two circles  $\\omega_1$  and  $\\omega_2$  with radii  $r_1$  and  $r_2$ ,  $r_2>r_1$ , are externally tangent. The  line  $t_1$  is tangent to the circles  $\\omega_1$  and  $\\omega_2$  at points  $A$  and  $D$  respectively. The parallel  line  $t_2$  to the line  $t_1$  is tangent to the circle  $\\omega_1$  and intersects the circle  $\\omega_2$  at points  $E$  and  $F$ . The line  $t_3$  passing through  $D$  intersects the line  $t_2$  and the circle  $\\omega_2$  in  $B$  and  $C$  respectively, both different of  $E$  and  $F$  respectively. Prove that the circumcircle of the triangle  $ABC$  is tangent to the line  $t_1$ .\r\n\r\n*Dinu Serbanescu*","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.06687,"p":[[0,16,0.0,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.06687,0.09431,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03562,"p":[[0,9,0.0,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4bf10f4faeeb0ae3","q":"Let  $ABCD$  be a cyclic quadrilateral satysfing  $AB=AD$  and  $AB+BC=CD$ . Determine  $\\measuredangle CDA$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.5,"p":[[0,28,0.0,0.5,0.45175,0.0,0.28571,1.0,0.0,1.0,10,14,0,10,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[4,28,0.1429,0.34821,0.42248,0.0,0.21428,1.0,0.0,1.0,15,9,0,15,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[8,28,0.2857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.36161,"x":1.0,"p":[[0,34,0.0,0.36161,0.41647,0.0,0.28571,1.0,0.0,1.0,14,9,0,14,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[4,34,0.1176,0.6875,0.38703,0.28571,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,19],[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],[12,34,0.3529,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[16,34,0.4706,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[20,34,0.5882,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,34,0.7059,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,34,0.8235,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[34,34,1.0,0.98214,0.09942,1.0,1.0,1.0,0.4286,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"ed755a0c36524706","q":"If $b$ is any element of $T$, prove that $b \\star b=b$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,79,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,79,0.0506,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,79,0.1013,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,79,0.1519,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,79,0.2025,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,79,0.2532,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,79,0.3038,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,79,0.3544,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,79,0.4051,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,79,0.4557,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,79,0.5063,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,79,0.557,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,79,0.6076,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,79,0.6582,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,79,0.7089,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,79,0.7595,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,79,0.8101,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,79,0.8608,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,79,0.9114,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,79,0.962,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"924b46dcc5e2a516","q":"On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.","t":[{"b":2,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,29,0.0,0.93304,0.15146,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,25],[4,29,0.1379,0.92411,0.14719,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,23],[8,29,0.2759,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[12,29,0.4138,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":4,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,6,0.0,0.89286,0.17128,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,21],[4,6,0.6667,0.92857,0.13363,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,7,0,22],[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":"ba4bb313c90c9719","q":"A quadratic polynomial  $p(x)$  has positive real coefficients with sum  $1$ . Show that given any positive real numbers with product  $1$ , the product of their values under  $p$  is at least  $1$ .","t":[{"b":5,"e":0.71429,"k":"volatile","v":0.58929,"x":0.76785,"p":[[0,4,0.0,0.58929,0.33455,0.57143,0.71429,0.71429,0.0,1.0,7,5,0,7,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,1,0,5],[4,4,1.0,0.76785,0.13716,0.71429,0.71429,0.85704,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,0,2,0,7]]},{"b":7,"e":0.71429,"k":"flat","v":0.71429,"x":0.78125,"p":[[0,6,0.0,0.71429,0.26,0.71429,0.71429,1.0,0.0,1.0,2,9,0,2,0,1,0,0,0,0,0,0,0,0,4,0,0,16,0,0,0,0,9],[4,6,0.6667,0.78125,0.15561,0.71429,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,17,0,0,0,0,10],[6,6,1.0,0.76786,0.15465,0.71429,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,17,0,0,0,0,9]]}]},{"i":"606eb1674cbf7a01","q":"In triangle  $ABC,$   $AB = 13,$   $BC = 14,$  and  $CA = 15.$  A circle of radius  $r$  passes through point  $A$  and is tangent to line  $BC$  at  $C.$  If  $r = m/n,$  where  $m$  and  $n$  are relatively prime positive integers, find  $100m + n.$ *Proposed by Michael Tang*","t":[{"b":0,"e":1.0,"k":"flat","v":0.91071,"x":0.99107,"p":[[0,39,0.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[4,39,0.1026,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[8,39,0.2051,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[12,39,0.3077,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,39,0.4103,0.91071,0.16269,0.82143,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,1,0,23],[20,39,0.5128,0.91964,0.12846,0.71429,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,23],[24,39,0.6154,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22],[28,39,0.7179,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[32,39,0.8205,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[36,39,0.9231,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[39,39,1.0,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25]]},{"b":4,"e":1.0,"k":"flat","v":0.91518,"x":1.0,"p":[[0,42,0.0,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[4,42,0.0952,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,42,0.1905,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[12,42,0.2857,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[16,42,0.381,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,42,0.4762,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25],[24,42,0.5714,0.92857,0.11845,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[28,42,0.6667,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[32,42,0.7619,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[36,42,0.8571,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[40,42,0.9524,0.93304,0.11837,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,24],[42,42,1.0,0.91518,0.12807,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,22]]}]},{"i":"0a39246c447acb63","q":"Let  $f$  be a function on defined on  $|x|<1$  such that   $f\\left (\\tfrac1{10}\\right )$  is rational and  $f(x)= \\sum_{i=1}^{\\infty} a_i x^i  $  where  $a_i\\in{\\{0,1,2,3,4,5,6,7,8,9\\}}$ . Prove that  $f$  can be written as  $f(x)= \\frac{p(x)}{q(x)}$   where  $p(x)$  and  $q(x)$  are polynomials with integer coefficients.","t":[{"b":0,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,23,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,23,0.1739,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,23,0.8696,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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":3,"e":1.0,"k":"flat","v":0.99107,"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],[8,27,0.2963,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,27,0.5926,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,27,0.7407,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"b7a6f9aea9141031","q":"Given a prime number  $p$  such that  $2p$  is equal to the sum of the squares of some four consecutive positive integers. Prove that  $p-7$  is divisible by 36.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.875,"x":0.95089,"p":[[0,37,0.0,0.92409,0.12873,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,2,0,23],[4,37,0.1081,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[8,37,0.2162,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[12,37,0.3243,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[16,37,0.4324,0.88839,0.14167,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,2,0,19],[20,37,0.5405,0.91964,0.13333,0.82143,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,23],[24,37,0.6486,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[28,37,0.7568,0.89731,0.15253,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,21],[32,37,0.8649,0.875,0.13243,0.71429,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,4,0,16],[36,37,0.973,0.90625,0.1411,0.71429,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,22],[37,37,1.0,0.87946,0.15198,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,1,0,19]]},{"b":5,"e":1.0,"k":"flat","v":0.90624,"x":0.99554,"p":[[0,57,0.0,0.92857,0.12877,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,1,0,24],[4,57,0.0702,0.93304,0.12869,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,25],[8,57,0.1404,0.93304,0.12364,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,24],[12,57,0.2105,0.92857,0.11845,0.85711,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,23],[16,57,0.2807,0.90624,0.13654,0.71429,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,21],[20,57,0.3509,0.91518,0.13767,0.71429,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,23],[24,57,0.4211,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[28,57,0.4912,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[32,57,0.5614,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[36,57,0.6316,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[40,57,0.7018,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[44,57,0.7719,0.94642,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[48,57,0.8421,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[52,57,0.9123,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,57,0.9825,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[57,57,1.0,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28]]}]},{"i":"5e66091d046e0ff6","q":"The roots of the polynomial  $x^4 - 4ix^3 +3x^2 -14ix - 44$  form the vertices of a parallelogram in the complex plane.  What is the area of the parallelogram?","t":[{"b":0,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,9,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,9,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":1,"e":0.85714,"k":"falling","v":0.57588,"x":0.99107,"p":[[0,38,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,38,0.1053,0.9375,0.16728,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,27],[8,38,0.2105,0.81696,0.23753,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,6,0,16],[12,38,0.3158,0.77232,0.24707,0.4286,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,5,0,0,4,0,14],[16,38,0.4211,0.70982,0.25376,0.42857,0.78564,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,6,0,0,1,0,0,5,0,0,8,0,8],[20,38,0.5263,0.6875,0.27301,0.42857,0.64286,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,3,0,0,10,0,0,3,0,0,1,0,0,4,0,11],[24,38,0.6316,0.72308,0.24207,0.42857,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,8,0,0,0,0,0,7,0,0,6,0,9],[28,38,0.7368,0.63393,0.27185,0.42857,0.4286,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,13,0,0,1,0,0,1,0,0,5,0,8],[32,38,0.8421,0.57588,0.21866,0.42857,0.42859,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,14,0,0,1,0,0,4,0,0,8,0,1],[36,38,0.9474,0.64732,0.25997,0.42857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,11,0,0,0,0,0,5,0,0,5,0,7],[38,38,1.0,0.66518,0.25155,0.42857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,5,0,0,7,0,0,1,0,0,5,0,0,9,0,5]]}]},{"i":"9b2a77e339cffdd2","q":"Let  $a$  and  $b$  be positive integers such that all but  $2009$  positive integers are expressible in the form  $ma + nb$ , where  $m$  and  $n$  are nonnegative integers. If  $1776 $ is one of the numbers that is not expressible, \ffind  $a + b$ .","t":[{"b":0,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,8,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,8,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":2,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,62,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,62,0.0645,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[8,62,0.129,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,62,0.1935,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,62,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,62,0.4516,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,62,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,62,0.5806,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[40,62,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[48,62,0.7742,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,62,0.8387,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[62,62,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"34e2017b241d1092","q":"Let  $a,b$  and  $c$  be the sides of a right angled triangle. Let  $\\theta$  be the smallest angle of this triangle. If   $\\frac{1}{a}, \\frac{1}{b}$  and  $\\frac{1}{c}$  are also the sides of a right angled triangle then show that  $\\sin\\theta=\\frac{\\sqrt{5}-1}{2}$","t":[{"b":0,"e":1.0,"k":"flat","v":0.94643,"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],[8,37,0.2162,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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],[16,37,0.4324,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,37,0.5405,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,37,0.8649,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,37,0.973,0.94643,0.15872,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,28],[37,37,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":1.0,"k":"flat","v":1.0,"x":1.0,"p":[[0,9,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,9,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"2c0df4a766e4ccd3","q":"$ABCD$  is a square with centre  $O$ . Two congruent isosceles triangle  $BCJ$  and  $CDK$  with base  $BC$  and  $CD$  respectively are constructed outside the square. let  $M$  be the midpoint of  $CJ$ . Show that  $OM$  and  $BK$  are perpendicular to each other.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.07143,"p":[[0,30,0.0,0.07143,0.24484,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,30,0.1333,0.05804,0.18851,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,30,0.5333,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,30,0.6667,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[24,30,0.8,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,30,0.9333,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[30,30,1.0,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]},{"b":7,"e":0.2857,"k":"rising","v":0.00893,"x":0.2366,"p":[[0,20,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.04463,0.1259,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.14286,0.14286,0.0,0.14285,0.28571,0.0,0.28571,16,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.15179,0.14258,0.0,0.28571,0.28571,0.0,0.28571,15,0,0,15,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.19642,0.13243,0.0,0.2857,0.28571,0.0,0.28571,10,0,0,10,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.2366,0.10479,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,1,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a66b6f2ed4636af0","q":"Let  $ABC$  be an acute triangle. Let $H$  and  $D$  be points on  $[AC]$  and  $[BC]$ , respectively, such that  $BH \\perp AC$  and  $HD \\perp BC$ . Let  $O_1$  be the circumcenter of  $\\triangle ABH$ , and  $O_2$  be the circumcenter of  $\\triangle BHD$ , and  $O_3$  be the circumcenter of  $\\triangle HDC$ . Find the ratio of area of  $\\triangle O_1O_2O_3$  and  $\\triangle ABH$ .","t":[{"b":1,"e":0.85714,"k":"falling","v":0.80804,"x":0.97768,"p":[[0,39,0.0,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[4,39,0.1026,0.92411,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,20],[8,39,0.2051,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[12,39,0.3077,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[16,39,0.4103,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[20,39,0.5128,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,39,0.6154,0.95535,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[28,39,0.7179,0.85714,0.12372,0.71429,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,8,0,12],[32,39,0.8205,0.90625,0.11071,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,17],[36,39,0.9231,0.875,0.1171,0.71429,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,10,0,13],[39,39,1.0,0.80804,0.09852,0.71429,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,13,0,4]]},{"b":5,"e":1.0,"k":"flat","v":0.89955,"x":0.99554,"p":[[0,56,0.0,0.89955,0.11691,0.83925,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,7,0,17],[4,56,0.0714,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[8,56,0.1429,0.90625,0.12169,0.82143,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,19],[12,56,0.2143,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[16,56,0.2857,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[20,56,0.3571,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[24,56,0.4286,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[28,56,0.5,0.9375,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,23],[32,56,0.5714,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,56,0.6429,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[40,56,0.7143,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[44,56,0.7857,0.95536,0.09062,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,25],[48,56,0.8571,0.95089,0.09852,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[52,56,0.9286,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[56,56,1.0,0.94642,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23]]}]},{"i":"84b2b4041291a090","q":"For every subset  $T$  of  $U = \\{ 1,2,3,\\ldots,18 \\}$ , let  $s(T)$  be the sum of the elements of  $T$ , with  $s(\\emptyset)$  defined to be  $0$ . If  $T$  is chosen at random among all subsets of  $U$ , the probability that  $s(T)$  is divisible by  $3$  is  $\\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m$ .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.92411,"x":0.98214,"p":[[0,52,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,52,0.0769,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[8,52,0.1538,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[12,52,0.2308,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[16,52,0.3077,0.92411,0.12364,0.82143,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,1,0,23],[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],[24,52,0.4615,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[28,52,0.5385,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[32,52,0.6154,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[36,52,0.6923,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[40,52,0.7692,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[44,52,0.8462,0.92857,0.12372,0.92857,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,24],[48,52,0.9231,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[52,52,1.0,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25]]},{"b":5,"e":0.71429,"k":"flat","v":0.90179,"x":0.98214,"p":[[0,50,0.0,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[4,50,0.08,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[8,50,0.16,0.9375,0.11811,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,25],[12,50,0.24,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[16,50,0.32,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,50,0.4,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[24,50,0.48,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,50,0.56,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[32,50,0.64,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[36,50,0.72,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,50,0.8,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[44,50,0.88,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26],[48,50,0.96,0.90179,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,21],[50,50,1.0,0.90179,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,21]]}]},{"i":"da6bb0298d0d4293","q":"Let  $x$  and  $y$  be positive real numbers where  $x \\neq y$ , such that  $\\frac{1-x^2}{x^3}=\\frac{1-y^2}{y^3}$ . Prove that  $xy > 3$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.92409,"x":0.99107,"p":[[0,25,0.0,0.92409,0.15969,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,3,0,24],[4,25,0.16,0.9375,0.13333,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,3,0,25],[8,25,0.32,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[12,25,0.48,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[16,25,0.64,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[20,25,0.8,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,25,0.96,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[25,25,1.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27]]},{"b":6,"e":1.0,"k":"flat","v":0.88392,"x":1.0,"p":[[0,32,0.0,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[4,32,0.125,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,32,0.25,0.88392,0.16148,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,2,0,20],[12,32,0.375,0.92411,0.13356,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,24],[16,32,0.5,0.89731,0.15666,0.82143,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,3,0,21],[20,32,0.625,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[24,32,0.75,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[28,32,0.875,0.96875,0.09932,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,29],[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":"16c5dc9d39069c23","q":"Alice, Bob, Cindy, David, and Emily sit in a circle. Alice refuses to sit to the right of Bob, and Emily sits next to Cindy. If David sits next to two girls, determine who could sit immediately to the right of Alice.","t":[{"b":0,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,32,0.0,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[4,32,0.125,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,32,0.25,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,32,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,32,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.97767,"x":1.0,"p":[[0,13,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,13,0.3077,0.97767,0.08834,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[8,13,0.6154,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[12,13,0.9231,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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":"b7d7271014137771","q":"The graphs of  $y=3(x-h)^2+j$  and  $y=2(x-h)^2+k$  have  $y$ -intercepts of  $2013$  and  $2014$ , respectively, and each graph has two positive integer  $x$ -intercepts. Find  $h$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"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],[4,69,0.058,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,69,0.1159,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,69,0.1739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,69,0.2319,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,69,0.2899,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,69,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,69,0.4058,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[36,69,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,69,0.5797,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[64,69,0.9275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":7,"e":1.0,"k":"flat","v":0.99553,"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],[8,56,0.1429,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,56,0.2143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,56,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"be76a27a8461a838","q":"From three boys and three girls, every boy knows exactly two girls and every girl knows exactly two boys. Prove that we can arrange boys and girls in pairs such that in every pair people know each other","t":[{"b":2,"e":1.0,"k":"flat","v":0.96429,"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],[4,29,0.1379,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[8,29,0.2759,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,29,0.4138,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[16,29,0.5517,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":7,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,45,0.0,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[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],[8,45,0.1778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,45,0.5333,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,45,0.6222,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,45,0.7111,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[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],[40,45,0.8889,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[44,45,0.9778,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[45,45,1.0,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30]]}]},{"i":"38e77757244b9ad5","q":"Let  $a$  and  $ b$  be integers and  $n$  a positive integer. Prove that\n\\[\\frac{b^{n-1}a(a + b)(a + 2b) \\cdots (a + (n - 1)b)}{n!}\\]\nis an integer.","t":[{"b":3,"e":1.0,"k":"flat","v":0.89286,"x":0.95536,"p":[[0,12,0.0,0.89286,0.17128,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,7,0,0,2,0,21],[4,12,0.3333,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[8,12,0.6667,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[12,12,1.0,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27]]},{"b":4,"e":0.71429,"k":"flat","v":0.94196,"x":0.98214,"p":[[0,35,0.0,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[4,35,0.1143,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[8,35,0.2286,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[12,35,0.3429,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[16,35,0.4571,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,35,0.5714,0.96429,0.09449,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[24,35,0.6857,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[28,35,0.8,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26],[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],[35,35,1.0,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25]]}]},{"i":"5b2efe9175bf7143","q":"Let  $ p\\equal{}4k\\plus{}1$  be a prime. Prove that  $ p$  has at least  $ \\frac{\\phi(p\\minus{}1)}2$  primitive roots.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,4,0.0,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.28571,"k":"flat","v":0.04455,"x":0.21866,"p":[[0,13,0.0,0.11607,0.1357,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,2,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.09375,0.13175,0.0,0.0,0.28571,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.04455,0.09731,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.21866,0.11842,0.24928,0.28571,0.28571,0.0,0.28571,7,0,0,7,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.08929,0.13245,0.0,0.0,0.28571,0.0,0.286,22,0,0,22,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"052eaaf847fc8074","q":"Consider a  $n \\times n$  grid divided into  $n ^ 2$  squares of  $1 \\times 1$ . Each of the  $(n + 1) ^ 2 $  vertices of the grid is colored red or blue. Find the number of coloring such that each unit square has two red and two blue vertices.","t":[{"b":5,"e":0.4286,"k":"flat","v":0.44647,"x":0.55356,"p":[[0,17,0.0,0.49098,0.2525,0.28571,0.50001,0.60714,0.0,1.0,2,1,0,2,0,3,0,0,4,0,0,7,0,0,8,0,0,3,0,0,4,0,1],[4,17,0.2353,0.50446,0.21424,0.39286,0.42857,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,6,0,0,9,0,0,6,0,0,5,0,0,4,0,0],[8,17,0.4706,0.55356,0.21354,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,3,0,0,2,0,0,9,0,0,4,0,0,11,0,0,2,0,1],[12,17,0.7059,0.48661,0.17445,0.42857,0.42859,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,10,0,0,10,0,0,3,0,0,2,0,0],[16,17,0.9412,0.44647,0.15464,0.42857,0.42857,0.46536,0.0,0.71429,1,0,0,1,0,0,0,0,6,0,0,17,0,0,3,0,0,5,0,0,0,0,0],[17,17,1.0,0.51338,0.13767,0.42857,0.57121,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,12,0,0,11,0,0,6,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.56695,"x":0.75446,"p":[[0,14,0.0,0.5759,0.21274,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,10,0,0,5,0,0,9,0,0,1,0,3],[4,14,0.2857,0.58036,0.25238,0.42857,0.57143,0.75,0.0,1.0,1,3,0,1,0,1,0,0,3,0,0,10,0,0,3,0,0,6,0,0,5,0,3],[8,14,0.5714,0.56695,0.24086,0.42857,0.57121,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,2,0,0,10,0,0,4,0,0,7,0,0,3,0,3],[12,14,0.8571,0.75446,0.18638,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,8,0,0,4,0,9],[14,14,1.0,0.74554,0.24932,0.57143,0.85714,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,2,0,0,4,0,0,2,0,0,6,0,0,7,0,10]]}]},{"i":"f7ca11e4a42d99ff","q":"Prove that each finite set of integers can be arranged without intersection.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,149,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,149,0.0268,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,149,0.0537,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,149,0.0805,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,149,0.1074,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,149,0.1342,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,149,0.1611,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,149,0.1879,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,149,0.2148,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,149,0.2416,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,149,0.2685,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,149,0.2953,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,149,0.3221,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,149,0.349,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,149,0.3758,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,149,0.4027,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,149,0.4295,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,149,0.4564,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,149,0.4832,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,149,0.5101,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,149,0.5369,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,149,0.5638,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,149,0.5906,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,149,0.6174,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,149,0.6443,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,149,0.6711,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,149,0.698,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,149,0.7248,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,149,0.7517,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,149,0.7785,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,149,0.8054,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,149,0.8322,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[128,149,0.8591,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[132,149,0.8859,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[136,149,0.9128,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[140,149,0.9396,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[144,149,0.9664,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[148,149,0.9933,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[149,149,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a6cf3723d45bc29b","q":"Santa Claus has at least  $n$  gifts for  $n$  children. For  $i\\in\\{1,2, ... , n\\}$ , the  $i$ -th child considers  $x_i > 0$  of these items to be desirable. Assume that\n\\[\\dfrac{1}{x_1}+\\cdots+\\dfrac{1}{x_n}\\le1.\\]\nProve that Santa Claus can give each child a gift that this child likes.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.96875,"x":1.0,"p":[[0,5,0.0,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[4,5,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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":0.95982,"x":1.0,"p":[[0,38,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,38,0.1053,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,38,0.2105,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,38,0.3158,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[20,38,0.5263,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,38,0.6316,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[28,38,0.7368,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,38,0.8421,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[36,38,0.9474,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[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":"39826b788e149e04","q":"Show that every consistent 2-configuration of order 4 on a finite set $A$ has a subset that is a consistent 2-configuration of order 2.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0534,"p":[[0,93,0.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,93,0.043,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,93,0.086,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,93,0.129,0.03562,0.0713,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,93,0.172,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,93,0.2151,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,93,0.2581,0.0534,0.06894,0.0,0.0,0.14286,0.0,0.143,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,93,0.3011,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,93,0.3441,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,93,0.3871,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,93,0.4301,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,93,0.4731,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,93,0.5161,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,93,0.5591,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,93,0.6022,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,93,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,93,0.6882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,93,0.7312,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,93,0.7742,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,93,0.8172,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,93,0.8602,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,93,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,93,0.9462,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,93,0.9892,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[93,93,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,3,0.0,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9ca4c81c5e3d3307","q":"At a party, every guest is a friend of exactly fourteen other guests (not including him or her). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party? Please explain your answer with proof.\n\n*Proposed by Alexander Slavik, Czech Republic*","t":[{"b":0,"e":0.85714,"k":"flat","v":0.77679,"x":0.88839,"p":[[0,11,0.0,0.80804,0.16213,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,0,8,0,10],[4,11,0.3636,0.88839,0.15458,0.82143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,5,0,19],[8,11,0.7273,0.82589,0.22227,0.82143,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,11,0,13],[11,11,1.0,0.77679,0.14698,0.67857,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,17,0,3]]},{"b":2,"e":0.85714,"k":"flat","v":0.84375,"x":0.94642,"p":[[0,8,0.0,0.84375,0.16115,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,6,0,14],[4,8,0.5,0.94642,0.09943,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,7,0,23],[8,8,1.0,0.9375,0.10062,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21]]}]},{"i":"62dc9417e9eeda42","q":"Let  $a_0 = 2$ ,  $a_1 = 5$ , and  $a_2 = 8$ , and for  $n>2$  define  $a_n$  recursively to be the remainder when  $4(a_{n-1} + a_{n-2} + a_{n-3})$  is divided by  $11$ .  Find  $a_{2018}\\cdot a_{2020}\\cdot a_{2022}$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.91964,"x":0.99107,"p":[[0,10,0.0,0.94643,0.18814,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,28],[4,10,0.4,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[8,10,0.8,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[10,10,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.89732,"x":0.99554,"p":[[0,53,0.0,0.91071,0.20124,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,23],[4,53,0.0755,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,53,0.1509,0.90179,0.24598,0.96429,1.0,1.0,0.0,1.0,2,24,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,24],[12,53,0.2264,0.94196,0.18161,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[16,53,0.3019,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,53,0.3774,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[24,53,0.4528,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,53,0.5283,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[32,53,0.6038,0.95536,0.10374,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[36,53,0.6792,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,53,0.7547,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[44,53,0.8302,0.91071,0.12753,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,21],[48,53,0.9057,0.89732,0.12993,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,19],[52,53,0.9811,0.91518,0.12299,0.82143,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,21],[53,53,1.0,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25]]}]},{"i":"848320939ad6c48d","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":1,"e":0.0,"k":"flat","v":0.17402,"x":0.31249,"p":[[0,28,0.0,0.31249,0.17654,0.2857,0.28571,0.28571,0.0,1.0,2,1,0,2,0,0,0,0,27,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[4,28,0.1429,0.27232,0.07457,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.29464,0.13803,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,28,0.4286,0.27678,0.04971,0.2857,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.27232,0.07457,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.24991,0.11302,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,2,0,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,28,0.8571,0.25892,0.09061,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,1,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.17402,0.12747,0.0,0.1429,0.2857,0.0,0.42857,9,0,0,9,0,8,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.25446,"x":0.34822,"p":[[0,58,0.0,0.30357,0.12753,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,58,0.069,0.28125,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,58,0.1379,0.29464,0.07936,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,58,0.2069,0.32589,0.18638,0.28571,0.28571,0.28571,0.0,1.0,1,2,0,1,0,1,0,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[16,58,0.2759,0.34822,0.17835,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,27,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[20,58,0.3448,0.32589,0.18293,0.2857,0.28571,0.28571,0.0,1.0,1,2,0,1,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[24,58,0.4138,0.25446,0.11143,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,1,0,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,58,0.4828,0.31695,0.14608,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,0,0,0,27,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[32,58,0.5517,0.29018,0.13113,0.2857,0.28571,0.28571,0.0,0.857,2,0,0,2,0,1,0,0,26,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[36,58,0.6207,0.29017,0.08364,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,58,0.6897,0.29018,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.33927,0.17766,0.28571,0.28571,0.28571,0.2857,1.0,0,2,0,0,0,0,0,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[48,58,0.8276,0.2991,0.13534,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,58,0.8966,0.27231,0.05486,0.2857,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.30357,0.12753,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[58,58,1.0,0.27678,0.04971,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"78b6b53fd5e71ffb","q":"Quadrilateral  $ABCD$  is a cyclic,  $AB = AD$ . Points  $M$  and  $N$  are chosen on sides  $BC$  and  $CD$  respectfully so that  $\\angle MAN =1/2 (\\angle BAD)$ . Prove that  $MN = BM + ND$ .\n\n*(5 points)*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.10268,"p":[[0,55,0.0,0.10268,0.12993,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.0625,0.12846,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,55,0.1455,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.03125,0.09933,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.0133,0.05465,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.10268,"p":[[0,96,0.0,0.10268,0.11425,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,12,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,96,0.0417,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,96,0.0833,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,96,0.125,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,96,0.1667,0.02678,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,96,0.2083,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,96,0.25,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,96,0.2917,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,96,0.3333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,96,0.375,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,96,0.4167,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,96,0.4583,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,96,0.5,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,96,0.5417,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,96,0.5833,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,96,0.625,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,96,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,96,0.7083,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,96,0.75,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,96,0.7917,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,96,0.8333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,96,0.875,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,96,0.9167,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,96,0.9583,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,96,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"34ff182b90e2b0a6","q":"Consider a $2018 \\times 2018$ table where each cell contains a non-zero natural number. No\u00e9mie modifies these numbers at her discretion, applying the following operations:\n$\\triangleright$ choose a row and then multiply by 2 all the integers in this row;\n$\\triangleright$ choose a column and then subtract 1 from all the integers in this column.\nShow that, by applying these operations, No\u00e9mie can manage to obtain a table where each cell contains the integer 0.","t":[{"b":1,"e":0.0,"k":"rising","v":0.26337,"x":0.68303,"p":[[0,57,0.0,0.26337,0.30534,0.0,0.14286,0.4642,0.0,1.0,15,2,2,15,0,3,0,0,1,0,0,5,0,0,5,0,0,1,0,0,0,0,2],[4,57,0.0702,0.47765,0.31054,0.28571,0.42857,0.71429,0.0,1.0,4,4,0,4,0,3,0,0,6,0,0,4,0,0,4,0,0,6,0,0,1,0,4],[8,57,0.1404,0.5714,0.30093,0.28571,0.57121,0.85714,0.0,1.0,1,6,0,1,0,3,0,0,6,0,0,4,0,0,4,0,0,5,0,0,3,0,6],[12,57,0.2105,0.43302,0.28455,0.2857,0.35714,0.71429,0.0,1.0,3,2,0,3,0,4,0,0,9,0,0,5,0,0,1,0,0,6,0,0,2,0,2],[16,57,0.2807,0.4732,0.28668,0.28571,0.4286,0.71429,0.0,1.0,5,2,0,5,0,2,0,0,2,0,0,8,0,0,4,0,0,8,0,0,1,0,2],[20,57,0.3509,0.51784,0.31288,0.2857,0.57143,0.71429,0.0,1.0,3,4,0,3,0,4,0,0,5,0,0,1,0,0,7,0,0,5,0,0,3,0,4],[24,57,0.4211,0.54016,0.30037,0.28571,0.42857,0.75,0.14286,1.0,0,7,0,0,0,4,0,0,8,0,0,5,0,0,4,0,0,3,0,0,1,0,7],[28,57,0.4912,0.55355,0.23076,0.42857,0.57143,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,6,0,0,5,0,0,11,0,0,1,0,2],[32,57,0.5614,0.68303,0.34206,0.42859,0.71429,1.0,0.0,1.0,3,13,0,3,0,2,0,0,1,0,0,3,0,0,3,0,0,5,0,0,2,0,13],[36,57,0.6316,0.56693,0.33595,0.39286,0.49979,0.89286,0.0,1.0,4,8,0,4,0,1,0,0,3,0,0,8,0,0,2,0,0,4,0,0,2,0,8],[40,57,0.7018,0.48212,0.28515,0.28571,0.4286,0.71429,0.0,1.0,5,2,0,5,0,0,0,0,4,0,0,9,0,0,4,0,0,5,0,0,3,0,2],[44,57,0.7719,0.53122,0.27487,0.28571,0.4998,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,7,0,0,6,0,0,6,0,0,4,0,0,1,0,5],[48,57,0.8421,0.55803,0.30379,0.2857,0.64286,0.85704,0.0,1.0,1,4,0,1,0,5,0,0,4,0,0,5,0,0,1,0,0,7,0,0,5,0,4],[52,57,0.9123,0.51771,0.22505,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,10,0,0,6,0,0,3,0,0,8,0,0,3,0,1],[56,57,0.9825,0.4999,0.2203,0.28571,0.5712,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,6,0,0,6,0,0,8,0,0,7,0,0,1,0,1],[57,57,1.0,0.5,0.25754,0.28571,0.50001,0.71429,0.0,1.0,2,2,0,2,0,3,0,0,4,0,0,7,0,0,5,0,0,8,0,0,1,0,2]]},{"b":3,"e":0.42857,"k":"rising","v":0.22767,"x":0.69641,"p":[[0,34,0.0,0.22767,0.25718,0.0,0.07143,0.42857,0.0,1.0,16,1,2,16,0,1,0,0,1,0,0,11,0,0,2,0,0,0,0,0,0,0,1],[4,34,0.1176,0.69641,0.29397,0.42857,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,5,0,0,4,0,0,2,0,0,5,0,0,4,0,11],[8,34,0.2353,0.52225,0.29365,0.28571,0.571,0.71429,0.0,1.0,3,3,0,3,0,3,0,0,4,0,0,2,0,0,10,0,0,3,0,0,4,0,3],[12,34,0.3529,0.44186,0.21249,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,10,0,0,8,0,0,4,0,0,6,0,0,0,0,1],[16,34,0.4706,0.41954,0.30717,0.14286,0.42857,0.60714,0.0,1.0,6,3,0,6,0,3,0,0,5,0,0,6,0,0,4,0,0,4,0,0,1,0,3],[20,34,0.5882,0.53125,0.23483,0.28571,0.42857,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,7,0,0,1,0,0,8,0,0,3,0,2],[24,34,0.7059,0.47319,0.22989,0.28571,0.42857,0.60714,0.14286,1.0,0,2,0,0,0,4,0,0,6,0,0,11,0,0,3,0,0,5,0,0,1,0,2],[28,34,0.8235,0.48217,0.24934,0.42857,0.42857,0.71429,0.0,1.0,3,2,0,3,0,1,0,0,3,0,0,13,0,0,3,0,0,6,0,0,1,0,2],[32,34,0.9412,0.45304,0.21037,0.28571,0.42857,0.66071,0.0,0.85714,1,0,0,1,0,4,0,0,4,0,0,12,0,0,2,1,0,7,0,0,1,0,0],[34,34,1.0,0.39286,0.18558,0.2857,0.42857,0.42857,0.0,0.85714,2,0,0,2,0,4,0,0,4,0,0,15,0,0,5,0,0,1,0,0,1,0,0]]}]},{"i":"280ff088aab744bf","q":"Let  $O$  be a point in the plane of the triangle  $ABC$ . A circle  $\\mathcal{C}$  which passes through  $O$  intersects the second time the lines  $OA,OB,OC$  in  $P,Q,R$  respectively. The circle  $\\mathcal{C}$  also intersects for the second time the circumcircles of the triangles  $BOC$ ,  $COA$  and  $AOB$  respectively in  $K,L,M$ .\r\n\r\nProve that the lines  $PK,QL$  and  $RM$  are concurrent.","t":[{"b":3,"e":0.57143,"k":"rising","v":0.35713,"x":0.64286,"p":[[0,41,0.0,0.35713,0.11292,0.2857,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,21,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[4,41,0.0976,0.60268,0.24932,0.53572,0.71429,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,4,0,0,1,0,0,2,0,0,19,0,0,1,0,2],[8,41,0.1951,0.61161,0.22654,0.53572,0.71429,0.71429,0.0,1.0,1,2,0,1,0,2,0,0,2,0,0,3,0,0,2,0,0,20,0,0,0,0,2],[12,41,0.2927,0.55797,0.22967,0.42859,0.64271,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,5,0,0,6,0,0,3,0,0,14,0,0,0,0,2],[16,41,0.3902,0.58477,0.17261,0.4286,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,2,0,0,6,0,0,6,0,0,17,0,0,0,0,0],[20,41,0.4878,0.57587,0.25376,0.28571,0.71429,0.71429,0.0,1.0,1,3,0,1,0,1,0,0,8,0,0,1,0,0,2,0,0,16,0,0,0,0,3],[24,41,0.5854,0.63838,0.22012,0.57132,0.71429,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,4,0,0,2,0,0,3,0,0,18,0,0,1,0,3],[28,41,0.6829,0.58914,0.20431,0.42857,0.71214,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,4,0,0,5,0,0,15,0,0,0,0,2],[32,41,0.7805,0.64271,0.17853,0.57132,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,3,0,0,2,0,0,21,0,0,0,0,2],[36,41,0.878,0.57587,0.17307,0.4286,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,5,0,0,3,0,0,18,0,0,0,0,0],[40,41,0.9756,0.64286,0.14725,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,4,0,0,1,0,0,25,0,0,0,0,0],[41,41,1.0,0.63823,0.14273,0.67525,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,24,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.35713,"x":0.92857,"p":[[0,47,0.0,0.35713,0.14724,0.2857,0.28571,0.32143,0.2857,0.85714,0,0,0,0,0,0,0,0,24,0,0,4,0,0,1,0,0,2,0,0,1,0,0],[4,47,0.0851,0.58482,0.24836,0.42857,0.71429,0.71429,0.0,1.0,2,2,0,2,0,1,0,0,4,0,0,3,0,0,1,0,0,19,0,0,0,0,2],[8,47,0.1702,0.58913,0.18463,0.5354,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,4,0,0,3,0,0,5,0,0,19,0,0,0,0,0],[12,47,0.2553,0.61606,0.20025,0.49968,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,1,0,0,21,0,0,1,0,1],[16,47,0.3404,0.51786,0.23623,0.28571,0.71429,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,10,0,0,2,0,0,0,0,0,18,0,0,0,0,0],[20,47,0.4255,0.58929,0.21943,0.39286,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,7,0,0,2,0,0,0,0,0,21,0,0,0,0,1],[24,47,0.5106,0.549,0.22914,0.28571,0.64286,0.71429,0.0,1.0,1,1,0,1,0,1,0,0,7,0,0,3,0,0,4,0,0,14,0,0,1,0,1],[28,47,0.5957,0.57142,0.18211,0.39286,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,2,0,0,4,0,0,18,0,0,0,0,0],[32,47,0.6809,0.58482,0.2,0.39286,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,3,0,0,1,0,0,19,0,0,0,0,1],[36,47,0.766,0.52679,0.24338,0.28571,0.71429,0.71429,0.0,0.71429,3,0,0,3,0,0,0,0,7,0,0,2,0,0,2,0,0,18,0,0,0,0,0],[40,47,0.8511,0.54456,0.16916,0.39286,0.57121,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,2,0,0,10,0,0,12,0,0,0,0,0],[44,47,0.9362,0.55354,0.18472,0.39286,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,5,0,0,2,0,0,17,0,0,0,0,0],[47,47,1.0,0.92857,0.15152,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,24]]}]},{"i":"c5d202e140a047e9","q":"Compute the number of ordered quadruples of positive integers  $(a,b,c,d)$  such that   $$ a!\\cdot b!\\cdot c!\\cdot d!=24! $$  [list=1]\n[*] 4\n[*] 4!\n[*]  $4^4$ [*] None of these\n[/list]","t":[{"b":2,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,74,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,74,0.0541,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[12,74,0.1622,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,74,0.2162,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,74,0.2703,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,74,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,74,0.3784,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,74,0.4324,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,74,0.4865,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[44,74,0.5946,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,74,0.6486,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[52,74,0.7027,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,74,0.7568,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[60,74,0.8108,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,74,0.8649,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,74,0.9189,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[72,74,0.973,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[74,74,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":7,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,35,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,35,0.1143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,35,0.2286,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,35,0.3429,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,35,0.4571,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,35,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,35,0.9143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"9e76f1fe18709bf9","q":"Let  $n>2$  be an integer. A deck contains  $\\frac{n(n-1)}{2}$  cards,numbered \\[1,2,3,\\cdots , \\frac{n(n-1)}{2}\\] Two cards form a *magic pair* if their numbers are consecutive , or if their numbers are  $1$  and  $\\frac{n(n+1)}{2}$ . For which  $n$  is it possible to distribute the cards into  $n$  stacks in such a manner that, among the cards in any two stacks , there is exactly one *magic pair*?","t":[{"b":2,"e":0.85714,"k":"falling","v":0.63836,"x":0.89732,"p":[[0,28,0.0,0.83928,0.23891,0.82132,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,2,0,0,6,0,18],[4,28,0.1429,0.89731,0.20279,0.85714,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,23],[8,28,0.2857,0.85267,0.24088,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,1,0,21],[12,28,0.4286,0.89732,0.212,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,22],[16,28,0.5714,0.80801,0.19437,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,0,4,0,14],[20,28,0.7143,0.71427,0.26487,0.5713,0.71429,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,5,0,0,5,0,0,6,0,0,4,0,10],[24,28,0.8571,0.80347,0.24961,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,0,0,0,1,0,0,3,0,0,6,0,0,6,0,14],[28,28,1.0,0.63836,0.34807,0.28571,0.78564,1.0,0.0,1.0,3,9,0,3,0,2,0,0,4,0,0,3,0,0,1,0,0,3,0,0,7,0,9]]},{"b":5,"e":0.57143,"k":"falling","v":0.63838,"x":0.93748,"p":[[0,37,0.0,0.93748,0.10682,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,22],[4,37,0.1081,0.63838,0.37625,0.42859,0.71429,1.0,0.0,1.0,7,10,0,7,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,5,0,10],[8,37,0.2162,0.74105,0.29546,0.57132,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,3,0,0,2,0,0,4,0,0,2,0,0,6,0,13],[12,37,0.3243,0.78125,0.3273,0.71429,0.92857,1.0,0.0,1.0,4,16,0,4,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,6,0,16],[16,37,0.4324,0.84375,0.21829,0.71429,1.0,1.0,0.1429,1.0,0,18,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,6,0,0,3,0,18],[20,37,0.5405,0.83482,0.15612,0.71429,0.85714,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,7,0,12],[24,37,0.6486,0.78122,0.19559,0.57143,0.78571,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,7,0,0,5,0,11],[28,37,0.7568,0.80802,0.12684,0.71429,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,14,0,5],[32,37,0.8649,0.71874,0.1262,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,16,0,0,8,0,1],[36,37,0.973,0.73657,0.14339,0.71429,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,16,0,0,9,0,2],[37,37,1.0,0.72764,0.14447,0.71429,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,13,0,0,11,0,1]]}]},{"i":"1f4cc521b2fcce3e","q":"Let  $ n$  be a positive integer, let  $ A$  be a subset of  $ \\{1, 2, \\cdots, n\\}$ , satisfying for any two numbers  $ x, y\\in A$ , the least common multiple of  $ x$ ,  $ y$  not more than  $ n$ . Show that  $ |A|\\leq 1.9\\sqrt {n} \\plus{} 5$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.14284,"p":[[0,38,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.09375,0.14555,0.0,0.0,0.14287,0.0,0.4286,20,0,0,20,0,7,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.06688,0.12359,0.0,0.0,0.14071,0.0,0.4286,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.14284,0.17493,0.0,0.0,0.28571,0.0,0.571,17,0,0,17,0,4,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,38,0.5263,0.10713,0.17125,0.0,0.0,0.14287,0.0,0.571,21,0,0,21,0,4,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,38,0.6316,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[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],[32,38,0.8421,0.05804,0.11769,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,38,0.9474,0.05357,0.10565,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.0401,0.07338,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.01339,"x":0.0625,"p":[[0,40,0.0,0.0267,0.08316,0.0,0.0,0.0,0.0,0.42857,28,0,1,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.05357,0.17768,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,40,0.6,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.05804,0.11214,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,40,0.8,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.04911,0.09852,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"92888aeb8d3e04d5","q":"Let $ABCD$ be a convex quadrilateral such that $AC = BD$ and such that the sides $AB$ and $CD$ are not parallel. Let $P$ be the intersection point of the diagonals $(AC)$ and $(BD)$. Let $E$ and $F$ be points on the segments $[BP]$ and $[AP]$ respectively such that $\\mathrm{PC} = \\mathrm{PE}$ and $\\mathrm{PD} = \\mathrm{PF}$. Show that the circumcircle of the triangle formed by the lines $(AB)$, $(CD)$, and $(EF)$ is tangent to the circumcircle of the triangle $ABP$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.11161,"p":[[0,66,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,66,0.1212,0.0625,0.15126,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,66,0.1818,0.0134,0.04165,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,66,0.2424,0.07143,0.14726,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,66,0.303,0.04018,0.13474,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,66,0.3636,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,66,0.4242,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,66,0.4848,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,66,0.5455,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,66,0.6061,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,66,0.6667,0.03571,0.11294,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,66,0.7273,0.10705,0.16364,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,7,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[52,66,0.7879,0.03571,0.10101,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,66,0.8485,0.05354,0.10575,0.0,0.0,0.035,0.0,0.43,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,66,0.9091,0.11161,0.1665,0.0,0.0,0.14287,0.0,0.57143,19,0,0,19,0,6,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[64,66,0.9697,0.03562,0.09439,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,34,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,34,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2ed9eeaddb8f4514","q":"Let  $ABC$  be an acute triangle with incenter  $I$  and circumcircle  $\\Omega$ . The line passing  $I$  and perpendicular to  $AI$  meets  $AB, AC$  at  $D, E$ , respectively.  $A$ -excircle of  $\\triangle{ABC}$  meets  $BC$  at  $T$ .  $AT$  meets  $\\Omega$  at  $P$ . The line passing  $P$  and parallel to  $BC$  meets  $\\Omega$  at  $Q$ . The intersection of  $QI$  and  $AT$  is  $K$ . Prove that  $Q,D,K,E$  are concyclic.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.03116,"p":[[0,94,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,94,0.0426,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,94,0.0851,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,94,0.1277,0.0267,0.06606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,94,0.1702,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,94,0.2128,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,94,0.2553,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,94,0.2979,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,94,0.3404,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,94,0.383,0.03116,0.07758,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,94,0.4255,0.01321,0.04109,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,94,0.4681,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,94,0.5106,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,94,0.5532,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,94,0.5957,0.02233,0.06299,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,94,0.6383,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,94,0.6809,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,94,0.7234,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,94,0.766,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,94,0.8085,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,94,0.8511,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,94,0.8936,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,94,0.9362,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,94,0.9787,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[94,94,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04009,"p":[[0,75,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,75,0.1067,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,75,0.16,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,75,0.2133,0.04009,0.0816,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,75,0.2667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,75,0.32,0.03116,0.06887,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,75,0.3733,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,75,0.4267,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,75,0.48,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,75,0.5333,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,75,0.64,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,75,0.6933,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,75,0.7467,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,75,0.8,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,75,0.8533,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[75,75,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"74ff5fddab7b6546","q":"Adalbert and Babette are playing dominoes on a rectangular grid that is 2 cells high and 2018 cells wide. Adalbert starts by placing a domino of size $1 \\times 2$ horizontally, covering exactly two cells of the grid. Then Babette plays a domino of size $1 \\times 2$ vertically, and so on. The first player who cannot place a domino without overlapping an already placed domino loses. Show that Adalbert has a winning strategy.","t":[{"b":4,"e":0.42857,"k":"rising","v":0.04911,"x":0.49554,"p":[[0,24,0.0,0.05357,0.18472,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,24,0.1667,0.04911,0.11633,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.10714,0.19562,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,2,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[12,24,0.5,0.11607,0.22428,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[16,24,0.6667,0.17857,0.26,0.0,0.0,0.32142,0.0,0.85714,19,0,0,19,0,3,0,0,2,0,0,3,0,0,2,0,0,2,0,0,1,0,0],[20,24,0.8333,0.41518,0.26812,0.14286,0.42859,0.71429,0.0,0.71429,7,0,0,7,0,2,0,0,1,0,0,8,0,0,5,0,0,9,0,0,0,0,0],[24,24,1.0,0.49554,0.1988,0.42857,0.50001,0.57143,0.0,0.85714,3,0,0,3,0,0,0,0,0,0,0,13,0,0,9,0,0,6,0,0,1,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.03125,"x":0.60714,"p":[[0,69,0.0,0.06696,0.2082,0.0,0.0,0.0,0.0,1.0,28,1,1,28,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[4,69,0.058,0.04911,0.13175,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,69,0.1159,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.08929,0.20438,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[16,69,0.2319,0.04911,0.15407,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[20,69,0.2899,0.05804,0.15916,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[24,69,0.3478,0.07589,0.2172,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[28,69,0.4058,0.08034,0.20803,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[32,69,0.4638,0.05804,0.17076,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[36,69,0.5217,0.04018,0.13474,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,69,0.5797,0.10714,0.23146,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,0],[44,69,0.6377,0.10268,0.24545,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0],[48,69,0.6957,0.08929,0.22232,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[52,69,0.7536,0.24554,0.32189,0.0,0.0,0.57143,0.0,1.0,18,2,0,18,0,2,0,0,0,0,0,3,0,0,5,0,0,2,0,0,0,0,2],[56,69,0.8116,0.41517,0.35239,0.0,0.42857,0.71429,0.0,1.0,11,1,0,11,0,2,0,0,0,0,0,4,0,0,3,0,0,6,0,0,5,0,1],[60,69,0.8696,0.48214,0.34947,0.0,0.57143,0.75,0.0,1.0,9,2,0,9,0,0,0,0,3,0,0,1,0,0,6,0,0,5,0,0,6,0,2],[64,69,0.9275,0.24553,0.32188,0.0,0.0,0.57143,0.0,0.85714,19,0,0,19,0,0,0,0,2,0,0,2,0,0,3,0,0,3,0,0,3,0,0],[68,69,0.9855,0.60714,0.26,0.42857,0.71429,0.71429,0.0,1.0,2,3,0,2,0,1,0,0,3,0,0,3,0,0,4,0,0,13,0,0,3,0,3],[69,69,1.0,0.46874,0.34299,0.0,0.57143,0.71429,0.0,1.0,10,2,0,10,0,0,0,0,0,0,0,2,0,0,9,0,0,5,0,0,4,0,2]]}]},{"i":"8a9ed5ec6be1ca5b","q":"A triangle  $ ABC$  is inscribed in a circle  $ C(O,R)$   and has incenter  $ I$ . Lines  $ AI,BI,CI$  meet the circumcircle  $ (O)$  of triangle  $ ABC$  at points  $ D,E,F$  respectively. The circles with diameter  $ ID,IE,IF$  meet the sides  $ BC,CA, AB$  at pairs of points  $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$  respectively.\n\nProve that the six points  $ A_1,A_2, B_1, B_2, C_1, C_2$  are concyclic.\n\n\nBabis","t":[{"b":1,"e":0.14286,"k":"flat","v":0.09822,"x":0.45976,"p":[[0,62,0.0,0.12945,0.21234,0.0,0.0,0.2857,0.0,0.71429,22,0,4,22,0,1,0,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[4,62,0.0645,0.45976,0.23069,0.42857,0.571,0.57143,0.0,0.85714,3,0,0,3,0,4,0,0,0,0,0,8,0,0,12,0,0,3,0,0,2,0,0],[8,62,0.129,0.27675,0.25485,0.0,0.2143,0.4642,0.0,0.71429,11,0,0,11,0,5,0,0,2,0,0,6,0,0,5,0,0,3,0,0,0,0,0],[12,62,0.1935,0.3571,0.30511,0.0,0.42857,0.60714,0.0,0.85714,10,0,0,10,0,4,0,0,1,0,0,4,0,0,5,0,0,6,0,0,2,0,0],[16,62,0.2581,0.34363,0.25223,0.14286,0.35714,0.57143,0.0,0.71429,6,0,0,6,0,7,0,0,3,0,0,5,0,0,6,0,0,5,0,0,0,0,0],[20,62,0.3226,0.33024,0.25116,0.105,0.42857,0.4642,0.0,0.71429,8,0,0,8,0,4,0,0,3,0,0,9,0,0,3,0,0,5,0,0,0,0,0],[24,62,0.3871,0.26757,0.25201,0.0,0.14286,0.4642,0.0,0.857,10,0,0,10,0,7,0,0,3,0,0,4,0,0,6,0,0,1,0,0,1,0,0],[28,62,0.4516,0.33926,0.20121,0.14286,0.42857,0.4286,0.0,0.71429,3,0,0,3,0,8,0,0,4,0,0,10,0,0,5,0,0,2,0,0,0,0,0],[32,62,0.5161,0.1875,0.16536,0.10714,0.14286,0.2857,0.0,0.71429,8,0,0,8,0,13,0,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[36,62,0.5806,0.19616,0.14184,0.14,0.14286,0.2857,0.0,0.4286,6,0,0,6,0,14,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.19178,0.14116,0.14,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,12,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.17849,0.14288,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,16,0,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[48,62,0.7742,0.14723,0.16165,0.0,0.14286,0.1786,0.0,0.57143,13,0,0,13,0,11,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[52,62,0.8387,0.09822,0.11539,0.0,0.14286,0.14286,0.0,0.4286,15,0,0,15,0,14,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,62,0.9032,0.13392,0.15539,0.0,0.14286,0.1429,0.0,0.571,14,0,0,14,0,11,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[60,62,0.9677,0.18287,0.16845,0.14214,0.14288,0.1786,0.0,0.71429,6,0,0,6,0,18,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[62,62,1.0,0.1517,0.11259,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,21,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.11161,"x":0.46872,"p":[[0,75,0.0,0.11161,0.18466,0.0,0.0,0.1429,0.0,0.57143,21,0,4,21,0,4,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[4,75,0.0533,0.46872,0.26056,0.42857,0.57143,0.57143,0.0,0.85714,6,0,0,6,0,1,0,0,0,0,0,5,0,0,13,0,0,5,0,0,2,0,0],[8,75,0.1067,0.20536,0.23128,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,6,0,0,2,0,0,6,0,0,2,0,0,2,0,0,0,0,0],[12,75,0.16,0.21425,0.26481,0.0,0.0,0.571,0.0,0.71429,18,0,0,18,0,2,0,0,0,0,0,3,0,0,8,0,0,1,0,0,0,0,0],[16,75,0.2133,0.24106,0.24854,0.0,0.2143,0.42858,0.0,0.71429,14,0,0,14,0,2,0,0,5,0,0,4,0,0,5,0,0,2,0,0,0,0,0],[20,75,0.2667,0.25426,0.2714,0.0,0.14286,0.42857,0.0,0.85714,11,0,0,11,0,8,0,0,2,0,0,5,0,0,3,0,0,0,0,0,3,0,0],[24,75,0.32,0.24105,0.24594,0.0,0.14288,0.42857,0.0,0.71429,13,0,0,13,0,5,0,0,1,0,0,7,0,0,4,0,0,2,0,0,0,0,0],[28,75,0.3733,0.20953,0.22015,0.0,0.14286,0.42858,0.0,0.71429,12,0,0,12,0,8,0,0,3,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[32,75,0.4267,0.1741,0.21937,0.0,0.07143,0.32143,0.0,0.85714,16,0,0,16,0,5,0,0,3,0,0,6,0,0,1,0,0,0,0,0,1,0,0],[36,75,0.48,0.14723,0.16935,0.0,0.14143,0.2857,0.0,0.57143,15,0,0,15,0,7,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[40,75,0.5333,0.22759,0.22267,0.0,0.2143,0.42857,0.0,0.71429,12,0,0,12,0,4,0,0,7,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[44,75,0.5867,0.20969,0.2395,0.0,0.14286,0.32143,0.0,0.85714,14,0,0,14,0,5,0,0,5,0,0,2,0,0,5,0,0,0,0,0,1,0,0],[48,75,0.64,0.28113,0.25125,0.0,0.2143,0.571,0.0,0.85714,9,0,0,9,0,7,0,0,4,0,0,3,0,0,7,0,0,1,0,0,1,0,0],[52,75,0.6933,0.1874,0.17655,0.0,0.14286,0.32143,0.0,0.57143,9,0,0,9,0,14,0,0,1,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[56,75,0.7467,0.27676,0.20493,0.14286,0.2143,0.4286,0.0,0.71429,4,0,0,4,0,12,0,0,6,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[60,75,0.8,0.32579,0.23759,0.14286,0.21431,0.4642,0.0,0.85714,2,0,0,2,0,14,0,0,3,0,0,5,0,0,3,0,0,4,0,0,1,0,0],[64,75,0.8533,0.4195,0.23135,0.1429,0.4286,0.57143,0.0,0.71429,2,0,0,2,0,7,0,0,3,0,0,6,0,0,7,0,0,7,0,0,0,0,0],[68,75,0.9067,0.3437,0.23376,0.14286,0.28571,0.57111,0.0,0.71429,4,0,0,4,0,9,0,0,4,0,0,3,0,0,9,0,0,3,0,0,0,0,0],[72,75,0.96,0.3214,0.21425,0.14286,0.28571,0.571,0.0,0.71429,3,0,0,3,0,11,0,0,4,0,0,5,0,0,7,0,0,2,0,0,0,0,0],[75,75,1.0,0.24554,0.18638,0.14286,0.14286,0.2857,0.0,0.71429,2,0,0,2,0,19,0,0,4,0,0,1,0,0,5,0,0,1,0,0,0,0,0]]}]},{"i":"3da277c4c7777c7c","q":"Let  $H$  be the orthocenter of an acute-angled triangle  $ABC$ ;  $A_1, B_1, C_1$  be the touching points of the incircle with  $BC, CA, AB$  respectively;  $E_A, E_B, E_C$  be the midpoints of  $AH, BH, CH$  respectively. The circle centered at  $E_A$  and passing through  $A$  meets for the second time the bisector of angle  $A$  at  $A_2$ ; points  $B_2, C_2$  are defined similarly. Prove that the triangles  $A_1B_1C_1$  and  $A_2B_2C_2$  are similar.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,96,0.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,96,0.0417,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,96,0.0833,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,96,0.125,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,96,0.1667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,96,0.2083,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,96,0.25,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,96,0.2917,0.03571,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,96,0.3333,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,96,0.375,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,96,0.4167,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,96,0.4583,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,96,0.5,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,96,0.5417,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,96,0.5833,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,96,0.625,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,96,0.6667,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,96,0.7083,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,96,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,96,0.7917,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,96,0.8333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,96,0.875,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,96,0.9167,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,96,0.9583,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,96,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.04464,"p":[[0,90,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,3,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,90,0.0444,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,90,0.0889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,90,0.1333,0.03571,0.11294,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,90,0.1778,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,90,0.2222,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,90,0.2667,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,90,0.3111,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,90,0.3556,0.04018,0.10853,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,90,0.4,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,90,0.4444,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,90,0.4889,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,90,0.5333,0.03571,0.15152,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[52,90,0.5778,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[56,90,0.6222,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,90,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,90,0.7111,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,90,0.7556,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,90,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,90,0.8444,0.03571,0.12372,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[80,90,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,90,0.9333,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[88,90,0.9778,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[90,90,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ef3a36a68f2b251c","q":"9. (IRN 1) Let $f(x)$ be a polynomial with rational coefficients and $\\alpha$ be a real number such that $\\alpha^{3}-\\alpha=f(\\alpha)^{3}-f(\\alpha)=33^{1992}$. Prove that for each $n \\geq 1$,  $$ \\left(f^{(n)}(\\alpha)\\right)^{3}-f^{(n)}(\\alpha)=33^{1992} $$  where $f^{(n)}(x)=f(f(\\ldots f(x)))$, and $n$ is a positive integer.","t":[{"b":2,"e":1.0,"k":"flat","v":0.99554,"x":1.0,"p":[[0,64,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,64,0.0625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,64,0.125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,64,0.1875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,64,0.25,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,64,0.3125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,64,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,64,0.4375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,64,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,64,0.5625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,64,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,64,0.6875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,64,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,64,0.8125,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,64,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,64,0.9375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,64,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":5,"e":1.0,"k":"flat","v":0.99554,"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],[4,54,0.0741,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,54,0.3704,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,54,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"1409c8d73b48125c","q":"How many ordered triples of integers  $(a, b, c)$  satisfy the following system?  $$ \\begin{cases} ab + c &= 17  a + bc &= 19 \\end{cases} $$    $$ \\mathrm a. ~ 2\\qquad \\mathrm b.~3\\qquad \\mathrm c. ~4 \\qquad \\mathrm d. ~5 \\qquad \\mathrm e. ~6 $$","t":[{"b":0,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,71,0.0,0.95536,0.10972,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[4,71,0.0563,0.94196,0.09355,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[8,71,0.1127,0.95982,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[12,71,0.169,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,71,0.2254,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[20,71,0.2817,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[24,71,0.338,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[28,71,0.3944,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[32,71,0.4507,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[36,71,0.507,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,71,0.5634,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":7,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,69,0.0,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[4,69,0.058,0.94196,0.13296,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,25],[8,69,0.1159,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[12,69,0.1739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,69,0.2319,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,69,0.2899,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,69,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,69,0.4058,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[36,69,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,69,0.5797,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[64,69,0.9275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"eac49d7f3f2992f2","q":"Find the smallest positive value of  $36^k - 5^m$ , where  $k$  and  $m$  are positive integers.","t":[{"b":4,"e":1.0,"k":"flat","v":0.97768,"x":1.0,"p":[[0,7,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,7,0.5714,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[7,7,1.0,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.90625,"x":0.99107,"p":[[0,36,0.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[4,36,0.1111,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[8,36,0.2222,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[12,36,0.3333,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,36,0.4444,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,36,0.5556,0.90625,0.19434,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,21],[24,36,0.6667,0.95536,0.10971,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,25],[28,36,0.7778,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,36,0.8889,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[36,36,1.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27]]}]},{"i":"37dbd4f89c67ce86","q":"In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the remaining 8 socks at random and on Wednesday two of the reaining 6 socks at random. The probability that Wednesday is the first day Sandy selects matching socks is  $\\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.875,"x":1.0,"p":[[0,17,0.0,0.9375,0.19541,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,29],[4,17,0.2353,0.875,0.26184,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,26],[8,17,0.4706,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,17,0.7059,0.95536,0.1729,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[16,17,0.9412,0.875,0.26184,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,26],[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]]},{"b":5,"e":1.0,"k":"flat","v":0.91964,"x":1.0,"p":[[0,8,0.0,0.95089,0.17353,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[4,8,0.5,0.91964,0.2111,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[8,8,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"2fd7ea6c49be608e","q":"Twenty distinct points are marked on a circle and labeled  $1$  through  $20$  in clockwise order.  A line segment is drawn between every pair of points whose labels differ by a prime number.  Find the number of triangles formed whose vertices are among the original  $20$  points.","t":[{"b":0,"e":1.0,"k":"flat","v":0.96875,"x":1.0,"p":[[0,79,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,79,0.0506,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,79,0.1013,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,79,0.1519,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,79,0.2025,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,79,0.2532,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,79,0.3038,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,79,0.3544,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,79,0.4051,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,79,0.4557,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,79,0.5063,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,79,0.557,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[52,79,0.6582,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,79,0.7089,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,79,0.7595,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,79,0.8101,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,79,0.8608,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[72,79,0.9114,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[76,79,0.962,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[79,79,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,57,0.0,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[4,57,0.0702,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,57,0.1404,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,57,0.2105,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[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],[20,57,0.3509,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[24,57,0.4211,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,57,0.4912,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,57,0.5614,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,57,0.6316,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,57,0.7018,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,57,0.7719,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,57,0.8421,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,57,0.9123,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[56,57,0.9825,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[57,57,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"95964d3580da041c","q":"Consider a plane  $\\epsilon$  and three non-collinear points  $A,B,C$  on the same side of  $\\epsilon$ ; suppose the plane determined by these three points is not parallel to  $\\epsilon$ . In plane  $\\epsilon$  take three arbitrary points  $A',B',C'$ . Let  $L,M,N$  be the midpoints of segments  $AA', BB', CC'$ ; Let  $G$  be the centroid of the triangle  $LMN$ . (We will not consider positions of the points  $A', B', C'$  such that the points  $L,M,N$  do not form a triangle.) What is the locus of point  $G$  as  $A', B', C'$  range independently over the plane  $\\epsilon$ ?","t":[{"b":2,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,37,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,37,0.1081,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,37,0.2162,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":0.85714,"k":"flat","v":0.9866,"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],[8,44,0.1818,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,44,0.2727,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,44,0.3636,0.9867,0.04136,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,44,0.4545,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,44,0.5455,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[32,44,0.7273,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,44,0.8182,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,44,0.9091,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[44,44,1.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"d74497c1a22e2aa5","q":"$\\vartriangle ABC$  has  $AB = 5$ ,  $BC = 12$ , and  $AC = 13$ . A circle is inscribed in  $\\vartriangle ABC$ , and  $MN$  tangent to the circle is drawn such that  $M$  is on  $\\overline{AC}$ ,  $N$  is on  $\\overline{BC}$ , and  $\\overline{MN} \\parallel \\overline{AB}$ . The area of  $\\vartriangle  MNC$  is  $m/n$  , where  $m$  and  $n $ are relatively prime positive integers. Find  $m + n$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,56,0.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[4,56,0.0714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[8,56,0.1429,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[12,56,0.2143,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[16,56,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,56,0.3571,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[24,56,0.4286,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,56,0.5,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,56,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,56,0.6429,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[40,56,0.7143,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[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],[48,56,0.8571,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[52,56,0.9286,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[56,56,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.96875,"x":0.99554,"p":[[0,40,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,40,0.1,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,40,0.2,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[12,40,0.3,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[16,40,0.4,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,40,0.5,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,40,0.6,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,40,0.7,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,40,0.8,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,40,0.9,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,40,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]}]},{"i":"2d571c2826c2fea3","q":"Let  $a_0 = 1$  and define the sequence  $\\{a_n\\}$  by \\[a_{n+1} = \\frac{\\sqrt{3}a_n - 1}{a_n + \\sqrt{3}}.\\] If  $a_{2017}$  can be expressed in the form  $a+b\\sqrt{c}$  in simplest radical form, compute  $a+b+c$ .\n\n*2016 CCA Math Bonanza Lightning #3.2*","t":[{"b":0,"e":1.0,"k":"flat","v":0.87277,"x":0.99107,"p":[[0,34,0.0,0.94866,0.10448,1.0,1.0,1.0,0.64286,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3,0,0,3,0,25],[4,34,0.1176,0.88393,0.14032,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,6,0,17],[8,34,0.2353,0.87277,0.12076,0.71429,0.85714,1.0,0.64286,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,8,0,0,10,0,13],[12,34,0.3529,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[16,34,0.4706,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[20,34,0.5882,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[24,34,0.7059,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[28,34,0.8235,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,34,0.9412,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[34,34,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,5,0.0,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[4,5,0.8,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[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":"ab392be2f5c6a4c4","q":"Write out the positive integers consisting of only  $1$ s,  $6$ s, and  $9$ s in ascending order as in:  $1,6,9,11,16,\\dots$ .\n\na. Find the order of  $1996$  in the sequence.\n\nb. Find the  $1996$ th term in the sequence.","t":[{"b":1,"e":1.0,"k":"flat","v":0.98214,"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],[4,22,0.1818,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,22,0.3636,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,22,0.5455,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,22,0.7273,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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":7,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,13,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,13,0.3077,0.96429,0.12877,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[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],[12,13,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"f3ac3163d70c3c43","q":"11. G5 (FRA) Let $A B C$ be a triangle, $\\Omega$ its incircle and $\\Omega_{a}, \\Omega_{b}, \\Omega_{c}$ three circles three circles orthogonal to $\\Omega$ passing through $B$ and $C, A$ and $C$, and $A$ and $B$ respectively. The circles $\\Omega_{a}, \\Omega_{b}$ meet again in $C^{\\prime}$; in the same way we obtain the points $B^{\\prime}$ and $A^{\\prime}$. Prove that the radius of the circumcircle of $A^{\\prime} B^{\\prime} C^{\\prime}$ is half the radius of $\\Omega$.","t":[{"b":5,"e":0.71429,"k":"flat","v":0.48659,"x":0.58481,"p":[[0,12,0.0,0.48659,0.2392,0.28571,0.57143,0.57143,0.0,1.0,3,1,0,3,0,2,0,0,5,0,0,0,0,0,15,0,0,6,0,0,0,0,1],[4,12,0.3333,0.54462,0.13091,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,2,0,0,23,0,0,4,0,0,0,0,0],[8,12,0.6667,0.58481,0.13534,0.57143,0.57143,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,1,0,0,15,0,0,12,0,0,0,0,0],[12,12,1.0,0.57136,0.12877,0.571,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,4,0,0,17,0,0,9,0,0,0,0,0]]},{"b":6,"e":0.1429,"k":"flat","v":0.43746,"x":0.49993,"p":[[0,6,0.0,0.49993,0.17872,0.42857,0.57143,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,1,0,0,8,0,0,15,0,0,5,0,0,0,0,0],[4,6,0.6667,0.43747,0.22568,0.28571,0.49979,0.57143,0.0,0.71429,3,0,0,3,0,4,0,0,3,0,0,6,0,0,10,0,0,6,0,0,0,0,0],[6,6,1.0,0.43746,0.16339,0.42857,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,4,0,0,12,0,0,12,0,0,1,0,0,0,0,0]]}]},{"i":"e4065c65b627dcd4","q":"Given the square  $ABCD$ . Let point  $M$  be the midpoint of the side  $BC$ , and  $H$  be the foot of the perpendicular from vertex  $C$  on the segment  $DM$ . Prove that  $AB = AH$ .\n\n(Danilo Hilko)","t":[{"b":0,"e":0.0,"k":"flat","v":0.34821,"x":0.57589,"p":[[0,28,0.0,0.46429,0.48313,0.0,0.14286,1.0,0.0,1.0,15,14,0,15,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14],[4,28,0.1429,0.38393,0.45937,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,11],[8,28,0.2857,0.44643,0.48936,0.0,0.07143,1.0,0.0,1.0,16,14,0,16,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[12,28,0.4286,0.57589,0.4758,0.0,1.0,1.0,0.0,1.0,11,17,0,11,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,17],[16,28,0.5714,0.44196,0.49275,0.0,0.0,1.0,0.0,1.0,17,14,0,17,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[20,28,0.7143,0.39286,0.4738,0.0,0.0,1.0,0.0,1.0,17,12,0,17,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[24,28,0.8571,0.34821,0.47237,0.0,0.0,1.0,0.0,1.0,20,11,0,20,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[28,28,1.0,0.36161,0.46837,0.0,0.0,1.0,0.0,1.0,19,11,0,19,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11]]},{"b":4,"e":0.0,"k":"falling","v":0.12946,"x":0.66964,"p":[[0,30,0.0,0.66964,0.45937,0.0,1.0,1.0,0.0,1.0,9,21,0,9,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,21],[4,30,0.1333,0.61161,0.47142,0.0,1.0,1.0,0.0,1.0,9,19,0,9,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[8,30,0.2667,0.54464,0.48765,0.0,1.0,1.0,0.0,1.0,13,17,0,13,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[12,30,0.4,0.47321,0.49544,0.0,0.07143,1.0,0.0,1.0,16,15,0,16,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[16,30,0.5333,0.59375,0.49113,0.0,1.0,1.0,0.0,1.0,13,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[20,30,0.6667,0.29464,0.44455,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[24,30,0.8,0.12946,0.32996,0.0,0.0,0.0,0.0,1.0,27,4,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[28,30,0.9333,0.22768,0.41011,0.0,0.0,0.14286,0.0,1.0,23,7,0,23,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[30,30,1.0,0.19196,0.38896,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6]]}]},{"i":"84715183d62ce46c","q":"Find the  $2019$ th strictly positive integer  $n$  such that  $\\binom{2n}{n}$  is not divisible by  $5$ .","t":[{"b":2,"e":0.85714,"k":"falling","v":0.77679,"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],[4,49,0.0816,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,49,0.1633,0.95536,0.0974,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,25],[12,49,0.2449,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[16,49,0.3265,0.86607,0.15947,0.85714,0.85714,1.0,0.5714,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,0,10,0,15],[20,49,0.4082,0.87946,0.14773,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,9,0,16],[24,49,0.4898,0.875,0.15465,0.85714,0.92857,1.0,0.5714,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,9,0,16],[28,49,0.5714,0.86607,0.12846,0.85714,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,18,0,10],[32,49,0.6531,0.83482,0.15612,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,11,0,11],[36,49,0.7347,0.80802,0.16215,0.57143,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,14,0,8],[40,49,0.8163,0.86161,0.145,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,14,0,12],[44,49,0.898,0.77679,0.13803,0.67857,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,16,0,3],[48,49,0.9796,0.82143,0.14725,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,14,0,8],[49,49,1.0,0.84375,0.13533,0.82143,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,15,0,9]]},{"b":6,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,28,0.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[4,28,0.1429,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"7ab1758d2e1b34d2","q":"In triangle  $ABC$ ,  $AB=2$ ,  $AC=1+\\sqrt{5}$ , and  $\\angle CAB=54^{\\circ}$ . Suppose  $D$  lies on the extension of  $AC$  through  $C$  such that  $CD=\\sqrt{5}-1$ . If  $M$  is the midpoint of  $BD$ , determine the measure of  $\\angle ACM$ , in degrees.","t":[{"b":0,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,55,0.0,0.92411,0.24996,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29],[4,55,0.0727,0.94643,0.21053,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[8,55,0.1455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,55,0.2182,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,55,0.2909,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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],[24,55,0.4364,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[28,55,0.5091,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":0.95536,"x":1.0,"p":[[0,51,0.0,0.95536,0.18707,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[4,51,0.0784,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[8,51,0.1569,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,51,0.2353,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[16,51,0.3137,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,51,0.3922,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,51,0.4706,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,51,0.549,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,51,0.6275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[40,51,0.7843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,51,0.8627,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,51,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[51,51,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"3a12aef181283832","q":"Find all pairs of integers  $a, b$  such that the following system of equations has a unique integral solution  $(x , y , z )$  : \n  $\\begin{cases}x + y = a - 1 \nx(y + 1) - z^2 = b \\end{cases}$","t":[{"b":5,"e":1.0,"k":"flat","v":0.96429,"x":1.0,"p":[[0,8,0.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[4,8,0.5,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[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":0.97321,"x":1.0,"p":[[0,5,0.0,0.97321,0.05576,1.0,1.0,1.0,0.8571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[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":"e21829979dee04cc","q":"A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is  $\\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.9375,"x":0.97768,"p":[[0,81,0.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[4,81,0.0494,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[8,81,0.0988,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[12,81,0.1481,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[16,81,0.1975,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[20,81,0.2469,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[24,81,0.2963,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[28,81,0.3457,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,81,0.3951,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,81,0.4444,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[40,81,0.4938,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[44,81,0.5432,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[48,81,0.5926,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[52,81,0.642,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[56,81,0.6914,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[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],[64,81,0.7901,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[68,81,0.8395,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[72,81,0.8889,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[76,81,0.9383,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[80,81,0.9877,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[81,81,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]},{"b":7,"e":0.85714,"k":"flat","v":0.94196,"x":0.96875,"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],[4,48,0.0833,0.96428,0.06187,0.96429,1.0,1.0,0.857,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[8,48,0.1667,0.94196,0.07017,0.85714,1.0,1.0,0.857,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[12,48,0.25,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[16,48,0.3333,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[20,48,0.4167,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[24,48,0.5,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[28,48,0.5833,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[32,48,0.6667,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[36,48,0.75,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[40,48,0.8333,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[44,48,0.9167,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[48,48,1.0,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20]]}]},{"i":"52830648d5696cc9","q":"In triangle $A B C$, the interior and exterior angle bisectors of $\\angle B A C$ intersect the line $B C$ in $D$ and $E$, respectively. Let $F$ be the second point of intersection of the line $A D$ with the circumcircle of the triangle $A B C$. Let $O$ be the circumcentre of the triangle $A B C$ and let $D^{\\prime}$ be the reflection of $D$ in $O$. Prove that $\\angle D^{\\prime} F E=90^{\\circ}$.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.00446,"x":0.03572,"p":[[0,19,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,36,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"123d82aafdcf687c","q":"Given any set $S$ of positive integers, show that at least one of the following two assertions holds:  (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\\sum_{x \\in F} 1 / x=\\sum_{x \\in G} 1 / x$;  (2) There exists a positive rational number $r<1$ such that $\\sum_{x \\in F} 1 / x \\neq r$ for all finite subsets $F$ of $S$.  (Luxembourg)","t":[{"b":1,"e":0.14286,"k":"flat","v":0.08036,"x":0.11607,"p":[[0,48,0.0,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.09813,0.06616,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.12054,"p":[[0,50,0.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[50,50,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"de2afbc241894817","q":"Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant?","t":[{"b":1,"e":0.0,"k":"flat","v":0.05357,"x":0.18749,"p":[[0,45,0.0,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.18749,0.20956,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,12,0,0,3,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[8,45,0.1778,0.15607,0.16507,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,9,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,45,0.2667,0.13393,0.14698,0.0,0.14286,0.1786,0.0,0.42857,14,0,0,14,0,10,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.10713,0.15564,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,11,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[20,45,0.4444,0.10714,0.13363,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,8,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.05357,0.11152,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.0625,0.1234,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.11598,0.14478,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,13,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,45,0.8889,0.07125,0.08734,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.05357,0.12752,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[45,45,1.0,0.08482,0.10012,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.24088,"p":[[0,25,0.0,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.24088,0.20031,0.14214,0.14286,0.42857,0.0,0.71429,6,0,0,6,0,13,0,0,4,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[8,25,0.32,0.08928,0.11152,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.14731,0.1493,0.0,0.14286,0.2857,0.0,0.571,12,0,0,12,0,11,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,25,0.64,0.07143,0.14725,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,25,0.8,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.0625,0.11812,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[25,25,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"31ebe4bd69eb01e7","q":"Show that there exists a degree  $58$  monic polynomial   $$ P(x) = x^{58} + a_1x^{57} + \\cdots + a_{58} $$  such that  $P(x)$  has exactly  $29$  positive real roots and  $29$  negative real roots and that  $\\log_{2017} |a_i|$  is a positive integer for all  $1 \\leq i \\leq 58$ .","t":[{"b":4,"e":0.1429,"k":"flat","v":0.00446,"x":0.13837,"p":[[0,46,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,13,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.09375,0.15815,0.0,0.0,0.14286,0.0,0.85714,17,0,0,17,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,46,0.1739,0.13837,0.20967,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,9,0,0,2,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[12,46,0.2609,0.04902,0.09173,0.0,0.0,0.14071,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.11161,0.23347,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[20,46,0.4348,0.08036,0.18536,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,46,0.5217,0.08036,0.14698,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[28,46,0.6087,0.08036,0.19541,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,46,0.6957,0.04911,0.13175,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,46,0.7826,0.12054,0.19597,0.0,0.0,0.14286,0.0,0.85714,17,0,0,17,0,10,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[40,46,0.8696,0.06696,0.18205,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,46,0.9565,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.1429,"k":"flat","v":0.01339,"x":0.15616,"p":[[0,49,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,9,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.15616,0.20935,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,15,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[8,49,0.1633,0.09375,0.17353,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,11,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[12,49,0.2449,0.11607,0.16917,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,16,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[16,49,0.3265,0.125,0.19805,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,11,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[20,49,0.4082,0.07589,0.13825,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,10,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,49,0.4898,0.08036,0.16342,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,5,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[28,49,0.5714,0.10714,0.22016,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,8,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[32,49,0.6531,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.06697,0.10705,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.05357,0.11152,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,49,0.898,0.06696,0.12364,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.08482,0.12299,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"13b40c997ba05621","q":"Define the function  $f: \\mathbb N \\cup \\{0\\} \\to \\mathbb{Q}$  as follows:  $f(0) = 0$  and \\[ f(3n+k) = -\\frac{3f(n)}{2} + k ,  \\] for  $k = 0, 1, 2$ . Show that  $f$  is one-to-one and determine the range of  $f$ .","t":[{"b":3,"e":1.0,"k":"rising","v":0.63838,"x":0.94196,"p":[[0,50,0.0,0.63838,0.27196,0.42857,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,5,0,0,4,0,0,4,0,0,4,0,0,8,0,5],[4,50,0.08,0.87946,0.14334,0.82143,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,8,0,16],[8,50,0.16,0.83929,0.1915,0.82143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,4,0,0,12,0,12],[12,50,0.24,0.86607,0.15947,0.82143,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,10,0,14],[16,50,0.32,0.83929,0.20124,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,8,0,0,4,0,16],[20,50,0.4,0.82589,0.17762,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,6,0,13],[24,50,0.48,0.77679,0.24984,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,4,0,0,8,0,12],[28,50,0.56,0.75893,0.26107,0.57143,0.78571,1.0,0.0,1.0,1,13,0,1,0,1,0,0,0,0,0,2,0,0,6,0,0,6,0,0,3,0,13],[32,50,0.64,0.74105,0.23539,0.67846,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,1,0,0,5,0,0,1,0,0,9,0,0,6,0,9],[36,50,0.72,0.91517,0.15094,0.85714,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,6,0,21],[40,50,0.8,0.94196,0.1063,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,24],[44,50,0.88,0.92856,0.12376,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,22],[48,50,0.96,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[50,50,1.0,0.86607,0.14698,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,4,0,16]]},{"b":5,"e":1.0,"k":"rising","v":0.64285,"x":0.98214,"p":[[0,54,0.0,0.64285,0.2342,0.42859,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,2,0,0,7,0,0,4,0,0,8,0,0,7,0,3],[4,54,0.0741,0.84375,0.17627,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,0,6,0,14],[8,54,0.1481,0.80357,0.17035,0.71429,0.85714,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,9,0,0,11,0,8],[12,54,0.2222,0.80356,0.2165,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,9,0,11],[16,54,0.2963,0.82143,0.14725,0.71429,0.85714,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,8,0,10],[20,54,0.3704,0.87054,0.19678,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,6,0,17],[24,54,0.4444,0.9375,0.15947,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,24],[28,54,0.5185,0.87054,0.20316,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0,8,0,18],[32,54,0.5926,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22],[36,54,0.6667,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,54,0.7407,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[44,54,0.8148,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[48,54,0.8889,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[52,54,0.963,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[54,54,1.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]}]},{"i":"21622b4b6e2a0022","q":"Written on a blackboard is the polynomial $x^{2}+x+2014$. Calvin and Hobbes take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or decrease the coefficient of $x$ by 1. And during his turn, Hobbes should either increase or decrease the constant coefficient by 1. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning strategy.","t":[{"b":4,"e":0.1429,"k":"rising","v":0.02232,"x":0.44195,"p":[[0,81,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,15,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,81,0.0494,0.30347,0.34398,0.0,0.14286,0.57111,0.0,1.0,11,3,0,11,0,7,0,0,5,0,0,0,0,0,3,0,0,0,0,0,3,0,3],[8,81,0.0988,0.29455,0.31735,0.0,0.14286,0.42857,0.0,1.0,10,3,0,10,0,7,0,0,5,0,0,3,0,0,2,0,0,1,0,0,1,0,3],[12,81,0.1481,0.27232,0.29744,0.0,0.14286,0.4286,0.0,1.0,13,1,0,13,0,4,0,0,3,0,0,5,0,0,3,0,0,1,0,0,2,0,1],[16,81,0.1975,0.29015,0.32825,0.0,0.21429,0.571,0.0,1.0,15,2,0,15,0,1,0,0,4,0,0,3,0,0,3,0,0,3,0,0,1,0,2],[20,81,0.2469,0.30354,0.35668,0.0,0.14286,0.57143,0.0,1.0,14,4,0,14,0,5,0,0,1,0,0,1,0,0,5,0,0,2,0,0,0,0,4],[24,81,0.2963,0.30357,0.32488,0.0,0.14288,0.46431,0.0,1.0,11,3,0,11,0,6,0,0,3,0,0,4,0,0,3,0,0,1,0,0,1,0,3],[28,81,0.3457,0.28123,0.29337,0.0,0.21428,0.4642,0.0,1.0,12,1,0,12,0,4,0,0,5,0,0,3,0,0,3,0,0,3,0,0,1,0,1],[32,81,0.3951,0.25,0.28346,0.0,0.14288,0.42858,0.0,1.0,14,1,0,14,0,4,0,0,2,0,0,6,0,0,3,0,0,1,0,0,1,0,1],[36,81,0.4444,0.35712,0.30092,0.0,0.42857,0.4642,0.0,1.0,9,3,0,9,0,3,0,0,1,0,0,11,0,0,4,0,0,1,0,0,0,0,3],[40,81,0.4938,0.28125,0.32827,0.0,0.14286,0.46429,0.0,1.0,14,2,0,14,0,3,0,0,5,0,0,2,0,0,2,0,0,2,0,0,2,0,2],[44,81,0.5432,0.27679,0.34244,0.0,0.14286,0.46429,0.0,1.0,12,3,0,12,0,10,0,0,0,0,0,2,0,0,2,0,0,1,0,0,2,0,3],[48,81,0.5926,0.30347,0.33267,0.0,0.14286,0.57111,0.0,1.0,12,2,0,12,0,6,0,0,2,0,0,3,0,0,2,0,0,3,0,0,2,0,2],[52,81,0.642,0.2499,0.32145,0.0,0.0,0.42857,0.0,1.0,17,2,0,17,0,1,0,0,3,0,0,5,0,0,2,0,0,0,0,0,2,0,2],[56,81,0.6914,0.26775,0.33074,0.0,0.07,0.57143,0.0,1.0,16,2,0,16,0,3,0,0,2,0,0,1,0,0,5,0,0,2,0,0,1,0,2],[60,81,0.7407,0.21427,0.27892,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,8,0,0,3,0,0,1,0,0,1,0,0,4,0,0,0,0,1],[64,81,0.7901,0.27668,0.3406,0.0,0.07,0.46418,0.0,1.0,16,3,0,16,0,3,0,0,0,0,0,5,0,0,2,0,0,3,0,0,0,0,3],[68,81,0.8395,0.14732,0.22442,0.0,0.0,0.1786,0.0,0.85714,18,0,0,18,0,6,0,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[72,81,0.8889,0.24998,0.29231,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,3,0,0,2,0,0,6,0,0,2,0,0,2,0,0,1,0,1],[76,81,0.9383,0.36383,0.3181,0.0,0.35714,0.60714,0.0,0.85714,9,0,0,9,0,5,0,1,1,0,0,4,0,0,4,0,0,3,0,0,5,0,0],[80,81,0.9877,0.41964,0.32915,0.14286,0.35714,0.71429,0.0,1.0,7,1,0,7,0,4,0,0,5,0,0,3,0,0,1,0,0,6,0,0,5,0,1],[81,81,1.0,0.44195,0.35239,0.0,0.4998,0.71429,0.0,1.0,9,2,0,9,0,3,0,0,1,0,0,3,0,0,4,0,0,5,0,0,5,0,2]]},{"b":5,"e":1.0,"k":"rising","v":0.01786,"x":0.55356,"p":[[0,57,0.0,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,20,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,57,0.0702,0.29018,0.29771,0.0,0.21428,0.42858,0.0,1.0,9,2,0,9,0,7,0,0,7,0,0,3,0,0,0,0,0,3,0,0,1,0,2],[8,57,0.1404,0.31695,0.34391,0.0,0.14286,0.71429,0.0,1.0,12,2,0,12,0,6,0,0,2,0,0,2,0,0,1,0,0,5,0,0,2,0,2],[12,57,0.2105,0.32587,0.31385,0.0,0.28571,0.571,0.0,1.0,11,3,0,11,0,3,0,0,3,0,0,6,0,0,5,0,0,1,0,0,0,0,3],[16,57,0.2807,0.23213,0.27373,0.0,0.14286,0.42858,0.0,0.85714,15,0,0,15,0,4,0,0,3,0,0,3,0,0,3,0,0,3,0,0,1,0,0],[20,57,0.3509,0.43755,0.3387,0.10714,0.4286,0.71429,0.0,1.0,8,1,0,8,0,3,0,0,3,0,0,4,0,0,0,0,0,8,0,0,5,0,1],[24,57,0.4211,0.47321,0.31427,0.24999,0.50001,0.74996,0.0,1.0,6,1,0,6,0,2,0,0,3,0,0,5,0,0,6,0,0,2,0,0,7,0,1],[28,57,0.4912,0.41963,0.34429,0.0,0.42857,0.71429,0.0,1.0,10,1,0,10,0,1,0,0,3,0,0,4,0,0,3,0,0,4,0,0,6,0,1],[32,57,0.5614,0.44641,0.33644,0.14286,0.42859,0.71429,0.0,1.0,7,3,0,7,0,3,0,0,4,0,0,3,0,0,4,0,0,5,0,0,3,0,3],[36,57,0.6316,0.35712,0.3388,0.0,0.14286,0.71429,0.0,1.0,10,2,0,10,0,7,0,0,0,0,0,2,0,0,2,0,0,9,0,0,0,0,2],[40,57,0.7018,0.38392,0.36671,0.0,0.28571,0.71429,0.0,1.0,12,1,0,12,0,3,0,0,2,0,0,1,0,0,2,0,0,5,0,0,6,0,1],[44,57,0.7719,0.40611,0.30957,0.105,0.42857,0.60714,0.0,0.85714,8,0,0,8,0,3,0,0,3,0,0,3,0,0,7,0,0,3,0,0,5,0,0],[48,57,0.8421,0.48661,0.33093,0.14286,0.64286,0.71429,0.0,1.0,6,1,0,6,0,5,0,0,0,0,0,3,0,0,2,0,0,10,0,0,5,0,1],[52,57,0.9123,0.55356,0.28515,0.42857,0.57143,0.71429,0.0,1.0,3,3,0,3,0,2,0,0,1,0,0,8,0,0,4,0,0,7,0,0,4,0,3],[56,57,0.9825,0.49107,0.29867,0.25,0.50001,0.71429,0.0,1.0,4,1,0,4,0,4,0,0,2,0,0,6,0,0,3,0,0,7,0,0,5,0,1],[57,57,1.0,0.55344,0.27619,0.39286,0.57143,0.74996,0.0,1.0,1,2,0,1,0,5,0,0,2,0,0,4,0,0,7,0,0,5,0,0,6,0,2]]}]},{"i":"7c3a38168b4411b7","q":"Evan has a simple graph with $v$ vertices and $e$ edges. Show that he can delete at least $\\frac{e-v+1}{2}$ edges so that each vertex still has at least half of its original degree.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,51,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,51,0.0784,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,51,0.1569,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,51,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,51,0.3137,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,51,0.3922,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,51,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,51,0.549,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,51,0.6275,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,51,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[44,51,0.8627,0.04018,0.11425,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[51,51,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,69,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,69,0.058,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,69,0.1159,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,69,0.2319,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[36,69,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,69,0.6377,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[56,69,0.8116,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"5634e4a19f824e13","q":"Let $m, n$ be positive integers with $m \\geq n$. Let $S$ be the set of pairs $(a, b)$ of relatively prime positive integers such that $a, b \\leq m$ and $a+b>m$.\n\nFor each pair $(a, b) \\in S$, consider the nonnegative integer solution $(u, v)$ to the equation $a u-b v=n$ chosen with $v \\geq 0$ minimal, and let $I(a, b)$ denote the (open) interval $(v / a, u / b)$.\nProve that $I(a, b) \\subseteq(0,1)$ for every $(a, b) \\in S$, and that any fixed irrational number $\\alpha \\in(0,1)$ lies in $I(a, b)$ for exactly $n$ distinct pairs $(a, b) \\in S$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13821,"x":0.38393,"p":[[0,109,0.0,0.13821,0.09094,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,109,0.0367,0.36607,0.17835,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,19,0,0,3,0,0,5,0,0,1,0,0,0,0,1],[8,109,0.0734,0.34821,0.21998,0.14286,0.28571,0.46429,0.0,1.0,1,1,0,1,0,8,0,0,13,0,0,2,0,0,4,0,0,3,0,0,0,0,1],[12,109,0.1101,0.37947,0.20079,0.28571,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,4,0,0,18,0,0,0,0,0,8,0,0,0,0,0,1,0,1],[16,109,0.1468,0.31695,0.26179,0.14286,0.28571,0.46418,0.0,1.0,5,1,0,5,0,8,0,0,10,0,0,1,0,0,3,0,0,3,0,0,1,0,1],[20,109,0.1835,0.38393,0.28221,0.24999,0.28571,0.57143,0.0,1.0,2,3,0,2,0,6,0,0,15,0,0,0,0,0,2,0,0,3,0,0,1,0,3],[24,109,0.2202,0.33036,0.20652,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,7,0,0,19,0,0,1,0,0,3,0,0,0,0,0,0,0,2],[28,109,0.2569,0.28125,0.12619,0.24999,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,19,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[32,109,0.2936,0.2991,0.20316,0.14286,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,10,0,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[36,109,0.3303,0.28125,0.19393,0.14286,0.2857,0.28571,0.0,1.0,1,1,0,1,0,12,0,0,14,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[40,109,0.367,0.32589,0.212,0.24999,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,8,0,0,18,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[44,109,0.4037,0.24107,0.0974,0.24999,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,5,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,109,0.4404,0.2991,0.16115,0.14286,0.2857,0.28571,0.14286,0.85714,0,0,0,0,0,9,0,0,18,0,0,0,0,0,4,0,0,0,0,0,1,0,0],[52,109,0.4771,0.24107,0.12078,0.14286,0.2143,0.28571,0.14286,0.57143,0,0,0,0,0,16,0,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[56,109,0.5138,0.28123,0.16933,0.14286,0.2857,0.28571,0.14286,1.0,0,1,0,0,0,11,0,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[60,109,0.5505,0.30348,0.23629,0.14286,0.28571,0.28571,0.14,1.0,0,2,0,0,0,14,0,0,13,0,0,0,0,0,2,0,0,0,0,0,1,0,2],[64,109,0.5872,0.33035,0.1729,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,5,0,0,21,0,0,0,0,0,5,0,0,0,0,0,0,0,1],[68,109,0.6239,0.23661,0.12169,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,14,0,0,14,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[72,109,0.6606,0.29464,0.15126,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,8,0,0,17,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[76,109,0.6972,0.28125,0.20666,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,17,0,0,9,0,0,0,0,0,4,0,0,1,0,0,0,0,1],[80,109,0.7339,0.29911,0.21829,0.14286,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,13,0,0,14,0,0,0,0,0,3,0,0,0,0,0,0,0,2],[84,109,0.7706,0.25446,0.13709,0.14286,0.2857,0.28571,0.14286,0.57143,0,0,0,0,0,15,0,0,13,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[88,109,0.8073,0.25446,0.18808,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,16,0,0,11,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[92,109,0.844,0.25893,0.19045,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,18,0,0,9,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[96,109,0.8807,0.21875,0.07129,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,109,0.9174,0.21875,0.07973,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,109,0.9541,0.21196,0.09703,0.14286,0.14286,0.28571,0.07143,0.57143,0,0,0,0,1,17,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[108,109,0.9908,0.22321,0.10677,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,12,0,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[109,109,1.0,0.16071,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.11607,"x":0.37946,"p":[[0,71,0.0,0.16053,0.10568,0.14214,0.14286,0.2857,0.0,0.42857,6,0,0,6,0,17,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.30803,0.17536,0.24999,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,8,0,0,19,0,0,1,0,0,2,0,0,0,0,0,2,0,0],[8,71,0.1127,0.27679,0.13803,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,9,0,0,17,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[12,71,0.169,0.29009,0.14943,0.14286,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,8,0,0,17,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[16,71,0.2254,0.37946,0.24643,0.2857,0.28571,0.28571,0.0,1.0,1,3,0,1,0,2,0,0,22,0,0,0,0,0,3,0,0,0,0,0,1,0,3],[20,71,0.2817,0.27232,0.16115,0.14286,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,11,0,0,16,0,0,0,0,0,2,0,0,2,0,0,0,0,0],[24,71,0.338,0.29461,0.15537,0.14286,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,9,0,0,18,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[28,71,0.3944,0.30804,0.17168,0.25,0.28571,0.32164,0.0,0.71429,2,0,0,2,0,6,0,0,16,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[32,71,0.4507,0.31695,0.20742,0.14286,0.28571,0.42857,0.0,1.0,3,1,0,3,0,6,0,0,14,0,0,2,0,0,6,0,0,0,0,0,0,0,1],[36,71,0.507,0.29911,0.17261,0.14286,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,9,0,0,15,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[40,71,0.5634,0.21873,0.1236,0.14286,0.21428,0.28571,0.0,0.571,3,0,0,3,0,13,0,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,71,0.6197,0.27223,0.13063,0.14286,0.28571,0.28571,0.14,0.57143,0,0,0,0,0,11,0,0,17,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[48,71,0.6761,0.21429,0.10715,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,17,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,71,0.7324,0.2366,0.13175,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,9,0,0,18,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[56,71,0.7887,0.25,0.12372,0.14286,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,11,0,0,17,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[60,71,0.8451,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.16062,0.07786,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,22,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,71,0.9577,0.1383,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3176a88b86ec6bc8","q":"Prove that there is no positive integer $n$ with the following property: For $k=1,2, \\ldots, 9$ the leftmost digit - in decimal notation - of $(n+k)!$ is $k$.","t":[{"b":3,"e":0.571,"k":"rising","v":0.10259,"x":0.79464,"p":[[0,138,0.0,0.10259,0.19958,0.0,0.0,0.14073,0.0,0.85714,23,0,3,23,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,138,0.029,0.6339,0.30709,0.28571,0.71429,0.85714,0.0,1.0,2,5,2,2,0,2,0,0,5,0,0,0,0,0,3,0,0,7,0,0,8,0,5],[8,138,0.058,0.50445,0.34807,0.14286,0.57143,0.71429,0.0,1.0,7,4,7,7,0,2,0,0,2,0,0,3,0,0,3,0,0,8,0,0,3,0,4],[12,138,0.087,0.6875,0.32818,0.42857,0.78571,1.0,0.0,1.0,2,12,2,2,0,2,0,0,2,0,0,4,0,0,2,0,0,4,0,0,4,0,12],[16,138,0.1159,0.54909,0.35912,0.25,0.57143,0.85714,0.0,1.0,6,7,4,6,0,2,0,0,3,0,0,0,0,0,6,0,0,6,0,0,2,0,7],[20,138,0.1449,0.69642,0.32878,0.53571,0.85714,1.0,0.0,1.0,1,12,0,1,0,4,0,0,2,0,0,1,0,0,5,0,0,1,0,0,6,0,12],[24,138,0.1739,0.6339,0.31529,0.42857,0.71429,0.85714,0.0,1.0,3,6,0,3,0,2,0,0,0,0,0,6,0,0,3,0,0,4,0,0,8,0,6],[28,138,0.2029,0.61607,0.31428,0.39286,0.64286,0.85714,0.0,1.0,1,7,0,1,0,5,0,0,2,0,0,2,0,0,6,0,0,4,0,0,5,0,7],[32,138,0.2319,0.65177,0.31327,0.39286,0.71429,1.0,0.0,1.0,1,10,0,1,0,2,0,0,5,0,0,3,0,0,3,0,0,5,0,0,3,0,10],[36,138,0.2609,0.58929,0.35129,0.28571,0.57143,1.0,0.0,1.0,2,9,1,2,0,5,0,0,4,0,0,2,0,0,4,0,0,2,0,0,4,0,9],[40,138,0.2899,0.64729,0.31132,0.42857,0.71429,0.89286,0.0,1.0,3,8,0,3,0,1,0,0,1,0,0,4,0,0,6,0,0,4,0,0,5,0,8],[44,138,0.3188,0.51768,0.36745,0.14286,0.5,0.89286,0.0,1.0,3,8,0,3,0,8,0,0,2,0,0,3,0,0,4,0,0,1,0,0,3,0,8],[48,138,0.3478,0.47766,0.34183,0.14286,0.42857,0.85714,0.0,1.0,3,5,0,3,0,7,0,0,5,0,0,3,0,0,3,0,0,2,0,0,4,0,5],[52,138,0.3768,0.55353,0.31894,0.28571,0.57143,0.85714,0.0,1.0,3,5,0,3,0,4,0,0,2,0,0,2,0,0,10,0,0,1,0,0,5,0,5],[56,138,0.4058,0.51339,0.28984,0.28571,0.50001,0.71429,0.0,1.0,2,5,0,2,0,2,0,0,8,0,0,4,0,0,5,0,0,6,0,0,0,0,5],[60,138,0.4348,0.51338,0.33476,0.14286,0.49979,0.85714,0.0,1.0,1,6,0,1,0,8,0,0,5,0,0,2,0,0,5,0,0,1,0,0,4,0,6],[64,138,0.4638,0.64732,0.34253,0.28571,0.64286,1.0,0.0,1.0,1,12,0,1,0,4,0,0,5,0,0,0,0,0,6,0,0,1,0,0,3,0,12],[68,138,0.4928,0.60938,0.30825,0.42857,0.64286,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,2,0,0,3,0,1,5,0,0,6,0,0,3,0,7],[72,138,0.5217,0.57143,0.35535,0.25,0.64286,1.0,0.0,1.0,2,9,0,2,0,6,0,0,4,0,0,2,0,0,2,0,0,5,0,0,2,0,9],[76,138,0.5507,0.68749,0.29974,0.42859,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,5,0,0,2,0,0,6,0,0,2,0,0,3,0,12],[80,138,0.5797,0.60267,0.30037,0.42857,0.57143,0.85714,0.0,1.0,1,4,0,1,0,5,0,0,1,0,0,4,0,0,6,0,0,2,0,0,9,0,4],[84,138,0.6087,0.58034,0.32525,0.42857,0.57143,0.85714,0.0,1.0,4,7,0,4,0,1,0,0,2,0,0,6,0,0,5,0,0,4,0,0,3,0,7],[88,138,0.6377,0.58925,0.30671,0.39286,0.57143,0.85714,0.0,1.0,1,7,0,1,0,5,0,0,2,0,0,3,0,0,7,0,0,5,0,0,2,0,7],[92,138,0.6667,0.74998,0.26001,0.57143,0.85714,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,3,0,0,4,0,0,4,0,0,7,0,11],[96,138,0.6957,0.71427,0.28795,0.67857,0.71429,1.0,0.0,1.0,1,9,0,1,0,3,0,0,1,0,0,1,0,0,2,0,0,9,0,0,6,0,9],[100,138,0.7246,0.62722,0.32866,0.39286,0.71429,1.0,0.0,1.0,3,9,0,3,0,1,0,0,4,0,0,3,0,0,3,1,0,5,0,0,3,0,9],[104,138,0.7536,0.6249,0.33468,0.42859,0.71429,1.0,0.0,1.0,3,9,0,3,0,3,0,0,1,0,0,3,0,0,5,0,0,5,0,0,3,0,9],[108,138,0.7826,0.72768,0.25843,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,3,0,0,5,0,0,7,0,0,3,0,11],[112,138,0.8116,0.74998,0.29015,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,0,0,0,1,0,0,6,0,0,3,0,0,4,0,14],[116,138,0.8406,0.79464,0.28557,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,2,0,0,0,0,0,2,0,0,2,0,0,5,0,0,3,0,17],[120,138,0.8696,0.69195,0.30328,0.57132,0.78564,1.0,0.14286,1.0,0,10,0,0,0,5,0,0,1,0,0,1,0,0,6,0,0,3,0,0,6,0,10],[124,138,0.8986,0.63837,0.30301,0.42857,0.64286,0.89286,0.0,1.0,2,8,0,2,0,1,0,0,4,0,0,2,0,0,7,0,0,4,0,0,4,0,8],[128,138,0.9275,0.74107,0.2889,0.67857,0.85707,1.0,0.0,1.0,2,11,0,2,0,1,0,0,1,0,0,1,0,0,3,0,0,7,0,0,6,0,11],[132,138,0.9565,0.58927,0.28959,0.28571,0.57143,0.74996,0.0,1.0,1,6,0,1,0,3,0,0,5,0,0,1,0,0,8,0,0,6,0,0,2,0,6],[136,138,0.9855,0.54461,0.26106,0.39286,0.57121,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,4,0,0,6,0,0,6,0,0,5,0,0,5,0,2],[138,138,1.0,0.56691,0.16164,0.42857,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,7,0,0,14,0,0,5,0,0,2,0,1]]},{"b":4,"e":0.85714,"k":"rising","v":0.16069,"x":0.79909,"p":[[0,135,0.0,0.16069,0.26422,0.0,0.0,0.17857,0.0,0.85714,21,0,1,21,0,3,0,0,1,0,0,1,0,0,3,0,0,2,0,0,1,0,0],[4,135,0.0296,0.6071,0.34993,0.42857,0.57143,1.0,0.0,1.0,6,9,3,6,0,0,0,0,0,0,0,3,0,0,9,0,0,2,0,0,3,0,9],[8,135,0.0593,0.68301,0.33642,0.42857,0.85707,1.0,0.0,1.0,3,10,2,3,0,2,0,0,1,0,0,3,0,0,3,0,0,2,0,0,8,0,10],[12,135,0.0889,0.55793,0.30601,0.28571,0.42859,0.85714,0.0,1.0,1,6,1,1,0,3,0,0,6,0,0,7,0,0,2,0,0,3,0,0,4,0,6],[16,135,0.1185,0.52678,0.32426,0.28592,0.57121,0.75,0.0,1.0,4,7,3,4,0,2,0,0,3,0,0,6,0,0,8,0,0,1,0,0,1,0,7],[20,135,0.1481,0.66516,0.27805,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,1,0,0,1,0,0,8,0,0,4,0,0,5,0,0,3,0,9],[24,135,0.1778,0.56696,0.35622,0.1429,0.57143,1.0,0.0,1.0,3,9,0,3,0,6,0,0,1,0,0,2,0,0,7,0,0,2,0,0,2,0,9],[28,135,0.2074,0.66516,0.35645,0.39286,0.78571,1.0,0.0,1.0,3,13,0,3,0,2,0,0,3,0,0,3,0,0,2,0,0,3,0,0,3,0,13],[32,135,0.237,0.57589,0.28004,0.39285,0.57143,0.75,0.0,1.0,1,4,0,1,0,3,0,0,4,0,0,5,0,0,4,0,0,7,0,0,4,0,4],[36,135,0.2667,0.62946,0.30275,0.42857,0.71429,0.85714,0.0,1.0,2,6,0,2,0,2,0,0,3,0,0,3,0,0,4,0,0,6,0,0,6,0,6],[40,135,0.2963,0.5982,0.27994,0.42857,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,5,0,0,2,0,0,4,0,0,6,0,0,6,0,0,4,0,5],[44,135,0.3259,0.66516,0.29582,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,2,0,0,5,0,0,5,0,0,3,0,0,5,0,9],[48,135,0.3556,0.67855,0.30094,0.57132,0.71429,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,0,0,0,2,0,0,3,0,0,8,0,0,7,0,7],[52,135,0.3852,0.6607,0.28739,0.53539,0.71429,0.85714,0.0,1.0,2,6,1,2,0,2,0,0,0,0,0,4,0,0,5,0,0,6,0,0,7,0,6],[56,135,0.4148,0.79909,0.20158,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,4,0,0,11,0,10],[60,135,0.4444,0.73213,0.19481,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,7,0,0,8,0,6],[64,135,0.4741,0.6964,0.23078,0.571,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,7,0,0,5,0,7],[68,135,0.5037,0.6607,0.2896,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,2,0,0,4,0,0,6,0,0,5,0,0,3,0,9],[72,135,0.5333,0.66959,0.23809,0.571,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,2,0,0,1,0,0,3,0,0,11,0,0,2,0,0,8,0,5],[76,135,0.563,0.62946,0.2854,0.53571,0.64286,0.85714,0.0,1.0,1,5,1,1,0,3,0,0,3,0,0,1,0,0,8,0,0,4,0,0,7,0,5],[80,135,0.5926,0.73658,0.22337,0.571,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,4,0,0,5,0,10],[84,135,0.6222,0.75443,0.22656,0.57143,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,8,0,0,5,0,10],[88,135,0.6519,0.68748,0.23267,0.571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,1,0,0,5,0,0,6,0,0,5,0,0,10,0,4],[92,135,0.6815,0.74997,0.18901,0.57143,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,7,0,0,9,0,6],[96,135,0.7111,0.75892,0.19703,0.71429,0.85707,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,3,0,0,3,0,0,7,0,0,13,0,5],[100,135,0.7407,0.73213,0.2442,0.4286,0.85707,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,8,0,0,2,0,0,4,0,0,8,0,9],[104,135,0.7704,0.66057,0.23644,0.571,0.71429,0.85714,0.14,1.0,0,6,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,8,0,0,3,0,6],[108,135,0.8,0.69643,0.24419,0.4286,0.71429,0.85714,0.1429,1.0,0,7,0,0,0,1,0,0,1,0,0,8,0,0,2,0,0,6,0,0,7,0,7],[112,135,0.8296,0.65618,0.22265,0.571,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,8,0,0,7,0,3],[116,135,0.8593,0.6607,0.22517,0.57132,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,1,0,0,4,0,0,7,0,0,7,0,0,8,0,3],[120,135,0.8889,0.63842,0.23682,0.4286,0.64286,0.85704,0.1429,1.0,0,5,0,0,0,1,0,0,3,0,0,6,0,0,6,0,0,7,0,0,4,0,5],[124,135,0.9185,0.63388,0.2447,0.5354,0.57143,0.85714,0.0,1.0,1,5,1,1,0,1,0,0,1,0,0,5,0,0,10,0,0,5,0,0,4,0,5],[128,135,0.9481,0.55801,0.21534,0.42859,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,1,0,0,8,0,0,8,0,0,7,0,0,5,0,0],[132,135,0.9778,0.62938,0.17078,0.571,0.57143,0.75,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,2,0,0,14,0,0,5,0,0,8,0,0],[135,135,1.0,0.63384,0.18879,0.571,0.64286,0.857,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,3,0,0,9,0,0,7,0,0,9,0,0]]}]},{"i":"0e9faee737998014","q":"Let  $n_1,n_2, \\cdots, n_{26}$  be pairwise distinct positive integers satisfying\n(1) for each  $n_i$ , its digits belong to the set  $\\{1,2\\}$ ;\n(2) for each  $i,j$ ,  $n_i$  can't be obtained from  $n_j$  by adding some digits on the right.\nFind the smallest possible value of  $\\sum_{i=1}^{26} S(n_i)$ , where  $S(m)$  denotes the sum of all digits of a positive integer  $m$ .","t":[{"b":1,"e":0.28571,"k":"rising","v":0.09366,"x":0.35714,"p":[[0,100,0.0,0.09366,0.10475,0.0,0.14143,0.14286,0.0,0.42857,15,0,7,15,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,100,0.04,0.10714,0.13832,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,9,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,100,0.08,0.14285,0.15564,0.0,0.14286,0.2857,0.0,0.571,14,0,1,14,0,8,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,100,0.12,0.10268,0.1394,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,8,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,100,0.16,0.24545,0.21503,0.14214,0.14286,0.42857,0.0,1.0,7,1,0,7,0,10,0,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[20,100,0.2,0.22767,0.1509,0.14286,0.2857,0.28571,0.0,0.57143,5,0,0,5,0,10,0,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[24,100,0.24,0.26339,0.19597,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,8,0,0,7,0,0,9,0,0,1,0,0,0,0,0,1,0,0],[28,100,0.28,0.30359,0.16268,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,9,0,0,8,0,0,9,0,0,4,0,0,0,0,0,0,0,0],[32,100,0.32,0.25446,0.15866,0.14286,0.2857,0.32143,0.0,0.57143,5,0,0,5,0,7,0,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[36,100,0.36,0.27678,0.14698,0.14286,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,6,0,0,16,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[40,100,0.4,0.25445,0.14607,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,12,0,0,12,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[44,100,0.44,0.25444,0.14604,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,12,0,0,12,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[48,100,0.48,0.22768,0.15916,0.14286,0.2857,0.42857,0.0,0.4286,7,0,0,7,0,8,0,0,8,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[52,100,0.52,0.25,0.12372,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,10,0,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[56,100,0.56,0.20981,0.11832,0.14286,0.14286,0.2857,0.0,0.571,2,0,0,2,0,17,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[60,100,0.6,0.25893,0.16536,0.14286,0.2143,0.42857,0.0,0.57143,3,0,0,3,0,13,0,0,6,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[64,100,0.64,0.30801,0.15608,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,10,0,0,8,0,0,9,0,0,4,0,0,0,0,0,0,0,0],[68,100,0.68,0.33036,0.12079,0.2857,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,9,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[72,100,0.72,0.26338,0.17166,0.14286,0.2857,0.42857,0.0,0.57143,4,0,0,4,0,10,0,0,9,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[76,100,0.76,0.29018,0.14054,0.14286,0.2857,0.42857,0.0,0.57143,2,0,0,2,0,7,0,0,13,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[80,100,0.8,0.35256,0.13366,0.2857,0.28571,0.42858,0.14,0.57143,0,0,0,0,0,5,0,0,12,0,0,10,0,0,5,0,0,0,0,0,0,0,0],[84,100,0.84,0.35712,0.13828,0.28571,0.42857,0.42858,0.14286,0.57143,0,0,0,0,0,6,0,0,9,0,0,12,0,0,5,0,0,0,0,0,0,0,0],[88,100,0.88,0.35714,0.14286,0.2857,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,10,0,0,10,0,0,6,0,0,0,0,0,0,0,0],[92,100,0.92,0.3125,0.15746,0.24999,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,6,0,0,13,0,0,6,0,0,5,0,0,0,0,0,0,0,0],[96,100,0.96,0.32588,0.13472,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,13,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[100,100,1.0,0.3392,0.13257,0.2857,0.35714,0.42857,0.14,0.57143,0,0,0,0,0,7,0,0,9,0,0,13,0,0,3,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.04018,"x":0.12945,"p":[[0,25,0.0,0.09375,0.12169,0.0,0.0,0.14286,0.0,0.42857,17,0,9,17,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.09375,0.12169,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.12945,0.16502,0.0,0.0,0.2857,0.0,0.57143,17,0,0,17,0,5,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,25,0.48,0.10714,0.15972,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,2,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b681be159cfd85ce","q":"A real number  $a$  is chosen randomly and uniformly from the interval  $[-20, 18]$ . The probability that the roots of the polynomial \\[x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2\\] are all real can be written in the form  $\\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .","t":[{"b":6,"e":0.57143,"k":"volatile","v":0.44643,"x":0.92857,"p":[[0,11,0.0,0.92857,0.18898,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,28],[4,11,0.3636,0.83929,0.24157,0.57143,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,6,0,0,4,0,0,0,0,0,0,0,22],[8,11,0.7273,0.46875,0.12492,0.42857,0.42857,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,20,0,0,8,0,0,0,0,0,0,0,1],[11,11,1.0,0.44643,0.07784,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,3,0,0,22,0,0,7,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.92857,"x":1.0,"p":[[0,55,0.0,0.92857,0.24484,1.0,1.0,1.0,0.0,1.0,2,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[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],[8,55,0.1455,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"b741092135531e2d","q":"In a party, each person knew exactly  $ 22$  other persons. For each two persons  $ X$  and  $ Y$ , if  $ X$  and  $ Y$  knew each other, there is no other person who knew both of them, and if  $ X$  and  $ Y$  did not know each other, there are exactly  $ 6$  persons who knew both of them. Assume that  $ X$  knew  $ Y$  iff  $ Y$  knew  $ X$ . How many people did attend the party?\r\n*Yudi Satria, Jakarta*","t":[{"b":0,"e":1.0,"k":"flat","v":0.97321,"x":1.0,"p":[[0,13,0.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[4,13,0.3077,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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]]},{"b":2,"e":1.0,"k":"flat","v":0.94196,"x":1.0,"p":[[0,12,0.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[4,12,0.3333,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[8,12,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,12,1.0,0.94196,0.07873,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20]]}]},{"i":"659cacdf8ddaa71d","q":"Prove the following inequality:\r\n\\[\\prod^k_{i=1} x_i \\cdot \\sum^k_{i=1} x^{n-1}_i \\leq \\sum^k_{i=1}\r\nx^{n+k-1}_i,\\] where  $x_i > 0,$   $k \\in \\mathbb{N}, n \\in\r\n\\mathbb{N}.$","t":[{"b":5,"e":1.0,"k":"rising","v":0.40179,"x":1.0,"p":[[0,31,0.0,0.50893,0.4642,0.0,0.57143,1.0,0.0,1.0,12,13,0,12,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,13],[4,31,0.129,0.64286,0.46702,0.0,1.0,1.0,0.0,1.0,10,20,0,10,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20],[8,31,0.2581,0.40179,0.47974,0.0,0.0,1.0,0.0,1.0,18,12,0,18,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12],[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],[24,31,0.7742,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[31,31,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.65179,"p":[[0,35,0.0,0.65179,0.46006,0.0,1.0,1.0,0.0,1.0,9,20,0,9,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,20],[4,35,0.1143,0.57588,0.46632,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,17],[8,35,0.2286,0.4732,0.47573,0.0,0.35693,1.0,0.0,1.0,15,13,0,15,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,13],[12,35,0.3429,0.5625,0.4642,0.0,0.85714,1.0,0.0,1.0,11,15,0,11,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,15],[16,35,0.4571,0.41964,0.48173,0.0,0.0,1.0,0.0,1.0,18,12,0,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,12],[20,35,0.5714,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,35,0.6857,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"76ad0244ecfae34f","q":"Rectangle  $ABCD$  has side lengths  $AB=84$  and  $AD=42$ . Point  $M$  is the midpoint of  $\\overline{AD}$ , point  $N$  is the trisection point of  $\\overline{AB}$  closer to  $A$ , and point  $O$  is the intersection of  $\\overline{CM}$  and  $\\overline{DN}$ . Point  $P$  lies on the quadrilateral  $BCON$ , and  $\\overline{BP}$  bisects the area of  $BCON$ . Find the area of  $\\triangle{CDP}$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.95536,"x":1.0,"p":[[0,31,0.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[4,31,0.129,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":4,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,62,0.0,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[4,62,0.0645,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,62,0.2581,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,62,0.3871,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,62,0.4516,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,62,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,62,0.5806,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,62,0.6452,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[48,62,0.7742,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[52,62,0.8387,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"0164a59670ad2185","q":"Four points  $B,E,A,F$  lie on line  $AB$  in order, four points  $C,G,D,H$  lie on line  $CD$  in order, satisfying:  $$ \\frac{AE}{EB}=\\frac{AF}{FB}=\\frac{DG}{GC}=\\frac{DH}{HC}=\\frac{AD}{BC}. $$  Prove that  $FH\\perp EG$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.04911,"p":[[0,20,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.02232,0.0724,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.04911,0.18074,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.02223,"p":[[0,19,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.02223,0.06281,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5f88fb413fd1f01d","q":"Suppose that  $X$  is a compact metric space and  $T: X\\rightarrow X$  is a continous function. Prove that  $T$  has a returning point. It means there is a strictly increasing sequence  $n_i$  such that  $\\lim_{k\\rightarrow \\infty} T^{n_k}(x_0)=x_0$  for some  $x_0$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.01777,"x":0.04911,"p":[[0,38,0.0,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,38,0.4211,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.07589,"p":[[0,32,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.07589,0.17491,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,7,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[8,32,0.25,0.07143,0.18898,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,32,0.375,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c0e18cd9c229ff9c","q":"Let  $ABC$  be a triangle such that  $|AB|=13 , |BC|=12$  and  $|CA|=5$ . Let the angle bisectors of  $A$  and  $B$  intersect at  $I$  and meet the opposing sides at  $D$  and  $E$ , respectively. The line passing through  $I$  and the midpoint of  $[DE]$  meets  $[AB]$  at  $F$ . What is  $|AF|$ ?     $ \n\\textbf{(A)}\\ \\dfrac{3}{2}\n\\qquad\\textbf{(B)}\\ 2\n\\qquad\\textbf{(C)}\\ \\dfrac{5}{2}\n\\qquad\\textbf{(D)}\\ 3\n\\qquad\\textbf{(E)}\\ \\dfrac{7}{2}\n$","t":[{"b":0,"e":1.0,"k":"flat","v":0.82589,"x":0.97321,"p":[[0,48,0.0,0.85268,0.27313,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,3,0,0,1,0,23],[4,48,0.0833,0.88393,0.26351,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,26],[8,48,0.1667,0.82589,0.25935,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,3,0,0,0,0,0,2,0,0,6,0,0,0,0,20],[12,48,0.25,0.92411,0.21424,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,27],[16,48,0.3333,0.92411,0.15966,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,25],[20,48,0.4167,0.875,0.27374,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,26],[24,48,0.5,0.9375,0.19541,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[28,48,0.5833,0.89732,0.26782,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,27],[32,48,0.6667,0.92411,0.20198,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,27],[36,48,0.75,0.93304,0.1988,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[40,48,0.8333,0.91964,0.21998,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,27],[44,48,0.9167,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[48,48,1.0,0.92411,0.20198,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,27]]},{"b":4,"e":1.0,"k":"flat","v":0.76786,"x":0.99107,"p":[[0,58,0.0,0.82143,0.32143,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[4,58,0.069,0.86161,0.29556,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,26],[8,58,0.1379,0.87946,0.25282,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,25],[12,58,0.2069,0.85714,0.25254,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,22],[16,58,0.2759,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,58,0.3448,0.92857,0.18898,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,27],[24,58,0.4138,0.90625,0.21609,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,26],[28,58,0.4828,0.87054,0.28203,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,26],[32,58,0.5517,0.86607,0.24206,0.71429,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,0,0,23],[36,58,0.6207,0.88393,0.26351,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,26],[40,58,0.6897,0.82589,0.28956,0.71429,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,4,0,0,0,0,22],[44,58,0.7586,0.76786,0.34023,0.28571,1.0,1.0,0.14286,1.0,0,21,0,0,0,3,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,21],[48,58,0.8276,0.84375,0.26573,0.71429,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,4,0,0,2,0,21],[52,58,0.8966,0.90625,0.21609,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,26],[56,58,0.9655,0.91517,0.19518,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,0,0,26],[58,58,1.0,0.85268,0.25874,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,3,0,0,2,0,22]]}]},{"i":"2e61d56a5eefd062","q":"Let $\\alpha, \\beta, \\gamma$ be the angles of a triangle opposite to its sides with lengths $a, b$ and $c$, respectively. Prove the inequality\n\n$$\na \\cdot\\left(\\frac{1}{\\beta}+\\frac{1}{\\gamma}\\right)+b \\cdot\\left(\\frac{1}{\\gamma}+\\frac{1}{\\alpha}\\right)+c \\cdot\\left(\\frac{1}{\\alpha}+\\frac{1}{\\beta}\\right) \\geq 2 \\cdot\\left(\\frac{a}{\\alpha}+\\frac{b}{\\beta}+\\frac{c}{\\gamma}\\right) \\cdot\n$$","t":[{"b":4,"e":1.0,"k":"rising","v":0.58928,"x":0.98661,"p":[[0,16,0.0,0.58928,0.4237,0.14286,0.85707,1.0,0.0,1.0,5,14,0,5,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,3,0,14],[4,16,0.25,0.60714,0.41342,0.14286,0.85714,1.0,0.0,1.0,5,12,0,5,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,0,6,0,12],[8,16,0.5,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[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],[16,16,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":7,"e":1.0,"k":"rising","v":0.49107,"x":0.99554,"p":[[0,13,0.0,0.49107,0.39599,0.14286,0.35714,1.0,0.0,1.0,3,10,0,3,0,11,0,0,2,0,0,3,0,0,1,0,0,0,0,0,2,0,10],[4,13,0.3077,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[8,13,0.6154,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,13,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[13,13,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"3c14dab8bbfe2cce","q":"Show that, given any 2 -configuration of a set $A$, every element of $A$ belongs to exactly one cell.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.32586,"p":[[0,87,0.0,0.13838,0.28899,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,1,0,1],[4,87,0.046,0.32586,0.39645,0.0,0.0,0.60714,0.0,1.0,18,5,0,18,0,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,1,0,5],[8,87,0.092,0.07589,0.23954,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[12,87,0.1379,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,87,0.1839,0.08929,0.25191,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[20,87,0.2299,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,87,0.2759,0.04017,0.15659,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[28,87,0.3218,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[32,87,0.3678,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,87,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,87,0.4598,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,87,0.5057,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,87,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,87,0.5977,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,87,0.6437,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,87,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,87,0.7356,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,87,0.7816,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,87,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,87,0.8736,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,87,0.9195,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,87,0.9655,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[87,87,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"volatile","v":0.12499,"x":0.68292,"p":[[0,11,0.0,0.12499,0.27604,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,2],[4,11,0.3636,0.13392,0.28331,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,1],[8,11,0.7273,0.68292,0.30053,0.5354,0.71429,1.0,0.0,1.0,2,10,0,2,0,1,0,0,2,0,0,3,0,0,4,0,0,7,0,0,3,0,10],[11,11,1.0,0.64058,0.33763,0.42859,0.71429,1.0,0.0,1.0,3,10,0,3,0,3,0,0,1,0,0,2,0,0,5,0,0,5,1,0,2,0,10]]}]},{"i":"b868558d226e52c8","q":"A set  $S$  is called perfect if it has the following two properties:\na)  $S$  has exactly four elements\nb) for every element  $x$  of  $S$ , at least one of the numbers  $x - 1$  or  $x+1$  belongs to  $S$ .\nFind the number of all perfect subsets of the set  $\\{1,2,... ,n\\}$","t":[{"b":1,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,64,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,64,0.0625,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,64,0.125,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,64,0.1875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,64,0.25,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,64,0.3125,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[24,64,0.375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,64,0.4375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,64,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,64,0.5625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,64,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,64,0.6875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[48,64,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[52,64,0.8125,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[56,64,0.875,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,64,0.9375,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,64,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.98661,"x":1.0,"p":[[0,11,0.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[4,11,0.3636,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[11,11,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"a83dfa6a64bd8301","q":"$AB$  and  $CD$  are two parallel chords of a parabola. Circle  $S_1$  passing through points  $A,B$  intersects circle  $S_2$  passing through  $C,D$  at points  $E,F$ . Prove that if  $E$  belongs to the parabola, then  $F$  also belongs to the parabola.\n\nI.Voronovich","t":[{"b":1,"e":1.0,"k":"flat","v":0.75444,"x":0.99107,"p":[[0,14,0.0,0.9017,0.25391,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,26],[4,14,0.2857,0.83482,0.30745,0.82132,1.0,1.0,0.0,1.0,3,21,3,3,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,3,0,21],[8,14,0.5714,0.75444,0.27718,0.5354,0.857,1.0,0.0,1.0,1,13,1,1,0,1,0,0,0,0,0,6,0,0,1,0,0,5,0,0,5,0,13],[12,14,0.8571,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[14,14,1.0,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":5,"e":1.0,"k":"flat","v":0.96428,"x":1.0,"p":[[0,22,0.0,0.96428,0.12878,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[4,22,0.1818,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[8,22,0.3636,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[12,22,0.5455,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,22,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,22,0.9091,0.98659,0.04168,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[22,22,1.0,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29]]}]},{"i":"08012bcf1b0bba9b","q":"Let's consider three pairwise non-parallel straight lines in the plane. Three points are moving along these lines with different non-zero velocities, one on each line (we consider the movement as having taken place for infinite time and continuing infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some \"zero\" moment in such a way that the points never were, are or will be collinear?","t":[{"b":0,"e":1.0,"k":"flat","v":0.93747,"x":1.0,"p":[[0,18,0.0,0.93747,0.12347,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[4,18,0.2222,0.95981,0.1143,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[8,18,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[12,18,0.6667,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":1.0,"k":"flat","v":0.90176,"x":1.0,"p":[[0,15,0.0,0.90176,0.15751,0.82143,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,22],[4,15,0.2667,0.96429,0.13363,1.0,1.0,1.0,0.2857,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[8,15,0.5333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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],[15,15,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"b5d3a5fdbb875083","q":"**1.** Let   $a_{v} $  and   $b_{v} $ ,   ${v= 1,2,\\dots,n} $ , be real numbers such that \n  $a_{1}\\geq a_{2} \\geq a_{3}\\geq\\dots\\geq a_{n}> 0 $  and \n  $b_{1}\\geq a_{1}, b_{1}b_{2}\\geq a_{1}a_{2},\\dots,b_{1}b_{2}\\dots b_{n}\\geq a_{1}a_{2}\\dots a_{n} $ Show that   $b_{1}+b_{2}+\\dots+b_{n}\\geq a_{1}+a_{2}+\\dots+a_{n} $  **(S. 4)**","t":[{"b":0,"e":0.28571,"k":"flat","v":0.28125,"x":0.40625,"p":[[0,16,0.0,0.34821,0.34244,0.0,0.28571,0.28571,0.0,1.0,9,6,0,9,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,6],[4,16,0.25,0.40625,0.27919,0.2857,0.28571,0.28571,0.0,1.0,1,5,0,1,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[8,16,0.5,0.29018,0.145,0.28571,0.28571,0.28571,0.0,1.0,2,1,0,2,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,16,0.75,0.28125,0.05629,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.28125,0.15355,0.28571,0.28571,0.28571,0.0,1.0,3,1,0,3,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":2,"e":0.28571,"k":"flat","v":0.23214,"x":0.625,"p":[[0,35,0.0,0.29911,0.37177,0.0,0.28571,0.28571,0.0,1.0,15,5,0,15,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,2,0,5],[4,35,0.1143,0.59373,0.36962,0.28571,0.35714,1.0,0.0,1.0,2,14,0,2,0,0,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,14],[8,35,0.2286,0.625,0.38091,0.28571,0.64286,1.0,0.0,1.0,2,16,0,2,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[12,35,0.3429,0.45535,0.3489,0.28571,0.28571,0.89275,0.0,1.0,4,8,0,4,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,8],[16,35,0.4571,0.4241,0.31234,0.2857,0.28571,0.28571,0.0,1.0,2,7,0,2,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[20,35,0.5714,0.41964,0.31529,0.28571,0.28571,0.28571,0.0,1.0,2,7,0,2,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[24,35,0.6857,0.41518,0.27976,0.28571,0.28571,0.28571,0.0,1.0,1,5,0,1,0,0,0,0,24,0,0,0,0,0,1,0,0,0,0,0,1,0,5],[28,35,0.8,0.23214,0.11152,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.26784,0.09938,0.2857,0.28571,0.28571,0.0,0.571,3,0,0,3,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"4019d3fe894168a1","q":"Suppose $0<a \\leq b$ and $4 \\nmid m n$. Prove that the number of ways in which an $m \\times n$ rectangle can be partitioned into dominoes of type ( $a, b$ ) is even.","t":[{"b":2,"e":0.42857,"k":"rising","v":0.0892,"x":0.45982,"p":[[0,25,0.0,0.0892,0.14171,0.0,0.0,0.17857,0.0,0.4286,22,0,1,22,0,2,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.45982,0.3707,0.10714,0.42857,0.71429,0.0,1.0,8,7,0,8,0,2,0,0,4,0,0,5,0,0,1,0,0,5,0,0,0,0,7],[8,25,0.32,0.42409,0.2912,0.2857,0.42857,0.57143,0.0,1.0,5,4,0,5,0,1,0,0,8,0,0,8,0,0,4,0,0,2,0,0,0,0,4],[12,25,0.48,0.44197,0.2,0.28571,0.42857,0.46431,0.0,1.0,1,2,0,1,0,0,0,0,10,0,0,13,0,0,4,0,0,2,0,0,0,0,2],[16,25,0.64,0.40625,0.24251,0.2857,0.42857,0.42858,0.0,1.0,3,1,0,3,0,3,0,0,7,0,0,12,0,0,1,0,0,3,0,0,2,0,1],[20,25,0.8,0.38838,0.15249,0.28571,0.42857,0.42858,0.0,0.71429,2,0,0,2,0,1,0,0,7,0,0,18,0,0,2,0,0,2,0,0,0,0,0],[24,25,0.96,0.41964,0.18877,0.28571,0.42857,0.4286,0.0,1.0,1,1,0,1,0,0,0,0,12,0,0,13,0,0,2,0,0,2,0,0,1,0,1],[25,25,1.0,0.40182,0.14032,0.28571,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,0,0,0,9,0,0,19,0,0,1,0,0,1,0,0,1,0,0]]},{"b":3,"e":0.42857,"k":"rising","v":0.11607,"x":0.66071,"p":[[0,32,0.0,0.11607,0.22428,0.0,0.0,0.17868,0.0,1.0,23,1,0,23,0,1,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[4,32,0.125,0.22772,0.31718,0.0,0.0,0.42857,0.0,1.0,18,3,0,18,0,1,0,0,3,0,0,6,0,0,0,0,0,1,0,0,0,0,3],[8,32,0.25,0.45982,0.34207,0.2857,0.42857,0.71429,0.0,1.0,7,6,0,7,0,0,0,0,6,0,0,8,0,0,0,0,0,5,0,0,0,0,6],[12,32,0.375,0.35714,0.33503,0.0,0.35714,0.46429,0.0,1.0,11,4,0,11,0,1,0,0,4,0,0,8,0,0,1,0,0,3,0,0,0,0,4],[16,32,0.5,0.51339,0.31715,0.42857,0.42857,0.71429,0.0,1.0,5,6,0,5,0,0,0,0,2,0,0,13,0,0,1,0,0,4,0,0,1,0,6],[20,32,0.625,0.53125,0.2321,0.42857,0.42859,0.71429,0.0,1.0,2,2,0,2,0,0,0,0,3,0,0,12,0,0,3,0,0,9,0,0,1,0,2],[24,32,0.75,0.56249,0.26711,0.42857,0.42857,0.71429,0.0,1.0,1,7,0,1,0,0,0,0,4,0,0,14,0,0,3,0,0,3,0,0,0,0,7],[28,32,0.875,0.55802,0.24836,0.42857,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,6,0,0,11,0,0,1,0,0,8,0,0,0,0,5],[32,32,1.0,0.66071,0.19805,0.42857,0.71429,0.71429,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,9,0,0,1,0,0,16,0,0,0,0,5]]}]},{"i":"c601206bc9ae14dc","q":"(F.Nilov) Two arcs with equal angular measure are constructed on the medians  $ AA'$  and  $ BB'$  of triangle  $ ABC$  towards vertex  $ C$ . Prove that the common chord of the respective circles passes through  $ C$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.23214,"x":0.28571,"p":[[0,41,0.0,0.23214,0.10564,0.2857,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,2,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.28571,1e-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],[8,41,0.1951,0.27678,0.04971,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.27679,0.04971,0.2857,0.28571,0.28571,0.0,0.286,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.28116,0.02535,0.2857,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.26339,0.0724,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.24108,0.10374,0.2857,0.28571,0.28571,0.0,0.286,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.23214,"x":0.28571,"p":[[0,47,0.0,0.25446,0.13236,0.2857,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,0,0,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,47,0.0851,0.25893,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.26339,0.06297,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.23214,0.11152,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.28571,0.0,0.28571,0.28571,0.28571,0.2857,0.28571,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.27231,0.05486,0.2857,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.26339,0.0724,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.24553,0.09606,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.27232,0.07457,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.25446,0.08552,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.23214,0.11151,0.2857,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"412fae6c28e20663","q":"The points in the plane are painted in 100 colors. Prove that there are three points of the same color that are the vertices of a triangle of area 1.\n\n*Proposed by V. Bragin*","t":[{"b":4,"e":0.2857,"k":"rising","v":0.0,"x":0.16518,"p":[[0,49,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.04464,0.13092,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,49,0.1633,0.03125,0.08552,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.09375,0.18073,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,7,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[28,49,0.5714,0.13393,0.15542,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,11,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[32,49,0.6531,0.08034,0.13328,0.0,0.0,0.14286,0.0,0.571,21,0,0,21,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,49,0.7347,0.08036,0.13803,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,7,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,49,0.8163,0.11139,0.15404,0.0,0.07,0.14286,0.0,0.71,16,0,0,16,0,11,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[44,49,0.898,0.12946,0.13054,0.0,0.14286,0.17857,0.0,0.42857,13,0,0,13,0,11,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.12505,0.13253,0.0,0.14286,0.17868,0.0,0.43,14,0,0,14,0,10,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.16518,0.14334,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,8,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.07588,"p":[[0,31,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.07588,0.13351,0.0,0.0,0.14286,0.0,0.571,21,0,0,21,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,31,0.2581,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.04464,0.13092,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,31,0.6452,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.05357,0.13243,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,31,0.9032,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.04022,0.08937,0.0,0.0,0.0,0.0,0.43,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7c6a52edc2302596","q":"Now suppose that before the game begins, the players choose the initial game state as follows:\n\n- 1. The first player chooses $s_{1}$ subject to the constraint that $2 \\leq s_{1} \\leq 5$; then\n- 2. the second player chooses $s_{2}$ subject to the constraint that $2 \\leq s_{2} \\leq 5$ and then specifies the board configuration.\n\nProve that the second player can always make her decisions so that she will win the game with probability strictly greater than $\\frac{1}{2}$.","t":[{"b":2,"e":0.0,"k":"rising","v":0.00446,"x":0.58482,"p":[[0,18,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.58482,0.41551,0.0,0.85714,0.85714,0.0,1.0,10,7,0,10,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,11,0,7],[8,18,0.4444,0.46875,0.38834,0.0,0.64286,0.85714,0.0,1.0,12,3,0,12,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,0,6,0,3],[12,18,0.6667,0.44642,0.40524,0.0,0.50001,0.85714,0.0,1.0,13,5,0,13,0,0,0,0,1,0,0,2,0,0,3,0,0,3,0,0,5,0,5],[16,18,0.8889,0.4107,0.38589,0.0,0.42857,0.75,0.0,1.0,13,3,0,13,0,1,0,0,1,0,0,2,0,0,3,0,0,4,0,0,5,0,3],[18,18,1.0,0.40179,0.36323,0.0,0.35714,0.71429,0.0,0.85714,11,0,0,11,0,4,0,0,1,0,0,1,0,0,1,0,0,7,0,0,7,0,0]]},{"b":5,"e":0.0,"k":"volatile","v":0.01339,"x":0.52677,"p":[[0,16,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.52677,0.39031,0.10714,0.71429,0.85714,0.0,1.0,8,6,0,8,0,2,0,0,3,0,0,1,0,0,1,0,0,5,0,0,6,0,6],[8,16,0.5,0.42411,0.42028,0.0,0.35714,0.85714,0.0,1.0,14,5,0,14,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,6,0,5],[12,16,0.75,0.51786,0.40994,0.0,0.71429,0.85714,0.0,1.0,11,5,0,11,0,0,0,0,2,0,0,0,0,0,2,0,0,3,0,0,9,0,5],[16,16,1.0,0.5,0.37965,0.0,0.57143,0.85714,0.0,1.0,9,4,0,9,0,1,0,0,3,0,0,1,0,0,3,0,0,4,0,0,7,0,4]]}]},{"i":"17dcced37a8cc8a6","q":"20. (ROM 3) Let $c_{1}, \\ldots, c_{n} \\in \\mathbb{R}(n \\geq 2)$ such that $0 \\leq \\sum_{i=1}^{n} c_{i} \\leq n$. Show that we can find integers $k_{1}, \\ldots, k_{n}$ such that $\\sum_{i=1}^{n} k_{i}=0$ and  $$ 1-n \\leq c_{i}+n k_{i} \\leq n \\quad \\text { for every } i=1, \\ldots, n $$","t":[{"b":0,"e":0.0,"k":"falling","v":0.07143,"x":0.72767,"p":[[0,52,0.0,0.58929,0.34947,0.28571,0.42859,1.0,0.14286,1.0,0,12,0,0,0,7,0,0,3,0,0,7,0,0,1,0,0,2,0,0,0,0,12],[4,52,0.0769,0.72767,0.3524,0.42857,1.0,1.0,0.0,1.0,1,19,0,1,0,4,0,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,19],[8,52,0.1538,0.54018,0.42368,0.14286,0.42857,1.0,0.0,1.0,6,13,0,6,0,6,0,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,13],[12,52,0.2308,0.54464,0.40789,0.14286,0.42857,1.0,0.0,1.0,6,12,0,6,0,4,0,0,3,0,0,4,0,0,0,0,0,2,0,0,1,0,12],[16,52,0.3077,0.33482,0.37048,0.14286,0.14286,0.5,0.0,1.0,7,6,0,7,0,14,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,6],[20,52,0.3846,0.23661,0.28928,0.14286,0.14286,0.14287,0.0,1.0,7,3,0,7,0,18,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,3],[24,52,0.4615,0.40615,0.41977,0.105,0.14286,1.0,0.0,1.0,8,10,0,8,0,11,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,10],[28,52,0.5385,0.21429,0.26726,0.0,0.14286,0.32142,0.0,1.0,11,2,0,11,0,12,0,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,2],[32,52,0.6154,0.17411,0.2618,0.0,0.14286,0.14286,0.0,1.0,14,2,0,14,0,11,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[36,52,0.6923,0.27679,0.35344,0.0,0.14286,0.28571,0.0,1.0,10,5,0,10,0,13,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,5],[40,52,0.7692,0.16964,0.20341,0.0,0.14286,0.17857,0.0,1.0,11,1,0,11,0,13,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[44,52,0.8462,0.18749,0.28218,0.0,0.07143,0.17857,0.0,1.0,16,1,0,16,0,8,0,0,2,0,0,2,0,0,0,0,0,1,0,0,2,0,1],[48,52,0.9231,0.10714,0.13363,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,8,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.07143,0.14286,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,7,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"falling","v":0.16509,"x":0.69643,"p":[[0,59,0.0,0.69643,0.39082,0.28571,1.0,1.0,0.0,1.0,3,18,0,3,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,0,2,0,18],[4,59,0.0678,0.64286,0.41802,0.14286,0.85714,1.0,0.0,1.0,6,16,0,6,0,4,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,16],[8,59,0.1356,0.49554,0.37113,0.14286,0.42857,0.89286,0.0,1.0,6,8,0,6,0,4,0,0,2,0,0,6,0,0,2,0,0,3,0,0,1,0,8],[12,59,0.2034,0.32589,0.33357,0.0,0.14288,0.46429,0.0,1.0,9,3,0,9,0,8,0,0,3,0,0,4,0,0,1,0,0,2,0,0,2,0,3],[16,59,0.2712,0.32143,0.32927,0.0,0.28571,0.42858,0.0,1.0,12,3,0,12,0,4,0,0,0,0,0,9,0,0,0,0,0,4,0,0,0,0,3],[20,59,0.339,0.44643,0.39082,0.14286,0.28571,1.0,0.0,1.0,7,9,0,7,0,5,0,0,5,0,0,4,0,0,0,0,0,2,0,0,0,0,9],[24,59,0.4068,0.42856,0.42257,0.10714,0.14286,1.0,0.0,1.0,8,10,0,8,0,9,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,10],[28,59,0.4746,0.29463,0.27878,0.10714,0.14286,0.42857,0.0,1.0,8,2,0,8,0,9,0,0,1,0,0,9,0,0,1,0,0,2,0,0,0,0,2],[32,59,0.5424,0.3125,0.32229,0.10714,0.14288,0.42857,0.0,1.0,8,4,0,8,0,9,0,0,4,0,0,5,0,0,0,0,0,2,0,0,0,0,4],[36,59,0.6102,0.20982,0.28118,0.0,0.14286,0.28571,0.0,1.0,14,2,0,14,0,7,0,0,5,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[40,59,0.678,0.18749,0.25109,0.0,0.14286,0.14286,0.0,1.0,11,2,0,11,0,14,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[44,59,0.7458,0.19188,0.19436,0.0,0.14286,0.42857,0.0,0.71429,12,0,0,12,0,8,0,0,3,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[48,59,0.8136,0.18304,0.21793,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,7,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[52,59,0.8814,0.21874,0.21422,0.0,0.21428,0.28571,0.0,1.0,10,1,0,10,0,6,0,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[56,59,0.9492,0.20536,0.19865,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,9,0,0,6,0,0,5,0,0,0,0,0,2,0,0,0,0,0],[59,59,1.0,0.16509,0.1717,0.0,0.14286,0.28571,0.0,0.71429,12,0,0,12,0,9,0,0,7,0,0,3,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"d8ad734b6da2d853","q":"We color some points in the plane with red, in such way that if  $P,Q$  are red and  $X$  is a point such that triangle  $\\triangle PQX$  has angles  $30\u00ba, 60\u00ba, 90\u00ba$  in some order, then  $X$  is also red. If we have vertices  $A, B, C$  all red, prove that the barycenter of triangle  $\\triangle ABC$  is also red.","t":[{"b":1,"e":0.0,"k":"flat","v":0.05357,"x":0.30793,"p":[[0,38,0.0,0.05357,0.10565,0.0,0.0,0.03571,0.0,0.42857,24,0,4,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.27677,0.31324,0.0,0.28571,0.57111,0.0,1.0,15,1,0,15,0,0,0,0,7,0,0,1,0,0,5,0,0,0,0,0,3,0,1],[8,38,0.2105,0.20982,0.27661,0.0,0.14286,0.28571,0.0,1.0,15,1,0,15,0,3,0,0,10,0,0,0,0,0,0,0,0,1,0,0,2,0,1],[12,38,0.3158,0.2232,0.29651,0.0,0.07143,0.32143,0.0,1.0,16,2,0,16,0,4,0,0,4,0,0,1,0,0,4,0,0,1,0,0,0,0,2],[16,38,0.4211,0.30793,0.2815,0.0,0.28571,0.4642,0.0,1.0,9,2,0,9,0,3,0,0,10,0,0,2,0,0,4,0,0,2,0,0,0,0,2],[20,38,0.5263,0.23214,0.24419,0.0,0.2143,0.32143,0.0,0.85714,13,0,0,13,0,3,0,0,8,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[24,38,0.6316,0.25893,0.2911,0.0,0.14286,0.57143,0.0,0.85714,15,0,0,15,0,3,0,0,2,0,0,2,0,0,7,0,0,1,0,0,2,0,0],[28,38,0.7368,0.25,0.26726,0.0,0.2857,0.32143,0.0,0.85714,14,0,0,14,0,0,0,0,10,0,0,1,0,0,4,0,0,1,0,0,2,0,0],[32,38,0.8421,0.17857,0.2369,0.0,0.0,0.28571,0.0,0.85714,18,0,0,18,0,1,0,0,7,0,0,1,0,0,4,0,0,0,0,0,1,0,0],[36,38,0.9474,0.24095,0.31019,0.0,0.0,0.35704,0.0,0.85714,17,0,0,17,0,1,0,0,6,0,0,0,0,0,3,0,0,1,0,0,4,0,0],[38,38,1.0,0.17857,0.21724,0.0,0.0,0.28571,0.0,0.85714,17,0,0,17,0,0,0,0,9,0,0,4,0,0,1,0,0,0,0,0,1,0,0]]},{"b":3,"e":0.2857,"k":"flat","v":0.05801,"x":0.32142,"p":[[0,33,0.0,0.09357,0.16597,0.0,0.0,0.14071,0.0,0.71429,22,0,1,22,0,3,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,33,0.1212,0.25892,0.29973,0.0,0.14288,0.4642,0.0,1.0,14,1,0,14,0,3,0,0,6,0,0,1,0,0,4,0,0,1,0,0,2,0,1],[8,33,0.2424,0.32142,0.31943,0.0,0.2857,0.57143,0.0,1.0,13,1,0,13,0,0,0,0,6,0,0,3,0,0,4,0,0,2,0,0,3,0,1],[12,33,0.3636,0.23213,0.23621,0.0,0.2857,0.42857,0.0,0.71429,14,0,0,14,0,1,0,0,8,0,0,2,0,0,6,0,0,1,0,0,0,0,0],[16,33,0.4848,0.24106,0.29109,0.0,0.07143,0.57111,0.0,0.85714,16,0,0,16,0,2,0,0,5,0,0,0,0,0,5,0,0,2,0,0,2,0,0],[20,33,0.6061,0.23659,0.28031,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,3,0,0,8,0,0,1,0,0,2,0,0,2,0,0,1,0,1],[24,33,0.7273,0.20525,0.27185,0.0,0.07,0.28571,0.0,1.0,16,1,0,16,0,3,0,0,7,0,0,1,0,0,2,0,0,1,0,0,1,0,1],[28,33,0.8485,0.27677,0.305,0.0,0.2857,0.4642,0.0,1.0,14,1,0,14,0,1,0,0,7,0,0,2,0,0,3,0,0,2,0,0,2,0,1],[32,33,0.9697,0.06695,0.20817,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[33,33,1.0,0.05801,0.15907,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"18cc4f9cb2d72890","q":"Sir Alex is coaching a soccer team of  $n$  players of distinct heights. He wants to line them up so that for each player  $P$ , the total number of players that are either to the left of  $P$  and taller than  $P$  or to the right of  $P$  and shorter than  $P$  is even. In terms of  $n$ , how many possible orders are there?\n\n*Michael Ren*","t":[{"b":0,"e":0.0,"k":"falling","v":0.01339,"x":0.60714,"p":[[0,63,0.0,0.60714,0.48049,0.0,1.0,1.0,0.0,1.0,12,19,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19],[4,63,0.0635,0.57143,0.47515,0.0,1.0,1.0,0.0,1.0,12,17,0,12,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,17],[8,63,0.127,0.5625,0.47372,0.0,1.0,1.0,0.0,1.0,11,17,0,11,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,17],[12,63,0.1905,0.5,0.4711,0.0,0.5,1.0,0.0,1.0,14,14,0,14,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,14],[16,63,0.254,0.49107,0.4935,0.0,0.35714,1.0,0.0,1.0,16,15,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15],[20,63,0.3175,0.32589,0.45769,0.0,0.0,1.0,0.0,1.0,20,10,0,20,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[24,63,0.381,0.27232,0.42911,0.0,0.0,0.57143,0.0,1.0,22,8,0,22,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[28,63,0.4444,0.25446,0.43114,0.0,0.0,0.35714,0.0,1.0,23,8,0,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[32,63,0.5079,0.20536,0.38949,0.0,0.0,0.03571,0.0,1.0,24,6,0,24,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[36,63,0.5714,0.32589,0.42892,0.0,0.0,0.78571,0.0,1.0,18,8,0,18,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,8],[40,63,0.6349,0.22768,0.41166,0.0,0.0,0.07143,0.0,1.0,24,7,0,24,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[44,63,0.6984,0.20536,0.38621,0.0,0.0,0.14286,0.0,1.0,23,6,0,23,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[48,63,0.7619,0.14286,0.3312,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[52,63,0.8254,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[56,63,0.8889,0.0625,0.2141,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[60,63,0.9524,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[63,63,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.62499,"p":[[0,65,0.0,0.41071,0.47481,0.0,0.0,1.0,0.0,1.0,17,12,5,17,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12],[4,65,0.0615,0.62499,0.46532,0.0,1.0,1.0,0.0,1.0,10,19,0,10,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,19],[8,65,0.1231,0.45982,0.49194,0.0,0.0,1.0,0.0,1.0,17,14,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,14],[12,65,0.1846,0.53125,0.49902,0.0,1.0,1.0,0.0,1.0,15,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[16,65,0.2462,0.40625,0.49113,0.0,0.0,1.0,0.0,1.0,19,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[20,65,0.3077,0.37054,0.46134,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[24,65,0.3692,0.26786,0.4282,0.0,0.0,0.46429,0.0,1.0,22,8,0,22,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[28,65,0.4308,0.28125,0.44961,0.0,0.0,1.0,0.0,1.0,23,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[32,65,0.4923,0.10714,0.27433,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[36,65,0.5538,0.11161,0.30249,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[40,65,0.6154,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.08036,0.24727,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[48,65,0.7385,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,65,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4ee39cc1fa403917","q":"Let  $ k$  be a circle and  $ k_{1},k_{2},k_{3},k_{4}$  four smaller circles with their centres  $ O_{1},O_{2},O_{3},O_{4}$  respectively, on  $ k$ . For  $ i \\equal{} 1,2,3,4$  and  $ k_{5}\\equal{} k_{1}$  the circles  $ k_{i}$  and  $ k_{i\\plus{}1}$  meet at  $ A_{i}$  and  $ B_{i}$  such that  $ A_{i}$  lies on  $ k$ . The points  $ O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}$  lie in that order on  $ k$  and are pairwise different.\r\n\r\nProve that  $ B_{1}B_{2}B_{3}B_{4}$  is a rectangle.","t":[{"b":1,"e":0.0,"k":"flat","v":0.125,"x":0.73651,"p":[[0,79,0.0,0.2499,0.21132,0.0,0.2857,0.42857,0.0,0.71429,10,0,2,10,0,4,0,0,7,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[4,79,0.0506,0.67855,0.39609,0.39286,0.85714,1.0,0.0,1.0,7,13,0,7,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,7,0,13],[8,79,0.1013,0.73651,0.36976,0.67857,0.92857,1.0,0.0,1.0,5,16,0,5,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,16],[12,79,0.1519,0.62945,0.41475,0.14286,0.85707,1.0,0.0,1.0,7,14,0,7,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,14],[16,79,0.2025,0.52677,0.46075,0.0,0.71429,1.0,0.0,1.0,13,13,0,13,0,0,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,13],[20,79,0.2532,0.58034,0.37104,0.25001,0.71429,0.85714,0.0,1.0,7,6,0,7,0,1,0,0,1,0,0,3,0,0,1,0,0,6,0,0,7,0,6],[24,79,0.3038,0.42837,0.44187,0.0,0.14143,0.85714,0.0,1.0,14,7,0,14,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,6,0,7],[28,79,0.3544,0.3616,0.43445,0.0,0.0,0.85714,0.0,1.0,17,7,0,17,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,7],[32,79,0.4051,0.48212,0.42219,0.0,0.57121,1.0,0.0,1.0,12,9,0,12,0,1,0,0,0,0,0,2,0,0,3,0,0,4,0,0,1,0,9],[36,79,0.4557,0.43279,0.41259,0.0,0.49979,0.85714,0.0,1.0,13,6,0,13,0,2,0,0,0,0,0,1,0,0,3,0,0,4,0,0,3,0,6],[40,79,0.5063,0.42856,0.42558,0.0,0.35714,0.85714,0.0,1.0,14,6,0,14,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,0,5,0,6],[44,79,0.557,0.49105,0.40553,0.0,0.64264,0.85714,0.0,1.0,11,6,0,11,0,0,0,0,3,0,0,1,0,0,1,0,0,5,0,0,5,0,6],[48,79,0.6076,0.55802,0.41397,0.0,0.71429,1.0,0.0,1.0,9,10,0,9,0,1,0,0,2,0,0,1,0,0,1,0,0,5,0,0,3,0,10],[52,79,0.6582,0.53115,0.43343,0.0,0.71429,1.0,0.0,1.0,11,10,0,11,0,1,0,0,1,0,0,1,0,0,0,0,0,5,0,0,3,0,10],[56,79,0.7089,0.4866,0.4132,0.0,0.71414,0.85714,0.0,1.0,11,7,0,11,0,1,0,0,3,0,0,0,0,0,0,0,0,7,0,0,3,0,7],[60,79,0.7595,0.45088,0.41204,0.0,0.4998,0.74996,0.0,1.0,12,7,0,12,0,2,0,0,1,0,0,1,0,0,1,0,0,7,0,0,1,0,7],[64,79,0.8101,0.50442,0.40873,0.0,0.57121,0.85714,0.0,1.0,11,7,0,11,0,1,0,0,0,0,0,1,0,0,4,0,0,4,0,0,4,0,7],[68,79,0.8608,0.54906,0.36789,0.25,0.571,0.89286,0.0,1.0,6,8,0,6,0,2,0,0,3,0,0,2,0,0,4,0,0,5,0,0,2,0,8],[72,79,0.9114,0.40623,0.37304,0.0,0.42859,0.74996,0.0,1.0,12,3,0,12,0,1,0,0,2,0,0,3,0,0,4,0,0,2,0,0,5,0,3],[76,79,0.962,0.49543,0.40411,0.0,0.64264,0.85714,0.0,1.0,11,5,0,11,0,1,0,0,0,0,0,3,0,0,1,0,0,4,0,0,7,0,5],[79,79,1.0,0.125,0.29827,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,2]]},{"b":7,"e":0.0,"k":"falling","v":0.0133,"x":0.58926,"p":[[0,68,0.0,0.16497,0.19594,0.0,0.14143,0.2857,0.0,0.71429,14,0,7,14,0,8,0,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[4,68,0.0588,0.45533,0.45095,0.0,0.35714,1.0,0.0,1.0,14,10,0,14,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0,3,0,10],[8,68,0.1176,0.51787,0.44138,0.0,0.64264,1.0,0.0,1.0,12,10,0,12,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,10],[12,68,0.1765,0.48214,0.46531,0.0,0.42859,1.0,0.0,1.0,14,12,0,14,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,12],[16,68,0.2353,0.4866,0.47091,0.0,0.57143,1.0,0.0,1.0,15,12,0,15,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,12],[20,68,0.2941,0.54014,0.43114,0.0,0.78564,0.85714,0.0,1.0,12,7,0,12,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,9,0,7],[24,68,0.3529,0.58926,0.4237,0.0,0.71429,1.0,0.0,1.0,9,12,0,9,0,1,0,0,1,0,0,1,0,0,1,0,0,4,0,0,3,0,12],[28,68,0.4118,0.58034,0.42097,0.0,0.71429,1.0,0.0,1.0,9,11,0,9,0,2,0,0,0,0,0,0,0,0,2,0,0,5,0,0,3,0,11],[32,68,0.4706,0.41963,0.43144,0.0,0.28571,0.89286,0.0,1.0,14,8,0,14,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,8],[36,68,0.5294,0.46866,0.43782,0.0,0.35714,1.0,0.0,1.0,12,10,0,12,0,2,0,0,2,0,0,1,0,0,0,0,0,4,0,0,1,0,10],[40,68,0.5882,0.53124,0.38997,0.0,0.71429,0.85714,0.0,1.0,9,5,0,9,0,1,0,0,2,0,0,1,0,0,1,0,0,6,0,0,7,0,5],[44,68,0.6471,0.45978,0.42216,0.0,0.49979,0.89275,0.0,1.0,12,8,0,12,0,2,0,0,1,0,0,1,0,0,2,0,0,4,0,0,2,0,8],[48,68,0.7059,0.48661,0.44155,0.0,0.71429,0.89286,0.0,1.0,13,8,0,13,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,5,0,8],[52,68,0.7647,0.33929,0.40049,0.0,0.14286,0.75,0.0,1.0,15,4,0,15,0,4,0,0,1,0,0,1,0,0,0,0,0,3,0,0,4,0,4],[56,68,0.8235,0.23214,0.38919,0.0,0.0,0.39285,0.0,1.0,23,4,0,23,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,4],[60,68,0.8824,0.13392,0.29218,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,2],[64,68,0.9412,0.04016,0.11965,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[68,68,1.0,0.0133,0.05465,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"beaa674a34d13f47","q":"Let  $U=\\{1, 2,\\ldots, 2014\\}$ . For positive integers  $a$ ,  $b$ ,  $c$  we denote by  $f(a, b, c)$  the number of ordered 6-tuples of sets  $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$  satisfying the following conditions: \n(i)  $Y_1 \\subseteq X_1 \\subseteq U$  and  $|X_1|=a$ ;\n(ii)  $Y_2 \\subseteq X_2 \\subseteq U\\setminus Y_1$  and  $|X_2|=b$ ;\n(iii)  $Y_3 \\subseteq X_3 \\subseteq U\\setminus (Y_1\\cup Y_2)$  and  $|X_3|=c$ .\nProve that  $f(a,b,c)$  does not change when  $a$ ,  $b$ ,  $c$  are rearranged. \n\n*Proposed by Damir A. Yeliussizov, Kazakhstan*","t":[{"b":0,"e":0.0,"k":"flat","v":0.88392,"x":1.0,"p":[[0,56,0.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[4,56,0.0714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,56,0.1429,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,56,0.2143,0.95536,0.1729,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[16,56,0.2857,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,56,0.3571,0.96429,0.15152,1.0,1.0,1.0,0.1429,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[24,56,0.4286,0.98213,0.07792,1.0,1.0,1.0,0.571,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[28,56,0.5,0.9508,0.19141,1.0,1.0,1.0,0.14,1.0,0,30,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[32,56,0.5714,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[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],[40,56,0.7143,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[44,56,0.7857,0.94642,0.19804,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,29],[48,56,0.8571,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[52,56,0.9286,0.88392,0.26351,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[56,56,1.0,0.95535,0.17656,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28]]},{"b":7,"e":1.0,"k":"flat","v":0.57589,"x":0.94196,"p":[[0,42,0.0,0.86607,0.33108,1.0,1.0,1.0,0.0,1.0,4,27,4,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,27],[4,42,0.0952,0.91518,0.26453,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[8,42,0.1905,0.875,0.31894,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,27],[12,42,0.2857,0.9375,0.24206,1.0,1.0,1.0,0.0,1.0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[16,42,0.381,0.94196,0.22548,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[20,42,0.4762,0.90625,0.29148,1.0,1.0,1.0,0.0,1.0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[24,42,0.5714,0.84821,0.3387,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,26],[28,42,0.6667,0.67847,0.42269,0.24928,1.0,1.0,0.0,1.0,7,19,0,7,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,19],[32,42,0.7619,0.57589,0.41263,0.14286,0.71429,1.0,0.0,1.0,7,12,0,7,0,2,0,0,3,0,0,3,0,0,0,0,0,2,0,0,3,0,12],[36,42,0.8571,0.72766,0.41398,0.49967,1.0,1.0,0.0,1.0,7,21,0,7,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,21],[40,42,0.9524,0.86161,0.30406,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,26],[42,42,1.0,0.80356,0.36553,0.92857,1.0,1.0,0.0,1.0,4,24,0,4,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,24]]}]},{"i":"6bac49f696559c5a","q":"The digits of the  $17$ -digit number are rearranged in the reverse order.  Prove that at list one digit of the sum of the new and the initial number is even.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.03124,"x":0.25446,"p":[[0,72,0.0,0.10267,0.22081,0.0,0.0,0.14286,0.0,1.0,23,1,8,23,0,4,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[4,72,0.0556,0.25446,0.32288,0.0,0.14286,0.42857,0.0,1.0,15,3,0,15,0,4,0,0,2,0,0,6,0,0,1,0,0,0,0,0,1,0,3],[8,72,0.1111,0.21429,0.27894,0.0,0.07143,0.32143,0.0,1.0,16,2,0,16,0,2,0,0,6,0,0,5,0,0,0,0,0,1,0,0,0,0,2],[12,72,0.1667,0.19188,0.27576,0.0,0.07,0.28571,0.0,1.0,16,2,0,16,0,6,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,2],[16,72,0.2222,0.16071,0.23351,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,1,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[20,72,0.2778,0.21875,0.31132,0.0,0.14286,0.2857,0.0,1.0,15,3,0,15,0,6,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[24,72,0.3333,0.19196,0.28259,0.0,0.07143,0.28571,0.0,1.0,16,2,0,16,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,0,1,0,2],[28,72,0.3889,0.24552,0.29715,0.0,0.2143,0.32143,0.0,1.0,14,3,0,14,0,2,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,3],[32,72,0.4444,0.2455,0.3411,0.0,0.0,0.42857,0.0,1.0,18,3,0,18,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,0,1,0,3],[36,72,0.5,0.11598,0.24855,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[40,72,0.5556,0.20981,0.29446,0.0,0.0,0.42857,0.0,1.0,19,2,0,19,0,0,0,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,2],[44,72,0.6111,0.20089,0.33093,0.0,0.0,0.28571,0.0,1.0,19,4,0,19,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[48,72,0.6667,0.1607,0.24154,0.0,0.0,0.28571,0.0,1.0,18,1,0,18,0,4,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[52,72,0.7222,0.21427,0.26724,0.0,0.14286,0.32143,0.0,1.0,15,1,0,15,0,4,0,0,5,0,0,3,0,0,3,0,0,0,0,0,1,0,1],[56,72,0.7778,0.23661,0.28033,0.0,0.14286,0.42857,0.0,1.0,14,2,0,14,0,4,0,0,3,0,0,8,0,0,0,0,0,1,0,0,0,0,2],[60,72,0.8333,0.24554,0.33547,0.0,0.07143,0.32143,0.0,1.0,16,4,0,16,0,3,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,4],[64,72,0.8889,0.24106,0.29972,0.0,0.14286,0.42857,0.0,1.0,14,3,0,14,0,4,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[68,72,0.9444,0.14286,0.26486,0.0,0.0,0.1786,0.0,1.0,21,2,0,21,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[72,72,1.0,0.03124,0.11136,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"flat","v":0.08927,"x":0.3482,"p":[[0,102,0.0,0.15625,0.24317,0.0,0.0,0.2857,0.0,1.0,19,1,6,19,0,4,0,0,2,0,0,5,0,0,0,0,0,1,0,0,0,0,1],[4,102,0.0392,0.24107,0.27302,0.0,0.14286,0.32143,0.0,1.0,12,2,0,12,0,5,0,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,2],[8,102,0.0784,0.20982,0.35532,0.0,0.0,0.28571,0.0,1.0,22,4,0,22,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,4],[12,102,0.1176,0.25447,0.35488,0.0,0.0,0.42857,0.0,1.0,18,4,0,18,0,2,0,0,1,0,0,5,0,0,1,0,0,0,0,0,1,0,4],[16,102,0.1569,0.14732,0.23003,0.0,0.0,0.2857,0.0,1.0,20,1,0,20,0,1,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[20,102,0.1961,0.30356,0.36552,0.0,0.14286,0.46418,0.0,1.0,15,5,0,15,0,3,0,0,1,0,0,5,0,0,2,0,0,1,0,0,0,0,5],[24,102,0.2353,0.21427,0.28346,0.0,0.14286,0.42857,0.0,1.0,15,2,0,15,0,6,0,0,2,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[28,102,0.2745,0.13393,0.25985,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0],[32,102,0.3137,0.31697,0.35307,0.0,0.28571,0.42858,0.0,1.0,13,5,0,13,0,2,0,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,5],[36,102,0.3529,0.32147,0.37629,0.0,0.14286,0.57143,0.0,1.0,15,5,0,15,0,2,0,0,2,0,0,4,0,0,2,0,0,1,0,0,1,0,5],[40,102,0.3922,0.3482,0.32327,0.10714,0.28571,0.42858,0.0,1.0,8,4,0,8,0,4,0,0,8,0,0,5,0,0,1,0,0,1,0,0,1,0,4],[44,102,0.4314,0.31695,0.37581,0.0,0.14286,0.57111,0.0,1.0,15,5,0,15,0,2,0,0,3,0,0,3,0,0,2,0,0,1,0,0,1,0,5],[48,102,0.4706,0.23213,0.36899,0.0,0.0,0.2857,0.0,1.0,19,5,0,19,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,5],[52,102,0.5098,0.16516,0.28144,0.0,0.0,0.2857,0.0,1.0,21,2,0,21,0,2,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,2],[56,102,0.549,0.20536,0.33491,0.0,0.0,0.32143,0.0,1.0,20,4,0,20,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[60,102,0.5882,0.23213,0.30876,0.0,0.0,0.42857,0.0,1.0,17,3,0,17,0,1,0,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,3],[64,102,0.6275,0.32586,0.36981,0.0,0.14288,0.571,0.0,1.0,15,4,0,15,0,2,0,0,0,0,0,6,0,0,2,0,0,1,0,0,2,0,4],[68,102,0.6667,0.19196,0.33237,0.0,0.0,0.2857,0.0,1.0,21,4,0,21,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,4],[72,102,0.7059,0.26786,0.36202,0.0,0.07143,0.42857,0.0,1.0,16,5,0,16,0,4,0,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,5],[76,102,0.7451,0.19641,0.29177,0.0,0.0,0.42857,0.0,1.0,19,2,0,19,0,2,0,0,2,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[80,102,0.7843,0.16963,0.23264,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,4,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[84,102,0.8235,0.15622,0.27512,0.0,0.0,0.17857,0.0,1.0,22,1,0,22,0,2,0,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,1],[88,102,0.8627,0.20088,0.30483,0.0,0.0,0.28571,0.0,1.0,18,3,0,18,0,3,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,3],[92,102,0.902,0.08927,0.15461,0.0,0.0,0.14286,0.0,0.571,22,0,0,22,0,4,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[96,102,0.9412,0.16963,0.30395,0.0,0.0,0.2857,0.0,1.0,21,3,0,21,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[100,102,0.9804,0.19643,0.34764,0.0,0.0,0.21429,0.0,1.0,22,4,0,22,0,2,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,4],[102,102,1.0,0.19642,0.3549,0.0,0.0,0.17857,0.0,1.0,22,4,0,22,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,4]]}]},{"i":"06f355e430a6bcb8","q":"Let $n$ be a positive integer, and let $a_{1}, a_{2}, \\ldots, a_{n}$ be a set of positive integers such that $a_{1}=2$ and $a_{m}=\\varphi\\left(a_{m+1}\\right)$ for all $1 \\leq m \\leq n-1$, where, for all positive integers $k, \\varphi(k)$ denotes the number of positive integers less than or equal to $k$ that are relatively prime to $k$. Prove that $a_{n} \\geq 2^{n-1}$.","t":[{"b":2,"e":1.0,"k":"flat","v":0.93302,"x":0.98661,"p":[[0,23,0.0,0.95089,0.15815,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,27],[4,23,0.1739,0.93302,0.17493,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[8,23,0.3478,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,23,0.5217,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[16,23,0.6957,0.98659,0.07464,1.0,1.0,1.0,0.571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[20,23,0.8696,0.96875,0.08552,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,28],[23,23,1.0,0.94643,0.10564,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,25]]},{"b":4,"e":0.57143,"k":"falling","v":0.67408,"x":0.94642,"p":[[0,13,0.0,0.94642,0.15872,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[4,13,0.3077,0.87498,0.22233,0.82132,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,4,0,0,2,0,22],[8,13,0.6154,0.85265,0.17676,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,3,0,17],[12,13,0.9231,0.74998,0.20517,0.57143,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,8,0,0,5,0,9],[13,13,1.0,0.67408,0.20277,0.57143,0.71414,0.85704,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,0,0,0,12,0,0,8,0,0,5,0,4]]}]},{"i":"0bd389b1e69b8b17","q":"Compute the sum of all positive integers  $50 \\leq n \\leq 100$  such that  $2n+3 \\nmid 2^{n!}-1$ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.875,"x":1.0,"p":[[0,52,0.0,0.93303,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,23],[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],[8,52,0.1538,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,52,0.2308,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,52,0.3077,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[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],[24,52,0.4615,0.96873,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[28,52,0.5385,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[32,52,0.6154,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[36,52,0.6923,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[40,52,0.7692,0.96429,0.11845,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,29],[44,52,0.8462,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[48,52,0.9231,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,28],[52,52,1.0,0.875,0.19149,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,6,0,19]]},{"b":7,"e":0.71429,"k":"flat","v":0.74999,"x":0.99107,"p":[[0,66,0.0,0.93303,0.15146,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,24],[4,66,0.0606,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,66,0.1212,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,66,0.1818,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[16,66,0.2424,0.93302,0.15564,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,3,0,25],[20,66,0.303,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[24,66,0.3636,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[28,66,0.4242,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[32,66,0.4848,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[36,66,0.5455,0.95982,0.15663,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,29],[40,66,0.6061,0.88393,0.30186,1.0,1.0,1.0,0.0,1.0,3,27,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[44,66,0.6667,0.88393,0.29329,1.0,1.0,1.0,0.0,1.0,3,26,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,26],[48,66,0.7273,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[52,66,0.7879,0.85714,0.15972,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,14,0,0,0,0,17],[56,66,0.8485,0.89732,0.17582,0.82132,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,2,0,22],[60,66,0.9091,0.83482,0.18935,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,12,0,0,1,0,16],[64,66,0.9697,0.74999,0.18212,0.71429,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,18,0,0,1,0,8],[66,66,1.0,0.8213,0.15577,0.71429,0.71429,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,13]]}]},{"i":"2beb6dd342f437e7","q":"Let $ABC$ be an isosceles triangle at $A$ but not a right triangle. Let $D$ be the point on $(BC)$ such that $(AD)$ is perpendicular to $(AB)$, and let $E$ be the orthogonal projection of $D$ onto $(AC)$. Finally, let $H$ be the midpoint of $[BC]$.\nShow that $AHE$ is isosceles at $H$.","t":[{"b":0,"e":1.0,"k":"falling","v":0.57589,"x":0.87497,"p":[[0,13,0.0,0.87497,0.21947,0.82143,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,22],[4,13,0.3077,0.76784,0.30463,0.57132,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,3,0,0,2,0,0,3,0,0,2,0,0,3,0,17],[8,13,0.6154,0.61606,0.35793,0.2857,0.64286,1.0,0.14286,1.0,0,13,0,0,0,5,0,0,9,0,0,0,0,0,2,0,0,2,0,0,1,0,13],[12,13,0.9231,0.57589,0.34899,0.2857,0.49999,1.0,0.0,1.0,1,11,0,1,0,3,0,0,11,0,0,1,0,0,1,0,0,4,0,0,0,0,11],[13,13,1.0,0.62051,0.32852,0.28571,0.71429,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,8,0,0,0,0,0,3,0,0,6,0,0,0,0,11]]},{"b":1,"e":1.0,"k":"flat","v":0.78125,"x":0.99107,"p":[[0,30,0.0,0.91518,0.19515,1.0,1.0,1.0,0.1429,1.0,0,25,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,25],[4,30,0.1333,0.84372,0.27051,0.67857,1.0,1.0,0.14286,1.0,0,23,0,0,0,2,0,0,1,0,0,1,0,0,4,0,0,1,0,0,0,0,23],[8,30,0.2667,0.78125,0.30719,0.67857,1.0,1.0,0.0,1.0,1,18,0,1,0,2,0,0,2,0,0,1,0,0,2,0,0,4,0,0,2,0,18],[12,30,0.4,0.92857,0.16366,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,26],[16,30,0.5333,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[20,30,0.6667,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,30,0.8,0.91072,0.23076,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,27],[28,30,0.9333,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[30,30,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]}]},{"i":"3371c2acf9f43324","q":"Given a scalene acute triangle  $ ABC$  with  $ AC>BC$  let  $ F$  be the foot of the altitude from  $ C$ . Let  $ P$  be a point on  $ AB$ , different from  $ A$  so that  $ AF\\equal{}PF$ . Let  $ H,O,M$  be the orthocenter, circumcenter and midpoint of  $ [AC]$ . Let  $ X$  be the intersection point of  $ BC$  and  $ HP$ . Let  $ Y$  be the intersection point of  $ OM$  and  $ FX$  and let  $ OF$  intersect  $ AC$  at  $ Z$ . Prove that  $ F,M,Y,Z$  are concyclic.","t":[{"b":6,"e":0.14286,"k":"flat","v":0.10688,"x":0.165,"p":[[0,11,0.0,0.12045,0.08071,0.14214,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.10688,0.06171,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.165,0.08833,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,28,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.15183,0.06141,0.14286,0.14286,0.14286,0.0,0.43,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.08929,"x":0.13813,"p":[[0,17,0.0,0.13813,0.06666,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.13394,0.07087,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.12947,0.07457,0.14286,0.14286,0.14286,0.0,0.4286,5,0,0,5,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.12938,0.04161,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.12491,0.04722,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dca89d2f4ac1e9a4","q":"Let  $a_1, a_2,...,a_{51}$  be non-zero elements of a field of characteristic  $p$ . We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence  $b_1, ... , b_{51}$ . If this new sequence is a permutation of the original one, find all possible values of  $p$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.89286,"x":1.0,"p":[[0,38,0.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[4,38,0.1053,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[8,38,0.2105,0.89286,0.24484,0.85714,1.0,1.0,0.0,1.0,2,22,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,22],[12,38,0.3158,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[16,38,0.4211,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[20,38,0.5263,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,38,0.6316,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[32,38,0.8421,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,38,0.9474,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[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":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,11,0.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[4,11,0.3636,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[8,11,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"b204ebed62805304","q":"Find the sum of first two integers  $n > 1$  such that  $3^n$  is divisible by  $n$  and  $3^n - 1$  is divisible by  $n - 1$ .","t":[{"b":5,"e":1.0,"k":"flat","v":0.98214,"x":0.9866,"p":[[0,5,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,5,0.8,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[5,5,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]},{"b":6,"e":1.0,"k":"flat","v":0.93304,"x":0.99554,"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],[4,52,0.0769,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,52,0.1538,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[12,52,0.2308,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[16,52,0.3077,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[20,52,0.3846,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,52,0.4615,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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],[32,52,0.6154,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[36,52,0.6923,0.95536,0.10374,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,27],[40,52,0.7692,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,52,0.8462,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[48,52,0.9231,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[52,52,1.0,0.93304,0.14719,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,26]]}]},{"i":"2afbd059c1887c52","q":"Let  $I$  denote the center of the circle inscribed in the right triangle  $ABC$  with right angle at the vertex  $A$ . Next, denote by  $M$  and  $N$  the midpoints of the lines  $AB$  and  $BI$ . Prove that the line  $CI$  is tangent to the circumscribed circle of triangle  $BMN$ . \n\n(Patrik Bak, Josef Tkadlec)","t":[{"b":1,"e":0.28571,"k":"flat","v":0.55793,"x":0.66964,"p":[[0,24,0.0,0.6473,0.30302,0.28571,0.64286,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,9,0,0,1,0,0,5,0,0,4,0,0,1,0,11],[4,24,0.1667,0.64732,0.33879,0.28571,0.64286,1.0,0.0,1.0,1,14,0,1,0,0,0,0,11,0,0,1,0,0,3,0,0,2,0,0,0,0,14],[8,24,0.3333,0.59372,0.28818,0.28571,0.57121,0.89275,0.2857,1.0,0,8,0,0,0,0,0,0,12,0,0,2,0,0,4,0,0,5,0,0,1,0,8],[12,24,0.5,0.55793,0.36143,0.28571,0.57121,1.0,0.0,1.0,3,10,2,3,0,3,0,0,8,0,0,1,0,0,3,0,0,3,0,0,1,0,10],[16,24,0.6667,0.55799,0.29957,0.28571,0.57121,0.74996,0.14286,1.0,0,5,0,0,0,5,0,0,8,0,0,0,0,0,4,0,0,7,0,0,3,0,5],[20,24,0.8333,0.6339,0.29868,0.28571,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,12,0,0,0,0,0,2,0,0,8,0,0,0,0,10],[24,24,1.0,0.66964,0.34523,0.28571,0.85714,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,12,0,0,0,0,0,2,0,0,1,0,0,0,0,16]]},{"b":2,"e":0.57143,"k":"flat","v":0.53569,"x":0.7589,"p":[[0,40,0.0,0.53569,0.28121,0.28571,0.57141,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,9,0,0,1,0,0,10,0,0,2,0,0,2,0,5],[4,40,0.1,0.74997,0.29882,0.571,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,6,0,0,0,0,0,4,0,0,4,0,0,0,0,17],[8,40,0.2,0.63838,0.3072,0.28571,0.64286,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,8,0,0,1,0,0,5,0,0,5,0,0,0,0,11],[12,40,0.3,0.58929,0.30878,0.28571,0.64286,1.0,0.1429,1.0,0,9,0,0,0,1,0,0,13,0,0,1,0,0,1,0,0,7,0,0,0,0,9],[16,40,0.4,0.67177,0.30471,0.28571,0.71429,1.0,0.14,1.0,0,12,0,0,0,2,0,0,7,0,0,0,0,0,5,0,0,5,1,0,0,0,12],[20,40,0.5,0.63837,0.3214,0.28571,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,10,0,0,1,0,0,1,0,0,4,0,0,4,0,10],[24,40,0.6,0.6116,0.31183,0.2857,0.71429,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,9,0,0,0,0,0,3,0,0,7,0,0,1,0,9],[28,40,0.7,0.65174,0.30293,0.28571,0.64286,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,7,0,0,1,0,0,6,0,0,4,0,0,1,0,11],[32,40,0.8,0.7589,0.29112,0.57142,0.9285,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,2,0,0,0,0,0,5,0,0,5,0,0,1,0,16],[36,40,0.9,0.65623,0.29637,0.28571,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,8,0,0,0,0,0,6,0,0,4,0,0,4,0,9],[40,40,1.0,0.66962,0.28669,0.28571,0.71429,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,9,0,0,0,0,0,6,0,0,5,0,0,1,0,11]]}]},{"i":"0588b2518103001d","q":"Let  $M$  be the midpoint of the side  $BC$  of triangle  $ABC$ . The bisector of the exterior angle of point  $A$  intersects the side  $BC$  in  $D$ . Let the circumcircle of triangle  $ADM$  intersect the lines  $AB$  and  $AC$  in  $E$  and  $F$  respectively. If the midpoint of  $EF$  is  $N$ , prove that  $MN\\parallel AD$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.02679,"x":0.0892,"p":[[0,31,0.0,0.05804,0.07017,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.08474,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.08464,0.07002,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.0892,0.17765,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,31,0.9032,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04018,"x":0.08482,"p":[[0,30,0.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.08482,0.17807,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,30,0.2667,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.04902,0.06774,0.0,0.0,0.14286,0.0,0.143,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"34b5509107991716","q":"Let  $A,B$  be two matrices with positive integer entries such that sum of entries of a row in  $A$  is equal to  sum of entries of the same row in  $B$  and  sum of entries of a column in  $A$  is equal to  sum of entries of the same column in  $B$ . Show that there exists a sequence of matrices  $A_1,A_2,A_3,\\cdots , A_n$  such that all entries of the matrix  $A_i$  are positive integers and in the sequence\n\\[A=A_0,A_1,A_2,A_3,\\cdots , A_n=B,\\]\nfor each index  $i$ , there exist indexes  $k,j,m,n$  such that\n\\[\\begin{array}{*{20}{c}}\n    \n  {{A_{i + 1}} - {A_{i}} = } \n\\end{array}\\begin{array}{*{20}{c}}\n  {\\begin{array}{*{20}{c}}\n \\quad \\quad \\ \\ j& \\ \\ \\ {k} \n\\end{array}}  \n  {\\begin{array}{*{20}{c}}\n  m  \n  n \n\\end{array}\\left( {\\begin{array}{*{20}{c}}\n  { + 1}&{ - 1}  \n  { - 1}&{ + 1} \n\\end{array}} \\right)} \n\\end{array} \\ \\text{or} \\ \\begin{array}{*{20}{c}}\n  {\\begin{array}{*{20}{c}}\n \\quad \\quad \\ \\ j& \\ \\ \\ {k} \n\\end{array}}  \n  {\\begin{array}{*{20}{c}}\n  m  \n  n \n\\end{array}\\left( {\\begin{array}{*{20}{c}}\n  { - 1}&{ + 1}  \n  { + 1}&{ - 1} \n\\end{array}} \\right)} \n\\end{array}.\\]\nThat is, all indices of  ${A_{i + 1}} - {A_{i}}$  are zero, except the indices  $(m,j), (m,k), (n,j)$ , and  $(n,k)$ .","t":[{"b":1,"e":0.28571,"k":"volatile","v":0.53571,"x":0.73428,"p":[[0,3,0.0,0.53571,0.35309,0.26785,0.53571,0.85714,0.0,1.0,4,6,0,4,0,3,0,1,5,0,0,2,0,1,1,0,0,4,0,0,5,0,6],[3,3,1.0,0.73428,0.20482,0.57143,0.71429,0.85714,0.14,1.0,0,7,0,0,0,1,0,0,0,0,0,2,0,1,6,0,0,9,0,0,6,0,7]]},{"b":6,"e":0.14286,"k":"volatile","v":0.37945,"x":0.74112,"p":[[0,9,0.0,0.51113,0.34121,0.2857,0.3928,0.85714,0.0,1.0,2,7,0,2,0,5,0,0,8,0,1,1,0,0,4,0,0,1,0,0,3,0,7],[4,9,0.4444,0.74112,0.30601,0.57143,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,1,0,0,3,0,0,3,0,0,3,0,0,6,0,13],[8,9,0.8889,0.40624,0.32558,0.14286,0.42857,0.71429,0.0,1.0,7,1,0,7,0,6,0,0,2,0,0,3,0,0,5,0,0,3,0,0,5,0,1],[9,9,1.0,0.37945,0.31462,0.14286,0.35714,0.60714,0.0,1.0,7,2,0,7,0,6,0,0,3,0,0,5,0,0,3,0,0,4,0,0,2,0,2]]}]},{"i":"547cfcca15fa0c12","q":"For every positive integer  $n$  determine the least possible value of the expression\n\\[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\\dots +|x_{1}+x_{2}+\\dots +x_{n-1}-x_{n}|\\]\ngiven that  $x_{1}, x_{2}, \\dots , x_{n}$  are real numbers satisfying  $|x_{1}|+|x_{2}|+\\dots+|x_{n}| = 1$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.15616,"x":0.3125,"p":[[0,27,0.0,0.16509,0.09525,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.15616,0.06548,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.3125,0.1357,0.14286,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,11,0,0,5,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[16,27,0.5926,0.26339,0.12931,0.14286,0.21429,0.42857,0.14286,0.42857,0,0,0,0,0,16,0,0,5,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.28116,0.13124,0.14286,0.28571,0.42857,0.14,0.57143,0,0,0,0,0,13,0,0,8,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[24,27,0.8889,0.2767,0.11822,0.14286,0.28571,0.42857,0.14,0.42857,0,0,0,0,0,12,0,0,10,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.28571,0.15152,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,14,0,0,7,0,0,9,0,0,1,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.12482,"x":0.21429,"p":[[0,35,0.0,0.19643,0.15047,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,25,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[4,35,0.1143,0.21429,0.12372,0.14286,0.14286,0.2857,0.0,0.4286,1,0,0,1,0,21,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.20536,0.2111,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,26,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[12,35,0.3429,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.14509,0.05782,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,1,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.143,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.12482,0.04718,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"654c94e872975732","q":"Let $a, b, c, d$ be positive real numbers such that  $$ a b c d=1 \\quad \\text { and } \\quad a+b+c+d>\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}{a} $$  Prove that  $$ a+b+c+d<\\frac{b}{a}+\\frac{c}{b}+\\frac{d}{c}+\\frac{a}{d} $$","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0491,"p":[[0,60,0.0,0.0491,0.1455,0.0,0.0,0.0,0.0,0.57143,28,0,2,28,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[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],[8,60,0.1333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,60,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,42,0.0,0.0625,0.15542,0.0,0.0,0.0,0.0,0.57143,27,0,3,27,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,42,0.0952,0.04463,0.1259,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,42,0.1905,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.0133,0.05465,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,42,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"95be1b5a86f9e3a6","q":"Let  $p \\geq 5$  be a prime number.  Prove that there exist at least 2 distinct primes  $q_1, q_2$  satisfying  $1 < q_i < p - 1$  and  $q_i^{p-1} \\not\\equiv 1 \\mbox{ (mod }p^2)$ , for  $i = 1, 2$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,17,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,40,0.0,0.00894,0.04976,0.0,0.0,0.0,0.0,0.286,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"53a76ea659a0756c","q":"$21$  distinct numbers are chosen from the set  $\\{1,2,3,\\ldots,2046\\}.$  Prove that we can choose three distinct numbers  $a,b,c$  among those  $21$  numbers such that\n\\[bc<2a^2<4bc\\]","t":[{"b":4,"e":0.2857,"k":"flat","v":0.08929,"x":0.38392,"p":[[0,53,0.0,0.15179,0.24206,0.0,0.0,0.2857,0.0,0.71429,21,0,1,21,0,2,0,0,2,0,0,3,0,0,1,0,0,3,0,0,0,0,0],[4,53,0.0755,0.37054,0.25719,0.14286,0.42857,0.42857,0.0,1.0,5,2,0,5,0,4,0,0,5,0,0,11,0,0,3,0,0,2,0,0,0,0,2],[8,53,0.1509,0.38392,0.23537,0.2857,0.42857,0.46429,0.0,0.857,6,0,0,6,0,1,0,0,4,0,0,13,0,0,3,0,0,4,0,0,1,0,0],[12,53,0.2264,0.35268,0.26483,0.0,0.42857,0.42857,0.0,1.0,9,1,0,9,0,0,0,0,4,0,0,12,0,0,2,0,0,4,0,0,0,0,1],[16,53,0.3019,0.29018,0.23279,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,4,0,0,4,0,0,10,0,0,2,0,0,3,0,0,0,0,0],[20,53,0.3774,0.37945,0.24381,0.25,0.42857,0.4642,0.0,0.85714,6,0,0,6,0,2,0,0,4,0,0,12,0,0,2,0,0,5,0,0,1,0,0],[24,53,0.4528,0.20973,0.22014,0.0,0.14286,0.42857,0.0,0.71429,14,0,0,14,0,3,0,0,6,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[28,53,0.5283,0.20088,0.22546,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,4,0,0,3,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[32,53,0.6038,0.18747,0.22706,0.0,0.0,0.42857,0.0,0.71429,17,0,0,17,0,2,0,0,4,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[36,53,0.6792,0.20088,0.21681,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,2,0,0,6,0,0,6,0,0,2,0,0,1,0,0,0,0,0],[40,53,0.7547,0.1875,0.19704,0.0,0.14286,0.32143,0.0,0.71429,14,0,0,14,0,4,0,0,6,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[44,53,0.8302,0.14284,0.18555,0.0,0.0,0.28571,0.0,0.571,19,0,0,19,0,1,0,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[48,53,0.9057,0.15623,0.21232,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,3,0,0,6,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[52,53,0.9811,0.08929,0.15047,0.0,0.0,0.1786,0.0,0.42857,23,0,0,23,0,1,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[53,53,1.0,0.13393,0.18877,0.0,0.0,0.2857,0.0,0.71429,18,0,0,18,0,5,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.17857,"x":0.71429,"p":[[0,94,0.0,0.17857,0.27433,0.0,0.0,0.32143,0.0,0.71429,21,0,3,21,0,1,0,0,2,0,0,2,0,0,1,0,0,5,0,0,0,0,0],[4,94,0.0426,0.32588,0.24283,0.0,0.42857,0.42857,0.0,1.0,9,1,0,9,0,1,0,0,2,0,0,16,0,0,2,0,0,1,0,0,0,0,1],[8,94,0.0851,0.39732,0.28512,0.21429,0.42857,0.46431,0.0,1.0,8,2,0,8,0,0,0,0,2,0,0,14,0,0,2,0,0,3,0,0,1,0,2],[12,94,0.1277,0.44642,0.2714,0.28571,0.4998,0.71429,0.0,1.0,6,1,0,6,0,1,0,0,3,0,0,6,0,0,7,0,0,8,0,0,0,0,1],[16,94,0.1702,0.37054,0.28316,0.10714,0.42857,0.42857,0.0,1.0,8,2,0,8,0,1,0,0,4,0,0,12,0,0,2,0,0,2,0,0,1,0,2],[20,94,0.2128,0.35266,0.29876,0.0,0.35714,0.57143,0.0,1.0,10,2,0,10,0,1,0,0,5,0,0,5,0,0,6,0,0,3,0,0,0,0,2],[24,94,0.2553,0.48655,0.22261,0.42857,0.42859,0.57143,0.0,1.0,3,1,0,3,0,0,0,0,3,0,0,11,0,0,8,0,0,5,0,0,1,0,1],[28,94,0.2979,0.53125,0.24545,0.39286,0.71429,0.71429,0.0,1.0,2,1,0,2,0,2,0,0,4,0,0,6,0,0,1,0,0,16,0,0,0,0,1],[32,94,0.3404,0.54911,0.19597,0.39286,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,6,0,0,3,0,0,14,0,0,0,0,1],[36,94,0.383,0.55802,0.20628,0.42857,0.71429,0.71429,0.0,0.7143,1,0,0,1,0,2,0,0,2,0,0,7,0,0,2,0,0,18,0,0,0,0,0],[40,94,0.4255,0.53571,0.22868,0.42857,0.64286,0.71429,0.0,0.71429,3,0,0,3,0,1,0,0,1,0,0,7,0,0,4,0,0,16,0,0,0,0,0],[44,94,0.4681,0.47321,0.25614,0.28571,0.5,0.71429,0.0,0.71429,4,0,0,4,0,2,0,0,4,0,0,6,0,0,2,0,0,14,0,0,0,0,0],[48,94,0.5106,0.5223,0.22477,0.42857,0.57143,0.71429,0.0,0.71429,3,0,0,3,0,0,0,0,4,0,0,5,0,0,6,0,0,14,0,0,0,0,0],[52,94,0.5532,0.66515,0.13177,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,19,0,0,2,0,1],[56,94,0.5957,0.61607,0.20024,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,0,0,0,3,0,0,5,0,0,20,0,0,0,0,1],[60,94,0.6383,0.63393,0.13803,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,5,0,0,2,0,0,23,0,0,0,0,0],[64,94,0.6809,0.59821,0.21558,0.57143,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,4,0,0,1,0,0,3,0,0,21,0,0,1,0,0],[68,94,0.7234,0.59371,0.18936,0.571,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,5,0,0,1,0,0,5,0,0,20,0,0,0,0,0],[72,94,0.766,0.63838,0.11838,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,21,0,0,0,0,0],[76,94,0.8085,0.55357,0.21053,0.42857,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,5,0,0,6,0,0,0,0,0,19,0,0,0,0,0],[80,94,0.8511,0.65177,0.15127,0.71429,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,1,0,0,1,0,0,4,0,0,25,0,0,0,0,0],[84,94,0.8936,0.62946,0.12807,0.57142,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,21,0,0,0,0,0],[88,94,0.9362,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[92,94,0.9787,0.69196,0.05187,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0],[94,94,1.0,0.70981,0.02493,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0]]}]},{"i":"bc8ffd318b399d35","q":"Let  $ABC$  be a triangle with incenter  $I$ , and the incircle touches  $BC$  at  $D$ . The points  $E, F$  are such that  $BE \\parallel AI \\parallel CF$  and  $\\angle BEI=\\angle CFI=90^{\\circ}$ . If  $DE, DF$  meet the incircle at  $E', F'$ , show that  $E'F' \\perp AI$ .","t":[{"b":6,"e":0.28571,"k":"rising","v":0.08929,"x":0.5982,"p":[[0,117,0.0,0.09375,0.1411,0.0,0.0,0.14286,0.0,0.42857,20,0,1,20,0,6,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,117,0.0342,0.11598,0.14912,0.0,0.07,0.14286,0.0,0.4286,16,0,0,16,0,11,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,117,0.0684,0.08929,0.11152,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,117,0.1026,0.16063,0.15873,0.0,0.14286,0.1786,0.0,0.4286,11,0,0,11,0,13,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[16,117,0.1368,0.15625,0.21535,0.0,0.07143,0.2857,0.0,0.85714,16,0,0,16,0,7,0,0,4,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[20,117,0.1709,0.24107,0.20025,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,5,0,0,9,0,0,7,0,0,0,0,0,2,0,0,0,0,0],[24,117,0.2051,0.24998,0.21722,0.0,0.2857,0.32143,0.0,0.71429,10,0,0,10,0,3,0,0,11,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[28,117,0.2393,0.20089,0.18161,0.0,0.21428,0.28571,0.0,0.71429,11,0,0,11,0,5,0,0,10,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[32,117,0.2735,0.29911,0.20316,0.14286,0.28571,0.42857,0.0,0.71429,5,0,0,5,0,6,0,0,9,0,0,8,0,0,1,0,0,3,0,0,0,0,0],[36,117,0.3077,0.30801,0.25778,0.10714,0.28571,0.4286,0.0,1.0,8,1,0,8,0,4,0,0,8,0,0,5,0,0,3,0,0,3,0,0,0,0,1],[40,117,0.3419,0.29241,0.26266,0.0,0.28571,0.42857,0.0,1.0,10,1,0,10,0,1,0,1,7,0,0,10,0,0,0,0,0,0,0,0,2,0,1],[44,117,0.3761,0.16518,0.19269,0.0,0.14286,0.2857,0.0,0.71429,14,0,0,14,0,6,0,0,9,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[48,117,0.4103,0.34372,0.20156,0.2857,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,1,0,0,15,0,0,6,0,0,3,0,0,2,0,0,1,0,0],[52,117,0.4444,0.38393,0.21559,0.2857,0.28571,0.46431,0.0,0.71429,3,0,0,3,0,2,0,0,12,0,0,7,0,0,1,0,0,7,0,0,0,0,0],[56,117,0.4786,0.40175,0.22425,0.28571,0.28571,0.57143,0.0,0.71429,3,0,0,3,0,2,0,0,12,0,0,3,0,0,5,0,0,7,0,0,0,0,0],[60,117,0.5128,0.41515,0.24315,0.2857,0.28571,0.57143,0.0,1.0,2,2,0,2,0,1,0,0,16,0,0,3,0,0,3,0,0,5,0,0,0,0,2],[64,117,0.547,0.37053,0.26692,0.2857,0.28571,0.4286,0.0,1.0,4,2,0,4,0,2,0,0,16,0,0,3,0,0,0,0,0,4,0,0,1,0,2],[68,117,0.5812,0.35268,0.18553,0.2857,0.28571,0.42858,0.0,0.71429,3,0,0,3,0,1,0,0,15,0,0,8,0,0,1,0,0,4,0,0,0,0,0],[72,117,0.6154,0.38391,0.25362,0.2857,0.28571,0.60714,0.0,0.85714,3,0,0,3,0,3,0,0,16,0,0,0,0,0,2,0,0,5,0,0,3,0,0],[76,117,0.6496,0.44644,0.2569,0.28571,0.42857,0.71429,0.0,1.0,2,2,0,2,0,2,0,0,11,0,0,6,0,0,2,0,0,6,0,0,1,0,2],[80,117,0.6838,0.47309,0.25098,0.2857,0.28586,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,15,0,0,1,0,0,2,0,0,10,0,0,0,0,2],[84,117,0.7179,0.49105,0.21705,0.28571,0.4286,0.71429,0.0,0.857,1,0,0,1,0,0,0,0,12,0,0,4,0,0,2,0,0,12,0,0,1,0,0],[88,117,0.7521,0.59373,0.20237,0.42857,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,0,4,0,0,2,0,0,16,0,0,2,0,1],[92,117,0.7863,0.52228,0.22191,0.28571,0.57143,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,8,0,0,2,0,0,5,0,0,15,0,0,0,0,0],[96,117,0.8205,0.51339,0.20782,0.28571,0.50001,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,13,0,0,3,0,0,1,0,0,14,0,0,1,0,0],[100,117,0.8547,0.53571,0.21429,0.28571,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,7,0,0,4,0,0,2,0,0,17,0,0,0,0,0],[104,117,0.8889,0.49997,0.23419,0.28571,0.57121,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,11,0,0,1,0,0,4,0,0,13,0,0,1,0,0],[108,117,0.9231,0.52232,0.22759,0.28571,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,12,0,0,1,0,0,1,0,0,16,0,0,1,0,0],[112,117,0.9573,0.48658,0.19839,0.28571,0.4286,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,10,0,0,6,0,0,4,0,0,11,0,0,0,0,0],[116,117,0.9915,0.5982,0.20341,0.42857,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,6,0,0,1,0,0,15,0,0,3,0,1],[117,117,1.0,0.56696,0.18724,0.39286,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,8,0,0,4,0,0,1,0,0,19,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04464,"x":0.16518,"p":[[0,39,0.0,0.04911,0.11633,0.0,0.0,0.0,0.0,0.42857,26,0,4,26,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.1384,0.20973,0.0,0.07143,0.1429,0.0,0.85714,16,0,0,16,0,11,0,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[8,39,0.2051,0.13393,0.14258,0.0,0.14286,0.1429,0.0,0.42857,13,0,0,13,0,12,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.08482,0.13296,0.0,0.0,0.14286,0.0,0.4286,20,0,1,20,0,8,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.12054,0.16409,0.0,0.0,0.14287,0.0,0.4286,18,0,0,18,0,7,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.09375,0.12682,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.06696,0.14279,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.16518,0.2055,0.0,0.0,0.32143,0.0,0.71429,18,0,0,18,0,1,0,0,5,0,0,7,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"fe232be7cc50b290","q":"For a prime number $p$ and a positive integer $n$, denote by $f(p, n)$ the largest integer $k$ such that $p^{k} \\mid n$ !. Let $p$ be a given prime number and let $m$ and $c$ be given positive integers. Prove that there exist infinitely many positive integers $n$ such that $f(p, n) \\equiv c$ $(\\bmod m)$.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.14286,"x":0.45087,"p":[[0,41,0.0,0.40169,0.3082,0.14286,0.28571,0.57143,0.0,1.0,2,4,0,2,0,9,0,0,8,0,0,4,0,0,2,0,0,1,0,0,2,0,4],[4,41,0.0976,0.39286,0.19885,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,16,0,0,6,0,0,3,0,0,2,0,0,1,0,1],[8,41,0.1951,0.38393,0.20958,0.28571,0.35714,0.46429,0.0,0.85714,1,0,0,1,0,6,0,0,9,0,0,8,0,0,4,0,0,2,0,0,2,0,0],[12,41,0.2927,0.42857,0.21129,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,5,0,0,10,0,0,5,0,0,6,0,0,4,0,0,2,0,0],[16,41,0.3902,0.45087,0.22333,0.28571,0.35714,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,13,0,0,3,0,0,6,0,0,3,0,0,4,0,0],[20,41,0.4878,0.43749,0.22285,0.28571,0.42857,0.57111,0.14286,1.0,0,2,0,0,0,3,0,0,11,0,0,9,0,0,5,0,0,0,0,0,2,0,2],[24,41,0.5854,0.37499,0.2335,0.14286,0.28571,0.57111,0.14286,1.0,0,1,0,0,0,9,0,0,11,0,0,3,0,0,5,0,0,1,0,0,2,0,1],[28,41,0.6829,0.25884,0.18713,0.14286,0.14286,0.28571,0.14,0.85714,0,0,0,0,0,20,0,0,5,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[32,41,0.7805,0.24545,0.22658,0.14286,0.14286,0.17857,0.14,1.0,0,1,0,0,0,24,0,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,1],[36,41,0.878,0.15625,0.09689,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,25,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,41,0.9756,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.14286,1e-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":2,"e":0.57143,"k":"rising","v":0.41071,"x":0.57589,"p":[[0,24,0.0,0.41071,0.31693,0.14286,0.28571,0.60714,0.0,1.0,3,4,0,3,0,7,0,0,10,0,0,1,0,0,3,0,0,2,0,0,2,0,4],[4,24,0.1667,0.43746,0.23126,0.28571,0.42857,0.57111,0.14286,1.0,0,2,0,0,0,4,0,0,11,0,0,6,0,0,7,0,0,0,0,0,2,0,2],[8,24,0.3333,0.53125,0.22371,0.39286,0.50001,0.60714,0.14286,1.0,0,2,0,0,0,1,0,0,7,0,0,8,0,0,8,0,0,2,0,0,4,0,2],[12,24,0.5,0.50446,0.20198,0.39286,0.42857,0.60714,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,9,0,0,7,0,0,4,0,0,4,0,0],[16,24,0.6667,0.57589,0.2004,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,7,0,0,9,0,0,6,0,0,3,0,2],[20,24,0.8333,0.52679,0.2299,0.42857,0.42857,0.60714,0.14286,1.0,0,3,0,0,0,1,0,0,6,0,0,12,0,0,5,0,0,2,0,0,3,0,3],[24,24,1.0,0.57142,0.18558,0.42857,0.57143,0.57143,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,10,0,0,14,0,0,1,0,0,2,0,3]]}]},{"i":"fadb9d467f936abf","q":"Let  $ABC$  be an acute triangle and  $\\Gamma$  its circumcircle. The lines tangent to  $\\Gamma$  through  $B$  and  $C$  meet at  $P$ . Let  $M$  be a point on the arc  $AC$  that does not contain  $B$  such that  $M \\neq A$  and  $M \\neq C$ , and  $K$  be the point where the lines  $BC$  and  $AM$  meet. Let  $R$  be the point symmetrical to  $P$  with respect to the line  $AM$  and  $Q$  the point of intersection of lines  $RA$  and  $PM$ . Let  $J$  be the midpoint of  $BC$  and  $L$  be the intersection point of the line  $PJ$  and the line through  $A$  parallel to  $PR$ . Prove that  $L, J, A, Q,$  and  $K$  all lie on a circle.","t":[{"b":2,"e":0.0,"k":"falling","v":0.07589,"x":0.4062,"p":[[0,82,0.0,0.29464,0.04971,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,82,0.0488,0.4062,0.27686,0.14289,0.35714,0.57143,0.0,1.0,4,1,0,4,0,5,0,0,7,0,0,3,0,0,7,0,0,2,0,0,3,0,1],[8,82,0.0976,0.37947,0.30849,0.10714,0.42857,0.57143,0.0,1.0,8,2,0,8,0,4,0,0,3,0,0,5,0,0,5,0,0,4,0,0,1,0,2],[12,82,0.1463,0.28571,0.23957,0.14286,0.2857,0.42857,0.0,1.0,6,1,0,6,0,7,0,0,10,0,0,5,0,0,1,0,0,1,0,0,1,0,1],[16,82,0.1951,0.308,0.29901,0.0,0.2857,0.4642,0.0,1.0,10,2,0,10,0,5,0,0,4,0,0,5,0,0,4,0,0,1,0,0,1,0,2],[20,82,0.2439,0.23214,0.24679,0.0,0.2143,0.32143,0.0,0.85714,13,0,0,13,0,3,0,0,8,0,0,3,0,0,2,0,0,2,0,0,1,0,0],[24,82,0.2927,0.24967,0.2367,0.0,0.14288,0.42858,0.0,0.71429,10,0,0,10,0,7,0,0,5,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[28,82,0.3415,0.2232,0.23671,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,5,0,0,8,0,0,3,0,0,1,0,0,2,0,0,1,0,0],[32,82,0.3902,0.20526,0.2257,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,10,0,0,4,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[36,82,0.439,0.33034,0.29758,0.0,0.2857,0.57143,0.0,1.0,9,1,0,9,0,4,0,0,8,0,0,0,0,0,4,0,0,5,0,0,1,0,1],[40,82,0.4878,0.23643,0.26399,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,8,0,0,6,0,0,2,0,0,0,0,0,3,0,0,2,0,0],[44,82,0.5366,0.28558,0.24743,0.14286,0.2143,0.46418,0.0,0.85714,6,0,0,6,0,10,0,0,7,0,0,1,0,0,5,0,0,1,0,0,2,0,0],[48,82,0.5854,0.2366,0.25405,0.0,0.14288,0.42857,0.0,1.0,11,1,0,11,0,8,0,0,3,0,0,6,0,0,1,0,0,2,0,0,0,0,1],[52,82,0.6341,0.23205,0.25693,0.0,0.14286,0.32143,0.0,0.85714,10,0,0,10,0,11,0,0,3,0,0,3,0,0,1,0,0,2,0,0,2,0,0],[56,82,0.6829,0.25436,0.26665,0.0,0.2857,0.42857,0.0,1.0,13,1,0,13,0,2,0,0,7,0,0,4,0,0,3,0,0,2,0,0,0,0,1],[60,82,0.7317,0.28112,0.29121,0.0,0.2143,0.42858,0.0,1.0,9,3,0,9,0,7,0,0,7,0,0,3,0,0,3,0,0,0,0,0,0,0,3],[64,82,0.7805,0.19196,0.24898,0.0,0.14286,0.32143,0.0,1.0,15,1,0,15,0,7,0,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,1],[68,82,0.8293,0.16954,0.23537,0.0,0.0,0.2857,0.0,1.0,17,1,0,17,0,4,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[72,82,0.878,0.125,0.17405,0.0,0.0,0.14289,0.0,0.71429,17,0,0,17,0,8,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[76,82,0.9268,0.09374,0.15403,0.0,0.0,0.1429,0.0,0.714,20,0,0,20,0,6,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[80,82,0.9756,0.12938,0.16505,0.0,0.14143,0.1429,0.0,0.71429,15,0,0,15,0,10,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[82,82,1.0,0.07589,0.11836,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.14277,"x":0.5089,"p":[[0,101,0.0,0.26339,0.11903,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,3,0,0,24,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,101,0.0396,0.5089,0.29652,0.2857,0.42859,0.71429,0.0,1.0,1,5,0,1,0,4,0,0,8,0,0,4,0,0,5,0,0,3,0,0,2,0,5],[8,101,0.0792,0.42856,0.29014,0.28571,0.42859,0.57143,0.0,1.0,5,4,0,5,0,1,0,0,8,0,0,6,0,0,7,0,0,1,0,0,0,0,4],[12,101,0.1188,0.47765,0.27106,0.28571,0.571,0.71429,0.0,1.0,2,2,0,2,0,4,0,0,7,0,0,2,0,0,8,0,0,5,0,0,2,0,2],[16,101,0.1584,0.35704,0.23425,0.1429,0.28571,0.4642,0.0,1.0,2,1,0,2,0,7,0,0,11,0,0,4,0,0,4,0,0,2,0,0,1,0,1],[20,101,0.198,0.33928,0.26904,0.14286,0.28571,0.46431,0.0,1.0,5,2,0,5,0,7,0,0,7,0,0,5,0,0,5,0,0,0,0,0,1,0,2],[24,101,0.2376,0.42843,0.26492,0.2857,0.42857,0.57111,0.0,1.0,2,2,0,2,0,5,0,0,7,0,0,7,0,0,6,0,0,0,0,0,3,0,2],[28,101,0.2772,0.33479,0.27104,0.14286,0.2857,0.57111,0.0,0.85714,6,0,0,6,0,7,0,0,7,0,0,2,0,0,5,0,0,2,0,0,3,0,0],[32,101,0.3168,0.34819,0.305,0.14286,0.2857,0.57111,0.0,1.0,7,3,0,7,0,5,0,0,8,0,0,3,0,0,4,0,0,1,0,0,1,0,3],[36,101,0.3564,0.5089,0.32914,0.2857,0.4286,0.75,0.0,1.0,3,7,0,3,0,3,0,0,7,0,0,4,0,0,5,0,0,2,0,0,1,0,7],[40,101,0.396,0.44642,0.30876,0.14289,0.42859,0.71429,0.0,1.0,4,2,0,4,0,6,0,0,5,0,0,2,0,0,3,0,0,8,0,0,2,0,2],[44,101,0.4356,0.39284,0.25999,0.2857,0.28571,0.57111,0.0,1.0,2,2,0,2,0,5,0,0,12,0,0,4,0,0,4,0,0,1,0,0,2,0,2],[48,101,0.4752,0.39276,0.29672,0.25,0.28571,0.57111,0.0,1.0,5,3,0,5,0,3,0,0,10,0,0,5,0,0,3,0,0,1,0,0,2,0,3],[52,101,0.5149,0.33929,0.24679,0.25,0.28571,0.46431,0.0,1.0,6,1,0,6,0,2,0,0,12,0,0,4,0,0,5,0,0,1,0,0,1,0,1],[56,101,0.5545,0.35264,0.26956,0.14286,0.28571,0.571,0.0,1.0,7,1,0,7,0,3,0,0,7,0,0,6,0,0,4,0,0,3,0,0,1,0,1],[60,101,0.5941,0.42853,0.29879,0.14286,0.42857,0.71429,0.0,1.0,5,2,0,5,0,4,0,0,6,0,0,3,0,0,4,0,0,7,0,0,1,0,2],[64,101,0.6337,0.39284,0.2696,0.2857,0.28571,0.57111,0.0,1.0,4,1,0,4,0,2,0,0,13,0,0,4,0,0,3,0,0,1,0,0,4,0,1],[68,101,0.6733,0.31666,0.26194,0.14,0.2857,0.571,0.0,1.0,6,1,0,6,0,9,0,0,4,0,0,3,0,0,8,0,0,0,0,0,1,0,1],[72,101,0.7129,0.33923,0.25437,0.14286,0.28571,0.4642,0.0,1.0,5,1,0,5,0,6,0,0,8,0,0,5,0,0,4,0,0,2,0,0,1,0,1],[76,101,0.7525,0.38837,0.31181,0.14286,0.2857,0.71429,0.0,1.0,7,2,0,7,0,4,0,0,7,0,0,1,0,0,4,0,0,6,0,0,1,0,2],[80,101,0.7921,0.3125,0.25111,0.14286,0.2857,0.42858,0.0,0.85714,4,0,0,4,0,11,0,0,6,0,0,4,0,0,3,0,0,1,0,0,3,0,0],[84,101,0.8317,0.36595,0.31735,0.14214,0.28571,0.57143,0.0,1.0,7,2,0,7,0,7,0,0,4,0,0,2,0,0,6,0,0,1,0,0,3,0,2],[88,101,0.8713,0.34371,0.292,0.10714,0.28571,0.571,0.0,1.0,8,1,0,8,0,4,0,0,7,0,0,3,0,0,4,0,0,3,0,0,2,0,1],[92,101,0.9109,0.2366,0.26148,0.0,0.14288,0.32143,0.0,1.0,11,1,0,11,0,8,0,0,5,0,0,2,0,0,4,0,0,0,0,0,1,0,1],[96,101,0.9505,0.26785,0.24677,0.14286,0.2857,0.28571,0.0,1.0,7,1,0,7,0,8,0,0,10,0,0,2,0,0,2,0,0,1,0,0,1,0,1],[100,101,0.9901,0.2275,0.2048,0.105,0.14286,0.32143,0.0,0.85714,8,0,0,8,0,10,0,0,6,0,0,5,0,0,2,0,0,0,0,0,1,0,0],[101,101,1.0,0.14277,0.17496,0.0,0.14286,0.1786,0.0,0.71429,14,0,0,14,0,10,0,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"b21f139ec17b71a3","q":"Let $A_{n}$ be the set of partitions of the sequence $(1,2, \\ldots, n)$ into several subsequences such that every two neighbouring terms of each subsequence have different parity, and let $B_{n}$ be the set of partitions of the sequence $(1,2, \\ldots, n)$ into several subsequences such that all the terms of each subsequence have the same parity. 2\n\nProve that for every positive integer $n$ the sets $A_{n}$ and $B_{n+1}$ contain the same number of elements.","t":[{"b":3,"e":0.0,"k":"flat","v":0.00438,"x":0.09375,"p":[[0,33,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.07143,0.17496,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[28,33,0.8485,0.09375,0.20079,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,0,0,0,1,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[32,33,0.9697,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[33,33,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,36,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"19fa746b9c6921c5","q":"Let  $ P$  and  $ Q$  be the common points of two circles. The ray with origin  $ Q$  reflects from the first circle in points  $ A_1$ ,  $ A_2$ , $ \\ldots$  according to the rule ''the angle of incidence is equal to the angle of reflection''. Another ray with origin  $ Q$  reflects from the second circle in the points  $ B_1$ ,  $ B_2$ , $ \\ldots$  in the same manner. Points  $ A_1$ ,  $ B_1$  and  $ P$  occurred to be collinear. Prove that all lines  $ A_iB_i$  pass through P.","t":[{"b":0,"e":0.0,"k":"flat","v":0.2589,"x":0.51328,"p":[[0,41,0.0,0.34374,0.24183,0.14286,0.35714,0.4286,0.0,0.85714,3,0,1,3,0,10,0,0,3,0,0,10,0,0,2,0,0,1,0,0,3,0,0],[4,41,0.0976,0.51328,0.36053,0.24999,0.4286,0.85714,0.0,1.0,6,5,0,6,0,2,0,0,4,0,0,5,0,0,2,0,0,1,0,0,7,0,5],[8,41,0.1951,0.49099,0.29434,0.28571,0.571,0.71429,0.0,1.0,4,2,0,4,0,3,0,0,3,0,0,5,0,0,7,0,0,4,0,0,4,0,2],[12,41,0.2927,0.3884,0.33165,0.0,0.4286,0.71429,0.0,1.0,10,1,0,10,0,4,0,0,0,0,0,4,0,0,3,0,0,8,0,0,2,0,1],[16,41,0.3902,0.3883,0.29937,0.14286,0.28571,0.71429,0.0,0.85714,5,0,0,5,0,8,0,0,4,0,0,3,0,0,2,0,0,6,0,0,4,0,0],[20,41,0.4878,0.39285,0.30721,0.10714,0.42857,0.60714,0.0,0.85714,8,0,0,8,0,3,0,0,4,0,0,4,0,0,5,0,0,3,0,0,5,0,0],[24,41,0.5854,0.45534,0.29109,0.14289,0.4286,0.71429,0.0,1.0,4,1,0,4,0,5,0,0,3,0,0,5,0,0,4,0,0,7,0,0,3,0,1],[28,41,0.6829,0.44196,0.30169,0.14289,0.42857,0.71429,0.0,1.0,5,1,0,5,0,4,0,0,4,0,0,6,0,0,2,0,0,6,0,0,4,0,1],[32,41,0.7805,0.41514,0.33187,0.10714,0.42859,0.71429,0.0,1.0,8,2,0,8,0,4,0,0,3,0,0,2,0,0,4,0,0,7,0,0,2,0,2],[36,41,0.878,0.38839,0.30563,0.14286,0.28571,0.71429,0.0,0.85714,7,0,0,7,0,5,0,0,6,0,0,0,0,0,2,0,0,10,0,0,2,0,0],[40,41,0.9756,0.43301,0.32825,0.14286,0.42857,0.71429,0.0,1.0,7,2,0,7,0,3,0,0,5,0,0,3,0,0,2,0,0,7,0,0,3,0,2],[41,41,1.0,0.2589,0.29108,0.0,0.14286,0.4286,0.0,1.0,12,1,0,12,0,7,0,0,3,0,0,3,0,0,2,0,0,3,0,0,1,0,1]]},{"b":2,"e":0.2857,"k":"flat","v":0.35696,"x":0.59375,"p":[[0,58,0.0,0.36598,0.34804,0.105,0.21428,0.60714,0.0,1.0,8,4,3,8,0,8,0,0,3,0,0,1,0,0,4,0,0,3,0,0,1,0,4],[4,58,0.069,0.50443,0.3694,0.14286,0.4998,0.85714,0.0,1.0,6,6,0,6,0,4,0,0,3,0,0,3,0,0,2,0,0,4,0,0,4,0,6],[8,58,0.1379,0.36157,0.30716,0.0,0.35714,0.57111,0.0,1.0,9,1,0,9,0,3,0,0,4,0,0,6,0,0,3,0,0,3,0,0,3,0,1],[12,58,0.2069,0.59375,0.32164,0.42857,0.71429,0.85714,0.0,1.0,5,4,0,5,0,1,0,0,1,0,0,2,0,0,5,0,0,8,0,0,6,0,4],[16,58,0.2759,0.49997,0.3234,0.24999,0.571,0.71429,0.0,1.0,5,3,0,5,0,3,0,0,3,0,0,4,0,0,4,0,0,6,0,0,4,0,3],[20,58,0.3448,0.5357,0.34069,0.14286,0.71429,0.75,0.0,1.0,5,4,0,5,0,4,0,0,1,0,0,3,0,0,2,0,0,9,0,0,4,0,4],[24,58,0.4138,0.4642,0.37635,0.14214,0.42859,0.85714,0.0,1.0,7,5,0,7,0,6,0,0,1,0,0,4,0,0,0,0,0,5,0,0,4,0,5],[28,58,0.4828,0.51337,0.31512,0.2857,0.4998,0.71429,0.0,1.0,3,5,0,3,0,3,0,0,6,0,0,4,0,0,4,0,0,5,0,0,2,0,5],[32,58,0.5517,0.55351,0.28737,0.28571,0.57143,0.74996,0.0,1.0,2,1,0,2,0,5,0,0,2,0,0,1,0,0,7,0,0,7,0,0,7,0,1],[36,58,0.6207,0.52678,0.35072,0.14286,0.57143,0.85704,0.0,1.0,5,5,0,5,0,4,0,0,3,0,0,1,0,0,4,0,0,6,0,0,4,0,5],[40,58,0.6897,0.47754,0.33625,0.14286,0.49979,0.857,0.0,1.0,3,3,0,3,0,8,0,0,4,0,0,1,0,0,4,0,0,3,0,0,6,0,3],[44,58,0.7586,0.49098,0.34066,0.14286,0.5,0.74996,0.0,1.0,5,3,0,5,0,4,0,0,5,0,0,2,0,0,1,0,0,7,0,0,5,0,3],[48,58,0.8276,0.49999,0.30929,0.14286,0.42857,0.85714,0.0,1.0,2,1,0,2,0,7,0,0,3,0,0,5,0,0,1,0,0,5,0,0,8,0,1],[52,58,0.8966,0.50429,0.36085,0.14286,0.42857,0.85714,0.0,1.0,4,6,0,4,0,6,0,0,3,0,0,5,0,0,1,0,0,2,0,0,5,0,6],[56,58,0.9655,0.39276,0.35898,0.14214,0.21428,0.74996,0.0,1.0,7,2,0,7,0,9,0,0,3,0,0,1,0,0,0,0,0,4,0,0,6,0,2],[58,58,1.0,0.35696,0.32743,0.14286,0.2857,0.42858,0.0,1.0,5,4,0,5,0,10,0,0,5,0,0,5,0,0,0,0,0,1,0,0,2,0,4]]}]},{"i":"a2d8081be2f78b08","q":"Let $a_{1}, a_{2}, a_{3}, \\ldots$ be a sequence of integers, with the property that every consecutive group of $a_{i}$ 's averages to a perfect square. More precisely, for all positive integers $n$ and $k$, the quantity  $$ \\frac{a_{n}+a_{n+1}+\\cdots+a_{n+k-1}}{k} $$  is always the square of an integer. Prove that the sequence must be constant (all $a_{i}$ are equal to the same perfect square).","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.11152,"p":[[0,102,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,102,0.0392,0.10715,0.09449,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,102,0.0784,0.09821,0.09062,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,102,0.1176,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,102,0.1569,0.09366,0.07662,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,102,0.1961,0.09375,0.09182,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,102,0.2353,0.08929,0.09943,0.0,0.14286,0.14286,0.0,0.4286,15,0,0,15,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,102,0.2745,0.09822,0.10374,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,102,0.3137,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,102,0.3529,0.11152,0.10552,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,19,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,102,0.3922,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,102,0.4314,0.09813,0.1037,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,102,0.4706,0.10268,0.08918,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,20,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,102,0.5098,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,102,0.549,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,12,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,102,0.5882,0.10268,0.12993,0.0,0.14286,0.14286,0.0,0.57143,15,0,0,15,0,14,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,102,0.6275,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,102,0.6667,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,102,0.7059,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.143,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,102,0.7451,0.0625,0.13803,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[80,102,0.7843,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,102,0.8235,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,102,0.8627,0.03572,0.08749,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,102,0.902,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,102,0.9412,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,102,0.9804,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[102,102,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.03125,"x":0.14286,"p":[[0,84,0.0,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,84,0.0476,0.14286,0.14286,0.0,0.14286,0.14286,0.0,0.71429,9,0,0,9,0,18,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,84,0.0952,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,84,0.1429,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,84,0.1905,0.10266,0.1142,0.0,0.14286,0.14286,0.0,0.571,13,0,0,13,0,17,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,84,0.2381,0.13384,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,84,0.2857,0.11143,0.09262,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,84,0.3333,0.14277,0.08748,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,84,0.381,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,84,0.4286,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,84,0.4762,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,84,0.5238,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,84,0.5714,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,84,0.619,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,84,0.6667,0.06697,0.07973,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,84,0.7143,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,84,0.7619,0.07589,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,84,0.8095,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,84,0.8571,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,84,0.9048,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,84,0.9524,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,84,1.0,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"821ffdded2a657c4","q":"An $n$-term sequence $\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_{n}$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to $0,1,0$ in that order. Let $b_{n}$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to $0,0,1,1$ or $1,1,0,0$ in that order. Prove that $b_{n+1}=2 a_{n}$ for all positive integers $n$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.16964,"p":[[0,76,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,76,0.0526,0.16964,0.33012,0.0,0.0,0.2857,0.0,1.0,23,4,0,23,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[8,76,0.1053,0.08482,0.26693,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[12,76,0.1579,0.07143,0.21724,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[16,76,0.2105,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,76,0.2632,0.08482,0.26693,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[24,76,0.3158,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[28,76,0.3684,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,76,0.4211,0.10268,0.29284,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[36,76,0.4737,0.13839,0.33021,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[40,76,0.5263,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,76,0.5789,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[52,76,0.6842,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.0982,0.26105,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[64,76,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,76,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.19643,"p":[[0,66,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,66,0.1212,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,66,0.1818,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,66,0.2424,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,66,0.303,0.05804,0.18851,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,66,0.3636,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[28,66,0.4242,0.09374,0.2639,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[32,66,0.4848,0.08929,0.24936,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[36,66,0.5455,0.14286,0.3312,0.0,0.0,0.0,0.0,1.0,26,4,0,26,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[40,66,0.6061,0.16071,0.36202,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[44,66,0.6667,0.19643,0.38919,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[48,66,0.7273,0.08928,0.24679,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[52,66,0.7879,0.125,0.30252,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[56,66,0.8485,0.03571,0.11294,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[60,66,0.9091,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[66,66,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"867d592a82449400","q":"Let   $ABC$  be a triangle with  $m (\\angle C) = 90^\\circ$   and the points  $D \\in [AC], E\\in  [BC]$ . Inside the triangle we construct the semicircles  $C_1, C_2, C_3, C_4$  of diameters  $[AC], [BC], [CD], [CE]$  and let   $\\{C, K\\} = C_1 \\cap C_2,  \\{C, M\\} =C_3 \\cap C_4, \\{C, L\\} = C_2 \\cap C_3, \\{C, N\\} =C_1 \\cap C_4$ . Show that points  $K, L, M, N$  are concyclic.","t":[{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.08036,"p":[[0,32,0.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.04466,0.10378,0.0,0.0,0.0,0.0,0.286,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.05348,0.10557,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.0625,0.11259,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.08036,0.12846,0.0,0.0,0.2857,0.0,0.28571,23,0,0,23,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.07589,"p":[[0,13,0.0,0.0625,0.11811,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.05357,0.14617,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,13,0.6154,0.07589,0.19556,0.0,0.0,0.0,0.0,1.0,26,1,1,26,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,13,0.9231,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"03e2dcd13fd401d7","q":"The incircle of the triangle  $ABC$  touches the side  $AC$  at the point  $D$ . Another circle passes through  $D$  and touches the rays  $BC$  and  $BA$ , the latter at the point  $A$ . Determine the ratio  $\\frac{AD}{DC}$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"42003e75c30aab67","q":"For a given positive integer  $ k$  denote the square of the sum of its digits by  $ f_1(k)$  and let  $ f_{n\\plus{}1}(k) \\equal{} f_1(f_n(k)).$  Determine the value of  $ f_{1991}(2^{1990}).$","t":[{"b":1,"e":0.85714,"k":"falling","v":0.77232,"x":0.95982,"p":[[0,13,0.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[4,13,0.3077,0.875,0.17035,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,8,0,17],[8,13,0.6154,0.77232,0.12299,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,12,0,3],[12,13,0.9231,0.83482,0.12931,0.85713,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,19,0,6],[13,13,1.0,0.78125,0.11285,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,12,0,3]]},{"b":5,"e":1.0,"k":"flat","v":0.92411,"x":1.0,"p":[[0,42,0.0,0.92411,0.13356,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,8,0,21],[4,42,0.0952,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[8,42,0.1905,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,42,0.2857,0.99777,0.01243,1.0,1.0,1.0,0.92857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31],[16,42,0.381,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,42,0.4762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,42,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,42,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,42,0.7619,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,42,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,42,0.9524,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[42,42,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"35ef4c50f1a17a0f","q":"$\\boxed{\\text{A2}}$  Find the maximum value of  $|\\sqrt{x^2+4x+8}-\\sqrt{x^2+8x+17}|$  where  $x$  is a real number.","t":[{"b":3,"e":1.0,"k":"flat","v":0.90178,"x":0.9375,"p":[[0,31,0.0,0.91964,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,18],[4,31,0.129,0.91071,0.10565,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,17],[8,31,0.2581,0.90625,0.09182,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,14],[12,31,0.3871,0.91517,0.08645,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[16,31,0.5161,0.91964,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,17],[20,31,0.6452,0.90178,0.09062,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,13],[24,31,0.7742,0.92857,0.08748,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,12,0,18],[28,31,0.9032,0.91071,0.08565,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,14],[31,31,1.0,0.9375,0.08702,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20]]},{"b":4,"e":1.0,"k":"flat","v":0.88839,"x":0.9375,"p":[[0,34,0.0,0.88839,0.09933,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,15,0,12],[4,34,0.1176,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[8,34,0.2353,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[12,34,0.3529,0.91071,0.09279,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,15],[16,34,0.4706,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[20,34,0.5882,0.91517,0.09354,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,16],[24,34,0.7059,0.91071,0.11152,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,8,0,18],[28,34,0.8235,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[32,34,0.9412,0.9241,0.07974,0.85714,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,16],[34,34,1.0,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15]]}]},{"i":"91df2272c8d0c4e6","q":"Let  $n=2^{\\alpha} \\cdot q$  be a positive integer, where  $\\alpha$  is a nonnegative integer and  $q$  is an odd number. Show that for any positive integer  $m$ , the number of integer solutions to the equation  $x_1^2+x_2^2+\\cdots +x_n^2=m$  is divisible by  $2^{\\alpha +1}$ .","t":[{"b":4,"e":0.28571,"k":"flat","v":0.49107,"x":0.69196,"p":[[0,16,0.0,0.63393,0.27184,0.4286,0.64286,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,4,0,0,3,0,0,7,0,0,7,0,0,2,0,7],[4,16,0.25,0.58482,0.29528,0.42857,0.57143,0.74996,0.0,1.0,2,7,0,2,0,3,0,0,0,0,0,7,0,0,8,0,0,4,0,0,1,0,7],[8,16,0.5,0.69196,0.21756,0.57143,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,10,0,0,2,0,7],[12,16,0.75,0.5089,0.1954,0.42857,0.57121,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,3,0,0,8,0,0,13,0,0,2,0,0,2,0,1],[16,16,1.0,0.49107,0.23128,0.39286,0.42859,0.71429,0.0,1.0,1,1,1,1,0,4,0,0,3,0,0,9,0,0,5,0,0,8,0,0,1,0,1]]},{"b":6,"e":0.42857,"k":"flat","v":0.51339,"x":0.66069,"p":[[0,17,0.0,0.64284,0.29015,0.42857,0.64286,1.0,0.14286,1.0,0,9,0,0,0,3,0,0,4,0,0,3,0,0,6,0,0,5,0,0,2,0,9],[4,17,0.2353,0.66069,0.23624,0.53539,0.64286,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,4,0,0,8,0,0,8,0,0,0,0,8],[8,17,0.4706,0.51339,0.1984,0.42857,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,14,0,0,2,0,0,7,0,0,2,0,1],[12,17,0.7059,0.5625,0.26711,0.42857,0.57143,0.75,0.14286,1.0,0,3,0,0,0,6,0,0,0,0,0,7,0,0,7,0,0,4,0,0,5,0,3],[16,17,0.9412,0.51339,0.20473,0.42857,0.5,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,1,0,0,11,0,0,7,0,0,7,0,0,1,0,1],[17,17,1.0,0.52667,0.26603,0.28571,0.49979,0.71429,0.14,1.0,0,3,0,0,0,5,0,0,4,0,0,7,0,0,6,0,0,3,0,0,4,0,3]]}]},{"i":"47d937deadad322a","q":"There are given $2^{500}$ points on a circle labeled $1,2, \\ldots, 2^{500}$ in some order. Prove that one can choose 100 pairwise disjoint chords joining some of these points so that the 100 sums of the pairs of numbers at the endpoints of the chosen chords are equal.","t":[{"b":4,"e":0.0,"k":"flat","v":0.01339,"x":0.02232,"p":[[0,13,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.01786,"p":[[0,6,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bfd228cd42f11d40","q":"**2** Let there be an infinite geometric sequence  $\\{a_n\\}$ , where the common ratio  $0<|q|<1$ . Given that   $$ \\sum_{i=1}^\\infty a_n = \\sum_{i=1}^\\infty a_n^2 $$  find the largest possible range of  $a_2$ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.70536,"x":0.74553,"p":[[0,18,0.0,0.74553,0.08552,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,3],[4,18,0.2222,0.73214,0.09942,0.71429,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,3],[8,18,0.4444,0.71429,0.07143,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,1],[12,18,0.6667,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[16,18,0.8889,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[18,18,1.0,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.6875,"x":0.72321,"p":[[0,55,0.0,0.72321,0.09407,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,27,0,0,1,0,2],[4,55,0.0727,0.72321,0.08702,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,29,0,0,0,0,2],[8,55,0.1455,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[12,55,0.2182,0.6875,0.08328,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,29,0,0,0,0,0],[16,55,0.2909,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[20,55,0.3636,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[24,55,0.4364,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[28,55,0.5091,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[32,55,0.5818,0.69643,0.06916,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0],[36,55,0.6545,0.70982,0.05629,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,30,0,0,1,0,0],[40,55,0.7273,0.69643,0.06916,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0],[44,55,0.8,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[48,55,0.8727,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[52,55,0.9455,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[55,55,1.0,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0]]}]},{"i":"d2942c5a5d4f2f22","q":"A ray emanating from the vertex $A$ of the triangle $A B C$ intersects the side $B C$ at $X$ and the circumcircle of $A B C$ at $Y$. Prove that $\\frac{1}{A X}+\\frac{1}{X Y} \\geq \\frac{4}{B C}$.","t":[{"b":0,"e":0.71429,"k":"flat","v":0.70982,"x":0.84375,"p":[[0,11,0.0,0.76339,0.23854,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,15,0,0,0,0,12],[4,11,0.3636,0.84375,0.22119,0.71429,1.0,1.0,0.1429,1.0,0,19,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,9,0,0,0,0,19],[8,11,0.7273,0.70982,0.06667,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,1],[11,11,1.0,0.71429,0.06186,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,0,0,1]]},{"b":1,"e":0.71429,"k":"flat","v":0.66516,"x":0.84821,"p":[[0,29,0.0,0.80357,0.21053,0.71429,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,12,0,0,0,0,15],[4,29,0.1379,0.66516,0.23313,0.53539,0.71429,0.71429,0.2857,1.0,0,7,0,0,0,0,0,0,5,0,0,3,0,0,4,0,0,13,0,0,0,0,7],[8,29,0.2759,0.84821,0.14258,0.71429,0.71429,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,15],[12,29,0.4138,0.82589,0.1461,0.71429,0.71429,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,0,0,13],[16,29,0.5517,0.82143,0.17128,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,16,0,0,0,0,14],[20,29,0.6897,0.75446,0.12993,0.71429,0.71429,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,24,0,0,0,0,6],[24,29,0.8276,0.69196,0.1017,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,27,0,0,0,0,1],[28,29,0.9655,0.70982,0.10403,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,27,0,0,0,0,2],[29,29,1.0,0.69196,0.11355,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,28,0,0,0,0,1]]}]},{"i":"7eeea1329977a96d","q":"Let  $m$  be a positive integer and  $x_0, y_0$  integers such that  $x_0, y_0$  are relatively prime,  $y_0$  divides  $x_0^2+m$ , and  $x_0$  divides  $y_0^2+m$ . Prove that there exist positive integers  $x$  and  $y$  such that  $x$  and  $y$  are relatively prime,  $y$  divides  $x^2 + m$ ,  $x$  divides  $y^2 + m$ , and  $x + y \\leq m+ 1.$","t":[{"b":1,"e":0.0,"k":"falling","v":0.00438,"x":0.42411,"p":[[0,23,0.0,0.42411,0.45103,0.0,0.14286,1.0,0.0,1.0,16,9,0,16,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,2,0,9],[4,23,0.1739,0.29464,0.44022,0.0,0.0,0.85714,0.0,1.0,22,7,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,7],[8,23,0.3478,0.16063,0.36203,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[12,23,0.5217,0.17411,0.33069,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,1],[16,23,0.6957,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.04017,0.15661,0.0,0.0,0.0,0.0,0.857,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.00446,"x":0.30804,"p":[[0,13,0.0,0.30804,0.43022,0.0,0.0,0.85714,0.0,1.0,21,5,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,5],[4,13,0.3077,0.1875,0.38038,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[8,13,0.6154,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.05803,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ff0e6986304f35f1","q":"Let  $a$  and  $b$  be positive integers such that  $ab+1$  divides  $a^{2}+b^{2}$ . Show that \\[\\frac{a^{2}+b^{2}}{ab+1}\\] is the square of an integer.","t":[{"b":0,"e":0.71429,"k":"falling","v":0.60708,"x":0.8616,"p":[[0,25,0.0,0.84821,0.19541,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,0,4,0,17],[4,25,0.16,0.8616,0.19719,0.71429,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,5,0,0,5,0,18],[8,25,0.32,0.6875,0.2126,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,7,0,0,3,0,7],[12,25,0.48,0.60708,0.16753,0.5354,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,3,0,0,5,0,0,9,0,0,12,0,0,2,0,1],[16,25,0.64,0.6741,0.09605,0.67857,0.71429,0.71429,0.28571,0.857,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,23,0,0,1,0,0],[20,25,0.8,0.66959,0.10379,0.57143,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,15,0,0,2,0,1],[24,25,0.96,0.64729,0.10707,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,21,0,0,0,0,0],[25,25,1.0,0.67844,0.11841,0.67536,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,22,0,0,1,0,1]]},{"b":1,"e":1.0,"k":"rising","v":0.80356,"x":1.0,"p":[[0,22,0.0,0.80356,0.22799,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,3,0,14],[4,22,0.1818,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[8,22,0.3636,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,22,0.5455,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[20,22,0.9091,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[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":"f5284256ee5ea46b","q":"Point  $H$  is the orthocenter of the acute triangle  $\\Delta ABC$  and point  $K$ ,situated on the line  $(BC)$ , is the foot of the perpendicular from point  $A$  .The circle   $\\Omega$  passes through points  $A$  and  $K$  ,intersecting the sides  $(AB)$  and  $(AC)$  in points  $M$  and  $N$  .The line that passes through point  $A$  and is parallel with  $BC$  intersects again the circumcircles of triangles  $\\Delta AHM$  and  $\\Delta AHN$  in points  $X$  and  $Y$ .Prove that  $XY =BC$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,26,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.03125,"p":[[0,25,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.01777,0.04701,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,25,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"99b3bafc955f0d16","q":"Let  $\\omega_{1}$  be a circle with centre  $O$ .  $P$  is a point on  $\\omega_{1}$ .  $\\omega_{2}$  is a circle with centre  $P$ , with radius smaller than  $\\omega_{1}$ .  $\\omega_{1}$  meets  $\\omega_{2}$  at points  $T$  and  $Q$ . Let  $TR$  be a diameter of  $\\omega_{2}$ . Draw another two circles with  $RQ$  as the radius,  $R$  and  $P$  as the centres. These two circles meet at point  $M$ , with  $M$  and  $Q$  lie on the same side of  $PR$ . A circle with centre  $M$  and radius  $MR$  intersects  $\\omega_{2}$  at  $R$  and  $N$ . Prove that a circle with centre  $T$  and radius  $TN$  passes through  $O$ .","t":[{"b":0,"e":0.42857,"k":"rising","v":0.63835,"x":0.90625,"p":[[0,23,0.0,0.64731,0.34066,0.42859,0.71429,1.0,0.0,1.0,4,9,0,4,0,1,0,0,1,0,0,5,0,0,2,0,0,4,0,0,6,0,9],[4,23,0.1739,0.75444,0.2702,0.57132,0.85714,1.0,0.1429,1.0,0,12,0,0,0,1,0,0,4,0,0,2,0,0,2,0,0,4,0,0,7,0,12],[8,23,0.3478,0.63835,0.29231,0.42857,0.64286,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,5,0,0,4,0,0,5,0,0,4,0,0,4,0,8],[12,23,0.5217,0.8973,0.12493,0.857,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,17],[16,23,0.6957,0.81691,0.18982,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,3,0,0,11,0,11],[20,23,0.8696,0.90625,0.12682,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,19],[23,23,1.0,0.87944,0.13883,0.857,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,10,0,15]]},{"b":4,"e":0.71429,"k":"flat","v":0.62946,"x":0.82143,"p":[[0,16,0.0,0.62946,0.22547,0.42857,0.64286,0.85704,0.28571,1.0,0,3,0,0,0,0,0,0,4,0,0,8,0,0,4,0,0,6,0,0,7,0,3],[4,16,0.25,0.63837,0.32923,0.39286,0.64286,1.0,0.0,1.0,1,11,0,1,0,4,0,0,3,0,0,3,0,0,5,0,0,3,0,0,2,0,11],[8,16,0.5,0.82143,0.15152,0.71429,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,9,0,9],[12,16,0.75,0.7321,0.18816,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,11,0,0,6,0,6],[16,16,1.0,0.72762,0.15308,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,13,0,0,6,0,4]]}]},{"i":"c25226d5866b8018","q":"Find all functions  $f$  from the set  $\\mathbb{R}$  of real numbers into  $\\mathbb{R}$  which satisfy for all  $x, y, z \\in \\mathbb{R}$  the identity \\[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\\]","t":[{"b":2,"e":0.2857,"k":"flat","v":0.13839,"x":0.25446,"p":[[0,31,0.0,0.13839,0.11564,0.0,0.14286,0.17857,0.0,0.42857,10,0,0,10,0,14,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.15625,0.10326,0.14286,0.14286,0.17857,0.0,0.42857,6,0,0,6,0,18,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.18304,0.14827,0.14286,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,15,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,31,0.3871,0.1875,0.12595,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,15,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,31,0.5161,0.20982,0.13825,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,14,0,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[20,31,0.6452,0.25446,0.13236,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,13,0,0,12,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[24,31,0.7742,0.21875,0.10705,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,13,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,31,0.9032,0.21875,0.08736,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,11,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.24107,0.07523,0.14286,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,11,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.09812,"x":0.14737,"p":[[0,18,0.0,0.14737,0.11575,0.10714,0.14286,0.14286,0.0,0.43,8,0,0,8,0,17,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.13384,0.09406,0.14214,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,21,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.12937,0.09005,0.14214,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.11161,0.07771,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.14286,0.07143,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.09812,0.06616,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ed4d92ab8d65f3d3","q":"Let  $a, b, c$  be positive reals such that  $a+b+c=3$ . Prove that \\[18\\sum_{\\text{cyc}}\\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\\ge 15. \\]*Proposed by David Stoner*","t":[{"b":2,"e":0.42857,"k":"flat","v":0.44643,"x":0.62946,"p":[[0,24,0.0,0.54911,0.42274,0.0,0.71429,1.0,0.0,1.0,9,12,0,9,0,2,0,0,1,0,0,2,0,0,1,0,0,5,0,0,0,0,12],[4,24,0.1667,0.58036,0.3976,0.28571,0.50001,1.0,0.0,1.0,6,13,0,6,0,1,0,0,4,0,0,5,0,0,1,0,0,1,0,0,1,0,13],[8,24,0.3333,0.51335,0.38607,0.14286,0.571,1.0,0.0,1.0,7,9,0,7,0,4,0,0,0,0,0,4,0,0,5,0,0,2,0,0,1,0,9],[12,24,0.5,0.44643,0.32488,0.14286,0.42857,0.71429,0.0,1.0,4,4,0,4,0,6,0,0,4,0,0,7,0,0,0,0,0,5,0,0,2,0,4],[16,24,0.6667,0.57588,0.27312,0.42857,0.57143,0.75,0.14286,1.0,0,6,0,0,0,4,0,0,2,0,0,8,0,0,7,0,0,3,0,0,2,0,6],[20,24,0.8333,0.62946,0.33855,0.28571,0.71429,1.0,0.0,1.0,1,10,0,1,0,5,0,0,3,0,0,3,0,0,3,0,0,3,0,0,4,0,10],[24,24,1.0,0.59375,0.32362,0.39286,0.71429,0.85714,0.0,1.0,2,6,0,2,0,5,0,0,1,0,0,4,0,0,3,0,0,6,0,0,5,0,6]]},{"b":5,"e":0.85714,"k":"falling","v":0.25,"x":0.49106,"p":[[0,15,0.0,0.41517,0.40462,0.0,0.28571,0.85714,0.0,1.0,11,7,0,11,0,4,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,7],[4,15,0.2667,0.49106,0.40396,0.10714,0.42859,1.0,0.0,1.0,8,10,0,8,0,4,0,0,1,0,0,5,0,0,1,0,0,3,0,0,0,0,10],[8,15,0.5333,0.40622,0.41049,0.0,0.21429,0.89286,0.0,1.0,11,8,0,11,0,5,0,0,2,0,0,1,0,0,3,0,0,1,0,0,1,0,8],[12,15,0.8,0.2857,0.26962,0.14286,0.14286,0.46418,0.0,1.0,7,1,0,7,0,11,0,0,4,0,0,2,0,0,3,0,0,4,0,0,0,0,1],[15,15,1.0,0.25,0.2369,0.10714,0.14286,0.42857,0.0,0.85714,8,0,0,8,0,11,0,0,3,0,0,5,0,0,2,0,0,2,0,0,1,0,0]]}]},{"i":"7fed920a484a3ecb","q":"Given convex hexagon  $ABCDEF$ , inscribed in the circle. Prove that  $AC*BD*DE*CE*EA*FB \\geq 27 AB * BC * CD * DE * EF * FA$","t":[{"b":1,"e":0.0,"k":"flat","v":0.02679,"x":0.04902,"p":[[0,2,0.0,0.04902,0.10471,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,1.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0625,"x":0.15178,"p":[[0,16,0.0,0.0625,0.11811,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.14285,0.13832,0.0,0.14286,0.2857,0.0,0.28571,15,0,0,15,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.13391,0.1747,0.0,0.0,0.28571,0.0,0.57143,19,0,0,19,0,0,0,0,11,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,16,0.75,0.10713,0.15563,0.0,0.0,0.2857,0.0,0.571,21,0,0,21,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,16,1.0,0.15178,0.14258,0.0,0.2857,0.28571,0.0,0.28571,15,0,0,15,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3fdbe2507cea4907","q":"In the right triangle  $ABC$  with hypotenuse  $AB$ , the incircle touches  $BC$  and  $AC$  at points  ${{A}_{1}}$  and  ${{B}_{1}}$  respectively. The straight line containing the midline of  $\\Delta ABC$  , parallel to  $AB$ , intersects its circumcircle at points  $P$  and  $T$ . Prove that points  $P,T,{{A}_{1}}$  and  ${{B}_{1}}$  lie on one circle.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.09366,"p":[[0,26,0.0,0.09366,0.22757,0.0,0.0,0.14071,0.0,1.0,23,1,0,23,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,26,0.1538,0.04455,0.08318,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,26,0.6154,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[24,26,0.9231,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.00446,"x":0.08473,"p":[[0,22,0.0,0.05803,0.16311,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,22,0.1818,0.06696,0.12869,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.08473,0.2168,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[12,22,0.5455,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c6a4fec98540a570","q":"Is there exist four points on plane, such that distance between any two of them is integer odd number?\n\nMay be it is geometry or number theory or combinatoric, I don't know, so it here :blush:","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,31,0.0,0.00893,0.0346,0.0,0.0,0.0,0.0,0.143,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"95c3e4616bb4933a","q":"Determine whether there exists an infinite sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of positive integers which satisfies the equality\n\n$$\na_{n+2}=a_{n+1}+\\sqrt{a_{n+1}+a_{n}}\n$$\n\nfor every positive integer n.\n(Japan)","t":[{"b":1,"e":1.0,"k":"flat","v":0.87054,"x":0.9375,"p":[[0,11,0.0,0.87054,0.25092,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,0,0,24],[4,11,0.3636,0.9375,0.17835,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[8,11,0.7273,0.91518,0.22829,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,27],[11,11,1.0,0.91518,0.16698,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,25]]},{"b":2,"e":1.0,"k":"flat","v":0.87946,"x":1.0,"p":[[0,13,0.0,0.88838,0.24155,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,25],[4,13,0.3077,0.87946,0.25781,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,4,0,23],[8,13,0.6154,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[12,13,0.9231,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[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":"078a61f01ef7ff42","q":"Let  $a, b, c$  be integers such that \\[\\frac ab+\\frac bc+\\frac ca= 3\\] Prove that  $abc$  is a cube of an integer.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,12,0.0,0.10714,0.26,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,1],[4,12,0.3333,0.02678,0.10374,0.0,0.0,0.0,0.0,0.5714,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,12,0.6667,0.01116,0.05085,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"volatile","v":0.16518,"x":0.77677,"p":[[0,4,0.0,0.16518,0.29257,0.0,0.0,0.21429,0.0,0.85714,23,0,0,23,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0],[4,4,1.0,0.77677,0.14698,0.71429,0.857,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,10,0,0,13,0,4]]}]},{"i":"df4720f6ef826f0b","q":"Determine all odd primes  $p$  and  $q$  such that the equation  $x^p + y^q = pq$  at least one solution  $(x, y)$  where  $x$  and  $y$  are positive integers.","t":[{"b":0,"e":1.0,"k":"rising","v":0.6473,"x":0.9732,"p":[[0,23,0.0,0.74552,0.30668,0.5354,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,3,0,0,3,0,0,3,0,0,2,0,16],[4,23,0.1739,0.6473,0.29012,0.42859,0.71429,0.89286,0.0,1.0,1,8,0,1,0,2,0,0,4,0,0,2,0,0,4,0,0,9,0,0,2,0,8],[8,23,0.3478,0.89285,0.12877,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,18],[12,23,0.5217,0.91964,0.1234,0.82132,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[16,23,0.6957,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[20,23,0.8696,0.9732,0.09067,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29],[23,23,1.0,0.96428,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27]]},{"b":7,"e":0.2857,"k":"falling","v":0.36597,"x":0.77232,"p":[[0,40,0.0,0.77232,0.33285,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,2,0,0,2,0,0,1,0,0,0,0,0,4,0,0,3,0,18],[4,40,0.1,0.68303,0.3169,0.42859,0.71429,1.0,0.0,1.0,1,13,0,1,0,2,0,0,2,0,0,7,0,0,2,0,0,3,0,0,2,0,13],[8,40,0.2,0.60713,0.29015,0.42857,0.71429,0.75,0.0,1.0,2,6,0,2,0,2,0,0,2,0,0,5,0,0,4,0,0,9,0,0,2,0,6],[12,40,0.3,0.6875,0.28669,0.42859,0.71429,1.0,0.0,1.0,1,11,0,1,0,1,0,0,3,0,0,4,0,0,2,0,0,10,0,0,0,0,11],[16,40,0.4,0.49102,0.29435,0.24999,0.4998,0.71429,0.0,1.0,2,2,0,2,0,6,0,0,4,0,0,4,0,0,4,0,0,6,0,0,4,0,2],[20,40,0.5,0.5625,0.25238,0.39286,0.50001,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,7,0,0,8,0,0,4,0,0,6,0,0,1,0,5],[24,40,0.6,0.5445,0.25605,0.28571,0.4286,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,8,0,0,8,0,0,2,0,0,8,0,0,1,0,4],[28,40,0.7,0.49092,0.32516,0.14286,0.57143,0.71429,0.0,1.0,3,5,0,3,0,7,0,0,3,0,0,1,0,0,6,0,0,7,0,0,0,0,5],[32,40,0.8,0.46651,0.35083,0.14286,0.42857,0.71429,0.0,1.0,4,7,0,4,1,5,0,0,5,0,0,3,0,0,4,0,0,3,0,0,0,0,7],[36,40,0.9,0.38374,0.30827,0.14286,0.28571,0.60714,0.0,1.0,5,2,0,5,0,8,0,0,5,0,0,2,0,0,4,0,0,4,0,0,2,0,2],[40,40,1.0,0.36597,0.28563,0.14286,0.28571,0.57143,0.0,1.0,5,2,0,5,0,7,0,0,6,0,0,4,0,0,3,0,0,5,0,0,0,0,2]]}]},{"i":"3b59725089b86fe7","q":"Determine, with proof, the smallest positive integer  $c$  such that for any positive integer  $n$ , the decimal representation of the number  $c^n+2014$  has digits all less than  $5$ .\n\n*Proposed by Evan Chen*","t":[{"b":3,"e":0.71429,"k":"flat","v":0.83036,"x":0.94196,"p":[[0,62,0.0,0.94196,0.11214,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,1,0,25],[4,62,0.0645,0.91964,0.1234,0.82143,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,2,0,22],[8,62,0.129,0.87054,0.15303,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,3,0,17],[12,62,0.1935,0.86161,0.17307,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,11,0,0,1,0,18],[16,62,0.2581,0.83929,0.17405,0.71429,0.85714,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,14,0,0,0,0,16],[20,62,0.3226,0.83929,0.20124,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,18],[24,62,0.3871,0.85714,0.15567,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,2,0,16],[28,62,0.4516,0.85714,0.16751,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,2,0,17],[32,62,0.5161,0.83036,0.18363,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,12,0,0,2,0,15],[36,62,0.5806,0.90625,0.13651,0.71429,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,2,0,21],[40,62,0.6452,0.83929,0.15872,0.71429,0.71429,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,15],[44,62,0.7097,0.84821,0.14698,0.71429,0.78571,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,1,0,15],[48,62,0.7742,0.83927,0.14176,0.71429,0.78571,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,3,0,13],[52,62,0.8387,0.8482,0.1673,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,11,0,0,2,0,16],[56,62,0.9032,0.84375,0.14445,0.71429,0.78564,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,2,0,14],[60,62,0.9677,0.88393,0.1448,0.71429,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,1,0,19],[62,62,1.0,0.89286,0.13832,0.71429,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,20]]},{"b":4,"e":1.0,"k":"flat","v":0.94643,"x":1.0,"p":[[0,42,0.0,0.95982,0.11971,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,28],[4,42,0.0952,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[8,42,0.1905,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,26],[12,42,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,42,0.381,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,42,0.4762,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,42,0.5714,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[28,42,0.6667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"a7d03356b1bb1b40","q":"Let  $\\sum_{n=1}^\\infty a_n$  be a divergent series with positive nonincreasing terms. Prove that the series  $$ \\sum_{n=1}^\\infty\\frac{a_n}{1+na_n} $$  diverges.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.1338,"x":0.17858,"p":[[0,20,0.0,0.17858,0.14726,0.14286,0.14286,0.1786,0.0,0.4286,7,0,0,7,0,17,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.1384,0.12619,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,19,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.17411,0.18808,0.14286,0.14286,0.14286,0.0,1.0,7,1,0,7,0,19,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[12,20,0.6,0.13393,0.07087,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.1338,0.07104,0.14286,0.14286,0.14286,0.0,0.43,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.11152,"x":0.19197,"p":[[0,26,0.0,0.13839,0.12103,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,18,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.14286,0.13363,0.0,0.14286,0.14287,0.0,0.4286,10,0,0,10,0,16,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.12946,0.11495,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,17,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.125,0.09943,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.19197,0.13175,0.14286,0.14286,0.2857,0.0,0.4286,4,0,0,4,0,19,0,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.14732,0.14054,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,16,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"73816675dd7ee438","q":"A triangle  $ ABC$  is given with  $ \\left|AB\\right| > \\left|AC\\right|$ . Line  $ l$  tangents in a point  $ A$  the circumcirle of  $ ABC$ . A circle centered in  $ A$  with radius  $ \\left|AC\\right|$  cuts  $ AB$  in the point  $ D$  and the line  $ l$  in points  $ E, F$  (such that  $ C$  and  $ E$  are in the same halfplane with respect to  $ AB$ ). Prove that the line  $ DE$  passes through the incenter of  $ ABC$ .","t":[{"b":2,"e":0.14286,"k":"falling","v":0.18741,"x":0.43747,"p":[[0,17,0.0,0.43747,0.17103,0.28571,0.57121,0.57143,0.0,0.57143,1,0,0,1,0,3,0,0,7,0,0,3,0,0,18,0,0,0,0,0,0,0,0],[4,17,0.2353,0.33034,0.16534,0.14286,0.28571,0.4642,0.14286,0.57143,0,0,0,0,0,10,0,0,10,0,0,4,0,0,8,0,0,0,0,0,0,0,0],[8,17,0.4706,0.18741,0.12598,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,18,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,17,0.7059,0.20973,0.1129,0.14286,0.2857,0.28571,0.0,0.42857,4,0,0,4,0,11,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.22313,0.13339,0.14286,0.2143,0.28571,0.0,0.4286,4,0,0,4,0,12,0,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.22768,0.14664,0.14286,0.2143,0.32143,0.0,0.4286,5,0,0,5,0,11,0,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.17411,"x":0.41071,"p":[[0,34,0.0,0.41071,0.21354,0.24999,0.42859,0.57143,0.0,1.0,1,1,0,1,0,7,0,0,5,0,0,4,0,0,14,0,0,0,0,0,0,0,1],[4,34,0.1176,0.34819,0.16724,0.14289,0.28571,0.571,0.14286,0.57143,0,0,0,0,0,9,0,0,9,0,0,5,0,0,9,0,0,0,0,0,0,0,0],[8,34,0.2353,0.25447,0.19799,0.10714,0.28571,0.42857,0.0,0.57143,8,0,0,8,0,6,0,0,8,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[12,34,0.3529,0.35714,0.20825,0.2857,0.35714,0.4286,0.0,1.0,4,1,0,4,0,2,0,0,10,0,0,9,0,0,6,0,0,0,0,0,0,0,1],[16,34,0.4706,0.27677,0.15945,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,6,0,0,13,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[20,34,0.5882,0.32141,0.18209,0.14286,0.28571,0.4642,0.0,0.57143,2,0,0,2,0,9,0,0,8,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[24,34,0.7059,0.28571,0.17128,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,6,0,0,13,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[28,34,0.8235,0.17411,0.14167,0.0,0.14286,0.2857,0.0,0.4286,9,0,0,9,0,11,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.29909,0.16113,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,6,0,0,12,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[34,34,1.0,0.26339,0.17896,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,4,0,0,11,0,0,7,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"2d27e5e645473a37","q":"For integers  $m\\geq 3$ ,  $n$  and  $x_1,x_2, \\ldots , x_m$  if  $x_{i+1}-x_i \\equiv x_i-x_{i-1} (mod n) $  for every  $2\\leq i \\leq m-1$ , we say that the  $m$ -tuple  $(x_1,x_2,\\ldots , x_m)$  is an arithmetic sequence in  $(mod n)$ . Let  $p\\geq 5$  be a prime number and  $1<a<p-1$  be an integer. Let  ${a_1,a_2,\\ldots , a_k}$  be the set of all possible remainders when positive powers of  $a$  are divided by  $p$ . Show that if a permutation of  ${a_1,a_2,\\ldots , a_k}$  is an arithmetic sequence in  $(mod p)$ , then  $k=p-1$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.73659,"x":1.0,"p":[[0,25,0.0,0.83036,0.26351,0.57143,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,0,1,0,21],[4,25,0.16,0.73659,0.33714,0.53539,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,2,0,0,3,0,0,4,0,0,1,0,0,2,0,17],[8,25,0.32,0.76785,0.34581,0.5354,1.0,1.0,0.0,1.0,3,21,0,3,0,0,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,21],[12,25,0.48,0.83482,0.25282,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,1,0,0,1,0,21],[16,25,0.64,0.89732,0.22934,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,26],[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":5,"e":1.0,"k":"rising","v":0.80357,"x":1.0,"p":[[0,33,0.0,0.80357,0.26426,0.57143,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,1,0,0,1,0,0,8,0,0,2,0,0,0,0,19],[4,33,0.1212,0.87946,0.24772,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,3,0,23],[8,33,0.2424,0.93302,0.14722,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26],[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],[20,33,0.6061,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,33,0.7273,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,33,0.9697,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[33,33,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]}]},{"i":"8d31f2ecd2a29eb8","q":"Let  $a$  and  $b$  be real numbers. Define  $f_{a,b}\\colon R^2\\to R^2$  by  $f_{a,b}(x;y)=(a-by-x^2;x)$ . If  $P=(x;y)\\in R^2$ , define  $f^0_{a,b}(P) = P$  and  $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$  for all nonnegative integers  $k$ .\r\n\r\nThe set  $per(a;b)$  of the *periodic points* of  $f_{a,b}$  is the set of points  $P\\in R^2$  such that  $f_{a,b}^n(P) = P$  for some positive integer  $n$ .\r\n\r\nFix  $b$ . Prove that the set  $A_b=\\{a\\in R \\mid per(a;b)\\neq \\emptyset\\}$  admits a minimum. Find this minimum.","t":[{"b":3,"e":1.0,"k":"flat","v":0.84374,"x":0.94642,"p":[[0,6,0.0,0.84374,0.21239,0.85714,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,0,12,0,14],[4,6,0.6667,0.94642,0.09279,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[6,6,1.0,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17]]},{"b":6,"e":1.0,"k":"flat","v":0.9375,"x":0.98661,"p":[[0,61,0.0,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[4,61,0.0656,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[8,61,0.1311,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[12,61,0.1967,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,61,0.2623,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[20,61,0.3279,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[24,61,0.3934,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,61,0.459,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,61,0.5246,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,61,0.5902,0.95535,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[40,61,0.6557,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,61,0.7213,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[48,61,0.7869,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[52,61,0.8525,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[56,61,0.918,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24],[60,61,0.9836,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[61,61,1.0,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"3f8378296b68567c","q":"Given a scalene triangle  $ ABC$ . Let  $ A'$ ,  $ B'$ ,  $ C'$  be the points where the internal bisectors of the angles  $ CAB$ ,  $ ABC$ ,  $ BCA$  meet the sides  $ BC$ ,  $ CA$ ,  $ AB$ , respectively. Let the line  $ BC$  meet the perpendicular bisector of  $ AA'$  at  $ A''$ . Let the line  $ CA$  meet the perpendicular bisector of  $ BB'$  at  $ B'$ . Let the line  $ AB$  meet the perpendicular bisector of  $ CC'$  at  $ C''$ . Prove that  $ A''$ ,  $ B''$  and  $ C''$  are collinear.","t":[{"b":0,"e":0.0,"k":"flat","v":0.01786,"x":0.07589,"p":[[0,27,0.0,0.07589,0.13355,0.0,0.0,0.07143,0.0,0.42857,24,0,1,24,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.04911,0.12682,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,27,0.2963,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[27,27,1.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01786,"x":0.05804,"p":[[0,26,0.0,0.05804,0.12299,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.02232,0.0724,0.0,0.0,0.0,0.0,0.2857,29,0,2,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9446407141cc7c8b","q":"Fix an integer  $n\\geq 2$ . Consider  $2n$  real numbers  $a_1,\\ldots,a_n$  and  $b_1,\\ldots, b_n$ . Let  $S$  be the set of all pairs  $(x, y)$  of real numbers for which  $M_i = a_ix + b_iy$ ,  $i=1,2,\\ldots,n$  are pairwise distinct. For every such pair sort the corresponding values  $M_1, M_2, \\ldots, M_n$  increasingly and let  $M(i)$  be the  $i$ -th term in the list thus sorted. This de\ffines an index permutation of  $1,2,\\ldots,n$ . Let  $N$  be the number of all such permutations, as the pairs run through all of  $S$ . In terms of  $n$ , determine the largest value  $N$  may achieve over all possible choices of  $a_1,\\ldots,a_n,b_1,\\ldots,b_n$ .","t":[{"b":1,"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],[8,39,0.2051,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,39,0.3077,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,39,0.5128,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,39,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[39,39,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":1.0,"k":"flat","v":0.99554,"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],[4,60,0.0667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,60,0.1333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[16,60,0.2667,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,60,0.5333,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]}]},{"i":"e1afc0f2f478b716","q":"Let $A B C$ be an acute angled triangle with $\\angle B A C=60^{\\circ}$ and $A B>A C$. Let $I$ be the incenter, and $H$ the orthocenter of the triangle $A B C$. Prove that\n\n$$\n2 \\angle A H I=3 \\angle A B C .\n$$","t":[{"b":0,"e":0.57143,"k":"flat","v":0.29016,"x":0.47097,"p":[[0,29,0.0,0.41515,0.30588,0.10714,0.57121,0.57143,0.0,1.0,8,2,1,8,0,2,0,0,4,0,0,0,0,0,11,0,0,5,0,0,0,0,2],[4,29,0.1379,0.29016,0.30195,0.0,0.14286,0.57143,0.0,1.0,12,1,1,12,0,6,0,0,2,0,0,0,0,0,7,0,0,4,0,0,0,0,1],[8,29,0.2759,0.41515,0.23243,0.24999,0.57143,0.57143,0.0,0.71429,4,0,0,4,0,4,0,0,5,0,0,0,0,0,16,0,0,3,0,0,0,0,0],[12,29,0.4138,0.47097,0.23066,0.53539,0.57143,0.57143,0.0,0.71429,4,0,0,4,1,2,0,0,0,0,0,1,0,0,20,0,0,4,0,0,0,0,0],[16,29,0.5517,0.45534,0.25614,0.24999,0.57143,0.57143,0.0,0.71429,6,0,0,6,0,2,0,0,1,0,0,0,0,0,17,0,0,6,0,0,0,0,0],[20,29,0.6897,0.4531,0.23048,0.46396,0.57143,0.57143,0.0,0.71429,4,0,0,4,1,3,0,0,0,0,0,0,0,0,22,0,0,2,0,0,0,0,0],[24,29,0.8276,0.32587,0.27252,0.0,0.49979,0.57143,0.0,0.71429,12,0,0,12,0,1,0,0,2,0,0,1,0,0,15,0,0,1,0,0,0,0,0],[28,29,0.9655,0.39732,0.25935,0.10714,0.57143,0.57143,0.0,0.71429,8,0,0,8,0,2,0,0,1,0,0,1,0,0,18,0,0,2,0,0,0,0,0],[29,29,1.0,0.45084,0.23717,0.49968,0.57143,0.57143,0.0,0.71429,6,0,0,6,0,1,0,0,1,0,0,0,0,0,22,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.38392,"x":0.45311,"p":[[0,34,0.0,0.44196,0.28651,0.10714,0.57143,0.57143,0.0,1.0,8,1,0,8,0,1,0,0,1,0,0,0,0,0,16,0,0,5,0,0,0,0,1],[4,34,0.1176,0.39282,0.3444,0.0,0.571,0.57143,0.0,1.0,12,3,0,12,0,1,0,0,1,0,0,0,0,0,11,0,0,4,0,0,0,0,3],[8,34,0.2353,0.45311,0.24911,0.28571,0.57143,0.57143,0.0,0.71429,6,0,0,6,0,1,0,0,2,0,0,1,0,0,16,1,0,5,0,0,0,0,0],[12,34,0.3529,0.38392,0.2911,0.0,0.57143,0.57143,0.0,0.71429,10,0,0,10,0,2,0,0,1,0,0,0,0,0,13,0,0,6,0,0,0,0,0],[16,34,0.4706,0.41067,0.28513,0.0,0.57143,0.57143,0.0,0.71429,9,0,0,9,0,1,0,0,2,0,0,0,0,0,13,0,0,7,0,0,0,0,0],[20,34,0.5882,0.38838,0.26542,0.0,0.57143,0.57143,0.0,0.71429,9,0,0,9,0,1,0,0,2,0,0,0,0,0,18,0,0,2,0,0,0,0,0],[24,34,0.7059,0.44639,0.24155,0.28571,0.57143,0.57143,0.0,0.71429,6,0,0,6,0,1,0,0,2,0,0,0,0,0,20,0,0,3,0,0,0,0,0],[28,34,0.8235,0.41962,0.29436,0.0,0.57143,0.71429,0.0,0.71429,9,0,0,9,0,2,0,0,0,0,0,1,0,0,11,0,0,9,0,0,0,0,0],[32,34,0.9412,0.42408,0.29769,0.0,0.57143,0.71429,0.0,0.71429,10,0,0,10,0,0,0,0,1,0,0,0,0,0,12,0,0,9,0,0,0,0,0],[34,34,1.0,0.39729,0.27369,0.10714,0.57143,0.57143,0.0,0.71429,8,0,0,8,0,3,0,0,1,0,0,1,0,0,14,0,0,5,0,0,0,0,0]]}]},{"i":"1d004f95b875ae60","q":"Let  $n\\ge 4$  be a positive integer and let  $M$  be a set of  $n$  points in the plane, where no three points are collinear and not all of the  $n$  points being concyclic. Find all real functions  $f:M\\to\\mathbb{R}$  such that for any circle  $\\mathcal{C}$  containing at least three points from  $M$ , the following equality holds:\n\\[\\sum_{P\\in\\mathcal{C}\\cap M} f(P)=0\\]\n*Dorel Mihet*","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.03116,"p":[[0,37,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.03116,0.06887,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00893,"x":0.04465,"p":[[0,39,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.04464,0.0974,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.03116,0.05889,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a1509488ac59f072","q":"Let $A B C D$ be a cyclic quadrilateral, and let $E, F, G$, and $H$ be the midpoints of $A B, B C, C D$, and $D A$ respectively. Let $W, X, Y$ and $Z$ be the orthocenters of triangles $A H E, B E F, C F G$ and $D G H$, respectively. Prove that the quadrilaterals $A B C D$ and $W X Y Z$ have the same area.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.49107,"x":0.67856,"p":[[0,25,0.0,0.67411,0.22654,0.42857,0.64286,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,8,0,0,7,0,0,3,0,0,8,0,5],[4,25,0.16,0.67856,0.17497,0.57132,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,3,0,0,14,0,0],[8,25,0.32,0.58482,0.17261,0.42857,0.57143,0.60714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,11,0,0,1,0,0,6,0,1],[12,25,0.48,0.5982,0.21261,0.42857,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,12,0,0,7,0,0,4,0,0,3,0,4],[16,25,0.64,0.55355,0.15871,0.42857,0.4998,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,9,0,0,3,0,0,3,0,1],[20,25,0.8,0.53125,0.10248,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,13,0,0,16,0,0,2,0,0,1,0,0],[24,25,0.96,0.49107,0.08703,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,17,0,0,13,0,0,1,0,0,0,0,0],[25,25,1.0,0.53124,0.09605,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,13,0,0,15,0,0,4,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"falling","v":0.50892,"x":0.71875,"p":[[0,11,0.0,0.68304,0.22227,0.4286,0.57143,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,10,0,0,7,0,0,1,0,0,8,0,6],[4,11,0.3636,0.71875,0.1906,0.57143,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,2,0,0,16,0,2],[8,11,0.7273,0.53125,0.12492,0.42857,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,14,0,0,11,0,0,5,0,0,1,0,0],[11,11,1.0,0.50892,0.09406,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,12,0,0,3,0,0,0,0,0]]}]},{"i":"2fa3d2bb0b6accc8","q":"21. (IRE 1) The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $A B C$ meet at $X$. Let $M$ be the midpoint of $B C$. Prove that (a) $\\angle B A M=\\angle C A X$, and (b) $\\frac{A M}{A X}=\\cos \\angle B A C$.","t":[{"b":0,"e":0.71429,"k":"falling","v":0.50884,"x":0.7857,"p":[[0,31,0.0,0.7857,0.29233,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,1,0,0,3,0,0,1,0,0,1,0,0,4,0,0,5,0,16],[4,31,0.129,0.71872,0.30197,0.57143,0.71429,1.0,0.0,1.0,3,12,0,3,0,0,0,0,1,0,0,1,0,0,5,0,0,8,0,0,2,0,12],[8,31,0.2581,0.71871,0.25125,0.57132,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,1,0,0,2,0,0,7,0,0,7,0,0,3,0,10],[12,31,0.3871,0.74993,0.23425,0.57143,0.71429,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,0,0,0,8,0,0,9,0,0,2,0,11],[16,31,0.5161,0.61599,0.14035,0.571,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,15,0,0,13,0,0,0,0,1],[20,31,0.6452,0.50884,0.23939,0.5354,0.57143,0.71429,0.0,0.71429,5,0,0,5,0,0,0,0,1,0,0,2,0,0,14,0,0,10,0,0,0,0,0],[24,31,0.7742,0.57131,0.17857,0.571,0.57143,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,3,0,0,19,0,0,5,0,0,0,0,2],[28,31,0.9032,0.57585,0.26362,0.571,0.71429,0.71429,0.0,1.0,4,2,0,4,0,0,0,0,2,0,0,1,0,0,8,0,0,14,0,0,1,0,2],[31,31,1.0,0.58033,0.2257,0.57143,0.64286,0.71429,0.0,1.0,3,1,0,3,0,0,0,0,2,0,0,0,0,0,11,0,0,15,0,0,0,0,1]]},{"b":3,"e":0.42857,"k":"flat","v":0.54015,"x":0.86159,"p":[[0,20,0.0,0.64434,0.23389,0.5354,0.66714,0.71429,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,5,0,0,7,1,0,9,0,0,1,0,6],[4,20,0.2,0.86159,0.1731,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,5,0,17],[8,20,0.4,0.67855,0.33503,0.42859,0.71429,1.0,0.0,1.0,2,13,1,2,0,3,0,0,1,0,0,3,0,0,5,0,0,3,0,0,2,0,13],[12,20,0.6,0.65621,0.27167,0.571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,4,0,0,0,0,0,1,0,0,5,0,0,10,0,0,7,0,4],[16,20,0.8,0.74254,0.18595,0.71429,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,1,0,0,0,0,1,0,0,0,0,0,16,0,0,10,0,3],[20,20,1.0,0.54015,0.28956,0.14289,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,8,0,0,1,0,0,2,0,0,2,0,0,11,0,0,7,0,0]]}]},{"i":"8fa262b846b41966","q":"Let $\\mathbb{R}$ be the set of real numbers. Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a function such that  $$ f(x+y) f(x-y) \\geqslant f(x)^{2}-f(y)^{2} $$  for every $x, y \\in \\mathbb{R}$. Assume that the inequality is strict for some $x_{0}, y_{0} \\in \\mathbb{R}$. Prove that $f(x) \\geqslant 0$ for every $x \\in \\mathbb{R}$ or $f(x) \\leqslant 0$ for every $x \\in \\mathbb{R}$. (Malaysia)","t":[{"b":5,"e":0.0,"k":"falling","v":0.09821,"x":0.70536,"p":[[0,51,0.0,0.70536,0.39276,0.28571,1.0,1.0,0.0,1.0,3,20,0,3,0,2,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,20],[4,51,0.0784,0.6517,0.40565,0.14289,1.0,1.0,0.0,1.0,2,18,0,2,0,7,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,18],[8,51,0.1569,0.65179,0.44167,0.14286,1.0,1.0,0.0,1.0,7,19,0,7,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,19],[12,51,0.2353,0.69643,0.41611,0.2857,1.0,1.0,0.0,1.0,6,20,0,6,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,20],[16,51,0.3137,0.44643,0.43264,0.0,0.28571,1.0,0.0,1.0,12,11,1,12,0,0,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,11],[20,51,0.3922,0.5,0.41496,0.21427,0.28571,1.0,0.0,1.0,8,12,0,8,0,0,0,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,12],[24,51,0.4706,0.33482,0.41743,0.0,0.07143,0.78571,0.0,1.0,16,8,0,16,0,2,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,8],[28,51,0.549,0.32143,0.38796,0.0,0.28571,0.42857,0.0,1.0,15,7,0,15,0,0,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,7],[32,51,0.6275,0.30804,0.41513,0.0,0.0,0.57145,0.0,1.0,17,8,0,17,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[36,51,0.7059,0.38839,0.38338,0.0,0.28571,0.67857,0.0,1.0,10,8,0,10,0,1,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,8],[40,51,0.7843,0.22312,0.29656,0.0,0.14143,0.28571,0.0,1.0,15,3,0,15,0,3,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,3],[44,51,0.8627,0.23659,0.30009,0.0,0.14286,0.32143,0.0,1.0,15,3,0,15,0,2,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,3],[48,51,0.9412,0.09821,0.14032,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,1,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.10714,0.15568,0.0,0.0,0.28571,0.0,0.4286,21,0,0,21,0,1,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.79911,"x":1.0,"p":[[0,11,0.0,0.80357,0.35848,0.92857,1.0,1.0,0.0,1.0,3,24,0,3,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[4,11,0.3636,0.79911,0.38442,1.0,1.0,1.0,0.0,1.0,5,25,0,5,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,25],[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":"7b1019ac92d0fabe","q":"Let  $a,b,c$  be positive real numbers. Prove that  $\\frac{a^3+3b^3}{5a+b}+\\frac{b^3+3c^3}{5b+c}+\\frac{c^3+3a^3}{5c+a} \\geq \\frac{2}{3}(a^2+b^2+c^2)$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.29018,"x":1.0,"p":[[0,23,0.0,0.29018,0.44102,0.0,0.0,0.89286,0.0,1.0,22,8,0,22,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,8],[4,23,0.1739,0.58482,0.47429,0.0,1.0,1.0,0.0,1.0,12,17,0,12,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,17],[8,23,0.3478,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":4,"e":0.0,"k":"falling","v":0.0,"x":0.39732,"p":[[0,16,0.0,0.39732,0.47074,0.0,0.07143,1.0,0.0,1.0,16,12,0,16,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[4,16,0.25,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[8,16,0.5,0.27679,0.44311,0.0,0.0,0.89286,0.0,1.0,23,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,8],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"6fdaa8488baa7ef6","q":"Let  $p$  be a positive integer,  $p>1.$  Find the number of  $m\\times n$  matrices with entries in the set  $\\left\\{ 1,2,\\dots,p\\right\\} $  and such that the sum of elements on each row and each column is not divisible by  $p.$","t":[{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.20536,"p":[[0,48,0.0,0.12054,0.24513,0.0,0.0,0.14287,0.0,1.0,23,1,0,23,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[4,48,0.0833,0.125,0.26426,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,1],[8,48,0.1667,0.20536,0.37276,0.0,0.0,0.17857,0.0,1.0,23,5,0,23,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,5],[12,48,0.25,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.05357,0.12242,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,48,0.4167,0.08036,0.19212,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[24,48,0.5,0.04464,0.1357,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,48,0.5833,0.04463,0.12072,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,48,0.6667,0.04464,0.10374,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.04018,0.08917,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.0892,0.15044,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,6,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"rising","v":0.05804,"x":0.29463,"p":[[0,12,0.0,0.11161,0.18808,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,4,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[4,12,0.3333,0.16071,0.32878,0.0,0.0,0.14287,0.0,1.0,23,4,0,23,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[8,12,0.6667,0.05804,0.12807,0.0,0.0,0.0,0.0,0.42857,26,0,1,26,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.29463,0.20495,0.14286,0.14288,0.42857,0.0,0.85714,1,0,0,1,0,16,0,0,4,0,0,5,0,0,4,0,0,1,0,0,1,0,0]]}]},{"i":"0d916093c20abb33","q":"In a country, there are some cities and the city named *Ben Song* is capital. Each cities are connected with others by some two-way roads. One day, the King want to choose  $n$  cities to add up with *Ben Song* city to establish an *expanded capital* such that the two following condition are satisfied:\n\n(i) With every two cities in *expanded capital*, we can always find a road connecting them and this road just belongs to the cities of *expanded capital*.\n\n(ii) There are exactly  $k$  cities which do not belong to *expanded capital* have the direct road to at least one city of *expanded capital*.\n\nProve that there are at most  $\\binom{n+k}{k}$  options to expand the capital for the King.","t":[{"b":2,"e":0.0,"k":"flat","v":0.03125,"x":0.05804,"p":[[0,22,0.0,0.04902,0.09173,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.04465,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.05357,0.12242,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.05804,0.13296,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.04014,"x":0.06697,"p":[[0,7,0.0,0.04902,0.08449,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.06697,0.09439,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.04014,0.08927,0.0,0.0,0.0,0.0,0.43,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fe4fdbd276efb03c","q":"Let $a, b, c, d$ be four real numbers such that $a \\geqslant b \\geqslant c \\geqslant d>0$ and $a+b+c+d=1$. Prove that  $$ (a+2 b+3 c+4 d) a^{a} b^{b} c^{c} d^{d}<1 $$  (Belgium)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04463,"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],[4,26,0.1538,0.04463,0.14027,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":2,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.05804,0.21086,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cd0238911b2d9ef0","q":"Find all triples  $(a, b, c)$  of positive real numbers that satisfy the system of equations\n\\[\na + b + c = \\frac{1}{a^3} + \\frac{1}{b^3} + \\frac{1}{c^3}, \\quad ab + bc + ca = \\sqrt{a} + \\sqrt{b} + \\sqrt{c}.\n\\]","t":[{"b":1,"e":0.28571,"k":"rising","v":0.0,"x":0.30804,"p":[[0,34,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.25446,0.13709,0.14289,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,4,0,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.30804,0.09523,0.28571,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,2,0,0,20,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.28125,0.08364,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.29464,0.07087,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.30803,0.0724,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,9,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.04464,0.10972,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3ee5acd8d52c6412","q":"In triangle  $ ABC,H,I,O$  are orthocenter, incenter and circumcenter, respectively.  $ CI$  cuts circumcircle at  $ L$ . If  $ AB\\equal{}IL$  and  $ AH\\equal{}OH$ , find angles of triangle  $ ABC$ .","t":[{"b":5,"e":0.85714,"k":"flat","v":0.65622,"x":0.80356,"p":[[0,33,0.0,0.80356,0.11709,0.85711,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,26,0,0],[4,33,0.1212,0.71874,0.22157,0.57143,0.85714,0.85714,0.0,0.85714,2,0,2,2,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,19,0,0],[8,33,0.2424,0.72768,0.14445,0.57143,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,2,0,0,17,0,0],[12,33,0.3636,0.65624,0.15511,0.57143,0.57143,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,1,0,0,9,0,1],[16,33,0.4848,0.73659,0.19598,0.57143,0.85707,0.85714,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,14,0,3],[20,33,0.6061,0.71205,0.13656,0.57143,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,2,1,0,14,0,0],[24,33,0.7273,0.67411,0.12993,0.57143,0.57143,0.75,0.5714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,6,0,0,7,0,1],[28,33,0.8485,0.67857,0.14725,0.57143,0.71429,0.75,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,11,0,0,8,0,0],[32,33,0.9697,0.65622,0.13297,0.57143,0.57143,0.75,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,5,0,0,8,0,0],[33,33,1.0,0.67857,0.15152,0.57143,0.64286,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,5,0,0,11,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.79908,"x":0.87946,"p":[[0,35,0.0,0.79908,0.11774,0.85714,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,25,0,0],[4,35,0.1143,0.82589,0.09932,0.85714,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,24,0,2],[8,35,0.2286,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[12,35,0.3429,0.84821,0.09407,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,24,0,4],[16,35,0.4571,0.86161,0.06667,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,28,0,3],[20,35,0.5714,0.87946,0.06298,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,6],[24,35,0.6857,0.85714,2e-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],[28,35,0.8,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[32,35,0.9143,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[35,35,1.0,0.87054,0.04164,0.85714,0.85714,0.85714,0.85714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,3]]}]},{"i":"7cb24c736474ba1e","q":"For any set of points  $A_1, A_2,...,A_n$  on the plane, one defines  $r( A_1, A_2,...,A_n)$  as the radius of the smallest circle that contains all of these points. Prove that if  $n \\ge 3$ , there are indices  $i,j,k$  such that  $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"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],[16,32,0.5,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1ea41d88d4d285e4","q":"For $a, b, c>0$ prove that  $$ \\frac{a^{3}+3 b^{3}}{5 a+b}+\\frac{b^{3}+3 c^{3}}{5 b+c}+\\frac{c^{3}+3 a^{3}}{5 c+a} \\geq \\frac{2}{3}\\left(a^{2}+b^{2}+c^{2}\\right) $$","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.30804,"p":[[0,32,0.0,0.30804,0.38484,0.0,0.0,0.71429,0.0,1.0,19,3,0,19,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,3],[4,32,0.125,0.19196,0.34369,0.0,0.0,0.10714,0.0,1.0,24,2,0,24,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0,2],[8,32,0.25,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,32,0.375,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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],[32,32,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.32143,"x":0.7366,"p":[[0,23,0.0,0.40179,0.36498,0.0,0.71429,0.71429,0.0,1.0,13,1,0,13,0,2,0,0,0,0,0,0,0,0,0,0,0,15,0,0,1,0,1],[4,23,0.1739,0.40179,0.37362,0.0,0.71429,0.71429,0.0,1.0,14,1,0,14,0,1,0,0,0,0,0,0,0,0,0,0,0,14,0,0,2,0,1],[8,23,0.3478,0.51786,0.37754,0.0,0.71429,0.71429,0.0,1.0,10,4,0,10,0,1,0,0,0,0,0,0,0,0,0,0,0,15,0,0,2,0,4],[12,23,0.5217,0.32143,0.35535,0.0,0.07143,0.71429,0.0,1.0,16,1,0,16,0,2,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,1],[16,23,0.6957,0.72768,0.04164,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0],[20,23,0.8696,0.7366,0.06297,0.71429,0.71429,0.71429,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,3,0,1],[23,23,1.0,0.71875,0.02486,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0]]}]},{"i":"43455db4808bc8f1","q":"In triangle  $ ABC$ ,  $ AC \\equal{} 13, BC \\equal{} 14,$  and  $ AB\\equal{}15$ .  Points  $ M$  and  $ D$  lie on  $ AC$  with  $ AM\\equal{}MC$  and  $ \\angle ABD \\equal{} \\angle DBC$ .  Points  $ N$  and  $ E$  lie on  $ AB$  with  $ AN\\equal{}NB$  and  $ \\angle ACE \\equal{} \\angle ECB$ .  Let  $ P$  be the point, other than  $ A$ , of intersection of the circumcircles of  $ \\triangle AMN$  and  $ \\triangle ADE$ .  Ray  $ AP$  meets  $ BC$  at  $ Q$ .  The ratio  $ \\frac{BQ}{CQ}$  can be written in the form  $ \\frac{m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers.  Find  $ m\\minus{}n$ .","t":[{"b":5,"e":0.0,"k":"falling","v":0.24107,"x":0.66518,"p":[[0,26,0.0,0.64732,0.37962,0.35714,0.85714,1.0,0.0,1.0,3,13,0,3,0,5,0,0,0,0,0,5,0,0,1,0,0,0,0,0,5,0,13],[4,26,0.1538,0.66518,0.32263,0.4286,0.85714,0.85714,0.0,1.0,3,7,0,3,0,1,0,0,3,0,0,2,0,0,3,0,0,3,0,0,10,0,7],[8,26,0.3077,0.45536,0.3597,0.14286,0.42857,0.75,0.0,1.0,7,6,1,7,0,4,0,0,1,0,0,8,0,0,2,0,0,2,0,0,2,0,6],[12,26,0.4615,0.33929,0.21354,0.14286,0.28571,0.57143,0.0,0.57143,5,0,0,5,0,5,0,0,7,0,0,3,0,0,12,0,0,0,0,0,0,0,0],[16,26,0.6154,0.26338,0.17534,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,8,0,0,10,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[20,26,0.7692,0.24107,0.20958,0.0,0.14286,0.42857,0.0,0.57143,9,0,0,9,0,8,0,0,5,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[24,26,0.9231,0.27232,0.17261,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,8,0,0,6,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[26,26,1.0,0.30357,0.1915,0.14286,0.28571,0.42857,0.0,0.57143,5,0,0,5,0,6,0,0,7,0,0,8,0,0,6,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.5625,"x":0.98661,"p":[[0,67,0.0,0.62947,0.27861,0.42857,0.57143,0.85714,0.0,1.0,2,5,0,2,0,1,0,0,1,0,0,6,0,0,7,0,0,3,0,0,7,0,5],[4,67,0.0597,0.67857,0.32341,0.39286,0.78571,1.0,0.0,1.0,1,13,0,1,0,1,0,0,6,0,0,3,0,0,4,0,0,1,0,0,3,0,13],[8,67,0.1194,0.5625,0.37616,0.14286,0.57143,1.0,0.0,1.0,5,9,1,5,0,4,0,0,2,0,0,2,0,0,4,0,0,3,0,0,3,0,9],[12,67,0.1791,0.79463,0.29438,0.67846,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,4,0,0,2,0,0,1,0,0,2,0,0,4,0,18],[16,67,0.2388,0.85714,0.24484,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,8,0,18],[20,67,0.2985,0.89732,0.21498,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,7,0,21],[24,67,0.3582,0.93304,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[28,67,0.4179,0.95089,0.12169,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,26],[32,67,0.4776,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[36,67,0.5373,0.95536,0.08328,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,24],[40,67,0.597,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[44,67,0.6567,0.94643,0.08564,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,22],[48,67,0.7164,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21],[52,67,0.7761,0.95982,0.07349,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24],[56,67,0.8358,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[60,67,0.8955,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[64,67,0.9552,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[67,67,1.0,0.95536,0.09062,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,24]]}]},{"i":"8cfd7681591d8400","q":"Let  $a_1, \\dots, a_n, b_1, \\dots, b_n$  be  $2n$  positive integers such that the  $n+1$  products \n\\[a_1 a_2 a_3 \\cdots a_n, b_1 a_2 a_3 \\cdots a_n, b_1 b_2 a_3 \\cdots a_n, \\dots, b_1 b_2 b_3 \\cdots b_n\\]\nform a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.","t":[{"b":0,"e":1.0,"k":"flat","v":0.54462,"x":0.69195,"p":[[0,12,0.0,0.57589,0.36854,0.14286,0.57143,1.0,0.0,1.0,2,11,1,2,0,7,0,0,2,0,0,4,0,0,2,0,0,3,0,0,1,0,11],[4,12,0.3333,0.58481,0.35778,0.14286,0.57143,1.0,0.14286,1.0,0,11,0,0,0,9,0,0,3,0,0,2,0,0,3,0,0,3,0,0,1,0,11],[8,12,0.6667,0.54462,0.35072,0.24999,0.57121,0.89286,0.0,1.0,2,8,0,2,0,6,0,0,5,0,0,2,0,0,4,0,0,2,0,0,3,0,8],[12,12,1.0,0.69195,0.29258,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,3,0,0,3,0,0,3,0,0,5,0,0,5,0,10]]},{"b":7,"e":1.0,"k":"rising","v":0.54018,"x":0.92856,"p":[[0,49,0.0,0.54018,0.33832,0.2857,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,7,0,0,8,0,0,2,0,0,2,0,0,1,0,0,5,0,7],[4,49,0.0816,0.62054,0.34921,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,5,0,0,7,0,0,3,0,0,0,0,0,3,0,0,2,0,12],[8,49,0.1633,0.86161,0.22724,0.82143,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,3,0,21],[12,49,0.2449,0.83036,0.27993,0.82143,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,0,4,0,20],[16,49,0.3265,0.86161,0.20973,0.82143,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,5,0,19],[20,49,0.4082,0.85267,0.24351,0.82143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,4,0,20],[24,49,0.4898,0.87946,0.16793,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,4,0,19],[28,49,0.5714,0.89732,0.1931,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,22],[32,49,0.6531,0.92856,0.14289,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,6,0,23],[36,49,0.7347,0.875,0.21651,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,0,6,0,20],[40,49,0.8163,0.92411,0.17122,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,5,0,24],[44,49,0.898,0.88839,0.18808,0.85714,1.0,1.0,0.28571,1.0,0,21,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,0,4,0,21],[48,49,0.9796,0.88392,0.19045,0.82132,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,4,0,20],[49,49,1.0,0.75893,0.2635,0.53571,0.85714,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,1,0,0,6,0,0,3,0,0,3,0,0,4,0,14]]}]},{"i":"5cf557b0908dc07d","q":"In an acute triangle  $ABC$ , let  $H$  denote the orthocenter. Let  $D$  be the midpoint of minor arc  $BC$  and suppose ray  $AH$  meets the circumcircle again at  $X$ . Given  $AH = 2HX = HD = 2$ , compute  $(AB+AC)^2$ .\n\n*Proposed by Ritwin Narra*","t":[{"b":0,"e":0.71429,"k":"flat","v":0.63837,"x":0.73661,"p":[[0,27,0.0,0.73213,0.2442,0.57143,0.71429,1.0,0.0,1.0,1,9,0,1,0,0,0,0,1,0,0,3,0,0,6,0,0,6,0,0,6,0,9],[4,27,0.1481,0.73658,0.21462,0.57143,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,0,0,0,2,0,0,7,0,0,7,0,0,9,0,6],[8,27,0.2963,0.63837,0.18893,0.57132,0.57143,0.85704,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,5,0,0,12,0,0,4,0,0,7,0,2],[12,27,0.4444,0.68748,0.13092,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,16,0,0,3,0,2],[16,27,0.5926,0.73659,0.11357,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,17,0,0,9,0,1],[20,27,0.7407,0.71429,0.08748,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,23,0,0,5,0,0],[24,27,0.8889,0.68303,0.18118,0.57143,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,15,0,0,5,0,2],[27,27,1.0,0.73661,0.08073,0.71429,0.71429,0.75,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,21,0,0,8,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.65177,"x":0.98661,"p":[[0,26,0.0,0.65177,0.23129,0.53539,0.57143,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,5,0,0,10,0,0,3,0,0,6,0,5],[4,26,0.1538,0.90625,0.08459,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,13],[8,26,0.3077,0.89732,0.09606,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,13],[12,26,0.4615,0.88393,0.05576,0.85714,0.85714,0.85714,0.85714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,6],[16,26,0.6154,0.85268,0.07563,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,23,0,4],[20,26,0.7692,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[24,26,0.9231,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[26,26,1.0,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]}]},{"i":"a61c3dd46d35261a","q":"Find all bounded sequences  $(a_n)_{n=1}^\\infty$  of natural numbers such that for all  $n \\ge 3$ , \\[ a_n = \\frac{a_{n-1} + a_{n-2}}{\\gcd(a_{n-1}, a_{n-2})}. \\]","t":[{"b":1,"e":0.85714,"k":"flat","v":0.79907,"x":0.87053,"p":[[0,7,0.0,0.82589,0.21349,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,4,0,0,6,0,15],[4,7,0.5714,0.79907,0.26933,0.67836,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,6,0,15],[7,7,1.0,0.87053,0.14445,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,15]]},{"b":4,"e":0.85714,"k":"flat","v":0.75445,"x":0.95088,"p":[[0,56,0.0,0.75445,0.2506,0.57143,0.78564,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,2,0,0,7,0,0,5,0,0,4,0,12],[4,56,0.0714,0.77229,0.23922,0.57143,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,1,0,0,6,0,0,5,0,0,7,0,11],[8,56,0.1429,0.90625,0.12682,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,6,0,19],[12,56,0.2143,0.89732,0.1439,0.85714,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,18],[16,56,0.2857,0.95088,0.07669,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[20,56,0.3571,0.8973,0.12996,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,14,0,15],[24,56,0.4286,0.91071,0.1171,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,6,0,19],[28,56,0.5,0.9241,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[32,56,0.5714,0.89732,0.10853,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,11,0,15],[36,56,0.6429,0.93303,0.10706,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[40,56,0.7143,0.86604,0.14261,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,9,0,14],[44,56,0.7857,0.85713,0.17498,0.71429,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,7,0,16],[48,56,0.8571,0.89732,0.12993,0.82143,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,6,0,18],[52,56,0.9286,0.80801,0.16606,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,10,0,9],[56,56,1.0,0.86159,0.14503,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,5,0,15]]}]},{"i":"8c9aba2665188e4b","q":"Positive integers  $a_1, a_2, ... , a_7, b_1, b_2, ... , b_7$  satisfy  $2 \\leq a_i \\leq 166$  and  $a_i^{b_i} \\cong a_{i+1}^2$  (mod 167) for each  $1 \\leq i \\leq 7$  (where  $a_8=a_1$ ). Compute the minimum possible value of  $b_1b_2 ... b_7(b_1 + b_2 + ...+ b_7)$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.70088,"x":0.79018,"p":[[0,10,0.0,0.79018,0.15561,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,10,0,7],[4,10,0.4,0.70088,0.16888,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,2,0,0,6,0,0,14,0,0,7,0,2],[8,10,0.8,0.78125,0.10092,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,13,0,2],[10,10,1.0,0.78124,0.11285,0.71429,0.71429,0.85714,0.5714,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,12,0,3]]},{"b":4,"e":0.71429,"k":"flat","v":0.64284,"x":0.74552,"p":[[0,34,0.0,0.65625,0.21086,0.57143,0.71429,0.85714,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,6,0,0,8,0,0,8,0,0,6,0,3],[4,34,0.1176,0.74552,0.15042,0.71429,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,13,0,0,10,0,3],[8,34,0.2353,0.64284,0.20204,0.57132,0.71429,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,0,0,0,6,0,0,8,0,0,11,0,0,3,0,3],[12,34,0.3529,0.68302,0.11702,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,12,0,0,7,0,0],[16,34,0.4706,0.70536,0.11811,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,16,0,0,6,0,1],[20,34,0.5882,0.70089,0.13533,0.57143,0.71429,0.75,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,12,0,0,6,0,2],[24,34,0.7059,0.70089,0.12556,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,16,0,0,6,0,1],[28,34,0.8235,0.70982,0.12103,0.57143,0.71429,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,15,0,0,7,0,1],[32,34,0.9412,0.71429,0.12877,0.57143,0.71429,0.75,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,15,0,0,6,0,2],[34,34,1.0,0.71428,0.15567,0.57143,0.71429,0.85704,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,12,0,0,5,0,4]]}]},{"i":"ed72bd5ce4716e91","q":"On a $2012 \\times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011 .","t":[{"b":0,"e":1.0,"k":"flat","v":0.96875,"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],[4,24,0.1667,0.96875,0.10555,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,29],[8,24,0.3333,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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],[20,24,0.8333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]},{"b":2,"e":1.0,"k":"flat","v":0.98214,"x":1.0,"p":[[0,36,0.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[4,36,0.1111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"4c902330afa38eb9","q":"Prove that for any positive integer $k$,  $$ \\left(k^{2}\\right)!\\cdot \\prod_{j=0}^{k-1} \\frac{j!}{(j+k)!} $$  is an integer.","t":[{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.32143,"p":[[0,11,0.0,0.32143,0.37965,0.0,0.07143,0.60714,0.0,1.0,16,5,0,16,0,1,0,0,2,0,0,3,0,0,2,0,0,3,0,0,0,0,5],[4,11,0.3636,0.12053,0.24772,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,1],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"volatile","v":0.0,"x":0.41962,"p":[[0,6,0.0,0.41962,0.39759,0.0,0.35714,0.71429,0.0,1.0,13,6,0,13,0,0,0,0,3,0,0,1,0,0,3,0,0,5,0,0,1,0,6],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c525dab5d93b8a08","q":"Point  $ O $  is the center of the circumscribed circle of an acute triangle  $ Abc $ . A certain circle passes through the points  $ B $  and  $ C $  and intersects sides  $ AB $  and  $ AC $  of a triangle. On its arc lying inside the triangle, points  $ D $  and  $ E $  are chosen so that the segments  $ BD $  and  $ CE $  pass through the point  $ O $ . Perpendicular  $ DD_1 $  to  $ AB $  side and perpendicular  $ EE_1 $  to  $ AC $  side intersect at  $ M $ . Prove that the points  $ A $ ,  $ M $  and  $ O $  lie on the same straight line.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.03572,"p":[[0,12,0.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02223,"x":0.03125,"p":[[0,15,0.0,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"46daed39773815e3","q":"Prove the inequality\n\n$$\n\\frac{x-y}{x y+2 y+1}+\\frac{y-z}{y z+2 z+1}+\\frac{z-x}{z x+2 x+1} \\geqslant 0\n$$\n\nwhere $x, y$, and $z$ are non-negative real numbers.\n\n(Du\u0161an \u0110uki\u0107)","t":[{"b":1,"e":1.0,"k":"volatile","v":0.56241,"x":0.89732,"p":[[0,7,0.0,0.56241,0.34626,0.14286,0.71429,0.75,0.0,1.0,3,7,0,3,0,6,0,0,2,0,0,1,0,0,2,0,0,10,0,0,1,0,7],[4,7,0.5714,0.89732,0.21793,0.85714,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,6,0,22],[7,7,1.0,0.89732,0.16841,0.85714,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,8,0,19]]},{"b":4,"e":0.57143,"k":"rising","v":0.50446,"x":0.81696,"p":[[0,40,0.0,0.50446,0.33309,0.24999,0.57143,0.71429,0.0,1.0,5,5,0,5,0,3,0,0,4,0,0,3,0,0,2,0,0,10,0,0,0,0,5],[4,40,0.1,0.67854,0.22305,0.67857,0.71429,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,3,0,0,1,0,0,3,0,0,18,0,0,1,0,5],[8,40,0.2,0.70536,0.31931,0.67857,0.71429,1.0,0.0,1.0,1,13,0,1,0,4,0,0,2,0,0,0,0,0,1,0,0,11,0,0,0,0,13],[12,40,0.3,0.61607,0.25614,0.57143,0.71429,0.71429,0.0,1.0,1,3,0,1,0,3,0,0,3,0,0,0,0,0,4,0,0,16,0,0,2,0,3],[16,40,0.4,0.61607,0.32031,0.28571,0.71429,0.78571,0.0,1.0,2,8,0,2,0,4,0,0,3,0,0,0,0,0,3,0,0,12,0,0,0,0,8],[20,40,0.5,0.64732,0.26722,0.53571,0.71429,0.71429,0.0,1.0,1,6,0,1,0,2,0,0,3,0,0,2,0,0,2,0,0,15,0,0,1,0,6],[24,40,0.6,0.81696,0.1684,0.71429,0.71429,1.0,0.28571,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,13],[28,40,0.7,0.74107,0.08328,0.71429,0.71429,0.71429,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,3],[32,40,0.8,0.70089,0.07457,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[36,40,0.9,0.69642,0.10566,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,28,0,0,0,0,1],[40,40,1.0,0.73213,0.06916,0.71429,0.71429,0.71429,0.714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,2]]}]},{"i":"c18390af5f66940e","q":"Suppose that $S$ tiles the set of all integer cubes. Prove that $S$ has only one element.","t":[{"b":1,"e":1.0,"k":"flat","v":0.87947,"x":0.99107,"p":[[0,24,0.0,0.88393,0.23538,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,3,0,23],[4,24,0.1667,0.87947,0.1992,0.71429,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,5,0,0,1,0,22],[8,24,0.3333,0.92857,0.19233,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,26],[12,24,0.5,0.97767,0.06299,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[16,24,0.6667,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[20,24,0.8333,0.94643,0.14617,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,27],[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":0.57143,"k":"falling","v":0.57142,"x":0.83033,"p":[[0,25,0.0,0.79463,0.23942,0.67846,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,6,0,0,1,0,0,6,0,0,2,0,16],[4,25,0.16,0.80802,0.26872,0.67846,1.0,1.0,0.0,1.0,1,19,0,1,0,0,0,0,0,0,0,6,0,0,1,0,0,4,0,0,1,0,19],[8,25,0.32,0.83033,0.20026,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,3,0,16],[12,25,0.48,0.75446,0.25812,0.42857,0.85707,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,11,0,0,1,0,0,3,0,0,2,0,15],[16,25,0.64,0.60713,0.26726,0.42857,0.42859,0.89286,0.0,1.0,1,8,0,1,0,0,0,0,0,0,0,17,0,0,3,0,0,1,0,0,2,0,8],[20,25,0.8,0.67856,0.24485,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,14,0,0,1,0,0,5,0,0,3,0,9],[24,25,0.96,0.67415,0.24541,0.42857,0.57143,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,1,0,0,6,0,8],[25,25,1.0,0.57142,0.22015,0.42857,0.42857,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,21,0,0,2,0,0,2,0,0,2,0,5]]}]},{"i":"6673fef1c297aa21","q":"Find all  $f:\\mathbb{N}\\to\\mathbb{N}$  satisfying that for all  $m,n\\in\\mathbb{N}$ , the nonnegative integer  $|f(m+n)-f(m)|$  is a divisor of  $f(n)$ .\n*Proposed by usjl*","t":[{"b":0,"e":0.42857,"k":"flat","v":0.30803,"x":0.43308,"p":[[0,17,0.0,0.30803,0.1017,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,4,0,0,21,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[4,17,0.2353,0.32143,0.07143,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.32143,0.09449,0.28571,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,2,0,0,22,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[12,17,0.7059,0.33035,0.14031,0.2857,0.28571,0.42857,0.14286,0.857,0,0,0,0,0,4,0,0,19,0,0,6,0,0,2,0,0,0,0,0,1,0,0],[16,17,0.9412,0.43308,0.16554,0.28571,0.42857,0.4286,0.2857,1.0,0,2,0,0,0,0,0,0,9,0,0,19,0,0,2,0,0,0,0,0,0,0,2],[17,17,1.0,0.38838,0.10851,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,9,0,0,17,0,0,4,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.21428,"x":0.32143,"p":[[0,26,0.0,0.30357,0.08564,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,23,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[4,26,0.1538,0.32143,0.10715,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,2,0,0,23,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[8,26,0.3077,0.30357,0.06916,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.26339,0.0724,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,7,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.29463,0.1921,0.14286,0.2857,0.32143,0.14286,0.71429,0,0,0,0,0,15,0,0,9,0,0,3,0,0,1,0,0,4,0,0,0,0,0],[20,26,0.7692,0.21428,0.07986,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,17,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.23214,0.07783,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,13,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.21428,0.08748,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,18,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"291b6f105a361483","q":"Find all integers that can be written in the form  $\\frac{1}{a_1}+\\frac{2}{a_2}+...+\\frac{9}{a_9}$  where  $a_1,a_2, ...,a_9$  are nonzero digits, not necessarily different.","t":[{"b":4,"e":0.42857,"k":"falling","v":0.35267,"x":0.75893,"p":[[0,71,0.0,0.6741,0.31183,0.39286,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,7,0,0,5,0,0,2,0,0,2,0,0,2,0,13],[4,71,0.0563,0.75893,0.2889,0.53572,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,7,0,0,1,0,0,1,0,0,5,0,0,2,0,16],[8,71,0.1127,0.61161,0.28624,0.42857,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,6,0,0,12,0,0,2,0,0,1,0,0,1,0,10],[12,71,0.169,0.64286,0.32927,0.28571,0.5,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,9,0,0,6,0,0,1,0,0,1,0,0,0,0,14],[16,71,0.2254,0.64286,0.34626,0.28571,0.57143,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,15],[20,71,0.2817,0.66071,0.31894,0.28571,0.64287,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,9,0,0,7,0,0,0,0,0,0,0,0,3,0,13],[24,71,0.338,0.75445,0.31388,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,6,0,0,3,0,0,1,0,0,1,0,0,2,0,18],[28,71,0.3944,0.58929,0.2896,0.39286,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,8,0,0,10,0,0,4,0,0,0,0,0,0,0,10],[32,71,0.4507,0.5714,0.28571,0.28571,0.42857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,10,0,0,8,0,0,4,0,0,0,0,0,2,0,8],[36,71,0.507,0.51785,0.29828,0.28571,0.42857,0.78571,0.14286,1.0,0,8,0,0,0,1,0,0,14,0,0,6,0,0,2,0,0,1,0,0,0,0,8],[40,71,0.5634,0.43304,0.22442,0.28571,0.35714,0.4286,0.14286,1.0,0,3,0,0,0,1,0,0,15,0,0,9,0,0,3,0,0,0,0,0,1,0,3],[44,71,0.6197,0.38393,0.15335,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,10,0,0,2,0,0,1,0,0,0,0,1],[48,71,0.6761,0.41518,0.2,0.28571,0.35714,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,15,0,0,10,0,0,3,0,0,0,0,0,1,0,2],[52,71,0.7324,0.4107,0.15464,0.28571,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,14,0,0,12,0,0,4,0,0,1,0,0,0,0,1],[56,71,0.7887,0.37946,0.14555,0.28571,0.28571,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,18,0,0,10,0,0,3,0,0,0,0,0,0,0,1],[60,71,0.8451,0.36161,0.14279,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,18,0,0,11,0,0,1,0,0,0,0,0,0,0,1],[64,71,0.9014,0.39286,0.17857,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,18,0,0,10,0,0,2,0,0,0,0,0,0,0,2],[68,71,0.9577,0.38839,0.17215,0.28571,0.28571,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,17,0,0,13,0,0,0,0,0,0,0,0,0,0,2],[71,71,1.0,0.35267,0.10089,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,20,0,0,10,0,0,1,0,0,1,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.49107,"x":1.0,"p":[[0,34,0.0,0.70536,0.31122,0.39286,0.78571,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,7,0,0,2,0,0,4,0,0,2,0,0,1,0,15],[4,34,0.1176,0.60719,0.29664,0.39286,0.42929,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,8,0,0,9,0,0,4,0,0,0,0,0,0,0,11],[8,34,0.2353,0.57143,0.29881,0.28571,0.42857,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,8,0,0,8,0,0,2,0,0,2,0,0,2,0,8],[12,34,0.3529,0.6875,0.3163,0.39286,0.71429,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,7,0,0,4,0,0,3,0,0,2,0,0,0,0,15],[16,34,0.4706,0.73661,0.28146,0.42857,0.85714,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,3,0,0,8,0,0,2,0,0,3,0,0,0,0,16],[20,34,0.5882,0.60714,0.31944,0.28571,0.4286,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,8,0,0,7,0,0,1,0,0,2,0,0,1,0,11],[24,34,0.7059,0.49107,0.28333,0.28571,0.28571,0.60714,0.14286,1.0,0,6,0,0,0,1,0,0,16,0,0,4,0,0,3,0,0,1,0,0,1,0,6],[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":"089cd1f74d6d2929","q":"1. In acute triangle  $ABC$  point  $O$  is circumcenter, segment  $CD$  is a height, point  $E$  lies on side  $AB$  and point  $M$  is a midpoint of  $CE$ . Line through  $M$  perpendicular to  $OM$  cuts lines  $AC$  and  $BC$  respectively in  $K$ ,  $L$ . Prove that  $\\frac{LM}{MK}=\\frac{AD}{DB}$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,34,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,34,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.05804,0.18509,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,34,0.5882,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,50,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,50,0.16,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,50,0.24,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.03795,0.17495,0.0,0.0,0.0,0.0,1.0,29,1,0,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,50,0.4,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,50,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"3bdffffd9be01f37","q":"Let $A B C D$ be a cyclic convex quadrilateral and let $r_{a}, r_{b}, r_{c}, r_{d}$ be the radii of the circles inscribed in the triangles $B C D, A C D, A B D, A B C$ respectively. Prove that $r_{a}+r_{c}=r_{b}+r_{d}$.","t":[{"b":3,"e":0.85714,"k":"flat","v":0.86607,"x":0.91964,"p":[[0,7,0.0,0.91964,0.20497,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,26],[4,7,0.5714,0.86607,0.25489,0.85711,1.0,1.0,0.0,1.0,1,21,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,4,0,21],[7,7,1.0,0.87945,0.22901,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,21]]},{"b":5,"e":0.71429,"k":"flat","v":0.84374,"x":0.98214,"p":[[0,5,0.0,0.84374,0.29312,0.85711,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,6,0,20],[4,5,0.8,0.9598,0.11431,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[5,5,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"03bdb30c89d4c35a","q":"Let $A B C$ be a triangle with incenter $I$. Suppose the reflection of $A B$ across $C I$ and the reflection of $A C$ across $B I$ intersect at a point $X$. Prove that $X I$ is perpendicular to $B C$.\n(The incenter is the point where the three angle bisectors meet.)","t":[{"b":3,"e":0.571,"k":"falling","v":0.54006,"x":0.91517,"p":[[0,23,0.0,0.7857,0.34257,0.71429,1.0,1.0,0.0,1.0,3,20,0,3,0,1,0,0,2,0,0,0,0,0,1,0,0,3,0,0,2,0,20],[4,23,0.1739,0.8973,0.20279,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,6,0,21],[8,23,0.3478,0.91517,0.15919,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,6,0,22],[12,23,0.5217,0.60266,0.26179,0.39293,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,2,0,0,6,0,0,4,0,0,4,0,0,7,0,0,5,0,4],[16,23,0.6957,0.54006,0.27149,0.39286,0.4286,0.74996,0.0,1.0,1,4,0,1,0,2,0,0,5,0,0,9,0,0,5,0,0,2,0,0,4,0,4],[20,23,0.8696,0.59815,0.21851,0.42859,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,5,0,0,4,0,0,7,0,0,9,0,0,4,0,2],[23,23,1.0,0.54463,0.2271,0.42857,0.4286,0.71429,0.0,1.0,1,2,0,1,0,1,0,0,2,0,0,13,0,0,4,0,0,6,0,0,3,0,2]]},{"b":7,"e":0.85714,"k":"rising","v":0.71872,"x":0.99554,"p":[[0,33,0.0,0.71872,0.28902,0.42859,0.85707,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,2,0,0,5,0,0,5,0,0,1,0,0,4,0,13],[4,33,0.1212,0.76339,0.35285,0.53571,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,0,2,0,20],[8,33,0.2424,0.83036,0.29545,0.85714,1.0,1.0,0.0,1.0,2,19,0,2,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,6,0,19],[12,33,0.3636,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,33,0.4848,0.97321,0.06623,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[20,33,0.6061,0.99553,0.02486,1.0,1.0,1.0,0.8571,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,33,0.7273,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[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],[32,33,0.9697,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[33,33,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"38751e7dff21790f","q":"For each integer  $ n$  of the form  $ n\\equal{}p_1 p_2 p_3 p_4$ , where  $ p_1,p_2,p_3,p_4$  are distinct primes, let  $ 1\\equal{}d_1<d_2<...<d_{15}<d_{16}\\equal{}n$  be the divisors of  $ n$ . Prove that if  $ n<1995$ , then  $ d_9\\minus{}d_8 \\not\\equal{} 22$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.46423,"x":0.63839,"p":[[0,8,0.0,0.46423,0.26496,0.28571,0.42857,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,10,0,0,5,0,0,1,0,0,6,0,0,4,0,1],[4,8,0.5,0.63839,0.22864,0.42857,0.71429,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,4,0,0,1,0,0,11,0,0,9,0,1],[8,8,1.0,0.60266,0.22226,0.42857,0.64286,0.85704,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,10,0,0,1,0,0,5,0,0,11,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.36161,"x":0.87499,"p":[[0,35,0.0,0.36161,0.26723,0.14286,0.28571,0.57143,0.0,0.85714,6,0,0,6,0,4,0,0,7,0,0,6,0,0,3,0,0,3,0,0,3,0,0],[4,35,0.1143,0.62494,0.24937,0.53539,0.71429,0.85704,0.14286,1.0,0,3,0,0,0,4,0,0,1,0,0,3,0,0,7,0,0,8,0,0,6,0,3],[8,35,0.2286,0.67409,0.19311,0.5354,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,7,0,0,5,0,0,9,0,0,7,0,3],[12,35,0.3429,0.83034,0.0974,0.857,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,24,0,2],[16,35,0.4571,0.85266,0.09775,0.85714,0.85714,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,23,0,5],[20,35,0.5714,0.83035,0.1729,0.82132,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,5,0,0,15,0,9],[24,35,0.6857,0.85712,0.07986,0.857,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,22,0,5],[28,35,0.8,0.87499,0.08564,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,20,0,8],[32,35,0.9143,0.87053,0.10326,0.85714,0.85714,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,23,0,7],[35,35,1.0,0.87499,0.09943,0.85714,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,19,0,9]]}]},{"i":"925bae2f4226b783","q":"Let $A B C$ be an acute-angled triangle with $|A B| \\neq|A C|$. The midpoints of the sides $\\overline{A B}$ and $\\overline{A C}$ are $D$ and $E$, respectively. The circumcircles of triangles $B C D$ and $B C E$ intersect the circumcircle of triangle $A D E$ at $P$ and $Q$, respectively, where $P \\neq D$ and $Q \\neq E$.\nProve that $|A P|=|A Q|$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,36,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,4,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9db8ad814fc4c276","q":"Let  ${ABC}$  be an acute angled triangle whose shortest side is  ${BC}$ . Consider a variable point  ${P}$  on the side  ${BC}$ , and let  ${D}$  and  ${E}$  be points on  ${AB}$  and  ${AC}$ , respectively, such that  ${BD=BP}$  and  ${CP=CE}$ . Prove that, as  ${P}$  traces  ${BC}$ , the circumcircle of the triangle  ${ADE}$  passes through a fixed point.","t":[{"b":2,"e":0.5714,"k":"flat","v":0.32139,"x":0.53561,"p":[[0,21,0.0,0.48657,0.31714,0.14286,0.57143,0.57143,0.0,1.0,6,5,0,6,0,3,0,0,0,0,0,1,0,0,17,0,0,0,0,0,0,0,5],[4,21,0.1905,0.32139,0.27661,0.14286,0.14286,0.57143,0.0,1.0,4,2,0,4,0,15,0,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,2],[8,21,0.381,0.53561,0.15149,0.571,0.57141,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,0,0,0,2,0,0,26,0,0,0,0,0,0,0,1],[12,21,0.5714,0.4419,0.19014,0.14286,0.5712,0.57143,0.14286,0.57143,0,0,0,0,0,9,0,0,0,0,0,2,0,0,21,0,0,0,0,0,0,0,0],[16,21,0.7619,0.51327,0.14218,0.571,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,4,0,0,0,0,0,1,0,0,27,0,0,0,0,0,0,0,0],[20,21,0.9524,0.4598,0.18116,0.28571,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,7,0,0,2,0,0,0,0,0,23,0,0,0,0,0,0,0,0],[21,21,1.0,0.47766,0.17716,0.5713,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,7,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.31692,"x":0.49993,"p":[[0,28,0.0,0.47765,0.29796,0.14286,0.57143,0.57143,0.0,1.0,5,4,1,5,0,4,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,4],[4,28,0.1429,0.31692,0.23614,0.14286,0.14288,0.57143,0.0,0.57143,5,0,0,5,0,12,0,0,0,0,0,1,0,0,14,0,0,0,0,0,0,0,0],[8,28,0.2857,0.49993,0.15564,0.571,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,5,0,0,0,0,0,1,0,0,26,0,0,0,0,0,0,0,0],[12,28,0.4286,0.48658,0.16696,0.571,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,6,0,0,0,0,0,1,0,0,25,0,0,0,0,0,0,0,0],[16,28,0.5714,0.49102,0.16725,0.571,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,6,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0],[20,28,0.7143,0.42395,0.20363,0.14286,0.5712,0.57143,0.14,0.57143,0,0,0,0,0,11,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,0],[24,28,0.8571,0.47312,0.17649,0.5354,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,7,0,0,0,0,0,1,0,0,24,0,0,0,0,0,0,0,0],[28,28,1.0,0.43748,0.19863,0.14289,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,10,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0]]}]},{"i":"01e143c6ed93291e","q":"In right-angle triangle  $ABC$ ,  $\\angle C=90$ \u00b0, Point  $D$  is the midpoint of side  $AB$ . Points  $M$  and  $C$  lie on the same side of  $AB$  such that  $MB\\bot AB$ , line  $MD$  intersects side  $AC$  at  $N$ , line  $MC$  intersects side  $AB$  at  $E$ . Show that  $\\angle DBN=\\angle BCE$ .","t":[{"b":6,"e":0.14286,"k":"flat","v":0.04465,"x":0.1159,"p":[[0,19,0.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,2,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.06687,0.0712,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.1159,0.17288,0.0,0.14286,0.14286,0.0,1.0,12,1,0,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,19,0.6316,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.06241,"x":0.09376,"p":[[0,28,0.0,0.09376,0.06786,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8cd1c7c80ef20470","q":"Let  $g(x) = ax^2 + bx + c$  a quadratic function with real coefficients such that the equation  $g(g(x)) = x$  has four distinct real roots. Prove that there isn't a function  $f$ :  $R--R$  such that  $f(f(x)) = g(x)$  for all  $x$  real","t":[{"b":0,"e":1.0,"k":"rising","v":0.74107,"x":0.99553,"p":[[0,24,0.0,0.74107,0.35614,0.64286,0.92857,1.0,0.0,1.0,4,16,4,4,0,0,0,0,3,0,0,1,0,0,0,0,0,3,0,0,5,0,16],[4,24,0.1667,0.8616,0.20355,0.85714,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,13,0,14],[8,24,0.3333,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[12,24,0.5,0.9933,0.02743,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,30],[16,24,0.6667,0.98884,0.0362,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,29],[20,24,0.8333,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[24,24,1.0,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30]]},{"b":2,"e":0.71429,"k":"flat","v":0.78571,"x":0.90624,"p":[[0,27,0.0,0.90624,0.16216,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,21],[4,27,0.1481,0.85268,0.21275,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,14,0,14],[8,27,0.2963,0.78571,0.26245,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,4,0,0,10,0,12],[12,27,0.4444,0.85714,0.07143,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,4],[16,27,0.5926,0.87946,0.08828,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,9],[20,27,0.7407,0.82142,0.07143,0.71429,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,22,0,1],[24,27,0.8889,0.80803,0.06784,0.71429,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,21,0,0],[27,27,1.0,0.8058,0.10294,0.71429,0.85714,0.85714,0.4286,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,3,0,17,0,2]]}]},{"i":"84b69a827f2b1d9f","q":"Let  $n$  be a positive integer. Let  $P_n=\\{2^n,2^{n-1}\\cdot 3, 2^{n-2}\\cdot 3^2, \\dots, 3^n \\}.$  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":1,"e":1.0,"k":"rising","v":0.683,"x":0.99107,"p":[[0,33,0.0,0.70086,0.1968,0.57143,0.57143,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,1,0,0,1,0,9],[4,33,0.1212,0.683,0.17401,0.57143,0.57143,0.74996,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,19,0,0,4,0,0,2,0,6],[8,33,0.2424,0.95536,0.11538,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,27],[12,33,0.3636,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[16,33,0.4848,0.97767,0.07241,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[20,33,0.6061,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[24,33,0.7273,0.95088,0.13179,1.0,1.0,1.0,0.571,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,28],[28,33,0.8485,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[32,33,0.9697,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[33,33,1.0,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31]]},{"b":4,"e":1.0,"k":"rising","v":0.65177,"x":0.95982,"p":[[0,10,0.0,0.65177,0.2111,0.57143,0.57143,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,3,0,0,17,0,0,2,0,0,1,0,7],[4,10,0.4,0.69643,0.18472,0.57143,0.57143,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,7,0,0,1,0,7],[8,10,0.8,0.91963,0.14261,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,3,0,23],[10,10,1.0,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28]]}]},{"i":"ae30cd004778c0bc","q":"Let $A B C$ be a triangle with incentre $I$. The angle bisectors $A I, B I$ and $C I$ meet $[B C],[C A]$ and $[A B]$ at $D, E$ and $F$, respectively. The perpendicular bisector of $[A D]$ intersects the lines $B I$ and $C I$ at $M$ and $N$, respectively. Show that $A, I, M$ and $N$ lie on a circle.\n\n#","t":[{"b":0,"e":0.14286,"k":"flat","v":0.08929,"x":0.22767,"p":[[0,27,0.0,0.21429,0.25505,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,9,0,0,1,0,0,4,0,0,0,0,0,5,0,0,0,0,0],[4,27,0.1481,0.22767,0.22261,0.14286,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,17,0,0,2,0,0,2,0,0,1,0,0,4,0,0,0,0,0],[8,27,0.2963,0.12947,0.13997,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,19,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,27,0.4444,0.16063,0.15873,0.105,0.14286,0.14286,0.0,0.71429,8,0,0,8,0,19,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[16,27,0.5926,0.12483,0.05918,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.09813,0.07518,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.08929,0.06917,0.0,0.14286,0.14286,0.0,0.143,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.07134,"x":0.12947,"p":[[0,18,0.0,0.12947,0.14445,0.0,0.14286,0.14286,0.0,0.42857,13,0,1,13,0,14,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.11161,0.12745,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,20,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,18,0.4444,0.1025,0.09599,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.143,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b7f9d2ceed94c628","q":"Let  $n$  and  $k$  be relatively prime positive integers with  $k<n$ . Each number in the set  $M=\\{1,2,3,\\ldots,n-1\\}$  is colored either blue or white. For each  $i$  in  $M$ , both  $i$  and  $n-i$  have the same color. For each  $i\\ne k$  in  $M$  both  $i$  and  $|i-k|$  have the same color. Prove that all numbers in  $M$  must have the same color.","t":[{"b":1,"e":1.0,"k":"rising","v":0.59366,"x":1.0,"p":[[0,46,0.0,0.59366,0.39637,0.2857,0.57143,1.0,0.0,1.0,4,14,0,4,0,3,0,0,6,0,0,2,0,0,2,0,0,0,0,0,1,0,14],[4,46,0.087,0.83036,0.28891,0.78571,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,23],[8,46,0.1739,0.75893,0.3102,0.42859,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,3,0,0,4,0,0,3,0,0,1,0,0,0,0,19],[12,46,0.2609,0.73214,0.34022,0.42857,1.0,1.0,0.0,1.0,1,18,0,1,0,4,0,0,0,0,0,4,0,0,4,0,0,0,0,0,1,0,18],[16,46,0.3478,0.87498,0.23079,0.92857,1.0,1.0,0.1429,1.0,0,24,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,1,0,0,0,0,24],[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],[24,46,0.5217,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,46,0.6087,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,46,0.6957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[36,46,0.7826,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,46,0.8696,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[46,46,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]},{"b":6,"e":1.0,"k":"rising","v":0.5357,"x":0.98214,"p":[[0,33,0.0,0.56694,0.27776,0.28571,0.4998,0.78571,0.14286,1.0,0,8,0,0,0,1,0,0,8,0,0,7,0,0,7,0,0,1,0,0,0,0,8],[4,33,0.1212,0.74107,0.33964,0.39286,0.92857,1.0,0.0,1.0,2,16,0,2,0,0,0,0,6,0,0,2,0,0,0,0,0,0,0,0,6,0,16],[8,33,0.2424,0.55801,0.37348,0.28571,0.42857,1.0,0.0,1.0,4,12,0,4,0,2,0,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,12],[12,33,0.3636,0.5357,0.33881,0.28571,0.42857,1.0,0.0,1.0,2,9,0,2,0,4,0,0,5,0,0,8,0,0,2,0,0,1,0,0,1,0,9],[16,33,0.4848,0.80357,0.29396,0.67857,1.0,1.0,0.1429,1.0,0,20,0,0,0,1,0,0,5,0,0,1,0,0,1,0,0,2,0,0,2,0,20],[20,33,0.6061,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[24,33,0.7273,0.96415,0.09485,1.0,1.0,1.0,0.71,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,28],[28,33,0.8485,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,33,0.9697,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[33,33,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"3347af970aac84e7","q":"Find all functions  $ f: \\mathbb{Q}^{\\plus{}} \\mapsto \\mathbb{Q}^{\\plus{}}$  such that: \n\\[ f(x) \\plus{} f(y) \\plus{} 2xy f(xy) \\equal{} \\frac {f(xy)}{f(x\\plus{}y)}.\\]","t":[{"b":1,"e":1.0,"k":"rising","v":0.3817,"x":1.0,"p":[[0,51,0.0,0.50447,0.40087,0.14286,0.42857,1.0,0.0,1.0,7,11,0,7,0,2,0,0,6,0,0,4,0,0,0,0,0,2,0,0,0,0,11],[4,51,0.0784,0.52232,0.36353,0.24999,0.42857,1.0,0.0,1.0,4,9,0,4,0,4,0,0,4,0,0,7,0,0,0,0,0,3,0,0,1,0,9],[8,51,0.1569,0.3817,0.34792,0.14286,0.28571,0.5,0.0,1.0,7,6,0,7,0,4,0,1,8,0,0,4,0,0,0,0,0,2,0,0,0,0,6],[12,51,0.2353,0.58929,0.37415,0.39286,0.4286,1.0,0.0,1.0,5,12,0,5,0,1,0,0,2,0,0,9,0,0,0,0,0,2,0,0,1,0,12],[16,51,0.3137,0.72772,0.32604,0.42964,0.92857,1.0,0.0,1.0,2,16,0,2,0,1,0,0,2,0,0,4,0,0,2,0,0,4,0,0,1,0,16],[20,51,0.3922,0.74554,0.35488,0.39286,1.0,1.0,0.0,1.0,2,20,0,2,0,1,0,0,5,0,0,2,0,0,0,0,0,2,0,0,0,0,20],[24,51,0.4706,0.61607,0.39194,0.28571,0.71429,1.0,0.0,1.0,6,12,0,6,0,1,0,0,3,0,0,3,0,0,0,0,0,4,0,0,3,0,12],[28,51,0.549,0.9375,0.16728,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,27],[32,51,0.6275,0.75,0.30929,0.42857,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,4,0,0,5,0,0,1,0,0,2,0,0,2,0,17],[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],[40,51,0.7843,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,51,0.8627,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[48,51,0.9412,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":5,"e":0.4286,"k":"flat","v":0.41516,"x":0.59821,"p":[[0,9,0.0,0.49106,0.36235,0.14286,0.42857,0.89286,0.0,1.0,4,8,0,4,0,5,0,0,5,0,0,6,0,0,1,0,0,1,0,0,2,0,8],[4,9,0.4444,0.41965,0.32328,0.25,0.42857,0.4286,0.0,1.0,5,6,0,5,0,3,0,0,7,0,0,10,0,0,0,0,0,1,0,0,0,0,6],[8,9,0.8889,0.41516,0.30379,0.14286,0.42857,0.71429,0.0,1.0,7,2,0,7,0,3,0,0,2,0,0,8,0,0,3,0,0,6,0,0,1,0,2],[9,9,1.0,0.59821,0.24074,0.42857,0.64286,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,6,0,0,9,0,0,1,0,0,10,0,0,1,0,5]]}]},{"i":"3f26c5f7280098cf","q":"Find all functions  $f: R \\to  R$  such that  $f(x)f(yf(x)-1)=x^2f(y)-f(x)$  for all real  $x ,y$","t":[{"b":6,"e":0.57143,"k":"falling","v":0.54908,"x":0.84374,"p":[[0,78,0.0,0.77232,0.19516,0.57143,0.85714,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,5,0,0,8,0,9],[4,78,0.0513,0.68303,0.19799,0.57143,0.57143,0.85704,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,16,0,0,4,0,0,2,0,7],[8,78,0.1026,0.72766,0.19352,0.57143,0.64286,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,3,0,0,5,0,8],[12,78,0.1538,0.79015,0.18555,0.57143,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,5,0,0,8,0,10],[16,78,0.2051,0.70535,0.22286,0.57142,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,3,0,0,7,0,7],[20,78,0.2564,0.69642,0.21054,0.57143,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,1,0,0,16,0,0,4,0,0,2,0,8],[24,78,0.3077,0.60263,0.15866,0.571,0.57143,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,20,0,0,1,0,0,5,0,1],[28,78,0.359,0.69643,0.20438,0.57143,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,7,0,0,3,0,7],[32,78,0.4103,0.65624,0.19516,0.57143,0.57143,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,18,0,0,3,0,0,3,0,5],[36,78,0.4615,0.74552,0.198,0.57143,0.64286,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,2,0,0,4,0,10],[40,78,0.5128,0.84374,0.19353,0.67857,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,4,0,0,2,0,18],[44,78,0.5641,0.68302,0.17763,0.57143,0.64286,0.85704,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,13,0,0,7,0,0,5,0,4],[48,78,0.6154,0.59819,0.13092,0.57143,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,26,0,0,1,0,0,1,0,2],[52,78,0.6667,0.62944,0.13768,0.57143,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,3,0,0,1,0,3],[56,78,0.7179,0.58923,0.15465,0.57142,0.57143,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,1,0,0,24,0,0,2,0,0,1,0,2],[60,78,0.7692,0.57588,0.09771,0.57143,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,2,0,0,1,0,0],[64,78,0.8205,0.54908,0.10779,0.57143,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,0,0,0,27,0,0,2,0,0,0,0,0],[68,78,0.8718,0.57585,0.02486,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0],[72,78,0.9231,0.5982,0.09062,0.57143,0.57143,0.57143,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,29,0,0,1,0,0,1,0,1],[76,78,0.9744,0.61607,0.11538,0.57143,0.57143,0.57143,0.5714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,2,0,0,1,0,2],[78,78,1.0,0.57142,7e-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]]},{"b":7,"e":0.857,"k":"rising","v":0.66962,"x":0.91071,"p":[[0,43,0.0,0.72765,0.19021,0.57143,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,4,0,0,8,0,6],[4,43,0.093,0.6875,0.22142,0.57143,0.57143,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,2,0,0,4,0,0,11,0,0,3,0,0,5,0,7],[8,43,0.186,0.66962,0.19705,0.57143,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,1,0,0,15,0,0,3,0,0,8,0,3],[12,43,0.2791,0.70536,0.23403,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,3,0,0,8,0,0,6,0,0,3,0,9],[16,43,0.3721,0.72317,0.22001,0.5714,0.64286,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,3,0,0,3,0,10],[20,43,0.4651,0.75892,0.1729,0.57143,0.78564,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,0,10,0,6],[24,43,0.5581,0.79018,0.17122,0.57143,0.85714,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,9,0,9],[28,43,0.6512,0.82587,0.17401,0.57143,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,10,0,12],[32,43,0.7442,0.77232,0.18161,0.57143,0.71429,1.0,0.57143,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,5,0,0,5,0,10],[36,43,0.8372,0.75443,0.16843,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,6,0,0,9,0,6],[40,43,0.9302,0.75,0.18211,0.57143,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,5,0,0,4,0,9],[43,43,1.0,0.91071,0.09943,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,15,0,15]]}]},{"i":"f928c1a7846c4fe1","q":"For each positive integer  $k$ , let  $S(k)$   be the sum of its digits. For example,  $S(21) = 3$  and  $S(105) = 6$ . Let  $n$  be the smallest integer for which  $S(n) - S(5n) = 2013$ . Determine the number of digits in  $n$ .","t":[{"b":4,"e":0.28571,"k":"flat","v":0.37055,"x":0.45089,"p":[[0,47,0.0,0.42857,0.26,0.28571,0.28571,0.57143,0.0,1.0,1,4,0,1,0,3,0,0,13,0,0,6,0,0,4,0,0,1,0,0,0,0,4],[4,47,0.0851,0.41071,0.1948,0.28571,0.42857,0.42857,0.0,1.0,1,2,0,1,0,0,0,0,14,0,0,10,0,0,5,0,0,0,0,0,0,0,2],[8,47,0.1702,0.37055,0.12807,0.28571,0.35729,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,15,0,0,12,0,0,3,0,0,1,0,0,0,0,0],[12,47,0.2553,0.375,0.09279,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,15,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[16,47,0.3404,0.41516,0.10324,0.28571,0.42857,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,15,0,0,7,0,0,0,0,0,0,0,0],[20,47,0.4255,0.41072,0.16656,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,4,0,0,6,0,0,15,0,0,6,0,0,0,0,0,0,0,1],[24,47,0.5106,0.44196,0.12037,0.42857,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,20,0,0,4,0,0,1,0,0,1,0,0],[28,47,0.5957,0.4107,0.13715,0.28571,0.42857,0.42857,0.2857,1.0,0,1,0,0,0,0,0,0,11,0,0,17,0,0,3,0,0,0,0,0,0,0,1],[32,47,0.6809,0.44643,0.12242,0.42857,0.42857,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,19,0,0,5,0,0,1,0,0,1,0,0],[36,47,0.766,0.39286,0.09449,0.28571,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,11,0,0,19,0,0,1,0,0,1,0,0,0,0,0],[40,47,0.8511,0.44643,0.12753,0.42857,0.42857,0.46431,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,17,0,0,6,0,0,1,0,0,1,0,0],[44,47,0.9362,0.45089,0.17536,0.39286,0.42857,0.42858,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,18,0,0,1,0,0,3,0,0,1,0,1],[47,47,1.0,0.41516,0.17626,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,11,0,0,16,0,0,2,0,0,0,0,0,0,0,2]]},{"b":5,"e":0.71429,"k":"volatile","v":0.4241,"x":0.80357,"p":[[0,7,0.0,0.51785,0.34023,0.28571,0.42857,1.0,0.0,1.0,3,9,1,3,0,0,0,0,12,0,0,5,0,0,2,0,0,0,0,0,1,0,9],[4,7,0.5714,0.4241,0.29121,0.2857,0.28571,0.46429,0.0,1.0,2,5,1,2,0,3,0,0,14,0,0,5,0,0,1,0,0,2,0,0,0,0,5],[7,7,1.0,0.80357,0.18123,0.67857,0.71429,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,10,0,0,0,0,14]]}]},{"i":"6ba3a474c58f9734","q":"Given a positive integer  $n (n>2004)$ , we put 1, 2, 3, \u2026, $n^2$  into squares of an  $n\\times n$  chessboard with one number in a square. A square is called a \u201cgood square\u201d if the square satisfies following conditions:\n1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square.\n2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square.\nFind the maximum value of the number of the \u201cgood square\u201d.","t":[{"b":2,"e":1.0,"k":"flat","v":0.9107,"x":0.99554,"p":[[0,54,0.0,0.95982,0.10853,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,26],[4,54,0.0741,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[8,54,0.1481,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[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],[16,54,0.2963,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[20,54,0.3704,0.94643,0.09279,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,22],[24,54,0.4444,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[28,54,0.5185,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[32,54,0.5926,0.94642,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[36,54,0.6667,0.94197,0.14222,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,25],[40,54,0.7407,0.95089,0.09182,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,24],[44,54,0.8148,0.9375,0.1234,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,23],[48,54,0.8889,0.9107,0.14177,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,20],[52,54,0.963,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[54,54,1.0,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22]]},{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":1.0,"p":[[0,45,0.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[4,45,0.0889,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[8,45,0.1778,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,45,0.2667,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[20,45,0.4444,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,45,0.5333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,45,0.6222,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[44,45,0.9778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"6f8682ef47f2c258","q":"Find all functions  $f:\\mathbb {Z}\\to\\mathbb Z$ , satisfy that for any integer  ${a}$ ,  ${b}$ ,  ${c}$ ,  $$ 2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2 $$","t":[{"b":4,"e":0.28571,"k":"flat","v":0.16964,"x":0.31249,"p":[[0,16,0.0,0.30355,0.23073,0.14286,0.28571,0.42858,0.0,0.85714,4,0,0,4,0,10,0,0,8,0,0,3,0,0,3,0,0,3,0,0,1,0,0],[4,16,0.25,0.26783,0.19145,0.14286,0.2857,0.32143,0.0,0.71429,4,0,0,4,0,11,0,0,9,0,0,2,0,0,5,0,0,1,0,0,0,0,0],[8,16,0.5,0.2008,0.20161,0.0,0.14286,0.28571,0.0,0.85714,9,0,0,9,0,12,0,0,5,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[12,16,0.75,0.31249,0.20023,0.14286,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,12,0,0,12,0,0,2,0,0,4,0,0,1,0,0,0,0,1],[16,16,1.0,0.16964,0.14914,0.0,0.14286,0.2857,0.0,0.57143,9,0,0,9,0,13,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.16516,"x":0.31245,"p":[[0,26,0.0,0.20533,0.16722,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,17,0,0,5,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[4,26,0.1538,0.29911,0.16888,0.14286,0.28571,0.32143,0.0,0.85714,1,0,0,1,0,9,0,0,14,0,0,4,0,0,3,0,0,0,0,0,1,0,0],[8,26,0.3077,0.31245,0.20334,0.14286,0.28571,0.46418,0.0,0.71429,3,0,0,3,0,9,0,0,9,0,0,3,0,0,6,0,0,2,0,0,0,0,0],[12,26,0.4615,0.28571,0.20825,0.14286,0.2857,0.42857,0.0,0.71429,6,0,0,6,0,7,0,0,8,0,0,4,0,0,6,0,0,1,0,0,0,0,0],[16,26,0.6154,0.28569,0.223,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,3,0,0,11,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[20,26,0.7692,0.19634,0.19152,0.0,0.14286,0.28571,0.0,0.85714,9,0,0,9,0,11,0,0,7,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[24,26,0.9231,0.24999,0.20823,0.14286,0.21428,0.28571,0.0,0.85714,6,0,0,6,0,10,0,0,9,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[26,26,1.0,0.16516,0.15195,0.0,0.14286,0.28571,0.0,0.571,12,0,0,12,0,6,0,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"8ef6fa8170f4c1bd","q":"Find all functions  $f:\\mathbb{N} \\rightarrow \\mathbb{N}$  such that for all positive integers  $n$ , there exists an unique positive integer  $k$ , satisfying  $f^k(n)\\leq n+k+1$ .","t":[{"b":3,"e":0.14286,"k":"falling","v":0.28125,"x":0.67852,"p":[[0,30,0.0,0.67852,0.32145,0.53539,0.71429,1.0,0.0,1.0,1,13,0,1,0,3,0,0,3,0,0,1,0,0,7,0,0,3,0,0,1,0,13],[4,30,0.1333,0.55799,0.34322,0.1429,0.571,1.0,0.0,1.0,1,9,0,1,0,8,0,0,2,0,0,3,0,0,4,0,0,5,0,0,0,0,9],[8,30,0.2667,0.5223,0.33044,0.2857,0.4998,0.78571,0.0,1.0,2,8,0,2,0,5,0,0,5,0,0,4,0,0,6,0,0,2,0,0,0,0,8],[12,30,0.4,0.50442,0.28115,0.28571,0.42859,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,3,0,0,8,0,0,5,0,0,4,0,0,3,0,3],[16,30,0.5333,0.43735,0.34787,0.14286,0.35714,0.71429,0.0,1.0,6,4,0,6,0,6,0,0,4,0,0,2,0,0,2,0,0,6,0,0,2,0,4],[20,30,0.6667,0.50444,0.24995,0.28571,0.4998,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,6,0,0,6,0,0,6,0,0,6,0,0,2,0,2],[24,30,0.8,0.29902,0.24323,0.14286,0.1429,0.4286,0.0,1.0,4,1,0,4,0,13,0,0,3,0,0,5,0,0,4,0,0,2,0,0,0,0,1],[28,30,0.9333,0.28125,0.26603,0.0,0.14286,0.4286,0.0,0.71429,10,0,0,10,0,7,0,0,2,0,0,6,0,0,1,0,0,6,0,0,0,0,0],[30,30,1.0,0.39485,0.2895,0.14286,0.42857,0.57143,0.0,1.0,3,2,0,3,0,10,0,0,1,0,1,7,0,0,3,0,0,3,0,0,2,0,2]]},{"b":7,"e":0.71429,"k":"rising","v":0.38828,"x":0.76783,"p":[[0,26,0.0,0.38828,0.31188,0.14286,0.35714,0.57143,0.0,1.0,4,3,0,4,0,11,0,0,1,0,0,4,0,0,5,0,0,3,0,0,1,0,3],[4,26,0.1538,0.45979,0.30873,0.14286,0.4998,0.60714,0.0,1.0,2,4,0,2,0,10,0,0,0,0,0,4,0,0,8,0,0,3,0,0,1,0,4],[8,26,0.3077,0.5089,0.32525,0.24999,0.42859,0.75,0.0,1.0,2,7,0,2,0,6,0,0,3,0,0,6,0,0,6,0,0,1,0,0,1,0,7],[12,26,0.4615,0.58905,0.34771,0.14286,0.64071,1.0,0.0,1.0,1,10,0,1,0,8,0,0,1,0,0,2,0,0,4,0,0,6,0,0,0,0,10],[16,26,0.6154,0.61157,0.30564,0.39286,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,7,0,0,1,0,0,2,0,0,4,0,0,9,0,0,2,0,7],[20,26,0.7692,0.76783,0.19481,0.71429,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,14,0,0,4,0,9],[24,26,0.9231,0.75,0.16751,0.71429,0.71429,0.75,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,21,0,0,1,0,7],[26,26,1.0,0.71425,0.15154,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,20,0,0,2,0,4]]}]},{"i":"7433288d22833c48","q":"For  $n\\in\\mathbb{Z}_+$ , let  $\\Sigma_n$  be the set containing all the bijections:  $\\{1,2,\\cdots ,n\\}\\rightarrow \\{1,2,\\cdots ,n\\}$ . For  $\\sigma\\in\\Sigma_n$ , define  $g(n,\\sigma )$  as the number of all the possible remainders of  $\\sigma (1),\\sigma(1)+\\sigma (2),\\cdots ,\\sigma(1)+\\sigma(2)+\\cdots\\sigma (n)\\text{ }(\\text{mod}n)$ .\nDefine  $f(n)=\\underset{\\sigma\\in\\Sigma_n}{\\text{min}}g(n,\\sigma).$ Prove: there exists  $\\alpha \\in\\mathbb{R}$  and  $0<c_1<c_2$  s.t.  $\\forall n, c_1n^{\\alpha}\\leq f(n)\\leq c_2n^{\\alpha}$ .","t":[{"b":4,"e":0.85714,"k":"flat","v":0.6205,"x":0.78571,"p":[[0,24,0.0,0.65625,0.18161,0.57143,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,3,0,0,6,0,0,14,0,0,7,0,0],[4,24,0.1667,0.66964,0.16917,0.57143,0.71429,0.75,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,5,0,0,4,0,0,14,0,0,8,0,0],[8,24,0.3333,0.6205,0.17717,0.42857,0.57143,0.74996,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,8,0,0,5,0,0,7,0,1],[12,24,0.5,0.74552,0.13237,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,12,0,0,15,0,0],[16,24,0.6667,0.75893,0.10972,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,0,12,0,1],[20,24,0.8333,0.78571,0.10102,0.71429,0.78571,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,14,0,2],[24,24,1.0,0.77232,0.10012,0.71429,0.78571,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,16,0,0]]},{"b":5,"e":0.85714,"k":"flat","v":0.65177,"x":0.79911,"p":[[0,27,0.0,0.65177,0.17105,0.57132,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,6,0,0,5,0,0,13,0,0,7,0,0],[4,27,0.1481,0.67854,0.17498,0.57143,0.71429,0.74996,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,3,0,0,8,0,0,12,0,0,6,0,2],[8,27,0.2963,0.70534,0.08705,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,20,0,0,5,0,0],[12,27,0.4444,0.77232,0.08645,0.71429,0.71429,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,12,0,1],[16,27,0.5926,0.79911,0.07016,0.71429,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,19,0,0],[20,27,0.7407,0.76338,0.09181,0.71429,0.71429,0.85714,0.5714,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,15,0,0,14,0,0],[24,27,0.8889,0.75446,0.0817,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,11,0,0],[27,27,1.0,0.75446,0.08917,0.71429,0.71429,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,0,12,0,0]]}]},{"i":"58058480d8758baf","q":"Let  $a_{i,j}\\enspace(\\forall\\enspace 1\\leq i\\leq n, 1\\leq j\\leq n)$  be  $n^2$  real numbers such that  $a_{i,j}+a_{j,i}=0\\enspace\\forall i, j$  (in particular,  $a_{i,i}=0\\enspace\\forall i$ ). Prove that  $$ {1\\over n}\\sum_{i=1}^{n}\\left(\\sum_{j=1}^{n} a_{i,j}\\right)^2\\leq{1\\over2}\\sum_{i=1}^{n}\\sum_{j=1}^{n} a_{i,j}^2. $$","t":[{"b":0,"e":1.0,"k":"rising","v":0.64286,"x":0.92857,"p":[[0,20,0.0,0.67857,0.37287,0.35714,0.71429,1.0,0.0,1.0,3,15,0,3,0,5,0,0,0,0,0,1,0,0,1,0,0,7,0,0,0,0,15],[4,20,0.2,0.64286,0.38132,0.14286,0.71429,1.0,0.0,1.0,2,13,0,2,0,8,0,0,0,0,0,1,0,0,0,0,0,6,0,0,2,0,13],[8,20,0.4,0.70536,0.37276,0.35714,1.0,1.0,0.0,1.0,2,17,0,2,0,6,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,17],[12,20,0.6,0.78571,0.30093,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,8,0,0,0,0,18],[16,20,0.8,0.89286,0.17496,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,19],[20,20,1.0,0.92857,0.10714,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,21]]},{"b":3,"e":1.0,"k":"rising","v":0.70536,"x":1.0,"p":[[0,11,0.0,0.78125,0.31539,0.71429,1.0,1.0,0.0,1.0,1,18,0,1,0,4,0,0,0,0,0,0,0,0,1,0,0,7,0,0,1,0,18],[4,11,0.3636,0.83034,0.26352,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,0,0,0,2,0,0,1,0,0,7,0,0,0,0,20],[8,11,0.7273,0.70536,0.33491,0.67857,0.71429,1.0,0.0,1.0,1,13,0,1,0,6,0,0,0,0,0,0,0,0,1,0,0,9,0,0,2,0,13],[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":"6f70e7fcef585631","q":"Suppose that  $a$  and  $b$  are natural numbers such that \\[p=\\frac{b}{4}\\sqrt{\\frac{2a-b}{2a+b}}\\] is a prime number. What is the maximum possible value of  $p$ ?","t":[{"b":4,"e":0.57143,"k":"rising","v":0.47767,"x":0.7991,"p":[[0,68,0.0,0.47767,0.26149,0.28571,0.42857,0.60714,0.0,1.0,2,2,1,2,0,3,0,0,5,0,0,9,0,0,5,0,0,3,0,0,3,0,2],[4,68,0.0588,0.71426,0.18899,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,1,0,0,7,0,0,8,0,0,12,0,2],[8,68,0.1176,0.72321,0.15946,0.71429,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,12,0,0,12,0,1],[12,68,0.1765,0.76338,0.12684,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,13,0,2],[16,68,0.2353,0.7366,0.16408,0.67857,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,10,0,0,11,0,3],[20,68,0.2941,0.7991,0.19839,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,14,0,7],[24,68,0.3529,0.75446,0.14826,0.71429,0.78564,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,10,0,0,14,0,2],[28,68,0.4118,0.76786,0.12752,0.71429,0.85712,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,11,0,0,16,0,1],[32,68,0.4706,0.78568,0.13836,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,19,0,2],[36,68,0.5294,0.71873,0.17673,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,7,0,0,9,0,4],[40,68,0.5882,0.72329,0.12857,0.67857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,15,0,0,7,0,2],[44,68,0.6471,0.73212,0.16269,0.67857,0.78564,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,8,0,0,15,0,1],[48,68,0.7059,0.7589,0.1692,0.71429,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,0,15,0,3],[52,68,0.7647,0.6875,0.17655,0.57143,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,10,0,0,10,0,1],[56,68,0.8235,0.73213,0.17036,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,6,0,0,9,0,0,11,0,3],[60,68,0.8824,0.71874,0.18381,0.67857,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,10,0,0,12,0,2],[64,68,0.9412,0.71875,0.19393,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,7,0,0,12,0,3],[68,68,1.0,0.70087,0.19678,0.57132,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,6,0,0,11,0,3]]},{"b":5,"e":0.71429,"k":"flat","v":0.47765,"x":0.55351,"p":[[0,10,0.0,0.47765,0.22476,0.28571,0.571,0.60714,0.0,0.85714,1,0,1,1,0,5,0,0,3,0,0,6,0,0,9,0,0,6,0,0,2,0,0],[4,10,0.4,0.54911,0.26752,0.42857,0.57143,0.71429,0.0,1.0,3,1,2,3,0,0,0,0,4,0,0,8,0,0,2,0,0,8,0,0,6,0,1],[8,10,0.8,0.51783,0.21052,0.42857,0.571,0.60714,0.14286,0.85714,0,0,0,0,0,3,0,0,4,0,0,8,0,0,9,0,0,3,0,0,5,0,0],[10,10,1.0,0.55351,0.19148,0.42857,0.57121,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,2,0,0,8,0,0,11,0,0,4,0,0,5,0,0]]}]},{"i":"878adb547685efa6","q":"Consider all permutations of natural numbers from  $1$  to  $100$ . A permutation is called  $\\emph{double}$  when it has the following property: If you write this permutation twice in a row, then delete  $100$  numbers from them you get the remaining numbers  $1, 2, 3, \\ldots , 100$  in order. How many  $\\emph{double}$  permutations are there?","t":[{"b":2,"e":0.42857,"k":"flat","v":0.24554,"x":0.33929,"p":[[0,8,0.0,0.24554,0.15663,0.14286,0.14286,0.42857,0.0,0.42857,3,0,0,3,0,16,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.3259,0.13474,0.14286,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,11,0,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.33929,0.12753,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,9,0,0,2,0,0,21,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.29911,"x":0.32134,"p":[[0,5,0.0,0.30795,0.14345,0.14286,0.42857,0.42857,0.0,0.42857,1,0,0,1,0,11,0,0,2,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.29911,0.13997,0.14286,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,14,0,0,1,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.32134,0.13844,0.14286,0.42857,0.42857,0.14,0.4286,0,0,0,0,0,12,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"667c2a5415a4ad85","q":". In the convex quadrilateral $A B C D$ we have $\\angle B=\\angle C$ and $\\angle D=90^{\\circ}$. Suppose that $|A B|=2|C D|$. Prove that the angle bisector of $\\angle A C B$ is perpendicular to $C D$.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.00446,"x":0.08036,"p":[[0,25,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.04,0.06395,0.0,0.0,0.14,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,21,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8a73b6fccc41ca02","q":"In the interior of an equilateral triangle  $ ABC $  a point  $ P $  is chosen such that  $ PA ^2 = PB ^2 + PC ^2 $ . Find the measure of  $ \\angle BPC $ .","t":[{"b":0,"e":0.57143,"k":"flat","v":0.38619,"x":0.56471,"p":[[0,14,0.0,0.5,0.33312,0.24999,0.42859,0.64286,0.0,1.0,4,7,0,4,0,4,0,0,2,0,0,7,0,0,7,0,0,0,0,0,1,0,7],[4,14,0.2857,0.52454,0.32567,0.39285,0.42857,0.85714,0.0,1.0,4,6,0,4,0,2,0,0,2,0,0,10,0,2,1,0,0,1,1,0,3,0,6],[8,14,0.5714,0.56471,0.32554,0.28571,0.57121,0.85714,0.0,1.0,3,5,0,3,0,1,0,0,7,0,0,4,0,0,3,0,0,2,0,0,6,1,5],[12,14,0.8571,0.45089,0.21756,0.28571,0.42857,0.57143,0.0,1.0,1,1,0,1,0,2,0,0,8,0,0,11,0,0,4,0,0,3,0,0,2,0,1],[14,14,1.0,0.38619,0.21715,0.2857,0.357,0.4286,0.0,1.0,2,1,0,2,0,3,0,0,9,0,3,9,0,0,3,0,0,0,0,0,2,0,1]]},{"b":5,"e":0.85714,"k":"flat","v":0.46428,"x":0.5692,"p":[[0,15,0.0,0.48661,0.37091,0.14286,0.42859,0.85714,0.0,1.0,7,7,0,7,0,3,0,0,2,0,0,5,0,0,5,0,0,0,0,0,3,0,7],[4,15,0.2667,0.49777,0.23656,0.42857,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,3,0,0,4,0,0,15,0,1,2,0,0,0,0,0,5,0,2],[8,15,0.5333,0.56249,0.2647,0.28571,0.50001,0.85704,0.14286,1.0,0,3,0,0,0,2,0,0,7,0,0,7,0,0,4,0,0,2,0,0,7,0,3],[12,15,0.8,0.46428,0.19805,0.28571,0.42857,0.57143,0.21429,1.0,0,1,0,0,0,0,0,1,10,0,0,11,0,1,4,0,0,1,0,0,3,0,1],[15,15,1.0,0.5692,0.24123,0.42857,0.42859,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,1,12,0,0,4,0,0,1,0,0,5,0,4]]}]},{"i":"f21628e5f2b29a11","q":"In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two towns, such that we can't fly from one to other.\nProved, that we can divided all country on  $4$  regions, such that any flight connected towns from other regions.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,22,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6f10bfc845e2093d","q":"Let  $ABC$  be a triangle with  $D$  the midpoint of side  $AB, E$  the midpoint of side  $BC$ , and  $F$  the midpoint of side  $AC$ . Let  $k_1$  be the circle passing through points  $A, D$ , and  $F$ , let  $k_2$  be the circle passing through points  $B, E$ , and  $D$ , and let  $k_3$  be the circle passing through  $C, F$ , and  $E$ . Prove that circles  $k_1, k_2$ , and  $k_3$  intersect in a point.","t":[{"b":2,"e":1.0,"k":"rising","v":0.78571,"x":1.0,"p":[[0,31,0.0,0.83929,0.25939,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,20],[4,31,0.129,0.85268,0.24868,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,0,0,21],[8,31,0.2581,0.85268,0.16935,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,2,0,16],[12,31,0.3871,0.78571,0.12372,0.71429,0.71429,0.78571,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,8],[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],[24,31,0.7742,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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]]},{"b":6,"e":1.0,"k":"rising","v":0.78125,"x":1.0,"p":[[0,28,0.0,0.83928,0.29827,0.71429,1.0,1.0,0.0,1.0,2,22,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,0,1,0,22],[4,28,0.1429,0.80804,0.25407,0.71429,0.85714,1.0,0.0,1.0,2,16,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,0,0,16],[8,28,0.2857,0.78125,0.2474,0.71429,0.71429,1.0,0.0,1.0,2,13,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,0,0,13],[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],[20,28,0.7143,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[24,28,0.8571,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,28,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"d98038abf431707c","q":"An integer $n$ is perfect if the sum of its divisors is $2n$. Let $n$ be a perfect integer and $p$ its smallest prime divisor. Show that the exponent of $p$ in the prime factorization of $n$ is even.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.07143,"p":[[0,8,0.0,0.07143,0.14286,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,8,0.5,0.04463,0.1259,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.12499,"p":[[0,14,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0625,0.11811,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.12499,0.21351,0.0,0.0,0.28571,0.0,0.71429,23,0,0,23,0,0,0,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[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]]}]},{"i":"58ddda5d9d52c486","q":"Let  $S_n$  denote the sum of the first  $n$  prime numbers. Prove that for any  $n$  there exists the square of an integer between  $S_n$  and  $S_{n+1}$ .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.72764,"x":0.75446,"p":[[0,7,0.0,0.75446,0.10248,0.71429,0.71429,0.71429,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,2,0,4],[4,7,0.5714,0.74999,0.07986,0.71429,0.71429,0.71429,0.714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,0,4,0,2],[7,7,1.0,0.72764,0.05487,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,27,0,0,4,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.72321,"x":0.77677,"p":[[0,4,0.0,0.77677,0.10676,0.71429,0.71429,0.85704,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,7,0,4],[4,4,1.0,0.72321,0.04971,0.71429,0.71429,0.71429,0.714,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,1]]}]},{"i":"45c8bca38b408c7d","q":"Within a group of  $ 2009$  people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.78571,"x":0.95536,"p":[[0,10,0.0,0.9375,0.07087,0.85714,1.0,1.0,0.85714,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[4,10,0.4,0.78571,0.33503,0.85714,1.0,1.0,0.14286,1.0,0,18,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,0,18],[8,10,0.8,0.92857,0.15567,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,21],[10,10,1.0,0.95536,0.06622,0.85714,1.0,1.0,0.85714,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22]]},{"b":5,"e":1.0,"k":"flat","v":0.83929,"x":0.96875,"p":[[0,46,0.0,0.83929,0.28738,0.85714,1.0,1.0,0.0,1.0,1,18,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,18],[4,46,0.087,0.89286,0.18898,0.85714,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,17],[8,46,0.1739,0.9107,0.16659,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,19],[12,46,0.2609,0.88393,0.22142,0.85714,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,19],[16,46,0.3478,0.92411,0.15561,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,20],[20,46,0.4348,0.88839,0.23347,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,8,0,21],[24,46,0.5217,0.87946,0.18595,0.85714,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,14],[28,46,0.6087,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[32,46,0.6957,0.92411,0.15561,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,20],[36,46,0.7826,0.85714,0.27664,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,20],[40,46,0.8696,0.86161,0.27776,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,21],[44,46,0.9565,0.88839,0.24675,0.85714,1.0,1.0,0.14286,1.0,0,22,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,22],[46,46,1.0,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]}]},{"i":"afa5e1968eabebfd","q":"Find all positive integers  $ a$  and  $ b$  for which \r\n\r\n\\[ \\left \\lfloor \\frac{a^2}{b} \\right \\rfloor \\plus{} \\left \\lfloor \\frac{b^2}{a} \\right \\rfloor \\equal{} \\left \\lfloor \\frac{a^2 \\plus{} b^2}{ab} \\right \\rfloor \\plus{} ab.\\]","t":[{"b":0,"e":0.4286,"k":"flat","v":0.43297,"x":0.62044,"p":[[0,27,0.0,0.4866,0.26452,0.28571,0.42859,0.71407,0.0,1.0,1,3,0,1,0,5,0,0,4,0,0,8,0,0,5,0,0,5,0,0,1,0,3],[4,27,0.1481,0.50888,0.21996,0.39286,0.4998,0.60714,0.0,1.0,1,1,0,1,0,1,0,0,6,0,0,8,0,0,8,0,0,4,0,0,3,0,1],[8,27,0.2963,0.47767,0.27803,0.14289,0.5,0.71429,0.0,0.85714,3,0,0,3,0,6,0,0,0,0,0,7,0,0,5,0,0,6,0,0,5,0,0],[12,27,0.4444,0.43297,0.22151,0.28571,0.42857,0.57111,0.14286,1.0,0,1,0,0,0,6,0,0,8,0,0,5,0,0,8,0,0,3,0,0,1,0,1],[16,27,0.5926,0.46863,0.19973,0.2857,0.571,0.57143,0.14,0.85714,0,0,0,0,0,4,0,0,7,0,0,4,0,0,12,0,0,3,0,0,2,0,0],[20,27,0.7407,0.62044,0.21312,0.4286,0.57121,0.71429,0.1429,1.0,0,3,0,0,0,1,0,0,2,0,0,7,0,0,7,0,0,8,0,0,4,0,3],[24,27,0.8889,0.54458,0.25613,0.28571,0.571,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,7,0,0,4,0,0,6,0,0,6,0,0,3,0,3],[27,27,1.0,0.51784,0.23075,0.39286,0.4998,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,4,0,0,8,0,0,5,0,0,7,0,0,3,0,1]]},{"b":3,"e":0.42857,"k":"flat","v":0.3704,"x":0.63835,"p":[[0,27,0.0,0.60709,0.29014,0.28571,0.71429,0.857,0.14286,1.0,0,6,0,0,0,4,0,0,6,0,0,0,0,0,5,0,0,8,0,0,3,0,6],[4,27,0.1481,0.62046,0.20705,0.42859,0.57143,0.74996,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,6,0,0,7,0,0,7,0,0,6,0,2],[8,27,0.2963,0.53122,0.19308,0.42857,0.4998,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,11,0,0,5,0,0,8,0,0,3,0,0],[12,27,0.4444,0.56245,0.2394,0.28571,0.571,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,10,0,0,5,0,0,3,0,0,7,0,0,5,0,2],[16,27,0.5926,0.42849,0.22582,0.2857,0.42857,0.5711,0.0,1.0,1,1,0,1,0,5,0,0,7,0,0,7,0,0,7,0,0,3,0,0,1,0,1],[20,27,0.7407,0.3704,0.23399,0.14286,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,8,0,0,7,0,0,6,0,0,4,0,0,3,0,0,2,0,0],[24,27,0.8889,0.63835,0.20821,0.5354,0.64286,0.74996,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,4,0,0,8,0,0,8,0,0,5,0,3],[27,27,1.0,0.60262,0.198,0.5354,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,5,0,0,13,0,0,6,0,0,2,0,3]]}]},{"i":"d3b6be5e6c276c06","q":"For a natural number  $n{},$  determine the number of ordered pairs  $(S,T)$  of subsets of  $\\{1,2,\\ldots,n\\}$  for which  $s>|T|$  for any element  $s\\in S$  and  $t>|S|$  for any element  $t\\in T.$","t":[{"b":2,"e":0.57143,"k":"volatile","v":0.35713,"x":0.82143,"p":[[0,11,0.0,0.68303,0.27833,0.42857,0.57143,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,6,0,0,3,0,0,8,0,0,2,0,0,1,0,12],[4,11,0.3636,0.82143,0.25,0.67857,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,4,0,0,2,0,0,4,0,0,0,0,20],[8,11,0.7273,0.66964,0.27302,0.42857,0.64286,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,9,0,0,3,0,0,4,0,0,1,0,11],[11,11,1.0,0.35713,0.07983,0.28571,0.28571,0.42857,0.2857,0.571,0,0,0,0,0,0,0,0,17,0,0,14,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.571,"k":"flat","v":0.51339,"x":0.83481,"p":[[0,31,0.0,0.65625,0.25218,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,8,0,0,3,0,0,7,0,0,2,0,8],[4,31,0.129,0.64732,0.28344,0.42857,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,6,0,0,8,0,0,3,0,0,4,0,0,0,0,11],[8,31,0.2581,0.59375,0.29904,0.28571,0.42859,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,8,0,0,8,0,0,3,0,0,2,0,0,0,0,10],[12,31,0.3871,0.62054,0.32851,0.28571,0.42857,1.0,0.0,1.0,1,13,0,1,0,0,0,0,8,0,0,8,0,0,2,0,0,0,0,0,0,0,13],[16,31,0.5161,0.64284,0.30094,0.42857,0.64286,1.0,0.0,1.0,2,10,0,2,0,0,0,0,3,0,0,7,0,0,4,0,0,5,0,0,1,0,10],[20,31,0.6452,0.51339,0.25966,0.28571,0.42857,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,9,0,0,10,0,0,3,0,0,4,0,0,0,0,5],[24,31,0.7742,0.83481,0.19269,0.71421,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,4,0,16],[28,31,0.9032,0.8125,0.22428,0.57143,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,5,0,0,4,0,0,4,0,0,2,0,17],[31,31,1.0,0.69643,0.24936,0.42859,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,10,0,0,4,0,0,5,0,0,1,0,11]]}]},{"i":"f1d903e23b528680","q":"11. (NET) On a semicircle with unit radius four consecutive chords $A B, B C$, $C D, D E$ with lengths $a, b, c, d$, respectively, are given. Prove that  $$ a^{2}+b^{2}+c^{2}+d^{2}+a b c+b c d<4 $$","t":[{"b":5,"e":0.71429,"k":"flat","v":0.62945,"x":0.81695,"p":[[0,39,0.0,0.65172,0.20499,0.571,0.64286,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,4,0,0,9,0,0,8,0,0,4,0,4],[4,39,0.1026,0.62945,0.18846,0.53575,0.64271,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,5,0,0,8,0,0,10,0,0,4,0,2],[8,39,0.2051,0.66069,0.15871,0.5714,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,8,0,0,12,0,0,4,0,2],[12,39,0.3077,0.70087,0.15714,0.71429,0.71429,0.857,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,16,0,0,8,0,1],[16,39,0.4103,0.69189,0.15617,0.57132,0.71429,0.74996,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,13,0,0,6,0,2],[20,39,0.5128,0.77229,0.10635,0.71429,0.71429,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,9,0,3],[24,39,0.6154,0.81695,0.10853,0.71429,0.857,0.85714,0.714,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,11,0,6],[28,39,0.7179,0.74997,0.11292,0.71429,0.71429,0.857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,16,0,0,11,0,1],[32,39,0.8205,0.73658,0.12934,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,17,0,0,8,0,2],[36,39,0.9231,0.79014,0.13828,0.71429,0.71429,0.85714,0.4286,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,14,0,0,9,0,6],[39,39,1.0,0.7991,0.1063,0.71429,0.71429,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,9,0,5]]},{"b":6,"e":0.42857,"k":"flat","v":0.52226,"x":0.74553,"p":[[0,20,0.0,0.67404,0.15254,0.571,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,11,0,0,5,0,2],[4,20,0.2,0.68743,0.19049,0.571,0.71429,0.85704,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,9,0,0,4,0,5],[8,20,0.4,0.74553,0.21048,0.57142,0.85707,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,5,0,0,10,0,7],[12,20,0.6,0.52226,0.15813,0.42857,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,9,0,0,13,0,0,2,0,0,3,0,0],[16,20,0.8,0.72764,0.20629,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,9,0,0,4,0,8],[20,20,1.0,0.66067,0.20124,0.571,0.64271,0.857,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,5,0,0,9,0,0,7,0,0,5,0,4]]}]},{"i":"af00c111b672481b","q":"Let  $a,b,c$  be positive real numbers satisfying  $a^2+b^2+c^2 \\geq 3.$  Prove that \n\n\\[ \\frac{(a+1)(b+2)}{(b+1)(b+5)} + \\frac{(b+1)(c+2)}{(c+1)(c+5)}+\\frac{(c+1)(a+2)}{(a+1)(a+5)} \\geq \\frac{3}{2} \\]","t":[{"b":1,"e":0.2857,"k":"rising","v":0.04464,"x":0.39954,"p":[[0,33,0.0,0.08705,0.15845,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,1,4,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,33,0.1212,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.10714,0.18558,0.0,0.0,0.10714,0.0,0.42857,24,0,0,24,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.16518,0.21756,0.0,0.0,0.32143,0.0,0.71429,18,0,0,18,0,3,0,0,3,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[16,33,0.4848,0.29469,0.20186,0.14286,0.28571,0.42858,0.0,0.71429,5,0,0,5,0,5,0,0,13,0,0,4,0,0,2,0,0,3,0,0,0,0,0],[20,33,0.6061,0.39954,0.18021,0.28571,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,12,0,0,6,0,0,8,1,0,2,0,0,0,0,0],[24,33,0.7273,0.21195,0.17636,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,1,3,0,0,12,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[28,33,0.8485,0.30804,0.21162,0.14286,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,2,0,0,9,0,0,10,0,0,1,0,0,3,0,0,0,0,0],[32,33,0.9697,0.37276,0.1701,0.28571,0.28571,0.4286,0.0,0.71429,1,0,0,1,0,3,0,0,13,0,1,7,0,0,4,0,0,3,0,0,0,0,0],[33,33,1.0,0.32142,0.22588,0.24999,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,1,0,0,12,0,0,5,0,0,3,0,0,4,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.02232,"x":0.15625,"p":[[0,28,0.0,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.13389,0.17926,0.0,0.0,0.2857,0.0,0.71429,17,0,0,17,1,4,0,1,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[12,28,0.4286,0.12052,0.17165,0.0,0.0,0.2857,0.0,0.571,20,0,0,20,0,2,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,28,0.5714,0.11607,0.14914,0.0,0.0,0.2857,0.0,0.4286,18,0,0,18,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.10259,0.13937,0.0,0.0,0.17857,0.0,0.42857,19,0,0,19,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.12946,0.16409,0.0,0.0,0.2857,0.0,0.57143,17,0,0,17,0,5,0,1,6,0,0,1,0,1,1,0,0,0,0,0,0,0,0],[28,28,1.0,0.15625,0.15714,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,5,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7b0cb285898c31a4","q":"For a positive integer  $n$ , denote  $p(n)$  to be the number of nonnegative integer tuples  $(x,y,z,w)$  such that  $x+y+2z+3w=n-1$ . Also, denote  $q(n)$  to be the number of nonnegative integer tuples  $(a,b,c,d)$  such that\n\n(i).  $a+b+c+d=n$ .\n(ii).  $a \\ge b$ ,  $c \\ge d$ ,  $a \\ge d$ .\n(iii).  $b < c$ .\n\nProve that for all  $n$ ,  $p(n) = q(n)$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.43304,"x":0.53121,"p":[[0,33,0.0,0.50888,0.27417,0.28571,0.57143,0.71429,0.0,1.0,3,2,1,3,0,3,0,0,3,0,0,3,0,0,11,0,0,4,0,0,3,0,2],[4,33,0.1212,0.53121,0.27486,0.39286,0.57141,0.71429,0.0,1.0,2,3,0,2,0,4,0,0,2,0,0,4,0,0,9,0,0,6,0,0,2,0,3],[8,33,0.2424,0.4687,0.21196,0.28571,0.57141,0.57143,0.0,1.0,2,1,0,2,0,3,0,0,4,0,0,2,0,0,19,0,0,1,0,0,0,0,1],[12,33,0.3636,0.46874,0.1931,0.42857,0.57143,0.57143,0.0,0.71429,3,0,2,3,0,1,0,0,2,0,0,7,0,0,16,0,0,3,0,0,0,0,0],[16,33,0.4848,0.43304,0.12103,0.39286,0.42857,0.57141,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,15,0,0,8,0,0,1,0,0,0,0,0],[20,33,0.6061,0.47317,0.12592,0.42857,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,2,0,0,13,0,0,14,0,0,1,0,0,0,0,0],[24,33,0.7273,0.46868,0.15659,0.42857,0.571,0.57143,0.0,0.71429,1,0,1,1,0,1,0,0,5,0,0,8,0,0,15,0,0,2,0,0,0,0,0],[28,33,0.8485,0.45086,0.12424,0.42857,0.4286,0.57111,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,16,0,0,10,0,0,1,0,0,0,0,0],[32,33,0.9697,0.47764,0.13172,0.42857,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,13,0,0,11,0,0,3,0,0,0,0,0],[33,33,1.0,0.48209,0.12238,0.42857,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,10,0,0,16,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.4061,"x":0.70533,"p":[[0,25,0.0,0.55354,0.22232,0.571,0.57143,0.71429,0.0,1.0,1,2,1,1,0,2,0,0,4,0,0,0,0,0,16,0,0,6,0,0,1,0,2],[4,25,0.16,0.4061,0.29675,0.10714,0.57121,0.60607,0.0,1.0,8,1,1,8,0,3,0,0,2,0,0,2,0,0,9,0,0,7,0,0,0,0,1],[8,25,0.32,0.49554,0.28568,0.25,0.57143,0.71429,0.0,1.0,1,4,0,1,0,7,0,0,4,0,0,1,0,0,10,0,0,5,0,0,0,0,4],[12,25,0.48,0.66961,0.14481,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,8,0,0,5,0,2],[16,25,0.64,0.70533,0.11262,0.57143,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,18,0,0,3,0,2],[20,25,0.8,0.63834,0.10095,0.57143,0.57143,0.71429,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,17,0,0,13,0,0,0,0,1],[24,25,0.96,0.70085,0.15306,0.57143,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,13,0,0,3,0,4],[25,25,1.0,0.63389,0.10679,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,13,0,0,0,0,1]]}]},{"i":"5a8be6086609d511","q":"Let  $ABC$  be a triangle with  $\\angle A<90^o, AB \\ne AC$ . Denote   $H$  the orthocenter of triangle  $ABC$ ,  $N$  the midpoint of segment  $[AH]$ ,  $M$  the midpoint of segment  $[BC]$  and  $D$  the intersection point of the angle bisector of  $\\angle BAC$  with the segment  $[MN]$ . Prove that  $\\angle ADH=90^o$","t":[{"b":6,"e":0.14286,"k":"falling","v":0.1383,"x":0.45982,"p":[[0,10,0.0,0.42846,0.31549,0.14286,0.28571,0.71429,0.0,1.0,1,2,0,1,0,14,0,0,2,0,0,0,0,0,5,0,0,4,0,0,4,0,2],[4,10,0.4,0.45982,0.34944,0.14286,0.28571,0.75,0.14286,1.0,0,6,0,0,0,16,0,0,0,0,0,2,0,0,3,0,0,3,0,0,2,0,6],[8,10,0.8,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"rising","v":0.35231,"x":0.91517,"p":[[0,23,0.0,0.44633,0.32692,0.14286,0.28571,0.71429,0.14,1.0,0,4,0,0,0,14,0,0,3,0,0,2,0,0,2,0,0,4,0,0,3,0,4],[4,23,0.1739,0.35231,0.25278,0.14286,0.1429,0.57143,0.14,0.85714,0,0,0,0,0,17,0,0,2,0,0,2,0,0,5,0,0,4,0,0,2,0,0],[8,23,0.3478,0.36607,0.29653,0.14286,0.14286,0.57143,0.0,1.0,1,2,0,1,0,17,0,0,0,0,0,4,0,0,4,0,0,1,0,0,3,0,2],[12,23,0.5217,0.79463,0.2141,0.71429,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,8,0,0,11,0,9],[16,23,0.6957,0.91517,0.1063,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,18],[20,23,0.8696,0.84821,0.12339,0.71429,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,8,0,11],[23,23,1.0,0.87053,0.09689,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,17,0,9]]}]},{"i":"0ec98cdb253affd9","q":"Find all sets of positive integers  $\\{x_1, x_2, \\dots, x_{20}\\}$  such that   $$ x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1}) $$   for  $i=1, 2, \\dots, 20$  where  $x_0=x_{20}, x_{21}=x_1, x_{22}=x_2$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.03125,"x":0.10714,"p":[[0,34,0.0,0.08482,0.13296,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.08482,0.15093,0.0,0.0,0.07143,0.0,0.42857,24,0,0,24,0,0,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.10714,0.15152,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,3,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0625,0.08702,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.05357,"x":0.1517,"p":[[0,18,0.0,0.12054,0.16016,0.0,0.0,0.17857,0.0,0.4286,18,0,0,18,0,6,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.1517,0.19212,0.0,0.14143,0.2857,0.0,0.85714,15,0,0,15,0,7,0,0,6,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[8,18,0.4444,0.125,0.15872,0.0,0.0,0.2857,0.0,0.57143,17,0,0,17,0,6,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,18,0.6667,0.12946,0.19351,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,9,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[16,18,0.8889,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0c0fd32260a963ef","q":"In the quadrilateral  $ABCD$ , we have  $\\measuredangle BAD = 100^{\\circ}$ ,  $\\measuredangle BCD = 130^{\\circ}$ , and  $AB=AD=1$  centimeter. Find the length of diagonal  $AC$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[20,25,0.8,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[25,25,1.0,0.04018,0.10853,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.00893,"x":0.03125,"p":[[0,19,0.0,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0134,0.04166,0.0,0.0,0.0,0.0,0.143,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"882730f57a53d6ec","q":"Let  $a, b, c, d$  be non-negative real numbers such that \\[\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}+\\frac{1}{d+1}=3.\\]\nProve that \\[3(ab+bc+ca+ad+bd+cd)+\\frac{4}{a+b+c+d}\\leqslant 5.\\]*Vasile C\u00eertoaje and Leonard Giugiuc*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.02214,"x":0.13822,"p":[[0,22,0.0,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.02214,0.05146,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,16,0.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b09487b444835784","q":"In the non-isosceles triangle $A B C$ the altitude from $A$ meets side $B C$ in $D$. Let $M$ be the midpoint of $B C$ and let $N$ be the reflection of $M$ in $D$. The circumcircle of the triangle $A M N$ intersects the side $A B$ in $P \\neq A$ and the side $A C$ in $Q \\neq A$. Prove that $A N, B Q$ and $C P$ are concurrent.","t":[{"b":3,"e":0.5714,"k":"flat","v":0.55804,"x":0.6295,"p":[[0,21,0.0,0.6295,0.12292,0.53575,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,3,0,0,21,0,0,0,0,0],[4,21,0.1905,0.60714,0.16366,0.42859,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,8,0,0,3,0,0,20,0,0,0,0,0],[8,21,0.381,0.55804,0.13997,0.42857,0.42859,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,1,0,0,14,0,0,0,0,0],[12,21,0.5714,0.57143,0.13832,0.42857,0.57143,0.71429,0.42857,0.7143,0,0,0,0,0,0,0,0,0,0,0,15,0,0,2,0,0,15,0,0,0,0,0],[16,21,0.7619,0.5625,0.15542,0.42857,0.57144,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,14,0,0,0,0,0,16,0,0,0,0,0],[20,21,0.9524,0.62946,0.12807,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,22,0,0,0,0,0],[21,21,1.0,0.62499,0.12753,0.42857,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,21,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.5982,"x":0.68749,"p":[[0,23,0.0,0.66964,0.10374,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,26,0,0,0,0,0],[4,23,0.1739,0.61607,0.10374,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,15,0,0,0,0,0],[8,23,0.3478,0.5982,0.06622,0.57143,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,7,0,0,0,0,0],[12,23,0.5217,0.62497,0.08565,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,14,0,0,0,0,0],[16,23,0.6957,0.62049,0.09854,0.57142,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,15,0,0,0,0,0],[20,23,0.8696,0.68749,0.05579,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0],[23,23,1.0,0.68747,0.05581,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0]]}]},{"i":"02a0948472cf7a90","q":"Let $I$ be the center of the inscribed circle of triangle $ABC$. A line through $I$ intersects the interior of segment $AB$ at $M$ and the interior of segment $BC$ at $N$. We assume that $BMN$ is an acute triangle. Let $K$ and $L$ be points on segment $AC$ such that $\\angle BMI = \\angle ILA$ and $\\angle BNI = \\angle IKC$.\nProve that $|AM| + |KL| + |CN| = |AC|$.","t":[{"b":0,"e":0.2857,"k":"rising","v":0.12054,"x":0.54018,"p":[[0,43,0.0,0.12054,0.18935,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,43,0.093,0.23661,0.2412,0.10714,0.14288,0.28571,0.0,1.0,8,2,0,8,0,9,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[8,43,0.186,0.21875,0.20198,0.10714,0.14286,0.28571,0.0,1.0,8,1,0,8,0,9,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[12,43,0.2791,0.28125,0.26362,0.14286,0.14286,0.32143,0.0,1.0,4,3,0,4,0,13,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,3],[16,43,0.3721,0.26995,0.22713,0.14286,0.17645,0.32143,0.0,1.0,3,2,0,3,0,13,0,1,7,0,0,6,0,0,0,0,0,0,0,0,0,0,2],[20,43,0.4651,0.26786,0.23351,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,8,0,0,6,0,0,9,0,0,0,0,0,0,0,0,1,0,1],[24,43,0.5581,0.34821,0.3008,0.14286,0.28571,0.42857,0.0,1.0,3,4,0,3,0,12,0,0,4,0,0,8,0,0,0,0,0,0,0,0,1,0,4],[28,43,0.6512,0.27232,0.2212,0.14286,0.28571,0.32143,0.0,1.0,6,1,0,6,0,6,0,0,12,0,0,6,0,0,0,0,0,0,0,0,1,0,1],[32,43,0.7442,0.35268,0.2766,0.14286,0.28571,0.42857,0.0,1.0,3,4,0,3,0,7,0,0,10,0,0,8,0,0,0,0,0,0,0,0,0,0,4],[36,43,0.8372,0.45089,0.33902,0.14289,0.35714,0.57145,0.0,1.0,2,8,0,2,0,7,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,8],[40,43,0.9302,0.54018,0.33069,0.28571,0.42857,1.0,0.0,1.0,2,10,0,2,0,2,0,0,5,0,0,13,0,0,0,0,0,0,0,0,0,0,10],[43,43,1.0,0.53125,0.33737,0.28571,0.42857,1.0,0.0,1.0,2,10,0,2,0,3,0,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,10]]},{"b":2,"e":0.42857,"k":"flat","v":0.14286,"x":0.26339,"p":[[0,4,0.0,0.14286,0.24484,0.0,0.07143,0.14286,0.0,1.0,16,2,0,16,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,4,1.0,0.26339,0.11356,0.14286,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,7,0,0,17,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"17fc1632005b839f","q":"Let  $ABC$  be an acute triangle with circumcircle  $\\omega$  and circumcenter  $O$ . The perpendicular from  $A$  to  $BC$  intersects  $BC$  and  $\\omega$  at  $D$  and  $E$ , respectively. Let  $F$  be a point on the segment  $AE$ , such that  $2 \\cdot FD = AE$ . Let  $l$  be the perpendicular to  $OF$  through  $F$ . Prove that  $l$ , the tangent to  $\\omega$  at  $E$ , and the line  $BC$  are concurrent.\n\nProposed by *Stefan Lozanovski, Macedonia*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.09366,"p":[[0,22,0.0,0.09366,0.11067,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,22,0.1818,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0133,0.04137,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.03571,"x":0.18304,"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],[4,34,0.1176,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.18304,0.06423,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,23,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.16965,0.06622,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.15178,0.03458,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.16509,0.05191,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"37c6227da850b92e","q":"Given a triangle $A B C$, let $P$ and $Q$ be points on segments $\\overline{A B}$ and $\\overline{A C}$, respectively, such that $A P=A Q$. Let $S$ and $R$ be distinct points on segment $\\overline{B C}$ such that $S$ lies between $B$ and $R, \\angle B P S=\\angle P R S$, and $\\angle C Q R=\\angle Q S R$. Prove that $P, Q$, $R, S$ are concyclic.","t":[{"b":2,"e":1.0,"k":"rising","v":0.75,"x":1.0,"p":[[0,23,0.0,0.75,0.33693,0.42857,1.0,1.0,0.0,1.0,2,20,0,2,0,0,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,20],[4,23,0.1739,0.84822,0.24727,0.64286,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,23],[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],[23,23,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":4,"e":0.42857,"k":"falling","v":0.42411,"x":0.80808,"p":[[0,20,0.0,0.80808,0.33232,0.64321,1.0,1.0,0.0,1.0,3,23,0,3,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,23],[4,20,0.2,0.76339,0.33808,0.42857,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,0,0,0,7,0,0,0,0,0,2,0,0,0,0,20],[8,20,0.4,0.73214,0.36727,0.42857,1.0,1.0,0.0,1.0,4,20,0,4,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,20],[12,20,0.6,0.60272,0.37749,0.42857,0.42929,1.0,0.0,1.0,5,13,0,5,0,1,0,0,1,0,0,10,0,0,0,0,0,1,0,0,1,0,13],[16,20,0.8,0.56249,0.3976,0.35714,0.42857,1.0,0.0,1.0,7,13,0,7,0,1,0,0,0,0,0,10,0,0,1,0,0,0,0,0,0,0,13],[20,20,1.0,0.42411,0.07563,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,29,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"4b01041eb625ec8d","q":"Let $p$ be a prime number.\nProve that there exists a prime $q$ such that $\\mathfrak{n}^{p} \\not\\equiv p$ for all $n \\in \\mathbb{Z}$.","t":[{"b":1,"e":0.0,"k":"falling","v":0.04911,"x":0.48661,"p":[[0,14,0.0,0.48661,0.4473,0.14286,0.14286,1.0,0.0,1.0,6,13,0,6,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,13],[4,14,0.2857,0.42411,0.42028,0.14286,0.14286,1.0,0.0,1.0,4,11,0,4,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[8,14,0.5714,0.4375,0.44455,0.0,0.14286,1.0,0.0,1.0,10,10,0,10,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,10],[12,14,0.8571,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.25884,"x":1.0,"p":[[0,34,0.0,0.34804,0.37964,0.14286,0.14286,0.46427,0.0,1.0,3,8,0,3,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[4,34,0.1176,0.39733,0.40991,0.14286,0.14286,1.0,0.0,1.0,4,10,0,4,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[8,34,0.2353,0.45081,0.43176,0.14286,0.1429,1.0,0.0,1.0,6,12,0,6,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[12,34,0.3529,0.25884,0.34154,0.0,0.14286,0.14286,0.0,1.0,9,4,0,9,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[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":"ed81ffb3b6b460e9","q":"Find all functions  $f:\\mathbb{N}\\to \\mathbb{N}$  so that for every natural numbers  $m,n$  : $f(n)+2mn+f(m)$  is a perfect square.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.2008,"x":0.29902,"p":[[0,32,0.0,0.21866,0.17495,0.14286,0.14286,0.28571,0.0,0.85714,2,0,0,2,0,20,0,0,6,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[4,32,0.125,0.26331,0.20863,0.14286,0.14288,0.28571,0.0,1.0,1,1,0,1,0,17,0,0,9,0,0,1,0,0,1,0,0,2,0,0,0,0,1],[8,32,0.25,0.27006,0.16529,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,17,0,1,6,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[12,32,0.375,0.2008,0.11219,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,21,0,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,32,0.5,0.25436,0.15462,0.14286,0.14286,0.28571,0.14,0.71429,0,0,0,0,0,17,0,0,10,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[20,32,0.625,0.29009,0.2273,0.14286,0.14286,0.28571,0.14,0.85714,0,0,0,0,0,18,0,0,7,0,0,2,0,0,1,0,0,1,0,0,3,0,0],[24,32,0.75,0.29902,0.2325,0.14286,0.21428,0.28571,0.14,0.85714,0,0,0,0,0,16,0,0,10,0,0,1,0,0,1,0,0,0,0,0,4,0,0],[28,32,0.875,0.2767,0.23133,0.14286,0.14286,0.28571,0.14,0.85714,0,0,0,0,0,19,0,0,8,0,0,1,0,0,0,0,0,0,0,0,4,0,0],[32,32,1.0,0.27231,0.15712,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,15,0,0,10,0,0,3,0,0,3,0,0,1,0,0,0,0,0]]},{"b":2,"e":0.2857,"k":"flat","v":0.17848,"x":0.24107,"p":[[0,13,0.0,0.24107,0.15335,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,19,0,0,8,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[4,13,0.3077,0.17848,0.09453,0.14286,0.14286,0.14287,0.0,0.57143,1,0,0,1,0,24,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,13,0.6154,0.2142,0.15976,0.14286,0.14286,0.28571,0.0,0.85714,1,0,0,1,0,22,0,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[12,13,0.9231,0.21429,0.09449,0.14286,0.14286,0.2857,0.14286,0.4286,0,0,0,0,0,19,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.24107,0.19044,0.14286,0.14286,0.28571,0.0,0.85714,1,0,1,1,0,19,0,0,8,0,0,1,0,0,1,0,0,0,0,0,2,0,0]]}]},{"i":"d899989348f4dfd4","q":"If $x, y, z$ are positive real numbers, prove that\n\n$$\n(x+y+z)^{2}(y z+z x+x y)^{2} \\leq 3\\left(y^{2}+y z+z^{2}\\right)\\left(z^{2}+z x+x^{2}\\right)\\left(x^{2}+x y+y^{2}\\right)\n$$","t":[{"b":0,"e":1.0,"k":"rising","v":0.49553,"x":1.0,"p":[[0,37,0.0,0.5,0.5,0.0,0.5,1.0,0.0,1.0,16,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[4,37,0.1081,0.49553,0.46564,0.0,0.57141,1.0,0.0,1.0,14,13,2,14,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,13],[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],[12,37,0.3243,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,37,0.4324,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,37,0.5405,0.97767,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[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],[28,37,0.7568,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[32,37,0.8649,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,37,0.973,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[37,37,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":6,"e":0.0,"k":"volatile","v":0.0,"x":0.76339,"p":[[0,13,0.0,0.59821,0.47974,0.0,1.0,1.0,0.0,1.0,12,18,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,18],[4,13,0.3077,0.54017,0.4841,0.0,0.85714,1.0,0.0,1.0,14,16,1,14,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,16],[8,13,0.6154,0.76339,0.41127,0.78571,1.0,1.0,0.0,1.0,7,23,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,23],[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]]}]},{"i":"3c7b5d4303f5116d","q":"Given is an isosceles triangle  $ABC$  with  $CA=CB$  and angle bisector  $BD$ ,  $D \\in AC$ . The line through the center  $O$  of  $(ABC)$ , perpendicular to  $BD$ , meets  $BC$  at  $E$ . The line through  $E$ , parallel to  $BD$ , meets  $AC$  at  $F$ . Prove that  $CE=DF$ .","t":[{"b":1,"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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"740e633a79725bac","q":"Find all functions  $f:\\mathbb{R} \\rightarrow \\mathbb{R}$  which satisfies the followings. (Note that  $\\mathbb{R}$  stands for the set of all real numbers)\n(1) For each real numbers  $x$ ,  $y$ , the equality  $f(x+f(x)+xy) = 2f(x)+xf(y)$  holds.\n(2) For every real number  $z$ , there exists  $x$  such that  $f(x) = z$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.21875,"x":0.28576,"p":[[0,11,0.0,0.27683,0.14262,0.1429,0.28571,0.32143,0.0,0.57143,3,0,0,3,0,6,0,0,15,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[4,11,0.3636,0.28576,0.18561,0.14286,0.28571,0.32143,0.0,1.0,3,1,0,3,0,7,0,0,14,0,0,6,0,0,1,0,0,0,0,0,0,0,1],[8,11,0.7273,0.21875,0.14718,0.14286,0.2143,0.28571,0.0,0.42857,6,0,0,6,0,10,0,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.22321,0.16342,0.14286,0.14288,0.42857,0.0,0.42857,7,0,0,7,0,10,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.28561,"x":0.33928,"p":[[0,60,0.0,0.30357,0.16269,0.24999,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,4,0,0,11,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[4,60,0.0667,0.28561,0.12884,0.14286,0.28571,0.32143,0.0,0.57143,1,0,0,1,0,8,0,0,15,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[8,60,0.1333,0.30356,0.1587,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,12,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[12,60,0.2,0.32143,0.14285,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,6,0,0,13,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[16,60,0.2667,0.31696,0.11143,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,17,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[20,60,0.3333,0.33482,0.12682,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,2,0,0,18,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[24,60,0.4,0.33928,0.11152,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,19,0,0,9,0,0,1,0,0,1,0,0,0,0,0],[28,60,0.4667,0.28576,0.113,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,6,0,0,18,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[32,60,0.5333,0.32589,0.10853,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,20,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[36,60,0.6,0.30802,0.10167,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,4,0,0,21,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[40,60,0.6667,0.32143,0.07143,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.31696,0.12745,0.2857,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,3,0,0,14,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[48,60,0.8,0.32143,0.09449,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,19,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[52,60,0.8667,0.30362,0.08571,0.28571,0.28571,0.28571,0.0,0.43,1,0,0,1,0,1,0,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.29455,0.06144,0.28571,0.28571,0.28571,0.14,0.4286,0,0,0,0,0,2,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.28571,0.07986,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,2,0,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d63ad3e909e6245a","q":"At a party with  $n$  people, it is known that among any  $4$  people, there are either  $3$  people who all know one another or  $3$  people none of which knows another. Show that the  $n$  people can be separated into two rooms, so that everyone in one room knows one another and no two people in the other room know each other.","t":[{"b":4,"e":0.0,"k":"flat","v":0.01339,"x":0.04911,"p":[[0,30,0.0,0.04018,0.08918,0.0,0.0,0.0,0.0,0.42857,25,0,1,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.04902,0.08449,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.04018,0.06424,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.14286,"k":"flat","v":0.02679,"x":0.05357,"p":[[0,11,0.0,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.143,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"27e663510ad3d597","q":"Is it possible to partition all positive integers into disjoint sets $A$ and $B$ such that\n\n(i) no three numbers of $A$ form arithmetic progression,\n\n(ii) no infinite non-constant arithmetic progression can be formed by numbers of $B$ ?","t":[{"b":1,"e":0.85714,"k":"rising","v":0.59812,"x":0.8616,"p":[[0,40,0.0,0.67855,0.34255,0.82132,0.85714,0.85714,0.0,1.0,6,1,0,6,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,23,0,1],[4,40,0.1,0.70982,0.30406,0.82143,0.85714,0.85714,0.0,1.0,4,1,0,4,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,23,0,1],[8,40,0.2,0.59812,0.36158,0.14214,0.85707,0.85714,0.0,0.85714,7,0,0,7,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,0],[12,40,0.3,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[16,40,0.4,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[20,40,0.5,0.83482,0.09523,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,23,0,3],[24,40,0.6,0.8482,0.06121,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,26,0,2],[28,40,0.7,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[32,40,0.8,0.83482,0.05187,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,27,0,0],[36,40,0.9,0.85713,0.05051,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,2],[40,40,1.0,0.84374,0.04164,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0]]},{"b":4,"e":0.857,"k":"rising","v":0.66518,"x":0.85266,"p":[[0,8,0.0,0.66518,0.33429,0.71429,0.85714,0.85714,0.0,0.85714,5,0,0,5,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,22,0,0],[4,8,0.5,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[8,8,1.0,0.85266,0.02485,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0]]}]},{"i":"1f22c38a606f3ed6","q":"Let $n$ be a positive integer such that the sum of all the positive divisors of $n$ (except $n$ ) plus the number of these divisors is equal to $n$. Prove that $n=2 m^{2}$ for some integer $m$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.72321,"x":1.0,"p":[[0,9,0.0,0.82589,0.28956,0.67857,1.0,1.0,0.14286,1.0,0,22,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,22],[4,9,0.4444,0.72321,0.32131,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,5,0,0,4,0,0,1,0,0,3,0,0,0,0,17],[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]]},{"b":6,"e":0.42857,"k":"falling","v":0.3125,"x":0.8616,"p":[[0,53,0.0,0.8616,0.2435,0.857,1.0,1.0,0.2857,1.0,0,22,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,1,0,0,3,0,22],[4,53,0.0755,0.73659,0.30117,0.42859,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,5,0,0,3,0,0,4,0,0,1,0,0,2,0,16],[8,53,0.1509,0.54464,0.23808,0.42857,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,4,0,0,7,0,0,8,0,0,5,0,0,2,0,3],[12,53,0.2264,0.47321,0.22428,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,5,0,0,8,0,0,7,0,0,4,0,0,2,0,1],[16,53,0.3019,0.31695,0.14164,0.14289,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,9,0,0,11,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[20,53,0.3774,0.31696,0.18466,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,12,0,0,7,0,0,10,0,0,2,0,0,0,0,0,0,0,1],[24,53,0.4528,0.34373,0.17805,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,9,0,0,10,0,0,7,0,0,3,0,0,3,0,0,0,0,0],[28,53,0.5283,0.37945,0.21009,0.14286,0.42857,0.42858,0.14286,1.0,0,1,0,0,0,10,0,0,3,0,0,12,0,0,5,0,0,0,0,0,1,0,1],[32,53,0.6038,0.48214,0.1948,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,4,0,0,3,0,0,11,0,0,7,0,0,5,0,0,2,0,0],[36,53,0.6792,0.33036,0.19045,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,13,0,0,4,0,0,10,0,0,3,0,0,1,0,0,1,0,0],[40,53,0.7547,0.44642,0.22516,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,5,0,0,5,0,0,13,0,0,5,0,0,0,0,0,2,0,2],[44,53,0.8302,0.33927,0.17766,0.14286,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,11,0,0,5,0,0,12,0,0,1,0,0,3,0,0,0,0,0],[48,53,0.9057,0.33482,0.21011,0.14286,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,12,0,0,6,0,0,11,0,0,0,0,0,1,0,0,1,0,1],[52,53,0.9811,0.3125,0.16146,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,10,0,0,12,0,0,6,0,0,2,0,0,2,0,0,0,0,0],[53,53,1.0,0.32143,0.15972,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,10,0,0,9,0,0,10,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"8315d8218c726684","q":"Let $A B C$ be an acute triangle. The altitudes $B E$ and $C F$ intersect at the orthocenter $H$, and point $O$ denotes the circumcenter. Point $P$ is chosen so that $\\angle A P H=\\angle O P E=90^{\\circ}$, and point $Q$ is chosen so that $\\angle A Q H=\\angle O Q F=90^{\\circ}$. Lines $E P$ and $F Q$ meet at point $T$. Prove that points $A, T, O$ are collinear.","t":[{"b":1,"e":0.57143,"k":"falling","v":0.5089,"x":0.91071,"p":[[0,26,0.0,0.91071,0.24157,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,27],[4,26,0.1538,0.83033,0.30608,0.89275,1.0,1.0,0.0,1.0,1,24,1,1,0,0,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,24],[8,26,0.3077,0.88839,0.2618,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,25],[12,26,0.4615,0.63835,0.22584,0.42857,0.57143,0.85704,0.2857,1.0,0,5,0,0,0,0,0,0,4,0,0,5,0,0,9,0,0,5,0,0,4,0,5],[16,26,0.6154,0.5089,0.17834,0.42857,0.42857,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,6,0,0,13,0,0,5,0,0,6,0,0,1,0,1],[20,26,0.7692,0.61157,0.20897,0.4286,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,4,0,0,9,0,0,8,0,0,3,0,3],[24,26,0.9231,0.66066,0.21945,0.571,0.57143,0.78571,0.2857,1.0,0,8,0,0,0,0,0,0,2,0,0,4,0,0,14,0,0,4,0,0,0,0,8],[26,26,1.0,0.7053,0.18881,0.57143,0.71429,0.75,0.28571,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,11,0,0,1,0,7]]},{"b":5,"e":1.0,"k":"flat","v":0.88384,"x":1.0,"p":[[0,35,0.0,0.89732,0.22934,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,26],[4,35,0.1143,0.88839,0.22227,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,24],[8,35,0.2286,0.88384,0.27326,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,25],[12,35,0.3429,0.90179,0.22142,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,26],[16,35,0.4571,0.97321,0.07526,1.0,1.0,1.0,0.714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[20,35,0.5714,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,35,0.6857,0.94641,0.11158,1.0,1.0,1.0,0.571,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,25],[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],[32,35,0.9143,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[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":"376814dc8891a5dd","q":"Let $A B C D$ be a cyclic quadrilateral with the property that $\\angle A B D=\\angle D B C$. Let $E$ be the intersection of the diagonals $A C$ and $B D$. Let $M$ be the midpoint of $A E$ and $N$ be the midpoint of $D C$. Prove that $M B C N$ is a cyclic quadrilateral.","t":[{"b":0,"e":0.0,"k":"flat","v":0.03125,"x":0.17857,"p":[[0,31,0.0,0.08482,0.25719,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[4,31,0.129,0.15625,0.31209,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,2],[8,31,0.2581,0.0625,0.17474,0.0,0.0,0.0,0.0,0.71429,26,0,1,26,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[12,31,0.3871,0.07589,0.1988,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,31,0.5161,0.08464,0.21082,0.0,0.0,0.14,0.0,1.0,23,1,0,23,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[20,31,0.6452,0.17857,0.29451,0.0,0.0,0.21429,0.0,0.71429,22,0,0,22,0,2,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0],[24,31,0.7742,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.0759,0.1287,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,8,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"volatile","v":0.05804,"x":0.74999,"p":[[0,12,0.0,0.05804,0.15916,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,12,0.3333,0.32143,0.43154,0.0,0.0,0.78571,0.0,1.0,18,8,0,18,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,8],[8,12,0.6667,0.60714,0.42408,0.14286,0.71429,1.0,0.0,1.0,7,15,0,7,0,3,0,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,15],[12,12,1.0,0.74999,0.41188,0.49968,1.0,1.0,0.0,1.0,6,23,0,6,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,23]]}]},{"i":"bbf11ea3f31537de","q":"Let  $\\mathbb{R}$  be the set of real numbers. Determine all functions  $f: \\mathbb{R} \\rightarrow \\mathbb{R}$  such that  $f(0)+1=f(1)$  and for any real numbers  $x$  and  $y$ ,   $$  f(xy-x)+f(x+f(y))=yf(x)+3  $$","t":[{"b":3,"e":1.0,"k":"rising","v":0.52232,"x":0.99554,"p":[[0,47,0.0,0.66515,0.33429,0.39286,0.71429,1.0,0.0,1.0,1,11,0,1,0,5,0,0,2,0,0,1,0,0,4,0,0,4,0,0,4,0,11],[4,47,0.0851,0.73214,0.31288,0.53572,0.85714,1.0,0.14286,1.0,0,14,0,0,0,4,0,0,2,0,0,2,0,0,3,0,0,2,0,0,5,0,14],[8,47,0.1702,0.61156,0.36463,0.14286,0.71429,1.0,0.0,1.0,1,10,0,1,0,9,0,0,1,0,0,0,0,0,3,0,0,4,0,0,4,0,10],[12,47,0.2553,0.60265,0.3655,0.1429,0.57121,1.0,0.14286,1.0,0,13,0,0,0,9,0,0,2,0,0,3,0,0,3,0,0,2,0,0,0,0,13],[16,47,0.3404,0.52232,0.36875,0.14286,0.57144,0.89286,0.14286,1.0,0,8,0,0,0,14,0,0,1,0,0,1,0,0,0,0,0,6,0,0,2,0,8],[20,47,0.4255,0.65179,0.33491,0.39285,0.71429,1.0,0.14286,1.0,0,11,0,0,0,7,0,0,1,0,0,3,0,0,2,0,0,5,0,0,3,0,11],[24,47,0.5106,0.63391,0.35524,0.25,0.71429,1.0,0.0,1.0,1,11,0,1,0,7,0,0,2,0,0,1,0,0,2,0,0,5,0,0,3,0,11],[28,47,0.5957,0.8125,0.2683,0.71429,1.0,1.0,0.14286,1.0,0,17,0,0,0,3,0,0,0,0,0,1,0,0,3,0,0,3,0,0,5,0,17],[32,47,0.6809,0.92411,0.14279,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,2,0,24],[36,47,0.766,0.93301,0.1336,0.96429,1.0,1.0,0.571,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[40,47,0.8511,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[44,47,0.9362,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[47,47,1.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":5,"e":0.71429,"k":"falling","v":0.55791,"x":0.81249,"p":[[0,13,0.0,0.75893,0.31428,0.64286,0.85714,1.0,0.14286,1.0,0,15,0,0,0,5,0,0,0,0,0,3,0,0,0,0,0,3,0,0,6,0,15],[4,13,0.3077,0.81249,0.29975,0.82143,1.0,1.0,0.0,1.0,1,19,0,1,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,0,5,0,19],[8,13,0.6154,0.7455,0.2544,0.67846,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,9,0,0,4,0,11],[12,13,0.9231,0.55791,0.14905,0.42857,0.57141,0.71429,0.14,0.71429,0,0,0,0,0,1,0,0,1,0,0,10,0,0,8,0,0,12,0,0,0,0,0],[13,13,1.0,0.58476,0.14446,0.42857,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,9,0,0,10,0,0,11,0,0,0,0,1]]}]},{"i":"013a3029fcd1c46e","q":"Let $A B C D$ be a parallelogram in the plane. We draw two circles of radius $R$, one through the points $A$ and $B$, the other through $B$ and $C$. Let $E$ be the other point of\nintersection of the circles. We assume that $E$ is not a vertex of the parallelogram. Show that the circle passing through $A, D$, and $E$ also has radius $R$.","t":[{"b":4,"e":1.0,"k":"rising","v":0.53125,"x":0.87052,"p":[[0,24,0.0,0.53125,0.37667,0.14286,0.57143,0.85714,0.0,1.0,6,7,1,6,0,5,0,0,0,0,0,2,0,0,5,0,0,3,0,0,4,0,7],[4,24,0.1667,0.55348,0.38928,0.14286,0.71429,1.0,0.0,1.0,5,9,0,5,0,6,0,0,1,0,0,2,0,0,1,0,0,5,0,0,3,0,9],[8,24,0.3333,0.69643,0.3549,0.39286,0.78571,1.0,0.0,1.0,2,15,0,2,0,4,0,0,2,0,0,1,0,0,1,0,0,6,0,0,1,0,15],[12,24,0.5,0.70086,0.34324,0.49968,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,4,0,0,0,0,0,5,0,0,2,0,0,2,0,15],[16,24,0.6667,0.79909,0.26454,0.67857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,4,0,0,4,0,16],[20,24,0.8333,0.87052,0.18683,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0,0,4,0,19],[24,24,1.0,0.85713,0.2143,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,4,0,19]]},{"b":6,"e":0.14286,"k":"flat","v":0.49553,"x":0.70533,"p":[[0,19,0.0,0.67409,0.37498,0.42859,0.78571,1.0,0.0,1.0,5,15,0,5,0,1,0,0,1,0,0,2,0,0,4,0,0,3,0,0,1,0,15],[4,19,0.2105,0.49553,0.34439,0.24999,0.42857,0.85714,0.0,1.0,4,6,0,4,0,4,0,0,5,0,0,5,0,0,4,0,0,0,0,0,4,0,6],[8,19,0.4211,0.49993,0.34625,0.14286,0.571,0.71429,0.0,1.0,7,6,0,7,0,2,0,0,1,0,0,2,0,0,11,0,0,2,0,0,1,0,6],[12,19,0.6316,0.62047,0.30849,0.5354,0.57143,0.89286,0.0,1.0,4,8,0,4,0,0,0,0,0,0,0,4,0,0,12,0,0,1,0,0,3,0,8],[16,19,0.8421,0.70533,0.2695,0.4286,0.64286,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,8,0,0,7,0,0,2,0,0,2,0,12],[19,19,1.0,0.61158,0.32387,0.42857,0.57143,0.89286,0.0,1.0,3,8,0,3,0,3,0,0,0,0,0,4,0,0,7,0,0,4,0,0,3,0,8]]}]},{"i":"86802d8eec1b4055","q":"For each pair  $(\\alpha, \\beta)$  of non-negative reals with  $\\alpha+\\beta \\geq 2$ , determine all functions  $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ , such that   $$ f(x)f(y) \\leq f(xy)+\\alpha x+\\beta y $$   for all reals  $x, y$ .","t":[{"b":3,"e":0.28571,"k":"flat","v":0.35266,"x":0.57588,"p":[[0,30,0.0,0.35266,0.26958,0.14286,0.28571,0.46418,0.0,1.0,5,1,0,5,0,5,0,0,10,0,0,4,0,0,2,0,0,3,0,0,2,0,1],[4,30,0.1333,0.41964,0.24205,0.2857,0.28571,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,15,0,0,5,0,0,2,0,0,2,0,0,5,0,0],[8,30,0.2667,0.41957,0.23668,0.28571,0.42857,0.571,0.0,1.0,3,1,0,3,0,1,0,0,10,0,0,8,0,0,5,0,0,2,0,0,2,0,1],[12,30,0.4,0.56472,0.28028,0.28571,0.49979,0.85714,0.14286,1.0,0,5,0,0,0,1,0,1,9,0,0,5,0,0,4,0,0,2,0,0,5,0,5],[16,30,0.5333,0.57588,0.2868,0.28571,0.57143,0.71429,0.0,1.0,1,6,0,1,0,1,0,0,9,0,0,2,0,0,4,0,0,8,0,0,1,0,6],[20,30,0.6667,0.51338,0.26692,0.28571,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,13,0,0,5,0,0,3,0,0,3,0,0,3,0,4],[24,30,0.8,0.48213,0.24419,0.28571,0.42857,0.60714,0.0,1.0,1,1,0,1,0,2,0,0,10,0,0,4,0,0,7,0,0,3,0,0,4,0,1],[28,30,0.9333,0.36606,0.17104,0.2857,0.28571,0.32143,0.2857,1.0,0,1,0,0,0,0,0,0,24,0,0,3,0,0,3,0,0,0,0,0,1,0,1],[30,30,1.0,0.36605,0.13801,0.2857,0.28571,0.46418,0.14286,0.71429,0,0,0,0,0,1,0,0,21,0,0,2,0,0,7,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.21875,"x":0.73656,"p":[[0,78,0.0,0.43301,0.27544,0.2857,0.35714,0.571,0.0,1.0,2,3,0,2,0,3,0,0,11,0,0,7,0,0,3,0,0,0,0,0,3,0,3],[4,78,0.0513,0.34376,0.30275,0.14286,0.28571,0.57143,0.0,1.0,7,1,0,7,0,5,0,0,10,0,0,1,0,0,2,0,0,2,0,0,4,0,1],[8,78,0.1026,0.21875,0.24217,0.0,0.14286,0.28571,0.0,0.857,13,0,0,13,0,4,0,0,9,0,0,1,0,0,2,0,0,2,0,0,1,0,0],[12,78,0.1538,0.27677,0.25735,0.10714,0.2857,0.42857,0.0,0.857,8,0,0,8,0,7,0,0,8,0,0,4,0,0,1,0,0,1,0,0,3,0,0],[16,78,0.2051,0.28571,0.26,0.10714,0.2857,0.46431,0.0,0.85714,8,0,0,8,0,7,0,0,8,0,0,1,0,0,4,0,0,2,0,0,2,0,0],[20,78,0.2564,0.29455,0.14267,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,4,0,0,20,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[24,78,0.3077,0.41518,0.21535,0.28571,0.28571,0.42857,0.1429,1.0,0,2,0,0,0,1,0,0,18,0,0,6,0,0,2,0,0,2,0,0,1,0,2],[28,78,0.359,0.41071,0.20748,0.28571,0.28571,0.46429,0.0,1.0,1,1,0,1,0,1,0,0,15,0,0,7,0,0,3,0,0,3,0,0,1,0,1],[32,78,0.4103,0.73656,0.21165,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,7,0,0,6,0,8],[36,78,0.4615,0.68303,0.19799,0.57143,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,7,0,0,7,0,0,7,0,4],[40,78,0.5128,0.70982,0.22155,0.42859,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,5,0,0,9,0,6],[44,78,0.5641,0.66516,0.22477,0.42857,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,8,0,0,5,0,0,7,0,0,4,0,6],[48,78,0.6154,0.64286,0.24223,0.42857,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,10,0,0,4,0,0,4,0,0,5,0,6],[52,78,0.6667,0.70534,0.23129,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,8,0,0,5,0,0,4,0,0,6,0,8],[56,78,0.7179,0.69642,0.22799,0.42857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,8,0,0,2,0,0,5,0,0,10,0,5],[60,78,0.7692,0.61163,0.19955,0.42857,0.57143,0.74996,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,13,0,0,4,0,0,6,0,0,6,0,2],[64,78,0.8205,0.62053,0.21902,0.42857,0.5712,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,14,0,0,4,0,0,2,0,0,8,0,3],[68,78,0.8718,0.65625,0.22828,0.42857,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,10,0,0,3,0,0,6,0,0,6,0,5],[72,78,0.9231,0.62499,0.23352,0.42857,0.64286,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,5,0,0,7,0,0,4,0,0,7,0,0,5,0,4],[76,78,0.9744,0.66071,0.19149,0.42857,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,8,0,0,10,0,1],[78,78,1.0,0.66514,0.19759,0.42857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,6,0,0,10,0,2]]}]},{"i":"9882abbe86a94ea5","q":"Let $\\Gamma$ be the circumcircle of a triangle $ABC$ and let $D$ be a point on line segment $BC$. The circle through $B$ and $D$ that is tangent to $\\Gamma$ and the circle through $C$ and $D$ that is tangent to $\\Gamma$ intersect at a point $E \\neq D$. The line $DE$ intersects $\\Gamma$ at two points $X$ and $Y$. Prove that $|EX|=|EY|$.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.13393,"p":[[0,18,0.0,0.13393,0.23941,0.0,0.0,0.14286,0.0,0.85714,21,0,1,21,0,4,0,0,3,0,0,0,0,0,1,0,0,2,0,0,1,0,0],[4,18,0.2222,0.12491,0.24678,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.09821,"x":0.19196,"p":[[0,29,0.0,0.125,0.21354,0.0,0.0,0.17857,0.0,1.0,20,1,0,20,0,4,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[4,29,0.1379,0.1875,0.28669,0.0,0.07143,0.28571,0.0,1.0,16,1,1,16,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,3,0,1],[8,29,0.2759,0.17411,0.28287,0.0,0.0,0.28571,0.0,1.0,19,2,3,19,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[12,29,0.4138,0.15179,0.21998,0.0,0.0,0.2857,0.0,0.85714,17,0,1,17,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[16,29,0.5517,0.09821,0.13092,0.0,0.0,0.2857,0.0,0.28571,20,0,3,20,0,2,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.19196,0.14555,0.0,0.28571,0.28571,0.0,0.4286,11,0,0,11,0,1,0,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.15626,0.13534,0.0,0.21428,0.28571,0.0,0.286,13,0,0,13,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.14732,0.14053,0.0,0.21428,0.2857,0.0,0.28571,15,0,0,15,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.19196,0.1411,0.0,0.2857,0.28571,0.0,0.42857,11,0,0,11,0,0,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5c4ff281a02866a2","q":"Let  $p$  be a prime number and  $\\mathcal{A}$  be a finite set of integers, with at least  $p^k$  elements. Denote by  $N_{\\text{even}}$  the number of subsets of  $\\mathcal{A}$  with even cardinality and sum of elements divisible by  $p^k$ . Define  $N_{\\text{odd}}$  similarly. Prove that  $N_{\\text{even}}\\equiv N_{\\text{odd}}\\bmod{p}.$","t":[{"b":0,"e":0.0,"k":"flat","v":0.04911,"x":0.09812,"p":[[0,10,0.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.08483,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.09812,0.13567,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,8,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[10,10,1.0,0.05804,0.11214,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.0625,"x":0.15178,"p":[[0,9,0.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.08482,0.10012,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.12491,0.09941,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,16,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.15178,0.11811,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,13,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"103e5bbcfda8d570","q":"Let $S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $P$ whose vertices are in $S$, let $a(P)$ be the number of vertices of $P$, and let $b(P)$ be the number of points of $S$ which are outside $P$. Prove that for every real number $x$  $$ \\sum_{P} x^{a(P)}(1-x)^{b(P)}=1 $$  where the sum is taken over all convex polygons with vertices in $S$.  NB. A line segment, a point and the empty set are considered as convex polygons of 2,1 and 0 vertices, respectively.  (Colombia)","t":[{"b":2,"e":1.0,"k":"rising","v":0.63839,"x":1.0,"p":[[0,39,0.0,0.63839,0.42855,0.14286,1.0,1.0,0.0,1.0,4,18,1,4,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,18],[4,39,0.1026,0.6875,0.42474,0.14289,1.0,1.0,0.0,1.0,6,19,2,6,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,19],[8,39,0.2051,0.70982,0.42331,0.28571,1.0,1.0,0.0,1.0,7,21,0,7,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,21],[12,39,0.3077,0.82143,0.3481,0.96429,1.0,1.0,0.0,1.0,3,24,1,3,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,24],[16,39,0.4103,0.78572,0.36421,0.71425,1.0,1.0,0.0,1.0,1,23,0,1,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,23],[20,39,0.5128,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[24,39,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[36,39,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[39,39,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":6,"e":1.0,"k":"rising","v":0.70089,"x":1.0,"p":[[0,8,0.0,0.70089,0.39344,0.35714,1.0,1.0,0.0,1.0,3,19,1,3,0,5,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,19],[4,8,0.5,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[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":"c4734c41d33c3ed7","q":"Let  $ n$  be an odd positive integer. Prove that  $((n-1)^n+1)^2$  divides  $ n(n-1)^{(n-1)^n+1}+n$ .","t":[{"b":0,"e":1.0,"k":"flat","v":0.67857,"x":0.97321,"p":[[0,26,0.0,0.87054,0.17985,0.82143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,16],[4,26,0.1538,0.67857,0.31944,0.53571,0.78571,0.89286,0.0,1.0,2,8,0,2,0,3,0,0,2,0,0,1,0,0,2,0,0,6,0,0,8,0,8],[8,26,0.3077,0.74105,0.33396,0.5713,0.85714,1.0,0.0,1.0,1,15,0,1,0,4,0,0,2,0,0,0,0,0,3,0,0,1,0,0,6,0,15],[12,26,0.4615,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[16,26,0.6154,0.95982,0.08172,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[20,26,0.7692,0.92857,0.17496,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,26],[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],[26,26,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":2,"e":1.0,"k":"flat","v":0.75,"x":0.87053,"p":[[0,9,0.0,0.75,0.29014,0.53571,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,4,0,0,3,0,0,1,0,0,5,0,0,4,0,14],[4,9,0.4444,0.77677,0.21997,0.71429,0.85707,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,5,0,0,12,0,8],[8,9,0.8889,0.87053,0.10326,0.85711,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,15,0,10],[9,9,1.0,0.85266,0.15765,0.857,0.85714,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,17,0,9]]}]},{"i":"41a5a938e8889880","q":"Let  $r=0.d_0d_1d_2\\ldots$  be a real number. Let  $e_n$  denote the number formed by the digits  $d_n, d_{n-1}, \\ldots, d_0$  written from left to right (leading zeroes are permitted). Given that  $d_0=6$  and for each  $n \\geq 0$ ,  $e_n$  is equal to the number formed by the  $n+1$  rightmost digits of  $e_n^2$ . Show that  $r$  is irrational.","t":[{"b":0,"e":0.71429,"k":"falling","v":0.625,"x":0.82143,"p":[[0,19,0.0,0.82143,0.20516,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,11,0,0,2,0,15],[4,19,0.2105,0.80357,0.20748,0.71429,0.71429,1.0,0.0,1.0,1,13,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,1,0,13],[8,19,0.4211,0.73217,0.22512,0.53575,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,7,0,0,1,0,0,11,0,0,2,0,10],[12,19,0.6316,0.66963,0.18364,0.57143,0.71429,0.71429,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,2,0,0,10,0,0,13,0,0,0,0,5],[16,19,0.8421,0.66948,0.12593,0.67536,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,22,0,0,1,0,1],[19,19,1.0,0.625,0.12752,0.57143,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,14,0,0,0,0,1]]},{"b":5,"e":1.0,"k":"rising","v":0.79018,"x":0.95089,"p":[[0,17,0.0,0.79018,0.22299,0.71429,0.71429,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,13,0,0,0,0,14],[4,17,0.2353,0.79018,0.23686,0.71429,0.71429,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,1,0,0,2,0,0,12,0,0,1,0,14],[8,17,0.4706,0.91964,0.13803,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,4,0,22],[12,17,0.7059,0.93304,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,23],[16,17,0.9412,0.94196,0.11769,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,25],[17,17,1.0,0.95089,0.12169,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,27]]}]},{"i":"e485afb5bf823f80","q":"Let  $\\triangle ABC$  be an acute triangle, with  $O$  as its circumcenter. Point  $H$  is the foot of the perpendicular from  $A$  to line  $\\overleftrightarrow{BC}$ , and points  $P$  and  $Q$  are the feet of the perpendiculars from  $H$  to the lines  $\\overleftrightarrow{AB}$  and  $\\overleftrightarrow{AC}$ , respectively.\n\nGiven that   $$ AH^2=2\\cdot AO^2, $$   prove that the points  $O,P,$  and  $Q$  are collinear.","t":[{"b":0,"e":0.85714,"k":"flat","v":0.79909,"x":0.91068,"p":[[0,42,0.0,0.83035,0.27993,0.71429,1.0,1.0,0.0,1.0,2,19,1,2,0,0,0,0,1,0,0,1,0,0,1,0,0,4,0,0,4,0,19],[4,42,0.0952,0.79911,0.28089,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,3,0,0,3,0,18],[8,42,0.1905,0.91068,0.1704,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,23],[12,42,0.2857,0.88393,0.11538,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,16,0,12],[16,42,0.381,0.89732,0.11971,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,13,0,15],[20,42,0.4762,0.89285,0.09449,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,19,0,11],[24,42,0.5714,0.79909,0.16313,0.67857,0.85714,0.85714,0.4286,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,3,0,0,14,0,7],[28,42,0.6667,0.89283,0.14292,0.85714,1.0,1.0,0.571,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,10,0,17],[32,42,0.7619,0.87498,0.17407,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,10,0,16],[36,42,0.8571,0.81694,0.17217,0.71429,0.85714,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,15,0,8],[40,42,0.9524,0.87053,0.15303,0.85714,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,13,0,13],[42,42,1.0,0.85714,0.18211,0.85714,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,12,0,13]]},{"b":1,"e":1.0,"k":"rising","v":0.76339,"x":1.0,"p":[[0,27,0.0,0.76339,0.30642,0.71429,0.85714,1.0,0.0,1.0,3,12,2,3,0,0,0,0,2,0,0,0,0,0,1,0,0,5,0,0,9,0,12],[4,27,0.1481,0.875,0.23623,0.82132,1.0,1.0,0.0,1.0,1,22,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,22],[8,27,0.2963,0.76339,0.29582,0.4286,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,5,0,0,1,0,0,2,0,0,4,0,16],[12,27,0.4444,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[16,27,0.5926,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,27,0.7407,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,27,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[27,27,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"a137b19e8496ebdb","q":"Let  $ABC$  be a triangle, and points  $D,E$  are on  $BA,CA$  respectively such that  $DB=BC=CE$ . Let  $O,I$  be the circumcenter, incenter of  $\\triangle ABC$ . Prove that the circumradius of  $\\triangle ADE$  is equal to  $OI$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"02fe18d7bbd8b070","q":"Let  $M$  be the midpoint of side  $AC$  of triangle  $ABC$ ,  $MD$  and  $ME$  be the perpendiculars from  $M$  to  $AB$  and  $BC$  respectively. Prove that the distance between the circumcenters of triangles  $ABE$  and  $BCD$  is equal to  $AC/4$ *(Proposed by M.Volchkevich)*","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.0267,"p":[[0,15,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0267,0.06606,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.02679,"p":[[0,17,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b701129dffdfccca","q":"Let  $D$  be a point on the side  $AC$  of triangle  $ABC$ . Let  $E$  and  $F$  be points on the segments  $BD$  and  $BC$  respectively, such that  $\\angle BAE = \\angle CAF$ . Let  $P$  and  $Q$  be points on the segments  $BC$  and  $BD$  respectively, such that  $EP \\parallel CD$  and  $FQ \\parallel CD$ . Prove that  $\\angle BAP = \\angle  CAQ$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,41,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0133,"p":[[0,68,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,68,0.0588,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,68,0.1176,0.0133,0.05465,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,68,0.1765,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,68,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,68,0.2941,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,68,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,68,0.4118,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,68,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,68,0.5294,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,68,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,68,0.6471,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,68,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,68,0.7647,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,68,0.8824,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"464af058ad92d368","q":"Let  $ABCD$  be a trapezoid with bases  $AB$  and  $CD$ , inscribed in a circle of center  $O$ . Let  $P$  be the intersection of the lines  $BC$  and  $AD$ . A circle through  $O$  and  $P$  intersects the segments  $BC$  and  $AD$  at interior points  $F$  and  $G$ , respectively. Show that  $BF=DG$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,10,0.0,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,10,0.4,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,19,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"49761d8921cb4115","q":"Let  $ABC$  be an acute-angled triangle such that  $|AB|<|AC|$ . Let  $X$  and  $Y$  be points on the minor arc  ${BC}$  of the circumcircle of  $ABC$  such that  $|BX|=|XY|=|YC|$ . Suppose that there exists a point  $N$  on the segment  $\\overline{AY}$  such that  $|AB|=|AN|=|NC|$ . Prove that the line  $NC$  passes through the midpoint of the segment  $\\overline{AX}$ .   (Ivan Novak)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.13393,"x":0.22768,"p":[[0,29,0.0,0.16518,0.11904,0.10714,0.14286,0.2857,0.0,0.4286,8,0,2,8,0,12,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.13393,0.10677,0.0,0.14286,0.17857,0.0,0.28571,10,0,1,10,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.21428,0.07986,0.14286,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,14,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.21875,0.07973,0.14286,0.21428,0.28571,0.14286,0.42857,0,0,0,0,0,16,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.20536,0.07087,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,18,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.21429,0.07987,0.14286,0.2857,0.28571,0.0,0.286,1,0,0,1,0,14,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.19634,0.06923,0.14286,0.14286,0.28571,0.14,0.28571,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.22768,0.07016,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,13,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.19643,0.07784,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,18,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.15178,"x":0.19643,"p":[[0,25,0.0,0.15616,0.10927,0.105,0.14286,0.28571,0.0,0.28571,8,0,0,8,0,13,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.18303,0.08171,0.14286,0.14286,0.2857,0.0,0.28571,2,0,0,2,0,19,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.19643,0.08564,0.14286,0.14286,0.28571,0.0,0.28571,2,0,0,2,0,16,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.19196,0.10479,0.14286,0.21429,0.28571,0.0,0.28571,5,0,0,5,0,11,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.15178,0.11258,0.10714,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,15,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.16964,0.10374,0.14286,0.14286,0.28571,0.0,0.28571,6,0,0,6,0,14,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.18741,0.09744,0.14286,0.14286,0.28571,0.0,0.28571,4,0,0,4,0,14,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.19643,0.09942,0.14286,0.21428,0.28571,0.0,0.28571,4,0,0,4,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"44bc0d25d707f1c3","q":"Let  $\\Delta ABC$  be a triangle with orthocenter  $H$  and  $\\Gamma$  be the circumcircle of  $\\Delta ABC$  with center  $O$ . Consider  $N$  the center of the circle that passes through the feet of the heights of  $\\Delta ABC$  and  $P$  the intersection of the line  $AN$  with the circle  $\\Gamma$ . Suppose that the line  $AP$  is perpendicular to the line  $OH$ . Prove that  $P$  belongs to the reflection of the line  $OH$  by the line  $BC$ .","t":[{"b":1,"e":0.71429,"k":"flat","v":0.64729,"x":0.79464,"p":[[0,23,0.0,0.64729,0.24481,0.5713,0.71429,0.71429,0.14286,1.0,0,4,0,0,0,4,0,0,1,0,0,1,0,0,5,0,0,14,0,0,3,0,4],[4,23,0.1739,0.77217,0.2419,0.71429,0.85707,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,1,0,0,2,0,0,9,0,0,6,0,11],[8,23,0.3478,0.79464,0.22851,0.71429,0.85707,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,9,0,0,3,0,14],[12,23,0.5217,0.65625,0.33093,0.42857,0.71429,1.0,0.0,1.0,2,11,1,2,0,4,0,0,0,0,0,4,0,0,2,0,0,8,0,0,1,0,11],[16,23,0.6957,0.68737,0.14911,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,29,0,0,0,0,1],[20,23,0.8696,0.67396,0.16063,0.71429,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,25,0,0,1,0,1],[23,23,1.0,0.6875,0.16535,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,23,0,0,2,0,2]]},{"b":7,"e":0.4286,"k":"rising","v":0.71875,"x":1.0,"p":[[0,15,0.0,0.71875,0.17671,0.71429,0.71429,0.71429,0.1429,1.0,0,5,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,22,0,0,1,0,5],[4,15,0.2667,0.8125,0.19377,0.71429,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,15,0,0,2,0,13],[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],[12,15,0.8,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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":"aa9a433084863f76","q":"In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer  $n$ , is there a way to use  $n$  colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02009,"p":[[0,28,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.02009,0.04803,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.01339,"x":0.04018,"p":[[0,16,0.0,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.02004,0.04798,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.03116,0.08541,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"12153803ac6333ad","q":"Let  $ABC$  be an acute triangle. Points  $X$  and  $Y$  lie on the segments  $AB$  and  $AC$ , respectively, such that  $AX=AY$  and the segment  $XY$  passes through the orthocenter of the triangle  $ABC$ . Lines tangent to the circumcircle of the triangle  $AXY$  at points  $X$  and  $Y$  intersect at point  $P$ . Prove that points  $A, B, C, P$  are concyclic.","t":[{"b":0,"e":0.0,"k":"volatile","v":0.11375,"x":0.61383,"p":[[0,11,0.0,0.61383,0.32284,0.39286,0.64286,0.85714,0.0,1.0,2,7,1,2,0,3,0,1,2,0,0,3,0,0,5,0,0,3,0,0,6,0,7],[4,11,0.3636,0.18728,0.18025,0.14286,0.14286,0.14287,0.0,1.0,3,1,1,3,0,24,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[8,11,0.7273,0.17624,0.17219,0.14214,0.14286,0.14286,0.0,0.857,6,0,0,6,1,18,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[11,11,1.0,0.11375,0.06146,0.14214,0.14286,0.14286,0.0,0.21429,7,0,0,7,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.30356,"x":0.46411,"p":[[0,18,0.0,0.46411,0.31327,0.14286,0.571,0.71429,0.0,1.0,4,3,0,4,0,6,0,0,3,0,0,2,0,0,7,0,0,5,0,0,2,0,3],[4,18,0.2222,0.39506,0.3169,0.14286,0.28571,0.71429,0.0,1.0,3,3,1,3,0,11,0,0,4,0,1,2,0,0,2,0,0,4,0,0,2,0,3],[8,18,0.4444,0.30356,0.23621,0.14286,0.2857,0.46418,0.0,0.85714,4,0,0,4,0,10,0,0,9,0,0,1,0,0,5,0,0,1,0,0,2,0,0],[12,18,0.6667,0.37499,0.20747,0.2857,0.42857,0.42857,0.0,0.857,2,0,0,2,0,5,0,0,8,0,0,11,0,0,1,0,0,4,0,0,1,0,0],[16,18,0.8889,0.40625,0.17896,0.28571,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,5,0,0,3,0,0,16,0,0,3,0,0,4,0,0,0,0,0],[18,18,1.0,0.32812,0.18802,0.14286,0.35714,0.42857,0.0,0.71429,3,0,0,3,0,6,0,1,6,0,0,13,0,0,0,0,0,3,0,0,0,0,0]]}]},{"i":"d7c99c319242217c","q":"Let  $ABCD$  a convex quadrilateral. Suppose that the circumference with center  $B$  and radius  $BC$  is tangent to  $AD$  in  $F$  and the circumference with center  $A$  and radius  $AD$  is tangent to  $BC$  in  $E$ . Prove that  $DE$  and  $CF$  are perpendicular.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.12491,"x":0.16947,"p":[[0,19,0.0,0.16945,0.12078,0.14286,0.14286,0.14286,0.0,0.571,4,0,0,4,0,22,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,19,0.2105,0.14277,0.11294,0.14286,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,23,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,19,0.4211,0.16947,0.12082,0.14286,0.14286,0.14286,0.0,0.57143,4,0,1,4,0,22,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,19,0.6316,0.16509,0.0883,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,25,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.12491,0.06914,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.12947,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.12464,"x":0.15625,"p":[[0,38,0.0,0.15625,0.08268,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,24,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.13821,0.07562,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.14732,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.13839,0.06666,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.15179,0.07087,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.14732,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,22,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.12464,0.05913,0.14,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.14714,0.04354,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5783c94634811027","q":"Let  $ABCD$  be a cyclic quadrilateral such that the circles with diameters  $AB$  and  $CD$  touch at  $S$ . If  $M, N$  are the midpoints of  $AB, CD$ , prove that the perpendicular through  $M$  to  $MN$  meets  $CS$  on the circumcircle of  $ABCD$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.07125,"x":0.10705,"p":[[0,16,0.0,0.07125,0.07125,0.0,0.07,0.14286,0.0,0.14286,16,0,3,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.10705,0.08745,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.10259,0.07344,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.04464,"x":0.12491,"p":[[0,15,0.0,0.06688,0.07965,0.0,0.0,0.14286,0.0,0.28571,18,0,2,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.08911,0.06903,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.12491,0.04722,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ab22bf5e75391a4c","q":"Find all polynomials  $P$  with integer coefficients, for which there exists a number  $N$ , such that for every natural number  $n \\geq N$ , all prime divisors of  $n+2^{\\lfloor \\sqrt{n} \\rfloor}$  are also divisors of  $P(n)$ .","t":[{"b":0,"e":0.71429,"k":"rising","v":0.12945,"x":0.49103,"p":[[0,76,0.0,0.12945,0.2093,0.0,0.0,0.14286,0.0,0.857,19,0,0,19,0,6,0,0,3,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[4,76,0.0526,0.17411,0.198,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,4,0,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[8,76,0.1053,0.23658,0.28256,0.0,0.0,0.4642,0.0,0.85714,17,0,0,17,0,0,0,0,5,0,0,2,0,0,4,0,0,3,0,0,1,0,0],[12,76,0.1579,0.22768,0.28984,0.0,0.0,0.46431,0.0,0.85714,17,0,0,17,0,2,0,0,4,0,0,1,0,0,4,0,0,2,0,0,2,0,0],[16,76,0.2105,0.30356,0.21942,0.10717,0.28571,0.57111,0.0,0.71429,8,0,0,8,0,1,0,0,12,0,0,2,0,0,8,0,0,1,0,0,0,0,0],[20,76,0.2632,0.31697,0.35667,0.0,0.21431,0.60714,0.0,1.0,14,3,0,14,0,2,0,0,5,0,0,1,0,0,2,0,0,3,0,0,2,0,3],[24,76,0.3158,0.33034,0.28444,0.0,0.28571,0.57143,0.0,1.0,9,1,0,9,0,2,0,0,10,0,0,0,0,0,6,0,0,3,0,0,1,0,1],[28,76,0.3684,0.3125,0.24597,0.0,0.28571,0.46429,0.0,0.85714,9,0,0,9,0,0,0,0,11,0,0,4,0,0,6,0,0,0,0,0,2,0,0],[32,76,0.4211,0.49103,0.31121,0.25,0.571,0.71429,0.0,1.0,4,3,0,4,0,4,0,0,3,0,0,4,0,0,7,0,0,3,0,0,4,0,3],[36,76,0.4737,0.35714,0.30723,0.0,0.28571,0.57143,0.0,1.0,9,1,0,9,0,2,0,0,8,0,0,2,0,0,4,0,0,3,0,0,3,0,1],[40,76,0.5263,0.2589,0.28218,0.0,0.2857,0.42857,0.0,1.0,13,1,0,13,0,2,0,0,8,0,0,3,0,0,3,0,0,0,0,0,2,0,1],[44,76,0.5789,0.40177,0.27764,0.2857,0.35714,0.57143,0.0,1.0,7,1,0,7,0,0,0,0,9,0,0,1,0,0,9,0,0,4,0,0,1,0,1],[48,76,0.6316,0.34821,0.35163,0.0,0.28571,0.60714,0.0,1.0,14,1,0,14,0,0,0,0,4,0,0,1,0,0,5,0,0,2,0,0,5,0,1],[52,76,0.6842,0.33481,0.27573,0.14286,0.28571,0.57143,0.0,1.0,7,1,0,7,0,4,0,0,10,0,0,1,0,0,6,0,0,1,0,0,2,0,1],[56,76,0.7368,0.40624,0.27224,0.24999,0.42857,0.57143,0.0,0.85714,7,0,0,7,0,1,0,0,5,0,0,4,0,0,10,0,0,2,0,0,3,0,0],[60,76,0.7895,0.38839,0.36811,0.0,0.28571,0.71429,0.0,1.0,11,4,0,11,0,2,0,0,5,0,0,1,0,0,4,0,0,2,0,0,3,0,4],[64,76,0.8421,0.24551,0.29928,0.0,0.0,0.46418,0.0,0.85714,17,0,0,17,0,0,0,0,5,0,0,2,0,0,4,0,0,1,0,0,3,0,0],[68,76,0.8947,0.41514,0.3141,0.10714,0.42859,0.60714,0.0,1.0,8,1,0,8,0,1,0,0,5,0,0,5,0,0,5,0,0,2,0,0,5,0,1],[72,76,0.9474,0.31247,0.28667,0.0,0.2857,0.57143,0.0,1.0,11,1,0,11,0,2,0,0,6,0,0,2,0,0,8,0,0,1,0,0,1,0,1],[76,76,1.0,0.31694,0.27134,0.0,0.28571,0.57111,0.0,1.0,10,1,0,10,0,1,0,0,8,0,0,3,0,0,8,0,0,0,0,0,1,0,1]]},{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.125,"p":[[0,25,0.0,0.125,0.16656,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,10,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,25,0.16,0.08927,0.17764,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,2,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[8,25,0.32,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[16,25,0.64,0.04463,0.12073,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,25,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"40b0f00e9a5ca581","q":"A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, $a b a b a b$ and $a b c a b c$ are repetitive, but $a b a b a$ and $a a b b$ are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)\n\n## Origin. Romania (Dan Schwarz).","t":[{"b":1,"e":0.28571,"k":"flat","v":0.03116,"x":0.06696,"p":[[0,18,0.0,0.03116,0.06887,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.04456,0.07512,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.06696,0.12364,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.4286,"k":"flat","v":0.01786,"x":0.08927,"p":[[0,12,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.04902,0.09173,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.08927,0.14612,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,8,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,12,1.0,0.06027,0.07806,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,1,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c0d989ffe6722a07","q":"Let  $H$  and  $O$  be the orthocenter and circumcenter of an acute-angled triangle  $ABC$ , respectively. The perpendicular bisector of  $BH$  meets  $AB$  and  $BC$  at points  $A_1$  and  $C_1$ , respectively. Prove that  $OB$  bisects the angle  $A_1OC_1$ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.57589,"x":0.79463,"p":[[0,33,0.0,0.57589,0.22156,0.42857,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,3,0,0,3,0,0,2,0,0,18,0,0,2,0,0],[4,33,0.1212,0.63839,0.15966,0.71429,0.71429,0.71429,0.0,0.71429,1,0,1,1,0,0,0,0,0,0,0,6,0,0,0,0,0,25,0,0,0,0,0],[8,33,0.2424,0.63838,0.15562,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,25,0,0,0,0,0],[12,33,0.3636,0.66072,0.21943,0.4286,0.71429,0.85714,0.0,1.0,1,1,1,1,0,0,0,0,1,0,0,7,0,0,3,0,0,8,0,0,11,0,1],[16,33,0.4848,0.77679,0.08702,0.71429,0.78571,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,16,0,0],[20,33,0.6061,0.79463,0.07086,0.71429,0.85707,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,18,0,0],[24,33,0.7273,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[28,33,0.8485,0.70982,0.04351,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,1,0,0],[32,33,0.9697,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[33,33,1.0,0.69196,0.12428,0.71429,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.35714,"x":0.70981,"p":[[0,28,0.0,0.58474,0.2326,0.42857,0.71429,0.71429,0.14,1.0,0,1,0,0,0,3,0,0,4,0,0,5,0,0,0,0,0,16,0,0,3,0,1],[4,28,0.1429,0.61161,0.19638,0.42857,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,4,0,0,0,0,0,21,0,0,2,0,0],[8,28,0.2857,0.60267,0.19474,0.42857,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,3,0,0,4,0,0,2,0,0,19,0,0,2,0,0],[12,28,0.4286,0.5625,0.24206,0.42857,0.71429,0.71429,0.0,0.71429,2,0,0,2,0,3,0,0,2,0,0,3,0,0,0,0,0,22,0,0,0,0,0],[16,28,0.5714,0.35714,0.20203,0.14286,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,10,0,0,8,0,0,8,0,0,0,0,0,6,0,0,0,0,0],[20,28,0.7143,0.67396,0.10245,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,27,0,0,0,0,0],[24,28,0.8571,0.70981,0.04356,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,1,0,0],[28,28,1.0,0.68301,0.07775,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,27,0,0,0,0,0]]}]},{"i":"6d5dfa818d719baa","q":"9. (MON) Let $a_{1}, a_{2}, \\ldots, a_{n}$ be positive real numbers such that $a_{1}+a_{2}+$ $\\cdots+a_{n}<1$. Prove that  $$ \\frac{a_{1} a_{2} \\cdots a_{n}\\left[1-\\left(a_{1}+a_{2}+\\cdots+a_{n}\\right)\\right]}{\\left(a_{1}+a_{2}+\\cdots+a_{n}\\right)\\left(1-a_{1}\\right)\\left(1-a_{2}\\right) \\cdots\\left(1-a_{n}\\right)} \\leq \\frac{1}{n^{n+1}} . $$","t":[{"b":4,"e":1.0,"k":"volatile","v":0.0,"x":1.0,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.41964,0.2788,0.28571,0.28571,0.28571,0.2857,1.0,0,6,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,18,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"7bcb0df3e97f46d3","q":"Let $ABC$ be a triangle, $D, E, F$ the points of tangency of the incircle on the sides $(BC), (AC)$, and $(AB)$, $M$ the midpoint of $[BC]$, $I$ the incenter of $ABC$. $G$ and $H$ are defined as the reflections of $E$ and $F$ with respect to $I$. We denote $Q$ as the intersection between the lines $(BC)$ and $(GH)$. Show that the lines $(IQ)$ and $(IM)$ are perpendicular.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.44643,"x":0.62499,"p":[[0,31,0.0,0.45982,0.40046,0.0,0.57143,0.85714,0.0,1.0,13,1,0,13,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,12,0,1],[4,31,0.129,0.61604,0.32031,0.571,0.71429,0.85714,0.0,1.0,6,1,1,6,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,13,0,1],[8,31,0.2581,0.48661,0.40067,0.0,0.57143,0.85714,0.0,1.0,10,4,0,10,0,3,0,0,0,0,0,2,0,0,2,0,0,2,0,0,9,0,4],[12,31,0.3871,0.50891,0.41792,0.0,0.85707,0.85714,0.0,1.0,12,3,0,12,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,14,0,3],[16,31,0.5161,0.62499,0.35847,0.46429,0.85707,0.85714,0.0,1.0,7,1,0,7,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,0,18,0,1],[20,31,0.6452,0.44643,0.41149,0.0,0.57144,0.85714,0.0,1.0,13,1,0,13,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,13,0,1],[24,31,0.7742,0.56249,0.39759,0.0,0.857,0.85714,0.0,1.0,10,3,0,10,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,14,0,3],[28,31,0.9032,0.55803,0.39181,0.0,0.85712,0.85714,0.0,1.0,10,2,0,10,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,15,0,2],[31,31,1.0,0.58929,0.37754,0.0,0.85714,0.85714,0.0,1.0,9,1,0,9,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,17,0,1]]},{"b":5,"e":1.0,"k":"rising","v":0.57143,"x":0.85713,"p":[[0,22,0.0,0.57143,0.38631,0.10714,0.85714,0.85714,0.0,1.0,8,3,0,8,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,14,0,3],[4,22,0.1818,0.66518,0.34369,0.67857,0.85714,0.85714,0.0,1.0,6,3,0,6,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,17,0,3],[8,22,0.3636,0.79013,0.20824,0.71429,0.85714,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0,16,0,6],[12,22,0.5455,0.82588,0.23072,0.857,0.85714,1.0,0.0,1.0,2,9,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,9],[16,22,0.7273,0.85266,0.16937,0.857,0.85714,1.0,0.1429,1.0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,14,0,11],[20,22,0.9091,0.85713,0.15972,0.85714,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,18,0,10],[22,22,1.0,0.85268,0.09094,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,19,0,6]]}]},{"i":"340fb0784213079e","q":"5. A5 (THA) Let $a, b, c>0$ and $a b+b c+c a=1$. Prove the inequality  $$ \\sqrt[3]{\\frac{1}{a}+6 b}+\\sqrt[3]{\\frac{1}{b}+6 c}+\\sqrt[3]{\\frac{1}{c}+6 a} \\leq \\frac{1}{a b c} $$","t":[{"b":1,"e":1.0,"k":"rising","v":0.45972,"x":0.8482,"p":[[0,33,0.0,0.46426,0.31338,0.14286,0.42857,0.71429,0.0,1.0,3,4,0,3,0,7,0,0,2,0,0,7,0,0,2,0,0,6,0,0,1,0,4],[4,33,0.1212,0.52677,0.34522,0.14286,0.64286,0.71429,0.0,1.0,3,6,0,3,0,8,0,0,0,0,0,3,0,0,2,0,0,9,0,0,1,0,6],[8,33,0.2424,0.45972,0.31899,0.14286,0.42857,0.71429,0.0,1.0,2,4,0,2,0,10,0,0,1,0,0,6,0,0,1,0,0,7,0,0,1,0,4],[12,33,0.3636,0.54464,0.35434,0.14286,0.71429,0.75,0.0,1.0,4,7,0,4,0,6,0,0,0,0,0,4,0,0,1,0,0,9,0,0,1,0,7],[16,33,0.4848,0.79464,0.16728,0.71429,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,0,2,0,11],[20,33,0.6061,0.83929,0.15872,0.71429,0.78571,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,13,0,0,1,0,15],[24,33,0.7273,0.82589,0.13709,0.71429,0.71429,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,1,0,12],[28,33,0.8485,0.8482,0.14701,0.71429,0.78564,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,1,0,15],[32,33,0.9697,0.79464,0.16728,0.71429,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,19,0,0,0,0,11],[33,33,1.0,0.83036,0.14032,0.71429,0.71429,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,13]]},{"b":7,"e":0.571,"k":"rising","v":0.54017,"x":0.88393,"p":[[0,47,0.0,0.58482,0.34692,0.24999,0.57144,1.0,0.0,1.0,1,10,0,1,0,7,0,0,1,0,0,7,0,0,0,0,0,5,0,0,1,0,10],[4,47,0.0851,0.62054,0.32263,0.42857,0.71429,1.0,0.0,1.0,2,10,0,2,0,3,0,0,1,0,0,7,0,0,2,0,0,7,0,0,0,0,10],[8,47,0.1702,0.67856,0.29015,0.67846,0.71429,1.0,0.0,1.0,1,9,0,1,0,4,0,0,0,0,0,2,0,0,1,0,0,15,0,0,0,0,9],[12,47,0.2553,0.54017,0.27602,0.42857,0.4286,0.71429,0.0,1.0,2,5,0,2,0,2,0,0,1,0,0,13,0,0,3,0,0,5,0,0,1,0,5],[16,47,0.3404,0.85268,0.14934,0.71429,0.85714,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,0,0,16],[20,47,0.4255,0.83928,0.16269,0.71429,0.78571,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,1,0,15],[24,47,0.5106,0.87052,0.14883,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,0,0,18],[28,47,0.5957,0.88393,0.14032,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,19],[32,47,0.6809,0.8482,0.15949,0.71429,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,1,0,16],[36,47,0.766,0.81249,0.16538,0.71429,0.71429,1.0,0.4286,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,16,0,0,0,0,13],[40,47,0.8511,0.80356,0.16271,0.71429,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,17,0,0,0,0,12],[44,47,0.9362,0.8482,0.15128,0.71429,0.85714,1.0,0.571,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,2,0,15],[47,47,1.0,0.88393,0.14032,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,19]]}]},{"i":"face0f0165c4fabc","q":"Let  $M$  be the midpoint of the side  $AD$  of the square  $ABCD.$  Consider the equilateral triangles  $DFM{}$  and  $BFE{}$  such that  $F$  lies in the interior of  $ABCD$  and the lines  $EF$  and   $BC$  are concurrent. Denote by  $P{}$  the midpoint of  $ME.$  Prove that\"\n[list=a]\n[*]The point  $P$  lies on the line  $AC.$ [*]The halfline  $PM$  is the bisector of the angle  $APF.$ [/list]\n*Adrian Bud*","t":[{"b":1,"e":0.0,"k":"flat","v":0.13839,"x":0.19643,"p":[[0,6,0.0,0.15179,0.29437,0.0,0.0,0.14287,0.0,1.0,21,3,0,21,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[4,6,0.6667,0.13839,0.25626,0.0,0.0,0.2857,0.0,1.0,21,2,0,21,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[6,6,1.0,0.19643,0.29397,0.0,0.0,0.28571,0.0,1.0,19,2,0,19,0,1,0,0,5,0,0,3,0,0,0,0,0,2,0,0,0,0,2]]},{"b":3,"e":0.0,"k":"flat","v":0.12945,"x":0.1741,"p":[[0,11,0.0,0.1741,0.28286,0.0,0.0,0.28571,0.0,1.0,21,2,0,21,0,0,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,2],[4,11,0.3636,0.12945,0.24051,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[8,11,0.7273,0.13839,0.21274,0.0,0.0,0.2857,0.0,0.85714,21,0,0,21,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[11,11,1.0,0.15626,0.29312,0.0,0.0,0.2857,0.0,1.0,21,3,0,21,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,3]]}]},{"i":"1be678234e402a50","q":"A real number sequence  $a_1, \\cdots ,a_{2021}$  satisfies the below conditions.  $$ a_1=1, a_2=2, a_{n+2}=\\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\\leq n \\leq 2019) $$  Let the minimum of  $a_1, \\cdots ,a_{2021}$  be  $m$ , and the maximum of  $a_1, \\cdots ,a_{2021}$  be  $M$ . \nLet a 2021 degree polynomial    $$ P(x):=(x-a_1)(x-a_2) \\cdots (x-a_{2021}) $$    $|P(x)|$  is maximum in  $[m, M]$  when  $x=\\alpha$ . Show that  $1<\\alpha <2$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0892,"x":0.14286,"p":[[0,12,0.0,0.09821,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.14286,0.10101,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,19,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0892,0.06909,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.125,0.05923,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.09366,"x":0.20979,"p":[[0,20,0.0,0.09366,0.08454,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.17857,0.11845,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.20979,0.13818,0.14286,0.14286,0.2857,0.0,0.571,2,0,0,2,0,20,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,20,0.6,0.18303,0.10853,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,14,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.12946,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2222980017faa47c","q":"Find all the numbers of  $5$  non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.72767,"x":0.92857,"p":[[0,62,0.0,0.77237,0.24179,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,4,0,0,2,0,0,6,0,0,6,0,12],[4,62,0.0645,0.72767,0.28428,0.53572,0.78564,1.0,0.0,1.0,1,12,0,1,0,0,0,0,4,0,0,3,0,0,2,0,0,6,0,0,4,0,12],[8,62,0.129,0.76338,0.24384,0.5354,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,2,0,0,6,0,0,2,0,0,2,0,0,9,0,11],[12,62,0.1935,0.74106,0.2867,0.5354,0.85714,1.0,0.0,1.0,1,13,0,1,0,1,0,0,1,0,0,5,0,0,3,0,0,3,0,0,5,0,13],[16,62,0.2581,0.88392,0.1729,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,10,0,17],[20,62,0.3226,0.87054,0.14445,0.71429,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,8,0,15],[24,62,0.3871,0.79464,0.2257,0.67857,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,4,0,0,8,0,12],[28,62,0.4516,0.92857,0.09449,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,10,0,19],[32,62,0.5161,0.84375,0.15303,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,11,0,11],[36,62,0.5806,0.81247,0.12597,0.82132,0.85714,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,21,0,3],[40,62,0.6452,0.85267,0.09771,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,23,0,5],[44,62,0.7097,0.87945,0.11356,0.85714,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,19,0,10],[48,62,0.7742,0.85714,0.09449,0.85714,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,21,0,6],[52,62,0.8387,0.82152,0.11848,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,23,0,3],[56,62,0.9032,0.84808,0.07959,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,25,0,3],[60,62,0.9677,0.83928,0.1171,0.85714,0.85714,0.85714,0.57143,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,22,0,5],[62,62,1.0,0.84374,0.09006,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,25,0,3]]},{"b":7,"e":0.571,"k":"flat","v":0.67862,"x":0.81696,"p":[[0,14,0.0,0.67862,0.32138,0.42857,0.85714,1.0,0.0,1.0,2,10,0,2,0,0,0,0,5,0,0,5,0,0,1,0,0,1,0,0,8,0,10],[4,14,0.2857,0.81696,0.18977,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,8,0,0,9,0,11],[8,14,0.5714,0.75446,0.27254,0.57143,0.85714,1.0,0.0,1.0,2,12,0,2,0,0,0,0,0,0,0,3,0,0,4,0,0,6,0,0,5,0,12],[12,14,0.8571,0.79464,0.27185,0.71429,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,3,0,0,1,0,0,2,0,0,5,0,0,4,0,16],[14,14,1.0,0.79464,0.23128,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,8,0,0,4,0,14]]}]},{"i":"a7eadbdc99060bad","q":"10. (IND 5) A natural number $n$ is said to have the property $P$ if whenever $n$ divides $a^{n}-1$ for some integer $a, n^{2}$ also necessarily divides $a^{n}-1$. (a) Show that every prime number has property $P$. (b) Show that there are infinitely many composite numbers $n$ that possess property $P$.","t":[{"b":5,"e":0.71429,"k":"flat","v":0.6875,"x":0.75446,"p":[[0,22,0.0,0.75446,0.22654,0.67857,0.71429,1.0,0.0,1.0,1,11,1,1,0,0,0,0,0,0,0,2,0,0,5,0,0,12,0,0,1,0,11],[4,22,0.1818,0.71872,0.09099,0.71429,0.71429,0.71429,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,0,1,0,2],[8,22,0.3636,0.73212,0.12243,0.71429,0.71429,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,20,0,0,2,0,4],[12,22,0.5455,0.6875,0.0974,0.57143,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,19,0,0,2,0,1],[16,22,0.7273,0.70535,0.08701,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,23,0,0,2,0,1],[20,22,0.9091,0.69615,0.1056,0.57143,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,17,0,0,4,0,1],[22,22,1.0,0.69639,0.0857,0.71429,0.71429,0.71429,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,23,0,0,1,0,1]]},{"b":7,"e":1.0,"k":"falling","v":0.59812,"x":0.86607,"p":[[0,26,0.0,0.86607,0.18189,0.71429,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,0,0,20],[4,26,0.1538,0.6339,0.15543,0.57143,0.57143,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,2,0,0,1,0,4],[8,26,0.3077,0.62944,0.13768,0.57143,0.57143,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,24,0,0,3,0,0,1,0,3],[12,26,0.4615,0.6249,0.15875,0.571,0.57143,0.60714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,20,0,0,4,0,0,0,0,4],[16,26,0.6154,0.60262,0.0927,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,5,0,0,0,0,1],[20,26,0.7692,0.61155,0.11972,0.57143,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,5,0,0,0,0,2],[24,26,0.9231,0.59813,0.07526,0.57142,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,5,0,0,1,0,0],[26,26,1.0,0.59812,0.13093,0.571,0.57143,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,23,0,0,2,0,0,1,0,2]]}]},{"i":"4c2e11c7552518dd","q":"$a_1,a_2,\\ldots$  is a sequence of <u>nonzero integer</u> numbers that for all  $n\\in\\mathbb{N}$ , if  $a_n=2^\\alpha k$  such that  $k$  is an odd integer and  $\\alpha$  is a nonnegative integer then:  $a_{n+1}=2^\\alpha-k$ . Prove that if this sequence is periodic, then for all  $n\\in\\mathbb{N}$  we have:  $a_{n+2}=a_n$ . (The sequence  $a_1,a_2,\\ldots$  is periodic iff there exists natural number  $d$  that for all  $n\\in\\mathbb{N}$  we have:  $a_{n+d}=a_n$ )","t":[{"b":5,"e":1.0,"k":"rising","v":0.77232,"x":0.96428,"p":[[0,19,0.0,0.77232,0.25719,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,1,0,0,1,0,0,8,0,0,6,0,12],[4,19,0.2105,0.875,0.16269,0.82132,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,4,0,0,7,0,17],[8,19,0.4211,0.94196,0.12807,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,24],[12,19,0.6316,0.93304,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,22],[16,19,0.8421,0.91071,0.17405,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,3,0,23],[19,19,1.0,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25]]},{"b":7,"e":1.0,"k":"rising","v":0.76786,"x":1.0,"p":[[0,46,0.0,0.79908,0.2283,0.71429,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,5,0,0,9,0,12],[4,46,0.087,0.76786,0.20439,0.71429,0.78571,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,9,0,0,7,0,9],[8,46,0.1739,0.94643,0.13243,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,26],[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],[20,46,0.4348,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,46,0.5217,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,46,0.6087,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[44,46,0.9565,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"8bd86876cc08c578","q":"In  $\\triangle ABC$  with incenter  $I$ ,  $AB = 61$ ,  $AC = 51$ , and  $BC=71$ . The circumcircles of triangles  $AIB$  and  $AIC$  meet line  $BC$  at points  $D$  ( $D \\neq B$ ) and  $E$  ( $E \\neq C$ ), respectively. Determine the length of segment  $DE$ .\n\n*James Tao*","t":[{"b":1,"e":0.71429,"k":"falling","v":0.26339,"x":0.47322,"p":[[0,13,0.0,0.47322,0.25862,0.14286,0.42859,0.71429,0.0,1.0,1,1,1,1,0,8,0,0,1,0,0,7,0,0,2,0,0,12,0,0,0,0,1],[4,13,0.3077,0.43303,0.25873,0.14286,0.42857,0.71429,0.0,0.85714,2,0,1,2,0,7,0,0,3,0,0,9,0,0,1,0,0,7,0,0,3,0,0],[8,13,0.6154,0.40179,0.26107,0.14286,0.35714,0.71429,0.14286,0.85714,0,0,0,0,0,14,0,0,2,0,0,4,0,0,1,0,0,10,0,0,1,0,0],[12,13,0.9231,0.26339,0.19269,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,22,0,0,0,0,0,6,0,0,1,0,0,3,0,0,0,0,0],[13,13,1.0,0.27679,0.21998,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,20,0,0,3,0,0,6,0,0,0,0,0,1,0,0,1,0,1]]},{"b":6,"e":0.71429,"k":"flat","v":0.33036,"x":0.55357,"p":[[0,17,0.0,0.42857,0.26486,0.14286,0.42857,0.71429,0.0,0.71429,3,0,2,3,0,8,0,0,1,0,0,6,0,0,2,0,0,12,0,0,0,0,0],[4,17,0.2353,0.36161,0.26241,0.14286,0.21429,0.71429,0.14286,0.85714,0,0,0,0,0,16,0,0,4,0,0,2,0,0,1,0,0,7,0,0,2,0,0],[8,17,0.4706,0.33036,0.23808,0.14286,0.21431,0.42857,0.14286,1.0,0,1,0,0,0,16,0,0,4,0,0,6,0,0,0,0,0,5,0,0,0,0,1],[12,17,0.7059,0.42857,0.25254,0.14286,0.42857,0.71429,0.14286,0.71429,0,0,0,0,0,11,0,0,4,0,0,4,0,0,0,0,0,13,0,0,0,0,0],[16,17,0.9412,0.55357,0.24419,0.39286,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,7,0,0,1,0,0,3,0,0,0,0,0,20,0,0,1,0,0],[17,17,1.0,0.5357,0.22587,0.42857,0.57121,0.71429,0.14286,0.85714,0,0,0,0,0,5,0,0,1,0,0,9,0,0,2,0,0,12,0,0,3,0,0]]}]},{"i":"cd7e9fedd6d770fc","q":"Let $k \\geq 14$ be an integer, and let $p_{k}$ be the largest prime number which is strictly less than $k$. You may assume that $p_{k} \\geq 3 k / 4$. Let $n$ be a composite integer. Prove:\n(a) if $n=2 p_{k}$, then $n$ does not divide $(n-k)$ !;\n(b) if $n>2 p_{k}$, then $n$ divides $(n-k)$ !.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.38836,"x":0.64285,"p":[[0,18,0.0,0.44632,0.3026,0.14289,0.28571,0.60714,0.14,1.0,0,4,0,0,0,9,0,0,9,0,0,2,0,0,4,0,0,1,0,0,3,0,4],[4,18,0.2222,0.38836,0.23208,0.25,0.28571,0.57141,0.14286,1.0,0,1,0,0,0,8,0,0,12,0,0,1,0,0,6,0,0,3,0,0,1,0,1],[8,18,0.4444,0.625,0.27836,0.42857,0.64286,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,6,0,0,7,0,0,2,0,0,6,0,0,2,0,8],[12,18,0.6667,0.64285,0.27199,0.42857,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,5,0,0,4,0,0,2,0,0,8,0,0,5,0,6],[16,18,0.8889,0.49103,0.28332,0.2857,0.42857,0.71429,0.14286,1.0,0,4,0,0,0,5,0,0,10,0,0,3,0,0,4,0,0,4,0,0,2,0,4],[18,18,1.0,0.55791,0.22957,0.39286,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,6,0,0,5,0,0,5,0,0,10,0,0,2,0,2]]},{"b":4,"e":0.28571,"k":"flat","v":0.35714,"x":0.54014,"p":[[0,18,0.0,0.45973,0.31497,0.14289,0.28571,0.75,0.14,1.0,0,4,0,0,0,9,0,0,9,0,0,3,0,0,0,0,0,3,0,0,4,0,4],[4,18,0.2222,0.49999,0.26244,0.2857,0.57121,0.71429,0.14286,1.0,0,2,0,0,0,5,0,0,8,0,0,2,0,0,8,0,0,3,0,0,4,0,2],[8,18,0.4444,0.35714,0.25505,0.14286,0.2857,0.4286,0.14286,1.0,0,2,0,0,0,11,0,0,11,0,0,4,0,0,1,0,0,1,0,0,2,0,2],[12,18,0.6667,0.5089,0.23401,0.28571,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,11,0,0,1,0,0,9,0,0,4,0,0,4,0,1],[16,18,0.8889,0.54014,0.20432,0.39286,0.57143,0.71407,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,4,0,0,11,0,0,5,0,0,3,0,1],[18,18,1.0,0.48213,0.24156,0.2857,0.4998,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,11,0,0,1,0,0,5,0,0,7,0,0,4,0,0]]}]},{"i":"d3af38b8b643fab3","q":"6. (FRA 1) In the triangle $A B C$ let $B^{\\prime}$ and $C^{\\prime}$ be the midpoints of the sides $A C$ and $A B$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $A B^{\\prime} C^{\\prime}, B C^{\\prime} H$, and $B^{\\prime} C H$ have a common point $I$ and that the line $H I$ passes through the midpoint of the segment $B^{\\prime} C^{\\prime}$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.12499,"p":[[0,21,0.0,0.07141,0.14721,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,21,0.1905,0.10712,0.1785,0.0,0.0,0.17857,0.0,0.571,22,0,0,22,0,2,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,21,0.381,0.12499,0.16265,0.0,0.0,0.28571,0.0,0.571,19,0,0,19,0,1,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,21,0.5714,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.07135,0.12874,0.0,0.0,0.14071,0.0,0.42857,23,0,0,23,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.07588,0.15961,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,1,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.07143,"x":0.21866,"p":[[0,30,0.0,0.1517,0.15126,0.0,0.14145,0.28571,0.0,0.4286,15,0,0,15,0,2,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.07143,0.14285,0.0,0.0,0.0,0.0,0.5714,25,0,0,25,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,30,0.2667,0.10266,0.16061,0.0,0.0,0.2857,0.0,0.571,22,0,0,22,0,0,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,30,0.4,0.08481,0.16307,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,1,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[16,30,0.5333,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.21866,0.1675,0.0,0.28571,0.28571,0.0,0.57143,9,0,0,9,0,4,0,0,14,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[24,30,0.8,0.14286,0.15972,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,1,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.17858,0.15568,0.0,0.2857,0.28571,0.0,0.42857,13,0,0,13,0,1,0,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.16519,0.17897,0.0,0.14285,0.28571,0.0,0.57143,16,0,0,16,0,0,0,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"b2ada32659436f41","q":"Let  $ABC$  be a triangle,  $I$  its incenter,  $\\omega$  its incircle,  $P$  a point such that  $PI\\perp BC$  and  $PA\\parallel BC$ ,  $Q\\in (AB), R\\in (AC)$  such that  $QR\\parallel BC$  and  $QR$  tangent to  $\\omega$ . \nShow that  $\\angle QPB = \\angle CPR$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,17,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"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],[4,33,0.1212,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,33,0.4848,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[33,33,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1a0d2286b119ae88","q":"Prove that the number of ordered triples  $(x, y, z)$  such that  $(x+y+z)^2  \\equiv axyz \\mod{p}$ , where  $gcd(a, p) = 1$  and  $p$  is prime is  $p^2 + 1$ .","t":[{"b":1,"e":0.42857,"k":"falling","v":0.19634,"x":0.71436,"p":[[0,30,0.0,0.71436,0.33507,0.39286,0.93,1.0,0.14286,1.0,0,16,0,0,0,4,0,0,4,0,0,2,0,0,2,0,0,2,0,0,2,0,16],[4,30,0.1333,0.54464,0.37701,0.14286,0.42857,1.0,0.0,1.0,1,12,0,1,0,8,0,0,7,0,0,0,0,0,4,0,0,0,0,0,0,0,12],[8,30,0.2667,0.30348,0.2738,0.14286,0.21428,0.42858,0.0,1.0,6,2,0,6,0,10,0,0,4,0,0,6,0,0,3,0,0,0,0,0,1,0,2],[12,30,0.4,0.33035,0.23537,0.14286,0.2857,0.4286,0.0,1.0,1,1,0,1,0,14,0,0,3,0,0,9,0,0,1,0,0,2,0,0,1,0,1],[16,30,0.5333,0.24554,0.18292,0.14286,0.1429,0.28571,0.0,0.71429,2,0,0,2,0,17,0,0,7,0,0,3,0,0,0,0,0,3,0,0,0,0,0],[20,30,0.6667,0.27232,0.2,0.14286,0.14286,0.32143,0.0,0.71429,1,0,0,1,0,18,0,0,5,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[24,30,0.8,0.19634,0.09947,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,18,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.21867,0.1597,0.14286,0.1429,0.28571,0.0,0.71429,4,0,0,4,0,16,0,0,5,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[30,30,1.0,0.25445,0.19473,0.14286,0.14286,0.42857,0.0,1.0,2,1,0,2,0,17,0,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.14286,"k":"falling","v":0.36602,"x":0.58928,"p":[[0,8,0.0,0.58928,0.37244,0.24999,0.64286,1.0,0.0,1.0,2,11,0,2,0,6,0,0,4,0,0,3,0,0,1,0,0,2,0,0,3,0,11],[4,8,0.5,0.41959,0.37445,0.14286,0.35714,0.71429,0.0,1.0,7,7,0,7,0,8,0,0,1,0,0,3,0,0,4,0,0,2,0,0,0,0,7],[8,8,1.0,0.36602,0.28329,0.14286,0.2857,0.571,0.0,1.0,2,3,0,2,0,13,0,0,3,0,0,3,0,0,7,0,0,1,0,0,0,0,3]]}]},{"i":"e69de07cf673776a","q":"In the non-isosceles triangle  $ABC$  an altitude from  $A$  meets side  $BC$  in  $D$  . Let  $M$  be the midpoint of  $BC$  and let  $N$  be the reflection of  $M$  in  $D$  . The circumcirle of triangle  $AMN$  intersects the side  $AB$  in  $P\\ne A$  and the side  $AC$  in  $Q\\ne A$  . Prove that  $AN,BQ$  and  $CP$  are concurrent.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.0892,"x":0.31695,"p":[[0,31,0.0,0.31695,0.1776,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,7,0,0,8,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[4,31,0.129,0.28571,0.15152,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,11,0,0,11,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[8,31,0.2581,0.18303,0.18293,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,7,0,0,8,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[12,31,0.3871,0.17848,0.14728,0.105,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,13,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,31,0.5161,0.18304,0.14826,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,12,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,31,0.6452,0.12054,0.11904,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,12,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.08929,0.11152,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.13388,0.12856,0.0,0.14286,0.17857,0.0,0.43,12,0,0,12,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.0892,0.09274,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"rising","v":0.3393,"x":0.54907,"p":[[0,26,0.0,0.34375,0.17445,0.24999,0.42857,0.4286,0.0,0.57143,3,0,0,3,0,5,0,0,6,0,0,12,0,0,6,0,0,0,0,0,0,0,0],[4,26,0.1538,0.3393,0.15047,0.28571,0.42857,0.42857,0.0,0.71429,2,0,1,2,0,4,0,0,9,0,0,15,0,0,1,0,0,1,0,0,0,0,0],[8,26,0.3077,0.49548,0.10701,0.42857,0.42857,0.5711,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,19,0,0,8,0,0,4,0,0,0,0,0],[12,26,0.4615,0.54907,0.14772,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,10,0,0,11,0,0,8,0,0,1,0,0],[16,26,0.6154,0.50441,0.12361,0.42857,0.571,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,11,0,0,13,0,0,4,0,0,0,0,0],[20,26,0.7692,0.51321,0.11187,0.42859,0.571,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,10,0,0,16,0,0,3,0,0,0,0,0],[24,26,0.9231,0.54463,0.16146,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,7,0,0,11,0,0,7,0,0,2,0,0],[26,26,1.0,0.52231,0.15406,0.42857,0.4998,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,4,0,0,12,0,0,9,0,0,5,0,0,2,0,0]]}]},{"i":"bec22299aa050fad","q":"Let  $O$  be the intersection of the diagonals of convex quadrilateral  $ABCD$ . The circumcircles of  $\\triangle{OAD}$  and  $\\triangle{OBC}$  meet at  $O$  and  $M$ . Line  $OM$  meets the circumcircles of  $\\triangle{OAB}$  and  $\\triangle{OCD}$  at  $T$  and  $S$  respectively. \r\n\r\nProve that  $M$  is the midpoint of  $ST$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.00893,"x":0.04902,"p":[[0,36,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.03125,"p":[[0,13,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"809be0deda15d89a","q":"Let  $a,b,c$  be positive reals satisfying  $a^3+b^3+c^3+abc=4$ .  Prove that\n\\[ \\frac{(5a^2+bc)^2}{(a+b)(a+c)} + \\frac{(5b^2+ca)^2}{(b+c)(b+a)} + \\frac{(5c^2+ab)^2}{(c+a)(c+b)} \\ge \\frac{(a^3+b^3+c^3+6)^2}{a+b+c} \\] and determine the cases of equality.\n\n*Proposed by Evan Chen*","t":[{"b":1,"e":0.42857,"k":"rising","v":0.20536,"x":0.39285,"p":[[0,5,0.0,0.20536,0.18536,0.10714,0.14286,0.28571,0.0,0.57143,8,0,0,8,0,13,0,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[4,5,0.8,0.39285,0.17856,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,5,0,0,9,0,0,10,0,0,1,0,0,0,0,0],[5,5,1.0,0.37944,0.21309,0.2857,0.42857,0.57111,0.0,0.85714,3,0,0,3,0,4,0,0,8,0,0,7,0,0,7,0,0,2,0,0,1,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.28125,"x":0.39284,"p":[[0,13,0.0,0.29911,0.26812,0.14286,0.14286,0.42857,0.0,1.0,3,1,0,3,0,16,0,0,3,0,0,4,0,0,2,0,0,0,0,0,3,0,1],[4,13,0.3077,0.28125,0.19718,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,9,0,0,10,0,0,4,0,0,4,0,0,0,0,0,1,0,0],[8,13,0.6154,0.37499,0.18122,0.2857,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,9,0,0,7,0,0,9,0,0,1,0,0,0,0,0],[12,13,0.9231,0.39284,0.17495,0.2857,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,6,0,0,5,0,0,8,0,0,12,0,0,0,0,0,0,0,0],[13,13,1.0,0.35715,0.17857,0.1429,0.35714,0.57143,0.0,0.57143,1,0,0,1,0,8,0,0,7,0,0,6,0,0,10,0,0,0,0,0,0,0,0]]}]},{"i":"389b674c348783b7","q":"Let $n$ be a strictly positive integer and let $a, a_{1}, \\ldots, a_{n}$ be strictly positive integers. Suppose that for any integer $k$ for which the integer $a k+1$ is a perfect square, at least one of the integers $a_{1} k+1, \\ldots, a_{n} k+1$ is also a perfect square.\nShow that there exists an index $1 \\leqslant i \\leqslant n$ such that $a=a_{i}$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.79906,"x":0.91069,"p":[[0,29,0.0,0.83924,0.20753,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,3,0,0,8,0,15],[4,29,0.1379,0.88392,0.19704,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,6,0,20],[8,29,0.2759,0.91069,0.1462,0.85711,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,6,0,21],[12,29,0.4138,0.89284,0.16754,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,5,0,20],[16,29,0.5517,0.86159,0.14936,0.71429,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,9,0,14],[20,29,0.6897,0.87499,0.18125,0.82143,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,6,0,18],[24,29,0.8276,0.86159,0.16556,0.82132,0.85714,1.0,0.4286,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,9,0,15],[28,29,0.9655,0.79906,0.18855,0.71429,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,9,0,10],[29,29,1.0,0.8839,0.16151,0.71429,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,2,0,20]]},{"b":6,"e":1.0,"k":"flat","v":0.85709,"x":0.96874,"p":[[0,32,0.0,0.89286,0.17857,0.85714,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,6,0,20],[4,32,0.125,0.85709,0.15977,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,9,0,14],[8,32,0.25,0.93302,0.10093,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,7,0,21],[12,32,0.375,0.90175,0.14039,0.82132,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,4,0,20],[16,32,0.5,0.93304,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[20,32,0.625,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[24,32,0.75,0.94194,0.10637,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[28,32,0.875,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[32,32,1.0,0.93747,0.10068,0.85714,1.0,1.0,0.571,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,21]]}]},{"i":"d5e1304449ae1e5a","q":"Define a function  $f:\\mathbb{N}\\rightarrow\\mathbb{N}$ , \\[f(1)=p+1,\\] \\[f(n+1)=f(1)\\cdot f(2)\\cdots f(n)+p,\\] where  $p$  is a prime number. Find all  $p$  such that there exists a natural number  $k$  such that  $f(k)$  is a perfect square.","t":[{"b":0,"e":0.57143,"k":"falling","v":0.35713,"x":0.7366,"p":[[0,35,0.0,0.7366,0.27225,0.53571,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,7,0,0,1,0,0,2,0,0,1,0,0,12,0,9],[4,35,0.1143,0.63838,0.24481,0.42857,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,6,0,0,6,0,0,3,0,0,4,0,0,10,0,3],[8,35,0.2286,0.67855,0.26487,0.42857,0.78571,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,4,0,0,6,0,0,2,0,0,3,0,0,10,0,6],[12,35,0.3429,0.53125,0.284,0.28571,0.42857,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,13,0,0,6,0,0,0,0,0,2,0,0,6,0,4],[16,35,0.4571,0.65612,0.2645,0.42857,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,4,0,0,5,0,0,7,0,0,3,0,0,4,0,8],[20,35,0.5714,0.59817,0.25111,0.42857,0.57121,0.85704,0.14286,1.0,0,4,0,0,0,3,0,0,1,0,0,9,0,0,6,0,0,4,0,0,5,0,4],[24,35,0.6857,0.46427,0.21724,0.28571,0.42857,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,12,0,0,9,0,0,3,0,0,4,0,0,1,0,2],[28,35,0.8,0.41518,0.20316,0.2857,0.35714,0.42858,0.14286,1.0,0,2,0,0,0,1,0,0,15,0,0,11,0,0,1,0,0,1,0,0,1,0,2],[32,35,0.9143,0.35713,0.13361,0.28571,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,3,0,0,16,0,0,8,0,0,4,0,0,1,0,0,0,0,0],[35,35,1.0,0.39627,0.14122,0.28571,0.41429,0.4286,0.1429,0.71429,0,0,0,0,0,1,0,0,14,0,0,11,0,0,3,0,0,3,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"falling","v":0.44641,"x":0.72765,"p":[[0,31,0.0,0.72765,0.24317,0.57132,0.857,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,1,0,0,12,0,7],[4,31,0.129,0.67411,0.26058,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,9,0,0,3,0,0,2,0,0,8,0,7],[8,31,0.2581,0.60711,0.26244,0.39286,0.57143,0.85714,0.14286,1.0,0,3,0,0,0,2,0,0,6,0,0,3,0,0,7,0,0,2,0,0,9,0,3],[12,31,0.3871,0.62943,0.27632,0.28571,0.64286,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,9,0,0,3,0,0,4,0,0,6,0,0,2,0,8],[16,31,0.5161,0.61607,0.30813,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,8,0,0,4,0,0,3,0,0,4,0,0,1,0,10],[20,31,0.6452,0.55805,0.27746,0.28571,0.57143,0.75,0.14286,1.0,0,5,0,0,0,1,0,0,12,0,0,2,0,0,4,0,0,5,0,0,3,0,5],[24,31,0.7742,0.625,0.27141,0.42857,0.57143,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,7,0,0,7,0,0,3,0,0,5,0,0,2,0,8],[28,31,0.9032,0.50447,0.2448,0.28571,0.50001,0.60714,0.14286,1.0,0,3,0,0,0,1,0,0,13,0,0,2,0,0,8,0,0,3,0,0,2,0,3],[31,31,1.0,0.44641,0.16655,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,10,0,0,6,0,0,10,0,0,4,0,0,0,0,0]]}]},{"i":"bca54ce404170333","q":"Let be four distinct complex numbers  $ a,b,c,d $  chosen such that   $$  |a|=|b|=|c|=|d|=|b-c|=\\frac{|c-d|}{2}=1,  $$  and  $$  \\min_{\\lambda\\in\\mathbb{C}} |a-\\lambda d -(1-\\lambda )c| =\\min_{\\lambda\\in\\mathbb{C}} |b-\\lambda d -(1-\\lambda )c| .  $$  Calculate  $ |a-c| $  and  $ |a-d|. $ *Carmen Botea*","t":[{"b":3,"e":1.0,"k":"flat","v":0.9375,"x":0.97768,"p":[[0,20,0.0,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[4,20,0.2,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,20,0.4,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[12,20,0.6,0.94642,0.17768,1.0,1.0,1.0,0.0,1.0,1,26,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[16,20,0.8,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[20,20,1.0,0.95089,0.06785,0.85714,1.0,1.0,0.85714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21]]},{"b":4,"e":0.57143,"k":"flat","v":0.84375,"x":0.93304,"p":[[0,13,0.0,0.93304,0.12364,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,22],[4,13,0.3077,0.84375,0.17985,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,4,0,16],[8,13,0.6154,0.87499,0.15874,0.71429,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,4,0,18],[12,13,0.9231,0.84821,0.17474,0.71429,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,5,0,16],[13,13,1.0,0.89286,0.13832,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,9,0,17]]}]},{"i":"d7939c10a49d9652","q":"Show that there exists an integer divisible by 1996 such that the sum of the its decimal digits is 1996 .","t":[{"b":5,"e":1.0,"k":"rising","v":0.23661,"x":0.97768,"p":[[0,26,0.0,0.26786,0.3567,0.0,0.0,0.42857,0.0,1.0,17,3,1,17,0,3,0,0,1,0,0,4,0,0,0,0,0,2,0,0,2,0,3],[4,26,0.1538,0.23661,0.33429,0.0,0.0,0.42857,0.0,1.0,18,2,0,18,0,3,0,0,0,0,0,6,0,0,0,0,0,0,0,0,3,0,2],[8,26,0.3077,0.41071,0.44571,0.0,0.14285,1.0,0.0,1.0,16,9,1,16,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,9],[12,26,0.4615,0.97768,0.08828,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,30],[16,26,0.6154,0.92857,0.17128,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,27],[20,26,0.7692,0.95536,0.14032,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,29],[24,26,0.9231,0.96875,0.12234,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30],[26,26,1.0,0.93749,0.19544,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,28]]},{"b":7,"e":1.0,"k":"falling","v":0.0,"x":0.29464,"p":[[0,31,0.0,0.22321,0.34615,0.0,0.0,0.42857,0.0,1.0,21,4,1,21,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,4],[4,31,0.129,0.29464,0.37954,0.0,0.0,0.4286,0.0,1.0,17,5,0,17,0,2,0,0,0,0,0,6,0,0,0,0,0,1,0,0,1,0,5],[8,31,0.2581,0.28116,0.31033,0.0,0.14286,0.4286,0.0,0.85714,14,0,1,14,0,4,0,0,0,0,0,7,0,0,0,0,0,4,0,0,3,0,0],[12,31,0.3871,0.21429,0.26486,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,3,0,0,1,0,0,10,0,0,0,0,0,0,0,0,1,0,1],[16,31,0.5161,0.25446,0.32874,0.0,0.0,0.42858,0.0,1.0,17,1,1,17,0,3,0,0,0,0,0,5,0,0,1,0,0,2,0,0,3,0,1],[20,31,0.6452,0.23661,0.3126,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,5,0,0,1,0,0,4,0,0,0,0,0,3,0,0,2,0,1],[24,31,0.7742,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"0315e39967666120","q":"Let $a, b$ and $c$ be positive real numbers. Prove that\n\n$$\n\\frac{8}{(a+b)^{2}+4 a b c}+\\frac{8}{(b+c)^{2}+4 a b c}+\\frac{8}{(c+a)^{2}+4 a b c}+a^{2}+b^{2}+c^{2} \\geq \\frac{8}{a+3}+\\frac{8}{b+3}+\\frac{8}{c+3}\n$$","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,7,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,52,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,52,0.1538,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,52,0.2308,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,52,0.3077,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,52,0.3846,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,52,0.5385,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,52,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"fda493f0327bbe5b","q":"Prove or disprove that  $\\forall a,b,c,d \\in \\mathbb{R}^+$  we have the following inequality:\n\n\\[3 \\leq \\frac{4a+b}{a+4b} + \\frac{4b+c}{b+4c} + \\frac{4c+a}{c+4a} < \\frac{33}{4}\\]","t":[{"b":0,"e":0.28571,"k":"flat","v":0.16518,"x":0.24098,"p":[[0,26,0.0,0.16518,0.14334,0.14286,0.14286,0.14286,0.0,0.57143,7,0,0,7,0,18,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,26,0.1538,0.20982,0.10092,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,20,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,26,0.3077,0.18303,0.08917,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,20,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.20982,0.11285,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,0,19,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,26,0.6154,0.20982,0.09438,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,20,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.24098,0.14486,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,17,0,0,7,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[24,26,0.9231,0.16964,0.0974,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,22,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.17402,0.07775,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,24,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.12054,"x":0.17857,"p":[[0,18,0.0,0.17857,0.13363,0.14286,0.14286,0.14286,0.0,0.71429,3,0,0,3,0,23,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,18,0.2222,0.15625,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.16063,0.05925,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.14723,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.12054,0.10171,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,21,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fcc6bcdc621f9826","q":"Let  $f (x) = x^2 + ax + b$  be a quadratic function with real coefficients  $a, b$ . It is given that the equation  $f (f (x)) = 0$  has  $4$  distinct real roots and the sum of  $2$  roots among these roots is equal to  $-1$ . Prove that  $b \\le  -\\frac14$","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,16,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,16,1.0,0.00223,0.01242,0.0,0.0,0.0,0.0,0.0714,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03576,"p":[[0,5,0.0,0.03576,0.09468,0.0,0.0,0.0,0.0,0.43,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"40749e0abfff2c7a","q":"Let $a, b, c$ be real numbers such that $0 \\leq a \\leq b \\leq c$. Prove that if\n\n$$\na+b+c=a b+b c+c a>0,\n$$\n\nthen $\\sqrt{b c}(a+1) \\geq 2$. When does the equality hold?","t":[{"b":3,"e":0.2857,"k":"flat","v":0.11152,"x":0.18741,"p":[[0,11,0.0,0.18741,0.18366,0.105,0.14286,0.17857,0.0,0.71429,8,0,0,8,0,16,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[4,11,0.3636,0.17411,0.1461,0.14286,0.14286,0.1429,0.0,0.71429,5,0,0,5,0,20,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,11,0.7273,0.11152,0.0993,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.15178,0.10062,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,19,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.1517,"x":0.24545,"p":[[0,15,0.0,0.23661,0.21312,0.14286,0.14288,0.28571,0.0,1.0,4,1,0,4,0,15,0,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[4,15,0.2667,0.24545,0.23757,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,12,0,0,9,0,0,1,0,0,1,0,0,1,0,0,1,0,1],[8,15,0.5333,0.1517,0.11259,0.14214,0.14286,0.14287,0.0,0.4286,7,0,0,7,0,18,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.19197,0.17717,0.0,0.14286,0.42857,0.0,0.57143,10,0,0,10,0,11,0,0,2,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[15,15,1.0,0.16518,0.14334,0.0,0.14286,0.2857,0.0,0.4286,9,0,0,9,0,14,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dc43dd3b685c755e","q":"Determine the smallest real constant  $c$  such that\n\\[\\sum_{k=1}^{n}\\left ( \\frac{1}{k}\\sum_{j=1}^{k}x_j \\right )^2\\leq c\\sum_{k=1}^{n}x_k^2\\]\nfor all positive integers  $n$  and all positive real numbers  $x_1,\\cdots ,x_n$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.10715,"x":0.16964,"p":[[0,14,0.0,0.13384,0.07935,0.14286,0.14286,0.14286,0.0,0.42857,5,0,2,5,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.16964,0.08328,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,29,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,30,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.10268,"x":0.12947,"p":[[0,27,0.0,0.125,0.07784,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,2,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.12938,0.04161,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.12947,0.07457,0.14286,0.14286,0.14286,0.0,0.42857,5,0,1,5,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b0926cf07a253f14","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.\n\n*Proposed by Warut Suksompong, Thailand*","t":[{"b":4,"e":0.57143,"k":"falling","v":0.32143,"x":0.62053,"p":[[0,8,0.0,0.62053,0.34183,0.28571,0.57143,1.0,0.0,1.0,2,12,0,2,0,0,0,0,9,0,0,3,0,0,4,0,0,0,0,0,2,0,12],[4,8,0.5,0.32143,0.07143,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,25,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[8,8,1.0,0.33928,0.09943,0.28571,0.28571,0.32143,0.2857,0.57143,0,0,0,0,0,0,0,0,24,0,0,4,0,0,4,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"volatile","v":0.34821,"x":0.70534,"p":[[0,6,0.0,0.70534,0.31327,0.42857,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,7,0,0,6,0,0,2,0,0,0,0,0,1,0,16],[4,6,0.6667,0.34821,0.10062,0.28571,0.28571,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,22,0,0,6,0,0,4,0,0,0,0,0,0,0,0],[6,6,1.0,0.34824,0.09408,0.2857,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,21,0,0,8,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"61118947ee3fa618","q":"Let  $\\alpha\\neq0$  be a real number. Determine all functions  $f:\\mathbb R\\to\\mathbb R$  such that \\[f\\left(x^2+y^2\\right)=f(x-y)f(x+y)+\\alpha yf(y)\\] holds for all  $x, y\\in\\mathbb R.$","t":[{"b":3,"e":1.0,"k":"rising","v":0.70976,"x":0.89732,"p":[[0,16,0.0,0.71873,0.23552,0.571,0.71429,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,6,0,0,3,0,10],[4,16,0.25,0.82587,0.20121,0.67836,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,1,0,17],[8,16,0.5,0.78568,0.23694,0.57143,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,3,0,0,7,0,0,2,0,0,5,0,14],[12,16,0.75,0.70976,0.27081,0.57143,0.64286,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,2,0,0,1,0,0,11,0,0,2,0,0,2,0,12],[16,16,1.0,0.89732,0.16841,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,3,0,21]]},{"b":4,"e":0.71429,"k":"falling","v":0.52229,"x":0.83482,"p":[[0,76,0.0,0.83482,0.25532,0.82143,1.0,1.0,0.0,1.0,1,18,1,1,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,6,0,18],[4,76,0.0526,0.71427,0.21429,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,5,0,0,3,0,9],[8,76,0.1053,0.665,0.19101,0.57143,0.64071,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,1,0,0,13,0,0,9,0,0,3,0,4],[12,76,0.1579,0.66067,0.21652,0.57132,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,3,0,0,2,0,0,15,0,0,2,0,0,4,0,6],[16,76,0.2105,0.79018,0.24218,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,0,0,0,7,0,0,6,0,0,2,0,15],[20,76,0.2632,0.70991,0.27317,0.57143,0.71429,1.0,0.0,1.0,1,10,1,1,0,1,0,0,1,0,0,4,0,0,6,0,0,4,0,0,5,0,10],[24,76,0.3158,0.62945,0.20158,0.5354,0.57143,0.74996,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,5,0,0,11,0,0,5,0,0,5,0,3],[28,76,0.3684,0.79909,0.20474,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,3,0,0,6,0,13],[32,76,0.4211,0.69638,0.20126,0.57143,0.57143,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,13,0,0,3,0,0,6,0,6],[36,76,0.4737,0.75888,0.18712,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,4,0,0,6,0,9],[40,76,0.5263,0.78121,0.23145,0.57132,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,3,0,0,4,0,14],[44,76,0.5789,0.7723,0.17077,0.57143,0.78564,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,5,0,0,8,0,8],[48,76,0.6316,0.58471,0.13535,0.571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,2,0,0,23,0,0,2,0,0,2,0,1],[52,76,0.6842,0.52229,0.21311,0.42857,0.57143,0.57143,0.0,0.85714,2,0,2,2,0,0,0,0,4,0,0,6,0,0,14,0,0,1,0,0,5,0,0],[56,76,0.7368,0.58032,0.11259,0.57132,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,21,0,0,4,0,0,2,0,0],[60,76,0.7895,0.58478,0.13054,0.571,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,20,0,0,2,0,0,4,0,0],[64,76,0.8421,0.58481,0.12556,0.57143,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,22,0,0,1,0,0,4,0,0],[68,76,0.8947,0.58029,0.13804,0.5354,0.57143,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,7,0,0,17,0,0,3,0,0,4,0,0],[72,76,0.9474,0.55804,0.13997,0.4286,0.57143,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,10,0,0,16,0,0,1,0,0,4,0,0],[76,76,1.0,0.58926,0.15872,0.57142,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,2,0,0,2,0,0,18,0,0,5,0,0,4,0,0]]}]},{"i":"58690d9e85fa0817","q":"Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  be integers selected from the set  $\\{1,2,\\dots,100\\}$ , uniformly and at random with replacement. Set \\[ M = a + 2b + 4c + 8d + 16e + 32f. \\] What is the expected value of the remainder when  $M$  is divided by  $64$ ?","t":[{"b":0,"e":0.42857,"k":"flat","v":0.64732,"x":0.83929,"p":[[0,43,0.0,0.75446,0.28174,0.64286,0.85714,1.0,0.0,1.0,2,11,2,2,0,0,0,0,0,0,0,6,0,0,0,0,0,4,0,0,9,0,11],[4,43,0.093,0.83929,0.19805,0.85714,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,13,0,13],[8,43,0.186,0.81696,0.2321,0.82143,0.85714,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,12,0,12],[12,43,0.2791,0.69643,0.21053,0.42857,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,18,0,1],[16,43,0.3721,0.66518,0.21609,0.42857,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,3,0,0,13,0,2],[20,43,0.4651,0.67411,0.22084,0.42857,0.78571,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,1,0,0,13,0,3],[24,43,0.5581,0.70759,0.2317,0.42857,0.85714,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,14,0,5],[28,43,0.6512,0.66518,0.22477,0.42857,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15,0,2],[32,43,0.7442,0.71875,0.20666,0.42857,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,7,0,0,10,0,5],[36,43,0.8372,0.70536,0.2141,0.42857,0.78571,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,5,0,0,12,0,4],[40,43,0.9302,0.6875,0.22142,0.42857,0.78571,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,11,0,0,0,0,0,4,0,0,13,0,3],[43,43,1.0,0.64732,0.2172,0.42857,0.64286,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,15,0,0,1,0,0,2,0,0,12,0,2]]},{"b":2,"e":0.85714,"k":"flat","v":0.64286,"x":0.8125,"p":[[0,29,0.0,0.79018,0.23686,0.42857,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,1,0,0,9,0,13],[4,29,0.1379,0.79911,0.27632,0.71429,0.85714,1.0,0.0,1.0,2,14,1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,0,9,0,14],[8,29,0.2759,0.8125,0.26592,0.85714,0.85714,1.0,0.0,1.0,2,13,2,2,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,0,12,0,13],[12,29,0.4138,0.70089,0.38855,0.53571,0.92857,1.0,0.0,1.0,6,16,5,6,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,0,2,0,16],[16,29,0.5517,0.71429,0.19232,0.4286,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,2,0,0,1,0,0,20,0,0],[20,29,0.6897,0.65179,0.20183,0.42857,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,4,0,0,12,0,1],[24,29,0.8276,0.72768,0.20316,0.42857,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,21,0,1],[28,29,0.9655,0.74554,0.18466,0.64286,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,1,0,0,23,0,0],[29,29,1.0,0.64286,0.24223,0.42857,0.85714,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,4,0,0,11,0,0,0,0,0,0,0,0,16,0,1]]}]},{"i":"b442eaec98c02a67","q":"Prove that there is no function from positive real numbers to itself, $f:(0,+\\infty) \\rightarrow(0,+\\infty)$ such that:\n\n$$\nf(f(x)+y)=f(x)+3 x+y f(y) \\quad \\text {,for every } \\quad x, y \\in(0,+\\infty)\n$$","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,19,0.0,0.06696,0.13355,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.06696,0.14279,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.12951,"p":[[0,22,0.0,0.08036,0.15947,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,1,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.12951,0.18343,0.0,0.0,0.32142,0.0,0.43,20,0,0,20,0,3,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.04911,0.13175,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.03571,0.10101,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"2a8d02bd50e5e4e7","q":"Let  $n\\in\\mathbb{Z}$ ,  $n\\geq 2$ . Find all functions  $f:\\mathbb{R}_{>0}\\rightarrow\\mathbb{R}_{>0}$  such that   $$ f(x_1+\\dots +x_n)^2=\\sum_{i=1}^nf(x_i) ^2+ 2\\sum_{i<j}f(x_ix_j), $$   for all  $x_1,\\dots ,x_n\\in\\mathbb{R}_{>0}$ .\n\n*Proposed by Andrei Vila*","t":[{"b":0,"e":0.0,"k":"flat","v":0.02679,"x":0.09375,"p":[[0,25,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.08036,0.15126,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,8,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,25,0.32,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.09375,0.18766,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,6,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[16,25,0.64,0.05357,0.11152,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,25,0.8,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.01786,"x":0.05804,"p":[[0,12,0.0,0.05804,0.11214,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.04911,0.11071,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,12,1.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"76e906dd9e1abbd5","q":"Prove that there exists an infinite sequence of perfect squares with the following properties:\n(i) The arithmetic mean of any two consecutive terms is a perfect square,\n(ii) Every two consecutive terms are coprime,\n(iii) The sequence is strictly increasing.","t":[{"b":2,"e":1.0,"k":"rising","v":0.79016,"x":1.0,"p":[[0,6,0.0,0.79016,0.22865,0.71429,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,9,0,0,4,0,13],[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":0.857,"k":"flat","v":0.88838,"x":0.9375,"p":[[0,11,0.0,0.88838,0.19477,0.85714,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,6,0,20],[4,11,0.3636,0.9375,0.07936,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,19],[8,11,0.7273,0.93303,0.07974,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,18],[11,11,1.0,0.9375,0.09407,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,8,0,21]]}]},{"i":"098cef701cd5816d","q":"The altitudes  $AA_1$  and  $CC_1$  of an acute-angled triangle  $ABC$  intersect at point  $H$ . A straight line passing through  $H$  parallel to line  $A_1C_1$  intersects the circumscribed circles of triangles  $AHC_1$  and  $CHA_1$  at points  $X$  and  $Y$ , respectively. Prove that points  $X$  and  $Y$  are equidistant from the midpoint of segment  $BH$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.00893,"x":0.12491,"p":[[0,28,0.0,0.00893,0.0346,0.0,0.0,0.0,0.0,0.143,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.11143,0.05897,0.14,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.05339,"p":[[0,23,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.05339,0.06893,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a97c3438368218b1","q":"Let  $V$  be a  $n-$ dimensional vector space over a field  $F$  with a basis  $\\{e_1,e_2, \\cdots ,e_n\\}$ .Prove that for any  $m-$ dimensional linear subspace  $W$  of  $V$ , the number of elements of the set  $W \\cap P$  is less than or equal to  $2^m$  where  $P=\\{\\lambda_1e_1 + \\lambda_2e_2 + \\cdots + \\lambda_ne_n : \\lambda_i=0,1\\}$ .","t":[{"b":4,"e":0.1429,"k":"flat","v":0.08036,"x":0.18741,"p":[[0,31,0.0,0.18741,0.16148,0.0,0.14288,0.28571,0.0,0.4286,10,0,0,10,0,9,0,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.17402,0.1546,0.0,0.14286,0.28571,0.0,0.57143,10,0,0,10,0,10,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,31,0.2581,0.14732,0.16164,0.0,0.14286,0.2857,0.0,0.57143,15,0,0,15,0,5,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,31,0.3871,0.15625,0.20628,0.0,0.07143,0.2857,0.0,1.0,16,1,0,16,0,3,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,31,0.5161,0.0982,0.16142,0.0,0.0,0.17857,0.0,0.571,22,0,0,22,0,2,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,31,0.6452,0.125,0.15872,0.0,0.0,0.2857,0.0,0.4286,18,0,0,18,0,4,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,31,0.9032,0.13839,0.15355,0.0,0.14286,0.2857,0.0,0.4286,15,0,0,15,0,7,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.09366,0.13648,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0758,"x":0.25893,"p":[[0,33,0.0,0.25893,0.14032,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,6,0,0,15,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,33,0.1212,0.2232,0.16339,0.10714,0.28571,0.28571,0.0,0.571,8,0,0,8,0,6,0,0,11,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[8,33,0.2424,0.12945,0.15299,0.0,0.14286,0.1786,0.0,0.571,15,0,0,15,0,9,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,33,0.3636,0.13839,0.14054,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,7,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.11607,0.1357,0.0,0.07143,0.1786,0.0,0.42857,16,0,0,16,0,8,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.08482,0.15916,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,5,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,33,0.7273,0.0758,0.11831,0.0,0.0,0.14287,0.0,0.28571,22,0,0,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.12043,0.20854,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,4,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[32,33,0.9697,0.13839,0.16554,0.0,0.0,0.2857,0.0,0.4286,17,0,0,17,0,4,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.09374,0.15403,0.0,0.0,0.14286,0.0,0.571,21,0,0,21,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"08979cb8c0ea78f9","q":"Piotrek is playing with pebbles. He starts with an empty stack. In  $i$ -th move, Piotrek removes  $i$  pebbles from the stack if he can. If he can't, he adds  $i$  pebbles to the stack. For example, after first  $5$  moves, Piotrek has  $1, 3, 0, 4, 9$  pebbles respectively. Find all positive integers  $n$  such that Piotrek has  $0$  pebbles after  $n$  moves.","t":[{"b":2,"e":0.42857,"k":"falling","v":0.42857,"x":0.90179,"p":[[0,32,0.0,0.82143,0.25254,0.53572,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,8,0,0,2,0,0,1,0,0,0,0,21],[4,32,0.125,0.81696,0.25564,0.42857,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,2,0,20],[8,32,0.25,0.90179,0.20958,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,26],[12,32,0.375,0.44643,0.13243,0.42857,0.42857,0.42857,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,27,0,0,3,0,0,0,0,0,0,0,1],[16,32,0.5,0.45088,0.06295,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0],[20,32,0.625,0.43304,0.02486,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0],[24,32,0.75,0.42857,0.03571,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0],[28,32,0.875,0.45088,0.10169,0.42857,0.42857,0.42857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,1],[32,32,1.0,0.43304,0.07563,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"falling","v":0.33482,"x":0.87054,"p":[[0,17,0.0,0.78571,0.27433,0.42857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,2,0,0,2,0,18],[4,17,0.2353,0.87054,0.24053,0.92857,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,0,0,0,24],[8,17,0.4706,0.81696,0.27254,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0,21],[12,17,0.7059,0.62498,0.29179,0.42857,0.42857,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,0,0,0,17,0,0,1,0,0,0,0,0,1,0,11],[16,17,0.9412,0.40179,0.17655,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,5,0,0,4,0,0,20,0,0,1,0,0,0,0,0,1,0,1],[17,17,1.0,0.33482,0.11633,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,7,0,0,7,0,0,18,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"61c99b91ed30156a","q":"For positive integers  $ n$ ,  $ f_n \\equal{} \\lfloor2^n\\sqrt {2008}\\rfloor \\plus{} \\lfloor2^n\\sqrt {2009}\\rfloor$ . Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence  $ f_1,f_2,\\ldots$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.48196,"x":1.0,"p":[[0,57,0.0,0.48196,0.44009,0.14214,0.14286,1.0,0.0,1.0,6,13,0,6,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,13],[4,57,0.0702,0.61161,0.4549,0.0,1.0,1.0,0.0,1.0,9,17,0,9,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,17],[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],[36,57,0.6316,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[57,57,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":3,"e":1.0,"k":"volatile","v":0.44196,"x":1.0,"p":[[0,10,0.0,0.44196,0.44658,0.0,0.14286,1.0,0.0,1.0,10,12,0,10,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,12],[4,10,0.4,0.57143,0.44176,0.10714,0.85707,1.0,0.0,1.0,8,13,0,8,0,4,0,0,2,0,0,0,0,0,0,0,0,1,0,0,4,0,13],[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],[10,10,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"39490636c24487b4","q":"Let  $n$  be a positive integer. Alice writes  $n$  real numbers  $a_1, a_2,\\dots, a_n$  in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ( $a_n$  is not a neighbor of  $a_1$ , nor vice versa). A number *changes sign* if it changes from being nonnegative to negative or vice versa. In terms of  $n$ , determine the maximum number of times that  $a_1$  can change sign, across all possible values of  $a_1,a_2,\\dots, a_n$  and all possible sequences of moves Alice may make.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.11161,"x":0.2366,"p":[[0,8,0.0,0.2366,0.2639,0.0,0.14286,0.42858,0.0,0.85714,13,0,0,13,0,7,0,0,1,0,0,4,0,0,4,0,0,2,0,0,1,0,0],[4,8,0.5,0.17857,0.18898,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,12,0,0,2,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[8,8,1.0,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.571,"k":"rising","v":0.14274,"x":0.66963,"p":[[0,17,0.0,0.17846,0.28346,0.0,0.0,0.17857,0.0,1.0,19,1,1,19,0,5,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,1],[4,17,0.2353,0.14274,0.19227,0.0,0.14143,0.14286,0.0,0.71429,15,0,2,15,0,11,0,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[8,17,0.4706,0.52232,0.23037,0.28571,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,4,0,0,4,0,0,1,0,0,10,0,0,10,0,0,2,0,0],[12,17,0.7059,0.66963,0.14033,0.57143,0.64286,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,10,0,0,4,0,2],[16,17,0.9412,0.64283,0.14726,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,13,0,0,8,0,0,5,0,1],[17,17,1.0,0.64731,0.15966,0.57143,0.57143,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,11,0,0,9,0,0,4,0,2]]}]},{"i":"b5b7a061c712782a","q":"Let $A B C D E$ be a convex pentagon such that the five vertices lie on a circle and the five sides are tangent to another circle inside the pentagon. There are $\\binom{5}{3}=10$ triangles which can be formed by choosing 3 of the 5 vertices. For each of these 10 triangles, mark its incenter. Prove that these 10 incenters lie on two concentric circles.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,8,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04017,"p":[[0,13,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.04017,0.10846,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"8feaf629f8f43d8b","q":"Let  $\\omega$  be the circumcircle of a triangle  $ABC$  ( ${AB\\ne AC}$ ),  $I$  be the incenter,  $P$  be the point on  $\\omega$  for which  $\\angle API=90^\\circ,$   $S$  be the intersection point of lines  $AP$  and  $BC,$   $W$  be the intersection point of line  $AI$  and  $\\omega.$  Line which passes through point  $W$  orthogonally to  $AW$  meets  $AP$  and  $BC$  at points  $D$  and  $E$  respectively. Prove that  $SD=IE.$  (Ye. Azarov)","t":[{"b":3,"e":0.0,"k":"flat","v":0.0625,"x":0.19192,"p":[[0,7,0.0,0.0625,0.14258,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.19192,0.23578,0.0,0.0,0.42857,0.0,0.57143,19,0,0,19,0,0,0,0,0,0,0,9,0,0,4,0,0,0,0,0,0,0,0],[7,7,1.0,0.13839,0.21866,0.0,0.0,0.32142,0.0,0.57143,22,0,0,22,0,1,0,0,1,0,0,4,0,0,4,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.10713,"p":[[0,25,0.0,0.10713,0.19558,0.0,0.0,0.03571,0.0,0.5714,24,0,0,24,0,1,0,0,0,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[4,25,0.16,0.04911,0.12683,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.08929,0.18814,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[12,25,0.48,0.05804,0.1551,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,25,0.64,0.03125,0.10555,0.0,0.0,0.0,0.0,0.5714,28,0,0,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,25,0.8,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b4a67b21f01f215","q":"Each point in the plane is assigned a real number. Suppose that for any nondegenerate triangle, the number at its incenter is the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are equal to each other.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,35,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,38,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,38,0.5263,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"3b071b212d006698","q":"Let  $ABC$  be a lattice triangle.  Prove that there exist integers  $m$  and  $n$  such that\n\\[ \n  m^2 + n^2 = (AB \\cdot AC)^2 + (BC \\cdot BA)^2 + (CA \\cdot CB)^2. \n\\]\n\n(A *lattice point* is a point whose  $x$ -coordinate and  $y$ -coordinate are both integers.  A *lattice triangle* is a triangle whose vertices are lattice points.)","t":[{"b":0,"e":1.0,"k":"flat","v":0.85713,"x":0.98214,"p":[[0,11,0.0,0.87053,0.29312,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,26],[4,11,0.3636,0.85713,0.26488,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,5,0,0,1,0,0,1,0,0,0,0,24],[8,11,0.7273,0.93304,0.1988,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[11,11,1.0,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30]]},{"b":1,"e":1.0,"k":"flat","v":0.73214,"x":0.93304,"p":[[0,29,0.0,0.73214,0.40524,0.25001,1.0,1.0,0.0,1.0,4,21,3,4,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,21],[4,29,0.1379,0.91964,0.23673,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,28],[8,29,0.2759,0.87054,0.26573,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,25],[12,29,0.4138,0.93304,0.15966,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,27],[16,29,0.5517,0.87054,0.3038,1.0,1.0,1.0,0.0,1.0,1,27,0,1,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[20,29,0.6897,0.83479,0.27921,0.67846,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,23],[24,29,0.8276,0.87944,0.25536,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,25],[28,29,0.9655,0.87052,0.2705,1.0,1.0,1.0,0.0,1.0,1,25,0,1,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,25],[29,29,1.0,0.87052,0.2953,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,26]]}]},{"i":"6491a57e18ba68b7","q":"Suppose  $ a,c,d \\in N$  and  $ d|a^2b\\plus{}c$  and  $ d\\geq a\\plus{}c$ \r\nProve that  $ d\\geq a\\plus{}\\sqrt[2b] {a}$","t":[{"b":6,"e":1.0,"k":"rising","v":0.55357,"x":0.85262,"p":[[0,25,0.0,0.55357,0.4828,0.0,0.9285,1.0,0.0,1.0,13,16,0,13,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,16],[4,25,0.16,0.81249,0.29329,0.71429,1.0,1.0,0.0,1.0,3,17,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,17],[8,25,0.32,0.84374,0.15303,0.71429,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,10,0,12],[12,25,0.48,0.85262,0.17316,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,5,0,16],[16,25,0.64,0.84821,0.19212,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,5,0,0,6,0,16],[20,25,0.8,0.84374,0.17627,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,10,0,0,6,0,14],[24,25,0.96,0.81247,0.20028,0.71429,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,7,0,0,4,0,14],[25,25,1.0,0.76784,0.21055,0.57143,0.78564,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,7,0,0,6,0,10]]},{"b":7,"e":0.0,"k":"volatile","v":0.04018,"x":0.52677,"p":[[0,9,0.0,0.52677,0.47974,0.0,0.71407,1.0,0.0,1.0,14,15,0,14,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,15],[4,9,0.4444,0.5,0.48181,0.0,0.42856,1.0,0.0,1.0,14,15,0,14,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,15],[8,9,0.8889,0.04018,0.17941,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[9,9,1.0,0.09375,0.249,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]}]},{"i":"7b30cca1f933f578","q":"Let $\\left(a_{n}\\right)_{n \\geqslant 1}$ be an increasing sequence of strictly positive integers and $k$ a strictly positive integer. Suppose that for some $r \\geqslant 1$, we have $a \\frac{r}{a_{r}}=k+1$. Show that there exists an integer $s \\geqslant 1$ such that $\\frac{s}{a_{s}}=k$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,5,0.0,0.10714,0.29667,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[4,5,0.8,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.11161,"x":1.0,"p":[[0,63,0.0,0.11161,0.25439,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,63,0.0635,0.70536,0.34981,0.39286,0.85714,1.0,0.0,1.0,3,15,0,3,0,0,0,0,5,0,0,2,0,0,1,0,0,3,0,0,3,0,15],[8,63,0.127,0.77232,0.36919,0.71429,1.0,1.0,0.0,1.0,4,21,0,4,0,1,0,0,2,0,0,0,0,0,0,0,0,3,0,0,1,0,21],[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],[60,63,0.9524,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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":"bbd93dd03f1e5741","q":"Let  $a_{i}$  and  $b_{i}$ , where  $i \\in \\{1,2, \\dots, 2005 \\}$ , be real numbers such that the inequality  $(a_{i}x-b_{i})^{2} \\ge \\sum_{j=1, j \\not= i}^{2005} (a_{j}x-b_{j})$  holds for all  $x \\in \\mathbb{R}$  and all  $i \\in \\{1,2, \\dots, 2005 \\}$ . Find the maximum possible number of positive numbers amongst  $a_{i}$  and  $b_{i}$ ,  $i \\in \\{1,2, \\dots, 2005 \\}$ .","t":[{"b":4,"e":0.71429,"k":"flat","v":0.66071,"x":0.75447,"p":[[0,8,0.0,0.66071,0.33264,0.39286,0.85714,1.0,0.0,1.0,2,10,0,2,0,2,0,0,4,0,0,2,0,0,5,0,0,0,0,0,7,0,10],[4,8,0.5,0.75447,0.27018,0.57143,0.85714,1.0,0.143,1.0,0,14,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,0,0,0,5,0,14],[8,8,1.0,0.71875,0.26362,0.42857,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,4,0,0,6,0,0,2,0,0,3,0,0,7,0,10]]},{"b":6,"e":1.0,"k":"rising","v":0.7991,"x":0.97321,"p":[[0,54,0.0,0.79911,0.27632,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,2,0,0,4,0,0,3,0,0,1,0,0,2,0,19],[4,54,0.0741,0.7991,0.22829,0.67857,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,5,0,0,5,0,14],[8,54,0.1481,0.87945,0.18937,0.85714,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,7,0,19],[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],[16,54,0.2963,0.95982,0.08917,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,25],[20,54,0.3704,0.91518,0.11769,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,13,0,17],[24,54,0.4444,0.89286,0.14725,0.85714,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,10,0,17],[28,54,0.5185,0.92857,0.07986,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,17],[32,54,0.5926,0.88393,0.10972,0.85714,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,15,0,12],[36,54,0.6667,0.90625,0.11071,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,16],[40,54,0.7407,0.86607,0.15126,0.85714,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,15,0,12],[44,54,0.8148,0.91964,0.10062,0.85714,1.0,1.0,0.5714,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,13,0,17],[48,54,0.8889,0.85267,0.15765,0.85714,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,16,0,10],[52,54,0.963,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28],[54,54,1.0,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24]]}]},{"i":"8c4b0396d048d203","q":"For each integer  $n>1$ , let  $p(n)$  denote the largest prime factor of  $n$ . Determine all triples  $(x, y, z)$  of distinct positive integers satisfying  \n- $x, y, z$  are in arithmetic progression,\n- $p(xyz) \\le 3$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.71429,"x":0.83929,"p":[[0,12,0.0,0.71429,0.31542,0.42857,0.78571,1.0,0.14286,1.0,0,15,0,0,0,3,0,0,4,0,0,2,0,0,3,0,0,4,0,0,1,0,15],[4,12,0.3333,0.75892,0.2683,0.57143,0.85714,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,3,0,0,3,0,0,3,0,0,3,0,0,6,0,13],[8,12,0.6667,0.79464,0.29001,0.71429,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,4,0,0,2,0,0,0,0,0,3,0,0,5,0,17],[12,12,1.0,0.83929,0.19805,0.71429,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,5,0,0,5,0,16]]},{"b":7,"e":0.42857,"k":"falling","v":0.25,"x":0.71426,"p":[[0,40,0.0,0.59375,0.31564,0.28571,0.57143,0.89286,0.14286,1.0,0,8,0,0,0,5,0,0,5,0,0,4,0,0,3,0,0,4,0,0,3,0,8],[4,40,0.1,0.63391,0.32131,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,4,0,0,6,0,0,3,0,0,1,0,0,4,0,0,5,0,9],[8,40,0.2,0.71426,0.24224,0.57143,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,2,0,0,1,0,0,9,0,0,4,0,0,8,0,7],[12,40,0.3,0.52665,0.33404,0.28571,0.571,0.85704,0.0,1.0,4,4,0,4,0,2,0,0,8,0,0,0,0,0,4,0,0,4,0,0,6,0,4],[16,40,0.4,0.54901,0.35744,0.14286,0.57143,0.85714,0.0,1.0,2,7,0,2,0,9,0,0,1,0,0,1,0,0,4,0,0,4,0,0,4,0,7],[20,40,0.5,0.41067,0.31691,0.14286,0.42857,0.57143,0.0,1.0,6,2,0,6,0,6,0,0,2,0,0,5,0,0,6,0,0,1,0,0,4,0,2],[24,40,0.6,0.41059,0.29833,0.14286,0.28571,0.71429,0.0,1.0,2,2,0,2,0,11,0,0,4,0,0,2,0,0,4,0,0,5,0,0,2,0,2],[28,40,0.7,0.33035,0.29327,0.14286,0.28571,0.4642,0.0,1.0,7,3,0,7,0,6,0,0,7,0,0,4,0,0,4,0,0,1,0,0,0,0,3],[32,40,0.8,0.3883,0.26066,0.14286,0.42857,0.57143,0.0,0.85714,3,0,0,3,0,9,0,0,2,0,0,7,0,0,5,0,0,3,0,0,3,0,0],[36,40,0.9,0.2767,0.21711,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,16,0,0,7,0,0,1,0,0,1,0,0,5,0,0,0,0,0],[40,40,1.0,0.25,0.22304,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,12,0,0,5,0,0,6,0,0,1,0,0,1,0,0,0,0,1]]}]},{"i":"5cbf2e0cc3b300d0","q":"Starting with the sequence  $F_1 = (1,2,3,4, \\ldots)$  of the natural numbers further sequences are generated as follows:  $F_{n+1}$  is created from  $F_n$  by the following rule: the order of elements remains unchanged, the elements from  $F_n$  which are divisible by  $n$  are increased by 1 and the other elements from  $F_n$  remain unchanged. Example:  $F_2 = (2,3,4,5 \\ldots)$  and  $F_3 = (3,3,5,5, \\ldots)$ . Determine all natural numbers  $n$  such that exactly the first  $n-1$  elements of  $F_n$  take the value  $n.$","t":[{"b":2,"e":0.57143,"k":"flat","v":0.65623,"x":0.83482,"p":[[0,18,0.0,0.66962,0.37191,0.28571,0.85714,1.0,0.0,1.0,4,13,1,4,0,2,0,0,3,0,0,0,0,0,4,0,0,1,0,0,5,0,13],[4,18,0.2222,0.72321,0.2285,0.57143,0.71429,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,0,0,0,1,0,0,7,0,0,7,0,0,10,0,5],[8,18,0.4444,0.65623,0.33476,0.57132,0.71429,0.89286,0.0,1.0,5,8,1,5,0,0,0,0,1,0,0,1,0,0,5,0,0,6,0,0,6,0,8],[12,18,0.6667,0.83482,0.17169,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,5,0,0,6,0,14],[16,18,0.8889,0.79908,0.17079,0.67857,0.78571,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,8,0,0,5,0,11],[18,18,1.0,0.80357,0.15047,0.71429,0.78571,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,7,0,9]]},{"b":3,"e":1.0,"k":"flat","v":0.62049,"x":0.86607,"p":[[0,48,0.0,0.625,0.40681,0.14286,0.85707,1.0,0.0,1.0,7,13,1,7,0,2,0,0,0,0,0,1,0,0,5,0,0,0,0,0,4,0,13],[4,48,0.0833,0.73214,0.30462,0.57143,0.85714,1.0,0.0,1.0,2,13,0,2,0,1,0,0,1,0,0,2,0,0,6,0,0,2,0,0,5,0,13],[8,48,0.1667,0.77232,0.2547,0.57143,0.85714,1.0,0.0,1.0,1,14,0,1,0,0,0,0,1,0,0,2,0,0,6,0,0,5,0,0,3,0,14],[12,48,0.25,0.64732,0.30301,0.53572,0.71429,0.85714,0.0,1.0,2,7,0,2,0,3,0,0,0,0,0,3,0,0,7,0,0,4,0,0,6,0,7],[16,48,0.3333,0.86607,0.20806,0.71429,1.0,1.0,0.14286,1.0,0,20,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,6,0,0,2,0,20],[20,48,0.4167,0.78122,0.23688,0.71429,0.85714,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,8,0,0,7,0,11],[24,48,0.5,0.70982,0.23003,0.57143,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,5,0,0,8,0,0,6,0,0,3,0,9],[28,48,0.5833,0.77225,0.21981,0.57143,0.85705,1.0,0.0,1.0,1,10,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,7,0,10],[32,48,0.6667,0.74995,0.19888,0.57143,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,6,0,0,7,0,8],[36,48,0.75,0.66963,0.24598,0.42857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,4,0,0,6,0,0,5,0,0,5,0,0,5,0,7],[40,48,0.8333,0.62049,0.2008,0.42857,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,2,0,0,8,0,0,9,0,0,7,0,0,2,0,4],[44,48,0.9167,0.75445,0.23754,0.5354,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,5,0,0,1,0,0,6,0,12],[48,48,1.0,0.76784,0.17406,0.57143,0.78571,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,7,0,0,9,0,7]]}]},{"i":"645a813b20657e8e","q":"For positive numbers  $ a_1,a_2,\\dots,a_n$ , we define\n\\[ A\\equal{}\\frac{a_1\\plus{}a_2\\plus{}\\cdots\\plus{}a_n}{n}, \\quad G\\equal{}\\sqrt[n]{a_1\\cdots a_n}, \\quad H\\equal{}\\frac{n}{a_1^{\\minus{}1}\\plus{}\\cdots\\plus{}a_n^{\\minus{}1}}\\]\nProve that\n\n(i)  $ \\frac{A}{H}\\leq \\minus{}1\\plus{}2\\left(\\frac{A}{G}\\right)^n$ , for n even\n\n(ii)  $ \\frac{A}{H}\\leq \\minus{}\\frac{n\\minus{}2}{n}\\plus{}\\frac{2(n\\minus{}1)}{n}\\left(\\frac{A}{G}\\right)^n$ , for  $ n$  odd","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"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],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,17,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"ec485788a27c2c80","q":"Let  $\\vartriangle ABC$  be such that  $\\angle BAC$  is acute. The line perpendicular on side  $AB$  from  $C$  and the line perpendicular on  $AC$  from  $B$  intersect the circumscribed circle of   $\\vartriangle ABC$  at  $D$  and  $E$  respectively. If  $DE = BC$  , calculate   $\\angle BAC$ .","t":[{"b":6,"e":0.42857,"k":"flat","v":0.40625,"x":0.58482,"p":[[0,23,0.0,0.52679,0.22142,0.42857,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,0,0,0,18,0,0,0,0,0,8,0,0,0,0,3],[4,23,0.1739,0.58482,0.25595,0.42857,0.71429,0.71429,0.0,1.0,1,4,1,1,0,3,0,0,0,0,0,10,0,0,0,0,0,14,0,0,0,0,4],[8,23,0.3478,0.48214,0.13242,0.42857,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,24,0,0,0,0,0,7,0,0,0,0,0],[12,23,0.5217,0.44196,0.05482,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0],[16,23,0.6957,0.40625,0.0724,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.42858,0.07142,0.42857,0.42857,0.42857,0.1429,0.71429,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0],[23,23,1.0,0.43748,0.03448,0.42857,0.42857,0.42857,0.42857,0.571,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.41072,"x":0.46875,"p":[[0,28,0.0,0.46875,0.18638,0.42857,0.42857,0.42858,0.0,1.0,1,1,1,1,0,2,0,0,0,0,0,22,0,0,0,0,0,6,0,0,0,0,1],[4,28,0.1429,0.42857,0.11845,0.42857,0.42857,0.42857,0.0,0.71429,1,0,1,1,0,1,0,0,0,0,0,27,0,0,1,0,0,2,0,0,0,0,0],[8,28,0.2857,0.42858,0.09449,0.42857,0.42857,0.42857,0.1429,0.85714,0,0,0,0,0,1,0,0,1,0,0,29,0,0,0,0,0,0,0,0,1,0,0],[12,28,0.4286,0.42857,0.10102,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,28,0,0,0,0,0,2,0,0,0,0,0],[16,28,0.5714,0.41965,0.04971,0.42857,0.42857,0.42857,0.1429,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.41964,0.08703,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,29,0,0,0,0,0,1,0,0,0,0,0],[24,28,0.8571,0.43303,0.09092,0.42857,0.42857,0.42857,0.14286,0.857,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,0,0,0,1,0,0],[28,28,1.0,0.41072,0.06916,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b225de3a78aca64e","q":"In a language $,$  an alphabet with  $25$  letters is used $;$  words are exactly all sequences of  $($  not necessarily different  $)$  letters of length  $17.$  Two ends of a paper strip are glued so that the strip forms a ring $;$  the strip bears a sequence of  $5^{18}$  letters $.$  Say that a word is singular if one can cut a piece bearing exactly that word from the strip $,$  but one cannot cut out two such non-overlapping pieces $.$  It is known that one can cut out  $5^{16}$  non-overlapping pieces each containing the same word $.$  Determine the largest possible number of singular words $.$  \n*(Bogdanov I.)*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,36,0.0,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,6,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,36,0.1111,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,4,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.07812,"p":[[0,32,0.0,0.07812,0.17254,0.0,0.0,0.0,0.0,0.71429,25,0,6,25,0,2,0,0,2,0,0,1,0,1,0,0,0,1,0,0,0,0,0],[4,32,0.125,0.05357,0.15872,0.0,0.0,0.0,0.0,0.71429,28,0,4,28,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[8,32,0.25,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,3,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,32,0.5,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"ef64170289e54787","q":"In hexagon  $ABCDEF$ , which is nonconvex but not self-intersecting, no pair of opposite sides are parallel.  The internal angles satisfy  $\\angle A=3\\angle D$ ,  $\\angle C=3\\angle F$ , and  $\\angle E=3\\angle B$ .  Furthermore  $AB=DE$ ,  $BC=EF$ , and  $CD=FA$ .  Prove that diagonals  $\\overline{AD}$ ,  $\\overline{BE}$ , and  $\\overline{CF}$  are concurrent.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,46,0.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,46,0.4348,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.05348,"p":[[0,27,0.0,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,1,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.05348,0.09935,0.0,0.0,0.14071,0.0,0.4286,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"162e5d4de7f6ee55","q":"In a triangle  $ABC ~(\\overline{AB} < \\overline{AC})$ , points  $D (\\neq A, B)$  and  $E (\\neq A, C)$  lies on side  $AB$  and  $AC$  respectively. Point  $P$  satisfies  $\\overline{PB}=\\overline{PD}, \\overline{PC}=\\overline{PE}$ .  $X (\\neq A, C)$  is on the arc  $AC$  of the circumcircle of triangle  $ABC$  not including  $B$ . Let  $Y (\\neq A)$  be the intersection of circumcircle of triangle  $ADE$  and line  $XA$ . Prove that  $\\overline{PX} = \\overline{PY}$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.05348,"p":[[0,23,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0267,0.05558,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.0625,"p":[[0,10,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ad9a864be370bcf8","q":"We take 1008 distinct integers between 1 and 2014 (inclusive).\n(i) Show that there exist three integers $a, b, c$ such that the gcd of $a$ and $b$ divides $c$ (the gcd or Greatest Common Divisor of two integers is the largest natural number that divides both of them).\n(ii) Show that there exist three integers $a, b, c$ such that the gcd of $a$ and $b$ does not divide $c$.","t":[{"b":3,"e":1.0,"k":"flat","v":0.41522,"x":0.67411,"p":[[0,31,0.0,0.41522,0.31004,0.14286,0.42859,0.60714,0.0,1.0,7,3,0,7,0,3,0,0,3,0,0,6,0,0,5,0,0,5,0,0,0,0,3],[4,31,0.129,0.67411,0.25812,0.42857,0.57143,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,13,0,0,7,0,0,0,0,0,0,0,12],[8,31,0.2581,0.54027,0.18461,0.42857,0.42857,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,19,0,0,9,0,0,0,0,0,0,0,4],[12,31,0.3871,0.57143,0.19561,0.42857,0.57143,0.57143,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,15,0,0,12,0,0,0,0,0,0,0,5],[16,31,0.5161,0.51786,0.16656,0.42857,0.42857,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,21,0,0,8,0,0,0,0,0,0,0,3],[20,31,0.6452,0.60267,0.21939,0.42857,0.57143,0.57143,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,14,0,0,11,0,0,0,0,0,0,0,7],[24,31,0.7742,0.5491,0.22047,0.42857,0.42857,0.57111,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,23,0,0,3,0,0,0,0,0,0,0,6],[28,31,0.9032,0.53572,0.17496,0.42857,0.42859,0.57143,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,19,0,0,9,0,0,0,0,0,1,0,3],[31,31,1.0,0.5625,0.17835,0.42857,0.57143,0.57143,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,14,0,0,14,0,0,0,0,0,0,0,4]]},{"b":7,"e":0.0,"k":"falling","v":0.11143,"x":0.375,"p":[[0,20,0.0,0.375,0.28515,0.14286,0.35714,0.57143,0.0,1.0,4,2,0,4,0,9,0,0,3,0,0,6,0,0,4,0,0,3,0,0,1,0,2],[4,20,0.2,0.33919,0.2363,0.14286,0.28571,0.46431,0.0,1.0,3,1,0,3,0,7,0,0,11,0,0,3,0,0,5,0,0,1,0,0,1,0,1],[8,20,0.4,0.13839,0.23551,0.0,0.0,0.14287,0.0,1.0,19,1,0,19,0,6,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[12,20,0.6,0.12491,0.21943,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,10,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[16,20,0.8,0.12043,0.18591,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,9,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[20,20,1.0,0.11143,0.2307,0.0,0.0,0.14071,0.0,1.0,22,1,0,22,0,6,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1]]}]},{"i":"5654fdd0a22a2ed2","q":"Let  $ABC$  be an acute triangle and let  $\\omega$  be its circumcircle. Let the tangents to  $\\omega$  through  $B,C$  meet each other at point  $P$ . Prove that the perpendicular bisector of  $AB$  and the parallel to  $AB$  through  $P$  meet at line  $AC$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.03571,"x":0.19628,"p":[[0,24,0.0,0.19628,0.25879,0.0,0.14143,0.2857,0.0,1.0,13,1,1,13,2,4,0,0,10,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[4,24,0.1667,0.18295,0.28847,0.0,0.07,0.1786,0.0,1.0,16,2,0,16,0,8,0,0,4,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[8,24,0.3333,0.16956,0.24075,0.0,0.14286,0.2857,0.0,1.0,13,2,1,13,0,10,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,24,0.5,0.05357,0.12242,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,24,0.6667,0.06696,0.10705,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.04446,0.08307,0.0,0.0,0.035,0.0,0.2857,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"volatile","v":0.10705,"x":0.83482,"p":[[0,19,0.0,0.29009,0.38549,0.0,0.14286,0.28571,0.0,1.0,12,7,0,12,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[4,19,0.2105,0.30793,0.35558,0.0,0.14286,0.4642,0.0,1.0,12,4,1,12,0,5,0,0,6,0,0,1,0,0,1,0,0,1,0,0,2,0,4],[8,19,0.4211,0.29455,0.33874,0.0,0.2857,0.28571,0.0,1.0,11,4,0,11,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[12,19,0.6316,0.83482,0.31765,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,23],[16,19,0.8421,0.15625,0.32608,0.0,0.0,0.14286,0.0,1.0,22,4,0,22,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[19,19,1.0,0.10705,0.29013,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3]]}]},{"i":"a1d413e3b19c68df","q":"BLR   Let $a, b, c$ be positive real numbers such that $a b+b c+c a \\leq 3 a b c$. Prove that  $$ \\sqrt{\\frac{a^{2}+b^{2}}{a+b}}+\\sqrt{\\frac{b^{2}+c^{2}}{b+c}}+\\sqrt{\\frac{c^{2}+a^{2}}{c+a}}+3 \\leq \\sqrt{2}(\\sqrt{a+b}+\\sqrt{b+c}+\\sqrt{c+a}) . $$","t":[{"b":1,"e":0.0,"k":"falling","v":0.0134,"x":0.38839,"p":[[0,18,0.0,0.28572,0.37965,0.0,0.14286,0.50002,0.0,1.0,15,4,0,15,0,7,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,4],[4,18,0.2222,0.38839,0.42892,0.0,0.14286,0.85714,0.0,1.0,13,6,0,13,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,6],[8,18,0.4444,0.29022,0.38877,0.0,0.07143,0.50107,0.0,1.0,16,5,0,16,0,5,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,5],[12,18,0.6667,0.09375,0.20079,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[16,18,0.8889,0.11152,0.24414,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[18,18,1.0,0.0134,0.04165,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.09366,"x":0.90624,"p":[[0,41,0.0,0.38839,0.43188,0.0,0.14286,0.85714,0.0,1.0,14,7,0,14,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,4,0,7],[4,41,0.0976,0.17411,0.3209,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,2],[8,41,0.1951,0.15179,0.3008,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,2],[12,41,0.2927,0.09366,0.26391,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[16,41,0.3902,0.90624,0.14111,0.85714,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,12,0,17],[20,41,0.4878,0.87052,0.16117,0.85714,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,12,0,14],[24,41,0.5854,0.84822,0.24984,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,9,0,17],[28,41,0.6829,0.78569,0.29667,0.71429,0.85714,1.0,0.0,1.0,3,13,0,3,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,10,0,13],[32,41,0.7805,0.83928,0.21651,0.85714,0.85714,1.0,0.0,1.0,1,10,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,17,0,10],[36,41,0.878,0.79017,0.23141,0.71429,0.85714,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,5,0,0,16,0,7],[40,41,0.9756,0.71875,0.29984,0.71429,0.85714,0.89286,0.0,1.0,1,8,0,1,0,4,0,0,1,0,0,0,0,0,1,0,0,7,0,0,10,0,8],[41,41,1.0,0.74998,0.26726,0.71429,0.85714,0.85714,0.0,1.0,3,5,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,8,0,0,15,0,5]]}]},{"i":"241a03ca6f6f02ee","q":"Given an integer  $n\\ge\\ 3$ , find the least positive integer  $k$ , such that there exists a set  $A$  with  $k$  elements, and  $n$  distinct reals  $x_{1},x_{2},\\ldots,x_{n}$  such that  $x_{1}+x_{2}, x_{2}+x_{3},\\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$  all belong to  $A$ .","t":[{"b":2,"e":0.71429,"k":"flat","v":0.58697,"x":0.63393,"p":[[0,9,0.0,0.61607,0.1357,0.57143,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,1,0,0,15,0,0,15,0,0,0,0,0],[4,9,0.4444,0.62495,0.11711,0.57142,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,0,12,0,0,2,0,0],[8,9,0.8889,0.58697,0.15541,0.57132,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,0,0,0,21,0,0,7,1,0,1,0,0],[9,9,1.0,0.63393,0.07936,0.57143,0.57143,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,12,0,0,1,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.44201,"x":0.66513,"p":[[0,37,0.0,0.63836,0.07975,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,13,0,0,1,0,0],[4,37,0.1081,0.53121,0.2237,0.53539,0.57143,0.60714,0.0,1.0,3,1,0,3,0,0,0,0,3,0,0,2,0,0,16,0,0,6,0,0,1,0,1],[8,37,0.2162,0.44201,0.24053,0.2857,0.57143,0.57143,0.0,0.71429,4,0,1,4,0,3,0,0,4,0,0,3,0,0,11,0,0,7,0,0,0,0,0],[12,37,0.3243,0.64731,0.0713,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0],[16,37,0.4324,0.62942,0.08647,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,9,0,0,2,0,0],[20,37,0.5405,0.6294,0.07876,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,11,0,0,1,0,0],[24,37,0.6486,0.62721,0.08516,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,1,0,11,0,0,1,0,0],[28,37,0.7568,0.63386,0.08706,0.57143,0.57143,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,10,0,0,2,0,0],[32,37,0.8649,0.64284,0.07144,0.57143,0.64286,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0],[36,37,0.973,0.62499,0.06917,0.57143,0.57143,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,12,0,0,0,0,0],[37,37,1.0,0.66513,0.10482,0.57143,0.64286,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,11,0,0,5,0,0]]}]},{"i":"f1cc99ee87f3d3db","q":"Let $m, n \\geqslant 1$ be two odd integers such that $m$ divides $n^{2}+2$ and $n$ divides $m^{2}+2$. Prove that $m$ and $n$ are both terms of the sequence $\\left(u_{n}\\right)_{n \\geqslant 1}$ defined by\n\n$$\n\\mathfrak{u}_{1}=\\mathrm{u}_{2}=1, \\quad \\mathrm{u}_{\\mathrm{n}}=4 \\mathrm{u}_{n-1}-\\mathrm{u}_{n-2} \\quad \\text { if } n \\geqslant 3\n$$","t":[{"b":1,"e":0.0,"k":"flat","v":0.21429,"x":0.34821,"p":[[0,17,0.0,0.23437,0.16193,0.14286,0.2857,0.42857,0.0,0.5,7,0,0,7,0,7,0,0,9,0,0,8,0,1,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.28125,0.1394,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,11,0,0,10,0,0,7,0,2,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.34821,0.15231,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,4,0,1,11,0,1,9,0,0,4,0,0,1,0,0,0,0,0],[12,17,0.7059,0.25894,0.15335,0.14286,0.2857,0.32164,0.0,0.57143,2,0,0,2,0,13,0,0,9,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[16,17,0.9412,0.29686,0.15576,0.14289,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,6,0,0,11,0,1,8,0,0,3,0,0,0,0,0,0,0,0],[17,17,1.0,0.21429,0.13832,0.14286,0.14286,0.28571,0.0,0.4286,5,0,0,5,0,12,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.24098,"x":0.31478,"p":[[0,18,0.0,0.29017,0.20038,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,9,0,0,8,0,0,6,0,0,4,0,0,0,0,0,1,0,0],[4,18,0.2222,0.24098,0.13098,0.14286,0.2143,0.32143,0.0,0.42857,2,0,0,2,0,14,0,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.31478,0.12863,0.25,0.28571,0.42857,0.0,0.5,1,0,0,1,0,7,0,0,9,0,0,14,0,1,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.3125,0.16146,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,7,0,0,10,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[16,18,0.8889,0.29018,0.12103,0.1429,0.28571,0.42857,0.0,0.42857,1,0,0,1,0,8,0,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.29464,0.13333,0.24999,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,6,0,0,13,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"f6f60e8861106c07","q":"A diameter  $AK$  is drawn for the circumscribed circle  $\\omega$  of an acute-angled triangle  $ABC$ , an arbitrary point  $M$  is chosen on the segment  $BC$ , the straight line  $AM$  intersects  $\\omega$  at point  $Q$ . The foot of the perpendicular drawn from  $M$  on  $AK$  is  $D$ , the tangent drawn to the circle  $\\omega$  through the point  $Q$ , intersects the straight line  $MD$  at  $P$ . A point  $L$  (different from  $Q$ ) is chosen on  $\\omega$  such that  $PL$  is tangent to  $\\omega$ . Prove that points  $L$ ,  $M$  and  $K$  lie on the same line.","t":[{"b":1,"e":0.0,"k":"flat","v":0.02679,"x":0.08036,"p":[[0,12,0.0,0.04893,0.06761,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,11,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,12,0.6667,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.08036,"p":[[0,31,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.08036,0.11811,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,31,0.3871,0.04911,0.08458,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.06687,0.08729,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d57a74009492a5cb","q":"Is there an arithmetic sequence with\n\na.  $2003$  \n\nb. infinitely many \n\nterms such that each term is a power of a natural number with a degree greater than  $1$ ?","t":[{"b":3,"e":0.0,"k":"falling","v":0.06696,"x":0.24107,"p":[[0,30,0.0,0.24107,0.2299,0.0,0.42857,0.42857,0.0,0.57143,15,0,1,15,0,0,0,0,0,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[4,30,0.1333,0.19197,0.21312,0.0,0.0,0.42857,0.0,0.57143,17,0,1,17,0,1,0,0,1,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[8,30,0.2667,0.21429,0.20517,0.0,0.28571,0.42857,0.0,0.4286,15,0,1,15,0,0,0,0,3,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.13839,0.19393,0.0,0.0,0.42857,0.0,0.4286,21,0,0,21,0,0,0,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.125,0.17405,0.0,0.0,0.2857,0.0,0.4286,20,0,0,20,0,2,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.06696,0.14719,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,1,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.08934,0.15882,0.0,0.0,0.07143,0.0,0.43,24,0,0,24,0,0,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.11607,0.17655,0.0,0.0,0.28571,0.0,0.4286,22,0,0,22,0,0,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.07589,0.15146,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.20094,"x":0.42862,"p":[[0,25,0.0,0.20094,0.20164,0.0,0.14286,0.42857,0.0,0.43,15,0,1,15,0,2,0,0,2,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.27232,0.2,0.0,0.42857,0.42857,0.0,0.4286,11,0,0,11,0,0,0,0,2,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.42862,0.00025,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.42857,1e-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],[16,25,0.64,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.42862,0.00025,0.42857,0.42857,0.42858,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.42857,1e-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],[25,25,1.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"89ba68f2c8ccaf90","q":"HUN   Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^{n}(x)=x$ is equal to $P(n)$ for every $n \\geq 1$, where $T^{n}$ denotes the $n$-fold application of $T$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.91071,"x":1.0,"p":[[0,37,0.0,0.91071,0.20124,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,5,0,23],[4,37,0.1081,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[8,37,0.2162,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,37,0.3243,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,37,0.4324,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,37,0.5405,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,37,0.6486,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[36,37,0.973,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[37,37,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]},{"b":3,"e":1.0,"k":"flat","v":0.87053,"x":0.95982,"p":[[0,18,0.0,0.95982,0.10853,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,28],[4,18,0.2222,0.94196,0.12807,0.96429,1.0,1.0,0.4286,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,6,0,24],[8,18,0.4444,0.91964,0.15126,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,5,0,0,1,0,24],[12,18,0.6667,0.89732,0.2237,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,23],[16,18,0.8889,0.87053,0.1488,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,3,0,17],[18,18,1.0,0.92411,0.11837,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,3,0,22]]}]},{"i":"206888f4c2c0e79e","q":"Let  $P_0=(a_0,b_0),P_1=(a_1,b_1),P_2=(a_2,b_2)$  be points on the plane such that  $P_0P_1P_2\\Delta$  contains the origin  $O$ . Show that the areas of triangles  $P_0OP_1,P_0OP_2,P_1OP_2$  form a geometric sequence in that order if and only if there exists a real number  $x$ , such that  $$ a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0 $$","t":[{"b":3,"e":0.57143,"k":"flat","v":0.4732,"x":0.65624,"p":[[0,21,0.0,0.4732,0.25614,0.28571,0.42859,0.60714,0.0,1.0,3,2,0,3,0,2,0,0,4,0,0,9,0,0,6,0,0,5,0,0,1,0,2],[4,21,0.1905,0.64953,0.14216,0.57142,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,7,0,0,18,0,0,0,1,1],[8,21,0.381,0.65624,0.12808,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,16,0,0,2,0,1],[12,21,0.5714,0.54464,0.14032,0.42857,0.42857,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,18,0,0,3,0,0,10,0,0,1,0,0],[16,21,0.7619,0.61831,0.11513,0.571,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,1,0,17,0,0,0,0,0],[20,21,0.9524,0.62501,0.1224,0.53582,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,20,0,0,0,0,0],[21,21,1.0,0.59151,0.11742,0.4286,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,9,0,0,9,1,0,13,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.54908,"x":0.70087,"p":[[0,21,0.0,0.54908,0.19597,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,7,0,0,6,0,0,13,0,0,1,0,0],[4,21,0.1905,0.5937,0.20859,0.42857,0.57143,0.71429,0.0,1.0,1,3,0,1,0,0,0,0,1,0,0,8,0,0,11,0,0,6,0,0,2,0,3],[8,21,0.381,0.54908,0.14772,0.42857,0.571,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,14,0,0,5,0,0,12,0,0,0,0,0],[12,21,0.5714,0.67404,0.17584,0.571,0.64271,0.74996,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,8,0,0,4,0,4],[16,21,0.7619,0.66963,0.19046,0.42859,0.71429,0.75,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,11,0,0,4,0,4],[20,21,0.9524,0.66513,0.18767,0.5354,0.71429,0.74996,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,8,0,0,7,0,0,9,0,0,4,0,4],[21,21,1.0,0.70087,0.17261,0.57143,0.71429,0.857,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,12,0,0,5,0,4]]}]},{"i":"d31484fa685ce330","q":"Given a triangle  $ABC$  where  $AC > BC$ ,  $D$  is located on the circumcircle of  $ABC$  such that  $D$  is the midpoint of the arc  $AB$  that contains  $C$ .  $E$  is a point on  $AC$  such that  $DE$  is perpendicular to  $AC$ . Prove that  $AE = EC + CB$ .","t":[{"b":5,"e":1.0,"k":"rising","v":0.47322,"x":0.92857,"p":[[0,19,0.0,0.47322,0.24856,0.42857,0.42857,0.42858,0.0,1.0,1,5,0,1,0,2,0,0,3,0,0,21,0,0,0,0,0,0,0,0,0,0,5],[4,19,0.2105,0.92857,0.11294,0.85713,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[8,19,0.4211,0.89731,0.18641,0.82143,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,1,0,23],[12,19,0.6316,0.85714,0.23145,0.71429,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,4,0,0,2,0,21],[16,19,0.8421,0.86158,0.1731,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,4,0,17],[19,19,1.0,0.89731,0.1864,0.92857,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,0,0,24]]},{"b":7,"e":0.4286,"k":"flat","v":0.41955,"x":0.54465,"p":[[0,20,0.0,0.54465,0.24337,0.42857,0.42857,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,3,0,0,18,0,0,1,0,0,2,0,0,2,0,5],[4,20,0.2,0.51786,0.26666,0.42857,0.42857,0.53574,0.0,1.0,1,5,0,1,0,1,0,0,4,0,0,18,0,0,0,0,0,0,0,0,3,0,5],[8,20,0.4,0.43751,0.04971,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0],[12,20,0.6,0.42857,1e-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],[16,20,0.8,0.41955,0.05021,0.42857,0.42857,0.42857,0.14,0.4286,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.42862,0.00025,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4c19095e6583d10c","q":"A function  $f$  from the set of positive real numbers to itself satisfies  $$ f(x + f(y) + xy) = xf(y) + f(x + y) $$  for all positive real numbers  $x$  and  $y$ . Prove that  $f(x) = x$  for all positive real numbers  $x$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.06687,"x":0.12947,"p":[[0,27,0.0,0.09367,0.11067,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,27,0.1481,0.06687,0.07965,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.11161,0.07771,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.125,0.09942,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,16,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.09821,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.09375,0.10479,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.09812,"p":[[0,40,0.0,0.09812,0.1037,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.09375,0.08459,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.08036,0.08703,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.09366,0.09177,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.07589,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"17cf7c8b4598af3c","q":"If $x, y, z$ are non-negative real numbers such that $x^{2}+y^{2}+z^{2}=x+y+z$, then show that:\n\n$$\n\\frac{x+1}{\\sqrt{x^{5}+x+1}}+\\frac{y+1}{\\sqrt{y^{5}+y+1}}+\\frac{z+1}{\\sqrt{z^{5}+z+1}} \\geq 3\n$$\n\nWhen does the equality hold?","t":[{"b":3,"e":0.0,"k":"flat","v":0.06696,"x":0.12499,"p":[[0,26,0.0,0.09375,0.12682,0.0,0.0,0.14287,0.0,0.42857,19,0,0,19,0,6,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.06696,0.10092,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.11607,0.1729,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,1,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,26,0.6154,0.12499,0.15461,0.0,0.07143,0.1786,0.0,0.571,16,0,0,16,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,26,0.7692,0.06696,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.10714,0.12877,0.0,0.0,0.17857,0.0,0.42857,17,0,0,17,0,7,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.08929,0.1171,0.0,0.0,0.14287,0.0,0.42857,18,0,0,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.571,"k":"flat","v":0.07143,"x":0.13394,"p":[[0,23,0.0,0.125,0.1915,0.0,0.0,0.14287,0.0,0.71429,19,0,0,19,0,6,0,0,2,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[4,23,0.1739,0.07143,0.12877,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.125,0.13717,0.0,0.14286,0.2857,0.0,0.4286,15,0,0,15,0,8,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.13393,0.14698,0.0,0.07143,0.2857,0.0,0.4286,16,0,0,16,0,4,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.08929,0.12753,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.13394,0.12847,0.0,0.14286,0.2857,0.0,0.286,14,0,0,14,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.08036,0.15126,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,5,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"7a40f03f3d86d639","q":"Given that  $\\{a_n\\}$  is a sequence of integers satisfying the following condition for all positive integral values of  $n$ :  $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$ . Find all possible values of  $a_1$  and  $a_2$","t":[{"b":5,"e":0.42857,"k":"falling","v":0.41071,"x":0.69643,"p":[[0,24,0.0,0.69643,0.28959,0.42857,0.71429,1.0,0.0,1.0,1,13,0,1,0,0,0,0,2,0,0,8,0,0,4,0,0,3,0,0,1,0,13],[4,24,0.1667,0.50447,0.22299,0.42857,0.42857,0.71429,0.0,1.0,1,3,1,1,0,1,0,0,2,0,0,19,0,0,0,0,0,6,0,0,0,0,3],[8,24,0.3333,0.48661,0.20473,0.42857,0.42857,0.42857,0.14286,1.0,0,4,0,0,0,1,0,0,2,0,0,24,0,0,1,0,0,0,0,0,0,0,4],[12,24,0.5,0.41964,0.07087,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,4,0,0,27,0,0,0,0,0,1,0,0,0,0,0],[16,24,0.6667,0.41518,0.04164,0.42857,0.42857,0.42857,0.28571,0.42857,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.41964,0.03458,0.42857,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.41071,0.04725,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.64723,"x":1.0,"p":[[0,69,0.0,0.64723,0.30316,0.42857,0.57143,1.0,0.0,1.0,1,12,1,1,0,1,0,0,2,0,0,10,0,0,4,0,0,2,0,0,0,0,12],[4,69,0.058,0.69197,0.26752,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,10,0,0,2,0,0,5,0,0,2,0,11],[8,69,0.1159,0.94643,0.10564,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,5,0,24],[12,69,0.1739,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,69,0.2899,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,69,0.3478,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,69,0.4058,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,69,0.4638,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,69,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[60,69,0.8696,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[64,69,0.9275,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"f5dcd319c8afc6fa","q":"The circles  $\\omega_1$  and  $\\omega_2$   intersect at points  $A$  and  $B$ , point  $M$  is the midpoint of  $AB$ . On line  $AB$  select points  $S_1$  and  $S_2$ . Let   $S_1X_1$  and  $S_1Y_1$  be tangents drawn from  $S_1$  to circle  $\\omega_1$ , similarly  $S_2X_2$  and  $S_2Y_2$  are tangents drawn from  $S_2$  to circles  $\\omega_2$ . Prove that if the point  $M$  lies on the line  $X_1X_2$ , then it also lies on the line  $Y_1Y_2$ .","t":[{"b":4,"e":0.71429,"k":"flat","v":0.56696,"x":0.71429,"p":[[0,30,0.0,0.56696,0.27545,0.60714,0.71429,0.71429,0.0,0.857,5,0,0,5,0,0,0,0,3,0,0,0,0,0,0,0,0,23,0,0,1,0,0],[4,30,0.1333,0.64719,0.20194,0.71429,0.71429,0.71429,0.0,1.0,2,1,1,2,0,0,0,0,1,0,0,2,0,0,0,0,0,26,0,0,0,0,1],[8,30,0.2667,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[12,30,0.4,0.69629,0.06913,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,30,0,0,0,0,0],[16,30,0.5333,0.70089,0.07457,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[20,30,0.6667,0.70536,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0],[24,30,0.8,0.69196,0.12428,0.71429,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[28,30,0.9333,0.69196,0.08828,0.71429,0.71429,0.71429,0.28571,0.7143,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,30,0,0,0,0,0],[30,30,1.0,0.70535,0.04971,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"flat","v":0.5938,"x":0.69643,"p":[[0,17,0.0,0.5938,0.2451,0.71429,0.71429,0.71429,0.0,0.71429,4,0,0,4,0,0,0,0,1,0,0,2,0,0,0,0,0,25,0,0,0,0,0],[4,17,0.2353,0.61594,0.2326,0.71429,0.71429,0.71429,0.0,0.71429,3,0,0,3,0,1,0,0,1,0,0,0,0,0,0,0,0,27,0,0,0,0,0],[8,17,0.4706,0.62487,0.23618,0.71429,0.71429,0.71429,0.0,0.71429,4,0,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0],[12,17,0.7059,0.66054,0.13241,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,26,0,0,0,0,0],[16,17,0.9412,0.69183,0.12426,0.71429,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0],[17,17,1.0,0.69643,0.09942,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0]]}]},{"i":"68072d9b1888b6f3","q":"Is it possible to find  $100$  positive integers not exceeding  $25000$  such that all pairwise sums of them are different?","t":[{"b":2,"e":0.14286,"k":"flat","v":0.04455,"x":0.12045,"p":[[0,15,0.0,0.04455,0.09052,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.12045,0.05184,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.1159,0.05568,0.14214,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.11152,"p":[[0,15,0.0,0.11152,0.16261,0.0,0.14286,0.14286,0.0,0.85714,14,0,0,14,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,15,0.2667,0.06697,0.09439,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.05357,0.11152,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"079d3e69fe0384a0","q":"A finite number of stones are *good* when the weight of each of these stones is less than the total weight of the rest. It is known that arbitrary  $n-1$  of the given  $n$  stones is *good*. Prove that it is possible to choose a *good* triple from these stones.","t":[{"b":6,"e":0.57143,"k":"falling","v":0.58034,"x":0.77232,"p":[[0,9,0.0,0.77232,0.19516,0.57143,0.71429,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,3,0,0,3,0,12],[4,9,0.4444,0.65622,0.14224,0.57143,0.57143,0.74999,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,1,0,0,6,0,2],[8,9,0.8889,0.71421,0.17133,0.57143,0.57143,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,4,0,0,5,0,6],[9,9,1.0,0.58034,0.03458,0.57143,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.67856,"x":0.77228,"p":[[0,12,0.0,0.67856,0.16367,0.57143,0.57143,0.75,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,3,0,0,3,0,5],[4,12,0.3333,0.73658,0.17538,0.57143,0.71429,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,4,0,0,6,0,7],[8,12,0.6667,0.77228,0.17811,0.57143,0.78564,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,4,0,0,7,0,9],[12,12,1.0,0.683,0.13237,0.57143,0.71429,0.71429,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,12,0,0,2,0,3]]}]},{"i":"a061b636f0166dda","q":"Let  $ABC$  be an acute triangle,  $D$  be the midpoint of  $BC$ . Bisectors of angles  $ADB$  and  $ADC$  intersect the circles circumscribed around the triangles  $ADB$  and  $ADC$  at points  $E$  and  $F$ , respectively. Prove that  $EF\\perp AD$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.08705,"p":[[0,23,0.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0692,0.09696,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,8,0,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0558,0.10372,0.0,0.0,0.08929,0.0,0.4286,23,0,0,23,1,4,0,2,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.08705,0.099,0.0,0.03571,0.14286,0.0,0.28571,16,0,0,16,1,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.04464,0.07523,0.0,0.0,0.07143,0.0,0.2857,22,0,0,22,3,5,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.08483,"p":[[0,11,0.0,0.05134,0.08033,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.01116,0.05085,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.08483,0.12808,0.0,0.0,0.14292,0.0,0.4286,21,0,0,21,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ea024c648b3f3a52","q":"We say  $p(x,y)\\in \\mathbb{R}\\left[x,y\\right]$  is *good* if for any  $y \\neq 0$  we have  $p(x,y) = p\\left(xy,\\frac{1}{y}\\right)$  . Prove that there are good polynomials  $r(x,y) ,s(x,y)\\in \\mathbb{R}\\left[x,y\\right]$  such that for any good polynomial  $p$  there is a  $f(x,y)\\in \\mathbb{R}\\left[x,y\\right]$  such that \\[f(r(x,y),s(x,y))= p(x,y)\\]\n\n*Proposed by Mohammad Ahmadi*","t":[{"b":1,"e":1.0,"k":"volatile","v":0.58929,"x":1.0,"p":[[0,8,0.0,0.65625,0.33855,0.39286,0.71429,1.0,0.14286,1.0,0,14,0,0,0,5,0,0,3,0,0,5,0,0,2,0,0,3,0,0,0,0,14],[4,8,0.5,0.58929,0.33834,0.28571,0.42859,1.0,0.14286,1.0,0,11,0,0,0,6,0,0,3,0,0,9,0,0,0,0,0,2,0,0,1,0,11],[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":3,"e":1.0,"k":"rising","v":0.58036,"x":1.0,"p":[[0,10,0.0,0.58482,0.37348,0.24999,0.57143,1.0,0.0,1.0,1,13,0,1,0,7,0,0,5,0,0,2,0,0,3,0,0,1,0,0,0,0,13],[4,10,0.4,0.58036,0.34615,0.25,0.42857,1.0,0.14286,1.0,0,11,0,0,0,8,0,0,1,0,0,8,0,0,2,0,0,1,0,0,1,0,11],[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":"6438c0063610e763","q":"Let  $ABCD$  be a convex quadrilateral. The lines parallel to  $AD$  and  $CD$  through the orthocentre  $H$  of  $ABC$  intersect  $AB$  and  $BC$  Crespectively at  $P$  and  $Q$ . prove that the perpendicular through  $H$  to th eline  $PQ$  passes through th eorthocentre of triangle  $ACD$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,26,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"84e61c9969ef6702","q":"There is  $n\\times n$  chessboard. Each square has a number between  $0$  and  $k$ . There is a button for each row and column, which increases the number of  $n$  numbers of the row or column the button represents(if the number of the square is  $k$ , then it becomes  $0$ ). If certain button is pressed, call it 'operation.'\n\nAnd we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number  $0$  within  $kn$  'operation's.","t":[{"b":5,"e":1.0,"k":"rising","v":0.74107,"x":1.0,"p":[[0,33,0.0,0.74107,0.36672,0.42857,1.0,1.0,0.14286,1.0,0,21,0,0,0,7,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,21],[4,33,0.1212,0.83481,0.26754,0.53539,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,23],[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":7,"e":0.2857,"k":"falling","v":0.23214,"x":0.79464,"p":[[0,17,0.0,0.79464,0.3008,0.4286,1.0,1.0,0.0,1.0,1,21,1,1,0,0,0,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,21],[4,17,0.2353,0.36159,0.32729,0.14286,0.28571,0.4642,0.0,1.0,7,5,0,7,0,6,0,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,5],[8,17,0.4706,0.40178,0.3163,0.24999,0.28571,0.46429,0.0,1.0,3,6,0,3,0,5,0,0,13,0,0,3,0,0,2,0,0,0,0,0,0,0,6],[12,17,0.7059,0.29018,0.31841,0.0,0.21428,0.42857,0.0,1.0,11,4,0,11,0,5,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,4],[16,17,0.9412,0.23214,0.16269,0.14286,0.2857,0.28571,0.0,0.57143,7,0,0,7,0,6,0,0,13,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[17,17,1.0,0.27231,0.19998,0.14286,0.2857,0.42857,0.0,1.0,3,1,0,3,0,12,0,0,8,0,0,6,0,0,2,0,0,0,0,0,0,0,1]]}]},{"i":"1cad4016c8e01323","q":"$s, k$  are positive integers.  $\\Bbb F$   is a  family of sets of order  $k$ .\n\nDoes there exist positive integer  $c$ , such that if  $|\\Bbb F|\\geq c$ , then there exist \\(A_1\\), \\(A_2\\), \\(\\dotsc\\), \\(A_s\\in\\Bbb F \\) such that for  each  \\(1\\leq i\\), \\(j\\), \\(p\\), \\(q\\leq s\\), \\(i\\ne j\\), \\(p\\ne q\\), we have  \\(A_i\\cap A_j=A_p\\cap A_q\\)?","t":[{"b":2,"e":0.57143,"k":"volatile","v":0.00446,"x":0.57362,"p":[[0,18,0.0,0.09822,0.19045,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,5,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[4,18,0.2222,0.20533,0.26468,0.0,0.0,0.46418,0.0,0.71429,17,0,0,17,0,4,0,0,2,0,0,1,0,0,5,0,0,3,0,0,0,0,0],[8,18,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.57362,0.16218,0.571,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,1,3,0,0,1,0,0,13,0,0,13,0,0,0,0,0],[18,18,1.0,0.57132,0.18578,0.57142,0.57143,0.71429,0.14,0.71429,0,0,0,0,0,3,0,0,3,0,0,0,0,0,11,0,0,15,0,0,0,0,0]]},{"b":5,"e":0.571,"k":"rising","v":0.13393,"x":0.29682,"p":[[0,16,0.0,0.14506,0.23036,0.0,0.0,0.19639,0.0,0.71429,21,0,0,21,0,3,0,0,0,0,1,2,0,0,4,0,0,1,0,0,0,0,0],[4,16,0.25,0.2031,0.26308,0.0,0.0,0.42858,0.0,0.71429,17,0,0,17,1,3,0,0,1,0,0,3,0,0,4,0,0,3,0,0,0,0,0],[8,16,0.5,0.13393,0.25238,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,5,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,1],[12,16,0.75,0.26777,0.22804,0.105,0.2857,0.42858,0.0,0.71429,8,0,0,8,0,7,0,0,7,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[16,16,1.0,0.29682,0.17858,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,1,11,0,0,4,0,0,9,0,0,5,0,0,0,0,0,0,0,0]]}]},{"i":"503c8aca83cdcfb1","q":"Do there exist 1990 relatively prime numbers such that all possible sums of two or more of these numbers are composite numbers?","t":[{"b":4,"e":0.2857,"k":"flat","v":0.3571,"x":0.49999,"p":[[0,9,0.0,0.49999,0.30304,0.28571,0.57143,0.71429,0.0,1.0,3,5,0,3,0,2,0,0,9,0,0,0,0,0,8,0,0,5,0,0,0,0,5],[4,9,0.4444,0.42411,0.30196,0.2857,0.28571,0.71429,0.0,1.0,4,3,0,4,0,2,0,0,14,0,0,0,0,0,3,0,0,4,0,0,2,0,3],[8,9,0.8889,0.3571,0.11838,0.28571,0.28571,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,23,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[9,9,1.0,0.37052,0.12805,0.28571,0.28571,0.46418,0.14286,0.57143,0,0,0,0,0,1,0,0,19,0,0,4,0,0,8,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.38391,"x":0.68302,"p":[[0,39,0.0,0.44195,0.2633,0.28571,0.42857,0.57143,0.0,1.0,4,1,0,4,0,1,0,0,9,0,0,3,0,0,9,0,0,2,0,0,3,0,1],[4,39,0.1026,0.38391,0.27066,0.2857,0.28571,0.57143,0.0,1.0,6,1,0,6,0,1,0,0,11,0,0,3,0,0,4,0,0,5,0,0,1,0,1],[8,39,0.2051,0.61606,0.20959,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,3,0,0,6,0,0,11,0,0,4,0,2],[12,39,0.3077,0.55356,0.24679,0.42857,0.57143,0.71429,0.0,1.0,3,1,0,3,0,0,0,0,4,0,0,2,0,0,10,0,0,9,0,0,3,0,1],[16,39,0.4103,0.58022,0.28101,0.39286,0.57143,0.75,0.0,1.0,3,3,0,3,0,0,0,0,5,0,0,2,0,0,7,0,0,7,0,0,5,0,3],[20,39,0.5128,0.66961,0.09064,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,19,0,0,2,0,0],[24,39,0.6154,0.64729,0.12871,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,11,0,0,18,0,0,0,0,1],[28,39,0.7179,0.67411,0.11971,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,15,0,0,5,0,0],[32,39,0.8205,0.66067,0.07788,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,21,0,0,0,0,0],[36,39,0.9231,0.67411,0.08171,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,22,0,0,1,0,0],[39,39,1.0,0.68302,0.09934,0.57143,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,18,0,0,4,0,0]]}]},{"i":"432fadabac7b40c0","q":"Let  $ABCD$  be a cyclic quadrilateral. The lines  $AD$  and  $BC$  intersect at  $P$  and the lines  $AB$  and  $CD$  intersect at  $Q$ . If  $\\angle APQ = 90^{\\circ}$ , prove that the perpendicular from  $P$  to  $AB$  bisects the diagonal  $BD$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"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],[4,40,0.1,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"069e781d1e256a7f","q":"Given a positive integer  $ n $ , let  $ A, B $  be two co-prime positive integers such that   $$  \\frac{B}{A} = \\left(\\frac{n\\left(n+1\\right)}{2}\\right)!\\cdot\\prod\\limits_{k=1}^{n}{\\frac{k!}{\\left(2k\\right)!}}  $$   Prove that  $ A $  is a power of  $ 2 $ .","t":[{"b":2,"e":0.85714,"k":"rising","v":0.29016,"x":0.85714,"p":[[0,42,0.0,0.29016,0.28229,0.14286,0.14286,0.46418,0.0,0.85714,6,0,0,6,0,14,0,0,2,0,0,2,0,0,3,0,0,1,0,0,4,0,0],[4,42,0.0952,0.42409,0.30195,0.14286,0.42857,0.71429,0.0,0.85714,5,0,0,5,0,6,0,0,2,0,0,7,0,0,2,0,0,4,0,0,6,0,0],[8,42,0.1905,0.61607,0.31427,0.28571,0.85707,0.85714,0.0,0.85714,2,0,0,2,0,5,0,0,2,0,0,1,0,0,1,0,0,4,0,0,17,0,0],[12,42,0.2857,0.6741,0.30143,0.64286,0.85714,0.85714,0.0,0.85714,3,0,0,3,0,2,0,0,1,0,0,2,0,0,0,0,0,3,0,0,21,0,0],[16,42,0.381,0.80803,0.13175,0.85714,0.85714,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,27,0,0],[20,42,0.4762,0.84375,0.05486,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0],[24,42,0.5714,0.83482,0.1017,0.85714,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,30,0,0],[28,42,0.6667,0.83928,0.07783,0.85714,0.85714,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,30,0,0],[32,42,0.7619,0.85713,3e-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],[36,42,0.8571,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0],[40,42,0.9524,0.85714,2e-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],[42,42,1.0,0.85268,0.02486,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.05357,"x":0.39732,"p":[[0,15,0.0,0.39732,0.34761,0.14286,0.21429,0.74996,0.0,0.85714,7,0,0,7,0,9,0,0,2,0,0,0,0,0,2,0,0,4,0,0,8,0,0],[4,15,0.2667,0.08929,0.16269,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,15,0.5333,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9d7f6a2891e18470","q":"Let $a_{0}, a_{1}, \\ldots, a_{d}$ be integers such that $\\operatorname{GCD}\\left(a_{0}, a_{1}\\right)=1$. For any integer $n \\geqslant 1$, we define\n\n$$\nu_{n}=\\sum_{k=0}^{d} a_{k} \\varphi(n+k)\n$$\n\nProve that 1 is the only natural number that divides all integers $u_{n}$.\nWe recall that $\\varphi(n+k)$ is the number of natural numbers $\\ell<n+k$ such that $\\operatorname{GCD}(\\ell, n+k)=1$.","t":[{"b":6,"e":0.42857,"k":"flat","v":0.15179,"x":0.41517,"p":[[0,23,0.0,0.27232,0.27283,0.0,0.14286,0.42858,0.0,0.85714,9,0,0,9,0,10,0,0,2,0,0,4,0,0,2,0,0,3,0,0,2,0,0],[4,23,0.1739,0.41517,0.3376,0.14286,0.35714,0.71429,0.0,1.0,5,3,0,5,0,9,0,0,2,0,0,4,0,0,1,0,0,5,0,0,3,0,3],[8,23,0.3478,0.25445,0.25934,0.14286,0.14286,0.32143,0.0,1.0,7,1,0,7,0,13,0,0,4,0,0,3,0,0,1,0,0,2,0,0,1,0,1],[12,23,0.5217,0.26785,0.3149,0.0,0.14286,0.46418,0.0,1.0,11,2,0,11,0,10,0,0,2,0,0,1,0,0,2,0,0,3,0,0,1,0,2],[16,23,0.6957,0.36605,0.3498,0.14286,0.14286,0.74996,0.0,1.0,7,1,0,7,0,11,0,0,2,0,0,0,0,0,2,0,0,2,0,0,7,0,1],[20,23,0.8696,0.35712,0.35354,0.0,0.28571,0.60682,0.0,1.0,10,3,0,10,0,5,0,0,4,0,0,3,0,0,2,0,0,1,0,0,4,0,3],[23,23,1.0,0.15179,0.12339,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,14,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.14732,"x":0.45973,"p":[[0,51,0.0,0.375,0.37754,0.0,0.1429,0.71429,0.0,1.0,10,5,0,10,0,7,0,0,1,0,0,2,0,0,3,0,0,2,0,0,2,0,5],[4,51,0.0784,0.45973,0.38429,0.14286,0.35714,0.85714,0.0,1.0,6,6,0,6,0,8,0,0,2,0,0,2,0,0,1,0,0,3,0,0,4,0,6],[8,51,0.1569,0.25445,0.25933,0.14286,0.14286,0.42857,0.0,1.0,7,1,0,7,0,14,0,0,2,0,0,3,0,0,3,0,0,1,0,0,1,0,1],[12,51,0.2353,0.34366,0.36402,0.0,0.14286,0.50002,0.0,1.0,9,5,0,9,0,8,0,0,4,0,0,3,0,0,0,0,0,1,0,0,2,0,5],[16,51,0.3137,0.2499,0.26728,0.14214,0.14286,0.28571,0.0,1.0,7,1,0,7,0,14,0,0,4,0,0,2,0,0,1,0,0,1,0,0,2,0,1],[20,51,0.3922,0.23204,0.27376,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,14,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,1],[24,51,0.4706,0.16062,0.20749,0.0,0.14286,0.17857,0.0,1.0,13,1,0,13,0,11,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[28,51,0.549,0.25892,0.27533,0.0,0.14286,0.42858,0.0,0.85714,10,0,0,10,0,10,0,0,2,0,0,3,0,0,2,0,0,3,0,0,2,0,0],[32,51,0.6275,0.23661,0.23313,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,15,0,0,4,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[36,51,0.7059,0.19196,0.23854,0.10714,0.14286,0.14286,0.0,1.0,8,2,0,8,0,17,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[40,51,0.7843,0.14732,0.18029,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[44,51,0.8627,0.24989,0.13367,0.14286,0.14286,0.28571,0.14,0.57143,0,0,0,0,0,17,0,0,8,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[48,51,0.9412,0.19643,0.09942,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,21,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.20089,0.14223,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,15,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"489d062d95ad3de8","q":"We call a sequence of integers a *Fibonacci-type sequence* if it is infinite in both ways and  $a_{n}=a_{n-1}+a_{n-2}$  for any  $n\\in\\mathbb{Z}$ . How many *Fibonacci-type sequences* can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than  $N$  ? (two sequences are considered to be the same if they differ only by shifting of indices)\n\n*Proposed by I. Pevzner*","t":[{"b":1,"e":0.0,"k":"flat","v":0.04018,"x":0.18304,"p":[[0,5,0.0,0.18304,0.13474,0.0,0.28571,0.28571,0.0,0.28571,11,0,0,11,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.20535,"x":0.27232,"p":[[0,35,0.0,0.21875,0.11837,0.24999,0.28571,0.28571,0.0,0.28571,7,0,0,7,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.20535,0.12846,0.0,0.28571,0.28571,0.0,0.28571,9,0,0,9,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.26339,0.0724,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.25446,0.08552,0.28571,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.25446,0.08552,0.28571,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.20982,0.12364,0.10714,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.27232,0.05486,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.22768,0.11214,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.24107,0.0974,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,2,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ee97a47fb633a3f0","q":"Consider the following operation on positive real numbers written on a blackboard:\n\nChoose a number $r$ written on the blackboard, erase that number, and then write a pair of positive real numbers $a$ and $b$ satisfying the condition $2 r^{2}=a b$ on the board.\n\nAssume that you start out with just one positive real number $r$ on the blackboard, and apply this operation $k^{2}-1$ times to end up with $k^{2}$ positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed $k r$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.03571,"x":0.4375,"p":[[0,19,0.0,0.20536,0.38785,0.0,0.0,0.07143,0.0,1.0,24,6,1,24,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[4,19,0.2105,0.18304,0.36111,0.0,0.0,0.07143,0.0,1.0,24,5,3,24,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,19,0.4211,0.20536,0.38785,0.0,0.0,0.07143,0.0,1.0,24,6,3,24,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[12,19,0.6316,0.4375,0.49608,0.0,0.0,1.0,0.0,1.0,18,14,3,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[16,19,0.8421,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.25,"x":1.0,"p":[[0,20,0.0,0.25,0.43301,0.0,0.0,0.25,0.0,1.0,24,8,2,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[4,20,0.2,0.25894,0.43071,0.0,0.0,0.4645,0.0,1.0,23,8,2,23,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[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":"879faeb5dc3e2f5e","q":"Given a positive integer  $a$ , prove that  $\\sigma(am) < \\sigma(am + 1)$  for infinitely many positive integers  $m$ . (Here  $\\sigma(n)$  is the sum of all positive divisors of the positive integer number  $n$ .) \n\n*Vlad Matei*","t":[{"b":2,"e":1.0,"k":"rising","v":0.59371,"x":0.92856,"p":[[0,16,0.0,0.7098,0.28901,0.57132,0.85707,0.89286,0.0,1.0,1,8,1,1,0,3,0,0,0,0,0,3,0,0,4,0,0,3,0,0,10,0,8],[4,16,0.25,0.69195,0.29904,0.57143,0.78564,0.89286,0.0,1.0,2,8,1,2,0,2,0,0,1,0,0,1,0,0,6,0,0,4,0,0,8,0,8],[8,16,0.5,0.59371,0.30952,0.28571,0.64286,0.85714,0.0,1.0,2,5,2,2,0,3,0,0,4,0,0,2,0,0,5,0,0,5,0,0,6,0,5],[12,16,0.75,0.92856,0.07144,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[16,16,1.0,0.89284,0.06187,0.85714,0.85714,0.89286,0.857,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8]]},{"b":3,"e":0.85714,"k":"rising","v":0.63836,"x":0.93749,"p":[[0,25,0.0,0.63836,0.27196,0.42857,0.71429,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,3,0,0,3,0,0,6,0,0,3,0,0,11,0,3],[4,25,0.16,0.67409,0.24284,0.57132,0.71429,0.85714,0.14286,1.0,0,6,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,5,0,0,6,0,6],[8,25,0.32,0.84375,0.22689,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,0,5,0,18],[12,25,0.48,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[16,25,0.64,0.93302,0.07131,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[20,25,0.8,0.91069,0.07785,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[24,25,0.96,0.89731,0.0735,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10],[25,25,1.0,0.90623,0.06786,0.85714,0.85714,1.0,0.857,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,11]]}]},{"i":"f5331e6b6b976b9b","q":"Let  $AD$  be an altitude in a acute triangle  $ABC$ . Points  $E, F$  are the projections of  $D$  onto lines  $AB$  and  $AC$  respectively. Points  $K, L, M$  are the midpoints of  $BD, BC, DC$  respectively. Prove that the lines  $EK, AL, FM$  are concurrent.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,23,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[23,23,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,34,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2af47ba3cec984eb","q":"Let $a, b$ and $c$ be positive real numbers such that $\\min \\{a b, b c, c a\\} \\geqslant 1$. Prove that  $$ \\sqrt[3]{\\left(a^{2}+1\\right)\\left(b^{2}+1\\right)\\left(c^{2}+1\\right)} \\leqslant\\left(\\frac{a+b+c}{3}\\right)^{2}+1 $$","t":[{"b":2,"e":0.0,"k":"flat","v":0.04911,"x":0.11161,"p":[[0,26,0.0,0.11161,0.22513,0.0,0.0,0.14287,0.0,1.0,23,1,0,23,0,2,0,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[4,26,0.1538,0.08482,0.21387,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[8,26,0.3077,0.08482,0.1984,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[12,26,0.4615,0.09822,0.18364,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,1,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,26,0.6154,0.08482,0.17445,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[20,26,0.7692,0.04911,0.12169,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,26,0.9231,0.09375,0.12682,0.0,0.0,0.14292,0.0,0.42857,19,0,0,19,0,6,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.06696,0.11837,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.12938,"p":[[0,28,0.0,0.03571,0.12372,0.0,0.0,0.0,0.0,0.5714,29,0,0,29,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,28,0.1429,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,28,0.2857,0.05804,0.12807,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,28,0.4286,0.12938,0.19351,0.0,0.0,0.2857,0.0,0.57143,20,0,0,20,0,3,0,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[16,28,0.5714,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7702c35b41d6a6c0","q":"Let $ABC$ be a triangle and $\\Omega$ its circumcircle. Let $X$ be the point of intersection of the tangents to $\\Omega$ at $B$ and $C$. Let $\\varphi$ be the angle $\\widehat{B A X}$ and $\\mu$ be the angle $\\widehat{X A C}$. Let $Y$ be the point on the line $(A X)$ such that $\\widehat{A C Y}=\\varphi$. Show that $\\widehat{A B Y}=\\mu$.","t":[{"b":2,"e":0.85714,"k":"flat","v":0.89284,"x":0.99107,"p":[[0,11,0.0,0.8973,0.25062,0.96429,1.0,1.0,0.0,1.0,2,24,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,24],[4,11,0.3636,0.89284,0.1786,0.85713,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,5,0,21],[8,11,0.7273,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[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":4,"e":1.0,"k":"flat","v":0.86607,"x":0.96429,"p":[[0,23,0.0,0.86607,0.17105,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,6,0,17],[4,23,0.1739,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[8,23,0.3478,0.90179,0.16536,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,3,0,22],[12,23,0.5217,0.93304,0.18893,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,25],[16,23,0.6957,0.95982,0.11425,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,28],[20,23,0.8696,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26],[23,23,1.0,0.96429,0.07986,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26]]}]},{"i":"51c7a76e43a05cff","q":"Let  $n\\geq 1$  be a positive integer. We say that an integer  $k$  is a *fan*of  $n$  if  $0\\leq k\\leq n-1$  and there exist integers  $x,y,z\\in\\mathbb{Z}$  such that\n\\begin{align*}\nx^2+y^2+z^2 &\\equiv 0 \\pmod n;\nxyz &\\equiv k \\pmod n.\n\\end{align*}\nLet  $f(n)$  be the number of fans of  $n$ . Determine  $f(2020)$ .","t":[{"b":1,"e":0.42857,"k":"flat","v":0.80803,"x":0.99554,"p":[[0,57,0.0,0.82134,0.28592,0.67857,1.0,1.0,0.14,1.0,0,21,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,1,0,0,2,0,21],[4,57,0.0702,0.80803,0.29148,0.57143,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,4,0,0,2,0,0,2,0,0,1,0,0,1,0,21],[8,57,0.1404,0.87499,0.21653,0.82143,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,2,0,22],[12,57,0.2105,0.9107,0.18816,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,25],[16,57,0.2807,0.89732,0.18977,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,0,3,0,23],[20,57,0.3509,0.85713,0.24224,0.78571,1.0,1.0,0.2857,1.0,0,23,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,0,0,0,1,0,23],[24,57,0.4211,0.95536,0.12078,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[28,57,0.4912,0.92409,0.1923,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,27],[32,57,0.5614,0.95089,0.14987,1.0,1.0,1.0,0.28571,1.0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,28],[36,57,0.6316,0.94643,0.17768,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,29],[40,57,0.7018,0.95089,0.14556,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,28],[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],[48,57,0.8421,0.91964,0.18877,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,26],[52,57,0.9123,0.90625,0.23313,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,27],[56,57,0.9825,0.85713,0.23421,0.71429,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,22],[57,57,1.0,0.80804,0.26392,0.57143,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,1,0,0,4,0,0,4,0,0,1,0,0,2,0,19]]},{"b":4,"e":1.0,"k":"flat","v":0.78125,"x":1.0,"p":[[0,65,0.0,0.90179,0.22711,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,25],[4,65,0.0615,0.78125,0.33309,0.57143,1.0,1.0,0.0,1.0,2,20,1,2,0,1,0,0,3,0,0,1,0,0,2,0,0,1,0,0,2,0,20],[8,65,0.1231,0.91071,0.21053,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,26],[12,65,0.1846,0.92856,0.17499,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,27],[16,65,0.2462,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[20,65,0.3077,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[24,65,0.3692,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,65,0.4308,0.95089,0.15407,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,29],[32,65,0.4923,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[36,65,0.5538,0.97321,0.10374,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,30],[40,65,0.6154,0.97768,0.1017,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[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],[52,65,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[56,65,0.8615,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[60,65,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[64,65,0.9846,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"ac248609c8938125","q":"Let  $\\Delta ABC$  be a triangle with  $AB=AC$  and  $\\angle A = \\alpha$ , and let  $O,H$  be its circumcenter and orthocenter, respectively. If  $P,Q$  are points on  $AB$  and  $AC$ , respectively, such that  $APHQ$  forms a rhombus, determine  $\\angle POQ$  in terms of  $\\alpha$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.16955,"x":0.23661,"p":[[0,36,0.0,0.20982,0.11285,0.14286,0.14288,0.28571,0.0,0.4286,3,0,0,3,0,14,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.23661,0.09852,0.14286,0.2857,0.28571,0.0,0.4286,1,0,0,1,0,12,0,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.20518,0.08715,0.14286,0.14286,0.2857,0.14,0.42857,0,0,0,0,0,20,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.23661,0.08458,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,13,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.19169,0.07685,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,19,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.19634,0.0779,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,18,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.20973,0.08744,0.14286,0.14286,0.28571,0.14,0.4286,0,0,0,0,0,19,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.16955,0.06625,0.14286,0.14286,0.14287,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.21424,0.08766,0.14286,0.14288,0.2857,0.14,0.43,0,0,0,0,0,18,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.20987,0.11294,0.14286,0.14286,0.28571,0.0,0.43,2,0,0,2,0,17,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.20089,"x":0.27683,"p":[[0,42,0.0,0.21875,0.09438,0.14286,0.2143,0.28571,0.0,0.4286,1,0,0,1,0,15,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.21866,0.12369,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,0,16,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,42,0.1905,0.20089,0.1063,0.14286,0.14288,0.2857,0.0,0.4286,3,0,0,3,0,15,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.23661,0.12169,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,13,0,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.20982,0.14719,0.14286,0.2857,0.2857,0.0,0.57143,7,0,0,7,0,8,0,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,42,0.4762,0.24098,0.09749,0.14286,0.2857,0.2857,0.14,0.42857,0,0,0,0,0,14,0,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.24553,0.13474,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,12,0,0,13,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[28,42,0.6667,0.20982,0.10092,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,15,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.26339,0.11904,0.14286,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,8,0,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.25897,0.12085,0.14286,0.2857,0.28571,0.0,0.43,2,0,0,2,0,9,0,0,14,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.21875,0.08736,0.14286,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,14,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.27683,0.10075,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4edc5be52e31d67b","q":"17. (USS) ${ }^{\\text {IMO5 }}$ Three equal circles touch the sides of a triangle and have one common point $O$. Show that the center of the circle inscribed in and of the circle circumscribed about the triangle $A B C$ and the point $O$ are collinear.","t":[{"b":4,"e":0.42857,"k":"volatile","v":0.29893,"x":0.89286,"p":[[0,12,0.0,0.79018,0.38297,0.92857,1.0,1.0,0.0,1.0,5,24,0,5,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[4,12,0.3333,0.89286,0.27894,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,27],[8,12,0.6667,0.41964,0.27185,0.28571,0.28571,0.42858,0.14286,1.0,0,5,0,0,0,4,0,0,16,0,0,6,0,0,0,0,0,1,0,0,0,0,5],[12,12,1.0,0.29893,0.09038,0.2857,0.28571,0.32143,0.14,0.4286,0,0,0,0,0,5,0,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.72768,"x":1.0,"p":[[0,24,0.0,0.72768,0.41088,0.25,1.0,1.0,0.0,1.0,4,22,0,4,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,22],[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":"6ae0896288045f5b","q":"Let  $A_1A_2A_3A_4A_5$  be a convex pentagon. Suppose rays $A_2A_3$  and  $A_5A_4$  meet at the point  $X_1$ . Define  $X_2$ ,  $X_3$ ,  $X_4$ ,  $X_5$  similarly. Prove that  $$ \\displaystyle\\prod_{i=1}^{5} X_iA_{i+2} = \\displaystyle\\prod_{i=1}^{5} X_iA_{i+3} $$  where the indices are taken modulo 5.","t":[{"b":1,"e":1.0,"k":"flat","v":0.99107,"x":1.0,"p":[[0,10,0.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[4,10,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[8,10,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":4,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,22,0.0,0.95089,0.18766,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[4,22,0.1818,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[8,22,0.3636,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,22,0.5455,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[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],[20,22,0.9091,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"267cf259f4b1266e","q":"In triangle  $ABC$ ,  $\\angle B=2\\angle C$  and  $\\angle A>90^\\circ$ . Let  $D$  be the point on the line  $AB$  such that  $CD$  is perpendicular to  $AC$ , and let  $M$  be the midpoint of  $BC$ . Prove that  $\\angle AMB=\\angle DMC$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"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],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"7ebe1d18a2dacf96","q":"Let  $ABC$  be a triangle with  $\\angle A < \\angle B \\le  \\angle C$ ,  $M$  and  $N$  the midpoints of sides  $CA$  and  $AB$ , respectively, and  $P$  and  $Q$  the projections of  $B$  and  $C$  on the medians  $CN$  and  $BM$ , respectively. Prove that the quadrilateral  $MNPQ$  is cyclic.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.35264,"x":0.52228,"p":[[0,13,0.0,0.41065,0.22512,0.2857,0.42857,0.57143,0.0,1.0,3,1,0,3,0,2,0,0,9,0,0,5,0,0,11,0,0,0,0,0,1,0,1],[4,13,0.3077,0.35264,0.22295,0.14286,0.42857,0.57143,0.0,0.57143,5,0,0,5,0,6,0,0,4,0,0,3,0,0,14,0,0,0,0,0,0,0,0],[8,13,0.6154,0.45972,0.16643,0.42857,0.571,0.57143,0.0,0.57143,1,0,0,1,0,4,0,0,1,0,0,7,0,0,19,0,0,0,0,0,0,0,0],[12,13,0.9231,0.52228,0.09179,0.42859,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,21,0,0,1,0,0,0,0,0],[13,13,1.0,0.48213,0.15464,0.42857,0.57143,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,3,0,0,7,0,0,18,0,0,0,0,0,1,0,0]]},{"b":3,"e":0.71429,"k":"flat","v":0.375,"x":0.56239,"p":[[0,30,0.0,0.375,0.23622,0.14286,0.28571,0.57143,0.0,1.0,1,1,0,1,0,11,0,0,5,0,0,2,0,0,10,0,0,2,0,0,0,0,1],[4,30,0.1333,0.44636,0.16263,0.28571,0.571,0.57143,0.14286,0.57143,0,0,0,0,0,5,0,0,4,0,0,5,0,0,18,0,0,0,0,0,0,0,0],[8,30,0.2667,0.44635,0.21646,0.28571,0.571,0.57143,0.0,1.0,2,1,0,2,0,5,0,0,2,0,0,4,0,0,18,0,0,0,0,0,0,0,1],[12,30,0.4,0.50002,0.13359,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,7,0,0,20,0,0,1,0,0,0,0,0],[16,30,0.5333,0.56239,0.12339,0.571,0.57141,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,22,0,0,1,0,0,1,0,1],[20,30,0.6667,0.52224,0.11629,0.571,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,1,0,0,5,0,0,25,0,0,0,0,0,0,0,0],[24,30,0.8,0.54905,0.12427,0.571,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,3,0,0,24,0,0,1,0,0,0,0,1],[28,30,0.9333,0.51779,0.11149,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,6,0,0,20,0,0,2,0,0,0,0,0],[30,30,1.0,0.51777,0.1324,0.571,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,3,0,0,3,0,0,24,0,0,1,0,0,0,0,0]]}]},{"i":"a3d86d4f07b9e1a3","q":"Let  $A$  be point in the exterior of the circle  $\\mathcal C$ . Two lines passing through  $A$  intersect the circle  $\\mathcal C$  in points  $B$  and  $C$  (with  $B$  between  $A$  and  $C$ ) respectively in  $D$  and  $E$  (with  $D$  between  $A$  and  $E$ ). The parallel from  $D$  to  $BC$  intersects the second time the circle  $\\mathcal C$  in  $F$ . Let  $G$  be the second point of intersection between the circle  $\\mathcal C$  and the line  $AF$  and  $M$  the point in which the lines  $AB$  and  $EG$  intersect. Prove that \r\n\\[ \\frac 1{AM} = \\frac 1{AB} + \\frac 1{AC}. \\]","t":[{"b":3,"e":0.2857,"k":"rising","v":0.07589,"x":0.3125,"p":[[0,24,0.0,0.07589,0.15965,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,24,0.1667,0.08929,0.17768,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,24,0.3333,0.27678,0.06121,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,1,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.3125,0.06622,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[16,24,0.6667,0.30357,0.05923,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.3125,0.07523,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,25,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[24,24,1.0,0.30803,0.06298,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14,"k":"flat","v":0.00893,"x":0.08929,"p":[[0,39,0.0,0.0625,0.1234,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.08929,0.17768,0.0,0.0,0.14286,0.0,0.57143,23,0,1,23,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[8,39,0.2051,0.05357,0.1171,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,39,0.3077,0.05357,0.14173,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[16,39,0.4103,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.05804,0.10013,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.0267,0.06607,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"884535bc65a8b092","q":"9. (FRG 1) ${ }^{\\text {IMO6 }}$ Let $a$ and $b$ be two positive integers such that $a b+1$ divides $a^{2}+b^{2}$. Show that $\\frac{a^{2}+b^{2}}{a b+1}$ is a perfect square.","t":[{"b":0,"e":1.0,"k":"falling","v":0.55357,"x":0.84373,"p":[[0,21,0.0,0.84373,0.23519,0.82132,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,5,0,19],[4,21,0.1905,0.61604,0.24074,0.42857,0.57143,0.74996,0.14286,1.0,0,4,0,0,0,2,0,0,3,0,0,5,0,0,7,0,0,7,0,0,4,0,4],[8,21,0.381,0.5714,0.2369,0.42857,0.571,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,5,0,0,9,0,0,5,0,0,6,0,0,2,0,4],[12,21,0.5714,0.55357,0.24419,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,5,0,0,8,0,0,7,0,0,4,0,0,2,0,4],[16,21,0.7619,0.58927,0.23623,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,3,0,0,3,0,0,4,0,0,8,0,0,7,0,0,5,0,2],[20,21,0.9524,0.60265,0.26423,0.42857,0.57143,0.75,0.14286,1.0,0,7,0,0,0,2,0,0,4,0,0,6,0,0,8,0,0,4,0,0,1,0,7],[21,21,1.0,0.65177,0.24206,0.4286,0.57143,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,2,0,0,6,0,0,9,0,0,4,0,0,3,0,7]]},{"b":7,"e":0.85714,"k":"flat","v":0.65625,"x":0.95535,"p":[[0,25,0.0,0.86158,0.16939,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,7,0,16],[4,25,0.16,0.65625,0.274,0.42859,0.71429,0.85714,0.0,1.0,2,6,2,2,0,0,0,0,2,0,0,5,0,0,5,0,0,6,0,0,6,0,6],[8,25,0.32,0.85719,0.21419,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,3,0,0,1,0,21],[12,25,0.48,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[16,25,0.64,0.89731,0.08918,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,11],[20,25,0.8,0.91071,0.11152,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,15],[24,25,0.96,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[25,25,1.0,0.94642,0.11152,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,23]]}]},{"i":"3c507657bbe84ae1","q":"$\\omega$  is the circumcirle of an acute triangle  $ABC$ . The tangent line passing through  $A$  intersects the tangent lines passing through points  $B$  and  $C$  at points  $K$  and  $L$ , respectively. The line parallel to  $AB$  through  $K$  and the line parallel to  $AC$  through  $L$  intersect at point  $P$ . Prove that  $BP=CP$ . \n(Author: P. Kozhevnikov)","t":[{"b":1,"e":0.14286,"k":"flat","v":0.05804,"x":0.16517,"p":[[0,21,0.0,0.06688,0.08729,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.1025,0.11419,0.0,0.14143,0.14286,0.0,0.42857,14,0,0,14,0,15,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.12045,0.08826,0.105,0.14286,0.14286,0.0,0.4286,8,0,0,8,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.16517,0.16405,0.14286,0.14286,0.14286,0.0,0.85714,5,0,0,5,0,24,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[21,21,1.0,0.09813,0.08323,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.09822,"p":[[0,81,0.0,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,2,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,81,0.0494,0.08911,0.10556,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,81,0.0988,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,81,0.1481,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,81,0.1975,0.03563,0.06171,0.0,0.0,0.035,0.0,0.143,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,81,0.2469,0.0267,0.05558,0.0,0.0,0.0,0.0,0.1429,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,81,0.2963,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,81,0.3457,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,81,0.3951,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,81,0.4444,0.0892,0.06909,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,81,0.4938,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,81,0.5432,0.09822,0.10972,0.0,0.14286,0.14286,0.0,0.4286,14,0,0,14,0,16,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,81,0.5926,0.07134,0.11288,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,81,0.642,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,81,0.6914,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,81,0.7407,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,81,0.7901,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,81,0.8395,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,81,0.8889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,81,0.9383,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,81,0.9877,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[81,81,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"866351b972b155d8","q":"12. (IRE 2) Let $n, k$ be positive integers with $k \\leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be the set of all real numbers of the form $x_{1}+x_{2}+\\cdots+x_{k}$, where $x_{1}, x_{2}, \\ldots, x_{k}$ are distinct elements of $S$. Prove that $T$ contains at least $k(n-k)+1$ distinct elements.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.07143,"x":0.26328,"p":[[0,23,0.0,0.26328,0.21462,0.0,0.21431,0.42857,0.0,0.57143,9,0,0,9,0,7,0,0,1,0,0,10,0,0,5,0,0,0,0,0,0,0,0],[4,23,0.1739,0.15177,0.18533,0.0,0.07143,0.32142,0.0,0.571,16,0,0,16,0,7,0,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[8,23,0.3478,0.16518,0.18935,0.0,0.14286,0.1786,0.0,0.57143,13,0,0,13,0,11,0,0,1,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[12,23,0.5217,0.13843,0.19397,0.0,0.0,0.2857,0.0,0.57143,19,0,0,19,0,4,0,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[16,23,0.6957,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,23,0.8696,0.09596,0.14011,0.0,0.0,0.14286,0.0,0.571,18,0,0,18,0,10,0,0,1,0,1,1,0,0,1,0,0,0,0,0,0,0,0],[23,23,1.0,0.11607,0.17655,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,2,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.57143,"k":"flat","v":0.18302,"x":0.37503,"p":[[0,6,0.0,0.2768,0.2285,0.0,0.42857,0.42858,0.0,0.57143,11,0,0,11,0,3,0,0,1,0,0,11,0,0,6,0,0,0,0,0,0,0,0],[4,6,0.6667,0.18302,0.21197,0.0,0.0,0.42857,0.0,0.5714,17,0,0,17,0,2,0,0,2,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[6,6,1.0,0.37503,0.15465,0.42857,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,1,0,0,0,0,0,25,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"959f0d7b4e4418e8","q":"A positive integer  $n < 2017$  is given. Exactly  $n$  vertices of a regular 2017-gon are colored red, and the remaining vertices are colored blue. Prove that the number of isosceles triangles whose vertices are monochromatic does not depend on the chosen coloring (but does depend on  $n$ .)","t":[{"b":6,"e":0.14286,"k":"falling","v":0.01786,"x":0.27232,"p":[[0,11,0.0,0.27232,0.29093,0.0,0.21429,0.42857,0.0,1.0,11,3,0,11,0,5,0,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,3],[4,11,0.3636,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.04911,"x":0.20964,"p":[[0,25,0.0,0.20964,0.18899,0.0,0.14286,0.42857,0.0,0.57143,10,0,0,10,0,10,0,0,0,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[4,25,0.16,0.1875,0.22711,0.0,0.14286,0.42857,0.0,1.0,14,1,0,14,0,7,0,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[8,25,0.32,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cb591502e88c13cc","q":"26. (NET 1) Let $p$ be a prime number greater than 5 . Let $V$ be the collection of all positive integers $n$ that can be written in the form $n=k p+1$ or $n=k p-1(k=1,2, \\ldots)$. A number $n \\in V$ is called indecomposable in $V$ if it is impossible to find $k, l \\in V$ such that $n=k l$. Prove that there exists a number $N \\in V$ that can be factorized into indecomposable factors in $V$ in more than one way.","t":[{"b":3,"e":0.42857,"k":"flat","v":0.5533,"x":0.68295,"p":[[0,11,0.0,0.66071,0.39245,0.39285,1.0,1.0,0.0,1.0,4,17,4,4,0,3,0,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,17],[4,11,0.3636,0.68295,0.42229,0.14286,1.0,1.0,0.0,1.0,5,20,4,5,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,20],[8,11,0.7273,0.55357,0.20124,0.42857,0.64286,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,0,0,0,13,0,0,1,0,0,15,0,0,1,0,0],[11,11,1.0,0.5533,0.14144,0.42857,0.42857,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,14,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"rising","v":0.83482,"x":1.0,"p":[[0,32,0.0,0.83482,0.31361,0.92857,1.0,1.0,0.0,1.0,1,24,0,1,0,3,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,24],[4,32,0.125,0.85268,0.29339,0.96429,1.0,1.0,0.0,1.0,2,24,2,2,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,24],[8,32,0.25,0.96875,0.1504,1.0,1.0,1.0,0.14286,1.0,0,30,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[12,32,0.375,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,32,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,32,0.625,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,32,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"70ed185a76839b5e","q":"a) Prove that there exist integers $\\mathbf{a}, \\boldsymbol{b}$, $\\mathbf{c}$ such that $(\\mathbf{a}, \\mathbf{b}, \\mathbf{c}) \\neq (0,0,0)$ and $|\\mathbf{a}|,|\\mathbf{b}|,|\\mathbf{c}| < 10^{6}$ for which\n\n$$\n|a + b \\sqrt{2} + c \\sqrt{3}| < 10^{-11}\n$$\n\nb) Let $\\mathbf{a}, \\mathbf{b}, \\mathrm{c}$ be integers such that $(\\mathbf{a}, \\boldsymbol{b}, \\mathrm{c}) \\neq (0,0,0)$ and $|\\mathbf{a}|,|\\mathbf{b}|,|\\mathbf{c}| < 10^{6}$. Prove that\n\n$$\n|a + b \\sqrt{2} + c \\sqrt{3}| > 10^{-21}\n$$","t":[{"b":0,"e":1.0,"k":"flat","v":0.85268,"x":1.0,"p":[[0,49,0.0,0.86607,0.15947,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,12,0,14],[4,49,0.0816,0.85268,0.20666,0.82143,0.85714,1.0,0.0,1.0,1,15,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,9,0,15],[8,49,0.1633,0.86607,0.13333,0.85711,0.85714,1.0,0.57143,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,13,0,12],[12,49,0.2449,0.94196,0.1063,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,6,0,23],[16,49,0.3265,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[20,49,0.4082,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,49,0.4898,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,49,0.5714,0.95982,0.09606,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,26],[32,49,0.6531,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[36,49,0.7347,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[40,49,0.8163,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,49,0.898,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[48,49,0.9796,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[49,49,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":6,"e":1.0,"k":"flat","v":0.89732,"x":1.0,"p":[[0,9,0.0,0.89732,0.10249,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,14],[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],[8,9,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[9,9,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"fbdaca75a6fddc7c","q":"2. A2 (SWE) Let $a$ and $b$ be nonnegative integers such that $a b \\geq c^{2}$, where $c$ is an integer. Prove that there is a number $n$ and integers $x_{1}, x_{2}, \\ldots, x_{n}, y_{1}, y_{2}, \\ldots, y_{n}$ such that  $$ \\sum_{i=1}^{n} x_{i}^{2}=a, \\quad \\sum_{i=1}^{n} y_{i}^{2}=b, \\quad \\text { and } \\quad \\sum_{i=1}^{n} x_{i} y_{i}=c $$","t":[{"b":1,"e":0.28571,"k":"flat","v":0.15177,"x":0.38826,"p":[[0,41,0.0,0.21429,0.20516,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,12,0,0,5,0,0,5,0,0,0,0,0,1,0,0,1,0,0],[4,41,0.0976,0.15625,0.18336,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,18,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[8,41,0.1951,0.16515,0.19591,0.0,0.14286,0.14287,0.0,0.71429,12,0,0,12,0,13,0,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[12,41,0.2927,0.15177,0.17832,0.0,0.14286,0.14287,0.0,0.85714,11,0,0,11,0,14,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[16,41,0.3902,0.16051,0.14611,0.14214,0.14286,0.14286,0.0,0.57143,6,0,0,6,0,22,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[20,41,0.4878,0.18728,0.20337,0.14286,0.14286,0.14286,0.0,1.0,5,1,0,5,0,23,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[24,41,0.5854,0.24108,0.20651,0.14286,0.1429,0.2857,0.0,0.71429,3,0,0,3,0,18,0,0,6,0,0,0,0,0,1,0,0,4,0,0,0,0,0],[28,41,0.6829,0.37478,0.25951,0.14286,0.28571,0.571,0.0,0.85714,1,0,0,1,0,13,0,0,3,0,0,5,0,0,3,0,0,4,0,0,3,0,0],[32,41,0.7805,0.31245,0.16137,0.14286,0.2857,0.42857,0.14286,0.57143,0,0,0,0,0,12,0,0,8,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[36,41,0.878,0.38826,0.21205,0.1429,0.42857,0.571,0.14,0.85714,0,0,0,0,0,9,0,0,6,0,0,8,0,0,5,0,0,2,0,0,2,0,0],[40,41,0.9756,0.3348,0.19099,0.14289,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,9,0,0,10,0,0,5,0,0,4,0,0,3,0,0,0,0,0],[41,41,1.0,0.28541,0.14305,0.14286,0.2857,0.42857,0.14,0.571,0,0,0,0,0,13,0,0,9,0,0,7,0,0,3,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.4286,"k":"flat","v":0.15624,"x":0.33927,"p":[[0,27,0.0,0.22736,0.21685,0.14,0.14286,0.2857,0.0,0.71429,6,0,0,6,0,17,0,0,2,0,0,0,0,0,5,0,0,2,0,0,0,0,0],[4,27,0.1481,0.19632,0.18121,0.14214,0.14286,0.2857,0.0,0.857,7,0,0,7,0,14,0,0,7,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[8,27,0.2963,0.17399,0.17024,0.14214,0.14286,0.14287,0.0,0.71429,7,0,0,7,0,19,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[12,27,0.4444,0.2856,0.28348,0.0,0.14286,0.4286,0.0,0.85714,9,0,0,9,0,9,0,0,2,0,0,5,0,0,3,0,0,0,0,0,4,0,0],[16,27,0.5926,0.15624,0.14441,0.0,0.14286,0.1429,0.0,0.571,9,0,1,9,0,16,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,27,0.7407,0.33927,0.21942,0.14286,0.2857,0.42857,0.0,0.85714,1,0,0,1,0,9,0,0,12,0,0,4,0,0,1,0,0,3,0,0,2,0,0],[24,27,0.8889,0.33925,0.22228,0.14286,0.2857,0.571,0.14286,0.85714,0,0,0,0,0,14,0,0,6,0,0,3,0,0,6,0,0,1,0,0,2,0,0],[27,27,1.0,0.33915,0.19482,0.14286,0.2857,0.4642,0.14,0.85714,0,0,0,0,0,11,0,0,9,0,0,4,0,0,6,0,0,1,0,0,1,0,0]]}]},{"i":"6560601f8df3a5f1","q":"Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $\\mathbf{x}_{0}=(0,0)$, and for every $n \\geqslant 1$, his jump is\n\n$$\n\\mathbf{x}_{n}-\\mathbf{x}_{n-1}=( \\pm f(n), \\pm g(n)) \\text { or }( \\pm g(n), \\pm f(n))\n$$\n\nwith eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0,0)$ infinitely many times when\n(a) $f, g$ are polynomials with integer coefficients?\n(b) $f, g$ are any pair of functions from the positive integers to the integers?\n\n## Proposed by United Kingdom","t":[{"b":3,"e":0.42857,"k":"flat","v":0.28571,"x":0.46431,"p":[[0,50,0.0,0.35268,0.07974,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,18,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[4,50,0.08,0.30804,0.05187,0.2857,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.32142,0.07143,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,25,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[12,50,0.24,0.35268,0.11283,0.2857,0.28571,0.42857,0.2857,0.857,0,0,0,0,0,0,0,0,20,0,0,11,0,0,0,0,0,0,0,0,1,0,0],[16,50,0.32,0.31697,0.08552,0.28571,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.3125,0.06622,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,50,0.48,0.31251,0.07523,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,25,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[28,50,0.56,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.28571,1e-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],[36,50,0.72,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.40179,0.05576,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.43755,0.10677,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,23,0,0,1,0,0,3,0,0,0,0,0],[50,50,1.0,0.46431,0.11841,0.42857,0.42857,0.42895,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,22,0,0,4,0,0,2,0,0,1,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.29017,"x":0.31255,"p":[[0,13,0.0,0.29909,0.10923,0.2857,0.28571,0.28571,0.0,0.571,2,0,0,2,0,1,0,0,22,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,13,0.3077,0.31255,0.0663,0.2857,0.28571,0.28579,0.14286,0.43,0,0,0,0,0,1,0,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.29924,0.04161,0.2857,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.29465,0.03458,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8be4fb9914fa0684","q":"13. (POL 4) Given any integer $n \\geq 2$, assume that the integers $a_{1}, a_{2}, \\ldots, a_{n}$ are not divisible by $n$ and, moreover, that $n$ does not divide $a_{1}+a_{2}+$ $\\cdots+a_{n}$. Prove that there exist at least $n$ different sequences $\\left(e_{1}, e_{2}, \\cdots, e_{n}\\right)$ consisting of zeros or ones such that $e_{1} a_{1}+e_{2} a_{2}+\\cdots+e_{n} a_{n}$ is divisible by $n$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,66,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,66,0.1212,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,66,0.1818,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,66,0.2424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,66,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,66,0.4242,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[36,66,0.5455,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,66,0.6061,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[48,66,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,66,0.7879,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[60,66,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f84124b1f611e0d2","q":"7. (USS) Prove that a tetrahedron $S A B C$ has five different spheres that touch all six lines determined by its edges if and only if it is regular.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.10241,"x":0.14277,"p":[[0,22,0.0,0.1317,0.1186,0.14286,0.14286,0.14286,0.0,0.71429,6,0,0,6,1,24,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,22,0.1818,0.14277,0.07143,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,27,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.13813,0.06666,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.12474,0.0691,0.14,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.13599,0.04501,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,1,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.10241,0.06407,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.09804,"x":0.14286,"p":[[0,29,0.0,0.1159,0.05568,0.14214,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.14277,0.06186,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.09804,0.0661,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.13812,0.02482,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.14286,2e-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],[20,29,0.6897,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c736c23e387277ed","q":"6. (GER 1) ${ }^{\\mathrm{IMO} 5}$ Let $\\mathbb{N}=\\{1,2,3, \\ldots\\}$. Determine whether there exists a strictly increasing function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ with the following properties:  $$ \\begin{aligned} f(1) & =2 \\\\ f(f(n)) & =f(n)+n \\quad(n \\in \\mathbb{N}) \\end{aligned} $$","t":[{"b":3,"e":0.85714,"k":"rising","v":0.34821,"x":0.87946,"p":[[0,14,0.0,0.41516,0.36484,0.10714,0.28571,0.71429,0.0,1.0,8,5,0,8,0,5,0,0,4,0,0,2,0,0,3,0,0,3,0,0,2,0,5],[4,14,0.2857,0.3616,0.34806,0.14286,0.14286,0.71429,0.0,1.0,6,4,1,6,0,13,0,0,0,0,0,2,0,0,2,0,0,4,0,0,1,0,4],[8,14,0.5714,0.34821,0.37447,0.0,0.21429,0.60714,0.0,1.0,11,6,0,11,0,5,0,0,5,0,0,1,0,0,2,0,0,2,0,0,0,0,6],[12,14,0.8571,0.87946,0.2055,0.85714,1.0,1.0,0.28571,1.0,0,18,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,10,0,18],[14,14,1.0,0.87946,0.18595,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,10,0,17]]},{"b":4,"e":0.2857,"k":"falling","v":0.06696,"x":0.45089,"p":[[0,42,0.0,0.45089,0.35912,0.14286,0.28571,0.85714,0.0,1.0,4,6,0,4,0,8,0,0,5,0,0,2,0,0,3,0,0,1,0,0,3,0,6],[4,42,0.0952,0.41964,0.3443,0.14286,0.35714,0.71429,0.0,1.0,7,3,0,7,0,5,0,0,4,0,0,3,0,0,3,0,0,3,0,0,4,0,3],[8,42,0.1905,0.32589,0.30978,0.10714,0.14286,0.71429,0.0,1.0,8,1,0,8,0,10,0,0,1,0,0,3,0,0,1,0,0,7,0,0,1,0,1],[12,42,0.2857,0.41965,0.36235,0.14286,0.28571,0.71429,0.0,1.0,7,6,0,7,0,5,0,0,5,0,0,4,0,0,1,0,0,3,0,0,1,0,6],[16,42,0.381,0.42857,0.35175,0.14286,0.28571,0.71429,0.0,1.0,6,3,0,6,0,8,0,0,3,0,0,0,0,0,2,0,0,7,0,0,3,0,3],[20,42,0.4762,0.34821,0.37105,0.0,0.14286,0.71429,0.0,1.0,10,5,0,10,0,8,0,0,2,0,0,2,0,0,1,0,0,3,0,0,1,0,5],[24,42,0.5714,0.23214,0.27375,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,9,0,0,6,0,0,1,0,0,1,0,0,0,0,0,4,0,0],[28,42,0.6667,0.06696,0.13356,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,42,0.7619,0.06696,0.14718,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,42,0.8571,0.19643,0.13716,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[40,42,0.9524,0.23659,0.11629,0.14286,0.14286,0.28571,0.14286,0.571,0,0,0,0,0,17,0,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[42,42,1.0,0.20089,0.08645,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,21,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2cb29d9a26228617","q":"In a convex pentagon $A B C D E$, the sides $A E$ and $B C$ are parallel and $\\angle A D E=\\angle B D C$. The diagonals $A C$ and $B E$ intersect at $P$. Prove that $\\angle E A D=\\angle B D P$ and $\\angle C B D=\\angle A D P$.","t":[{"b":3,"e":0.71429,"k":"rising","v":0.33482,"x":0.70089,"p":[[0,23,0.0,0.33482,0.24643,0.10714,0.28571,0.57143,0.0,0.71429,8,0,2,8,0,2,0,0,8,0,0,2,0,0,9,0,0,3,0,0,0,0,0],[4,23,0.1739,0.48214,0.21943,0.28571,0.57143,0.71429,0.0,0.71429,2,0,2,2,0,0,0,0,11,0,0,1,0,0,7,0,0,11,0,0,0,0,0],[8,23,0.3478,0.40625,0.26752,0.2857,0.28571,0.71429,0.0,0.71429,6,0,5,6,0,1,0,0,10,0,0,0,0,0,5,0,0,10,0,0,0,0,0],[12,23,0.5217,0.55357,0.24419,0.28571,0.71429,0.71429,0.0,0.71429,3,0,2,3,0,0,0,0,6,0,0,1,0,0,1,0,0,21,0,0,0,0,0],[16,23,0.6957,0.53123,0.22934,0.28571,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,2,0,0,8,0,0,1,0,0,2,0,0,18,0,0,0,0,0],[20,23,0.8696,0.65606,0.11211,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,21,0,0,1,0,0],[23,23,1.0,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"rising","v":0.35268,"x":0.70088,"p":[[0,64,0.0,0.35268,0.22864,0.2857,0.28571,0.57143,0.0,0.71429,5,0,2,5,0,1,0,0,15,0,0,2,0,0,3,0,0,6,0,0,0,0,0],[4,64,0.0625,0.45088,0.23986,0.28571,0.57143,0.71429,0.0,0.71429,4,0,0,4,0,0,0,0,10,0,0,0,0,0,9,0,0,9,0,0,0,0,0],[8,64,0.125,0.36161,0.24218,0.25,0.28571,0.57143,0.0,0.71429,5,0,2,5,0,3,0,0,12,0,0,0,0,0,6,0,0,6,0,0,0,0,0],[12,64,0.1875,0.39955,0.28227,0.21427,0.35714,0.71429,0.0,0.71429,8,0,2,8,0,0,0,0,8,0,0,1,0,0,4,1,0,10,0,0,0,0,0],[16,64,0.25,0.48214,0.23351,0.28571,0.57143,0.71429,0.0,1.0,2,1,0,2,0,1,0,0,9,0,0,3,0,0,7,0,0,9,0,0,0,0,1],[20,64,0.3125,0.41071,0.25939,0.2857,0.28571,0.71429,0.0,0.71429,4,0,0,4,0,3,0,0,11,0,0,0,0,0,3,0,0,11,0,0,0,0,0],[24,64,0.375,0.44196,0.23517,0.2857,0.42857,0.71429,0.0,0.71429,3,0,1,3,0,1,0,0,11,0,0,2,0,0,5,0,0,10,0,0,0,0,0],[28,64,0.4375,0.52677,0.21558,0.28571,0.57143,0.71429,0.0,0.71429,2,0,0,2,0,0,0,0,7,0,0,2,0,0,7,0,0,14,0,0,0,0,0],[32,64,0.5,0.6918,0.06298,0.71429,0.71429,0.71429,0.4286,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,28,0,0,0,0,0],[36,64,0.5625,0.67408,0.14828,0.71429,0.71429,0.71429,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,1,0,0,4,0,0,25,0,0,0,0,1],[40,64,0.625,0.67634,0.09446,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,1,0,26,0,0,0,0,0],[44,64,0.6875,0.68079,0.05921,0.69643,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,24,0,0,0,0,0],[48,64,0.75,0.70088,0.04168,0.71429,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0],[52,64,0.8125,0.67857,0.09449,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,27,0,0,0,0,0],[56,64,0.875,0.66964,0.0974,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,26,0,0,0,0,0],[60,64,0.9375,0.69867,0.04276,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,1,0,28,0,0,0,0,0],[64,64,1.0,0.67409,0.10247,0.71429,0.71429,0.71429,0.42857,0.857,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,25,0,0,1,0,0]]}]},{"i":"bcb69069a0c2f108","q":"Points  $P,Q,R$  lie on the sides  $AB,BC,CA$  of triangle  $ABC$  in such a way that  $AP=PR, CQ=QR$ . Let  $H$  be the orthocenter of triangle  $PQR$ , and  $O$  be the circumcenter of triangle  $ABC$ .\nProve that   $$ OH||AC $$  .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.08482,"x":0.12045,"p":[[0,32,0.0,0.09367,0.06779,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.12045,0.05184,0.14286,0.14286,0.14286,0.0,0.143,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.10269,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.12045,0.05184,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.08911,"x":0.13831,"p":[[0,37,0.0,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.08911,0.06903,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.12501,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,1,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.1026,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.10706,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.11153,0.05902,0.14214,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.10697,0.06176,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.09813,0.06616,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"faa3e8ae408ea18d","q":"Determine all pairs  $(n,p)$  of positive integers such that \n- $p$  is a prime,  $n>1$ ,\n- $(p-1)^{n} + 1$  is divisible by  $n^{p-1}$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.61158,"x":1.0,"p":[[0,19,0.0,0.61158,0.23484,0.42857,0.57143,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,5,0,0,6,0,0,10,0,0,2,0,0,4,0,5],[4,19,0.2105,0.692,0.23171,0.53575,0.71429,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,4,0,0,8,0,6],[8,19,0.4211,0.75891,0.24599,0.57142,0.78571,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,4,0,0,2,0,14],[12,19,0.6316,0.71872,0.22443,0.57143,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,1,0,0,9,0,0,7,0,0,3,0,9],[16,19,0.8421,0.68304,0.21048,0.57143,0.71429,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,3,0,0,3,0,0,6,0,0,12,0,0,2,0,6],[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":5,"e":0.571,"k":"flat","v":0.55352,"x":0.69641,"p":[[0,27,0.0,0.57588,0.22156,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,7,0,0,5,0,0,9,0,0,5,0,0,3,0,3],[4,27,0.1481,0.66515,0.23314,0.5713,0.57143,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,5,0,0,1,0,0,11,0,0,4,0,0,5,0,6],[8,27,0.2963,0.69641,0.17406,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,8,0,0,4,0,5],[12,27,0.4444,0.60265,0.18809,0.57143,0.57143,0.57143,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,22,0,0,1,0,0,1,0,4],[16,27,0.5926,0.55801,0.08268,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,1,0,0,27,0,0,2,0,0,0,0,0],[20,27,0.7407,0.58034,0.07085,0.57143,0.57143,0.57143,0.42857,0.857,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,0,2,0,0,1,0,0],[24,27,0.8889,0.55352,0.06915,0.5713,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,27,0,0,1,0,0,0,0,0],[27,27,1.0,0.56243,0.03456,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0]]}]},{"i":"05884138af6bc3b6","q":"Compute the number of rearrangements  $a_1, a_2, \\dots, a_{2018}$  of the sequence  $1, 2, \\dots, 2018$  such that  $a_k > k$  for  $\\textit{exactly}$  one value of  $k$ .","t":[{"b":1,"e":0.57143,"k":"flat","v":0.36598,"x":0.5625,"p":[[0,12,0.0,0.46429,0.29233,0.28571,0.42857,0.57143,0.0,1.0,4,3,0,4,0,3,0,0,3,0,0,9,0,0,6,0,0,1,0,0,3,0,3],[4,12,0.3333,0.5625,0.2111,0.42857,0.50001,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,3,0,0,12,0,0,4,0,0,7,0,0,3,0,2],[8,12,0.6667,0.54018,0.21349,0.42857,0.5,0.60714,0.0,1.0,1,2,0,1,0,0,0,0,3,0,0,12,0,0,8,0,0,3,0,0,3,0,2],[12,12,1.0,0.36598,0.18547,0.14286,0.42857,0.57143,0.0,0.57143,2,0,0,2,0,7,0,0,4,0,0,9,0,0,10,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.40625,"x":0.86607,"p":[[0,32,0.0,0.40625,0.21756,0.28571,0.42857,0.46429,0.0,1.0,2,1,0,2,0,3,0,0,8,0,0,11,0,0,4,0,0,2,0,0,1,0,1],[4,32,0.125,0.47321,0.27765,0.28571,0.42857,0.71429,0.0,1.0,4,1,0,4,0,3,0,0,2,0,0,8,0,0,6,0,0,4,0,0,4,0,1],[8,32,0.25,0.45982,0.14166,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,17,0,0,6,0,0,4,0,0,0,0,0],[12,32,0.375,0.46427,0.18557,0.39286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,10,0,0,9,0,0,4,0,0,1,0,0],[16,32,0.5,0.58036,0.18189,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,13,0,0,8,0,0,5,0,0,3,0,2],[20,32,0.625,0.73214,0.21651,0.53571,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,8,0,0,4,0,0,3,0,0,10,0,7],[24,32,0.75,0.77679,0.18877,0.67857,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,6,0,0,10,0,8],[28,32,0.875,0.7232,0.19542,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,8,0,0,7,0,6],[32,32,1.0,0.86607,0.07087,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,24,0,5]]}]},{"i":"ce5963ea881cfed8","q":"Consider the sequence  $(a_k)_{k\\ge 1}$  of positive rational numbers defined by  $a_1 = \\frac{2020}{2021}$  and for  $k\\ge 1$ , if  $a_k = \\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , then \n\n\\[a_{k+1} = \\frac{m + 18}{n+19}.\\]\n\nDetermine the sum of all positive integers  $j$  such that the rational number  $a_j$  can be written in the form  $\\frac{t}{t+1}$  for some positive integer  $t$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.875,"x":0.99107,"p":[[0,13,0.0,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[4,13,0.3077,0.875,0.29397,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,25],[8,13,0.6154,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,13,0.9231,0.9375,0.1234,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,6,0,23],[13,13,1.0,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21]]},{"b":7,"e":0.85714,"k":"flat","v":0.88839,"x":1.0,"p":[[0,47,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[4,47,0.0851,0.95536,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[8,47,0.1702,0.96429,0.11294,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,28],[12,47,0.2553,0.95089,0.11071,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,24],[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],[20,47,0.4255,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[24,47,0.5106,0.88839,0.19145,0.85714,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,3,0,0,6,0,20],[28,47,0.5957,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[32,47,0.6809,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[36,47,0.766,0.94252,0.07819,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,20],[40,47,0.8511,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[44,47,0.9362,0.89732,0.10248,0.85714,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,14],[47,47,1.0,0.92411,0.08737,0.85714,1.0,1.0,0.71429,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,17]]}]},{"i":"e9fadb8c30a6de0d","q":"Let  $n$  be a positive integer. The numbers  $1, 2, \\dots, 2n+1$  are arranged in a circle in that order, and some of them are *marked*.\nWe define, for each  $k$  such that  $1\\leq k \\leq 2n+1$  , the interval  $I_k$  to be the closed circular interval starting at  $k$  and ending in  $k+n$  (taking remainders mod(2n+1)). We call in interval *magical* if it contains strictly more than half of all the marked elements.\nProve that the following two statements are equivalent:\n1. At least  $n+1$  of the intervals  $I_1, I_2, \\dots, I_{2n+1}$   are magical\n2. The number of marked numbers is odd","t":[{"b":4,"e":1.0,"k":"flat","v":0.62053,"x":0.95536,"p":[[0,52,0.0,0.79897,0.3348,0.71429,1.0,1.0,0.0,1.0,4,20,1,4,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,2,0,20],[4,52,0.0769,0.66071,0.42969,0.0,1.0,1.0,0.0,1.0,9,17,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,17],[8,52,0.1538,0.62053,0.43243,0.0,0.85707,1.0,0.0,1.0,10,13,0,10,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,13],[12,52,0.2308,0.90625,0.15407,0.85714,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,21],[16,52,0.3077,0.89286,0.17128,0.85714,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,4,0,21],[20,52,0.3846,0.86159,0.18382,0.71429,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,4,0,18],[24,52,0.4615,0.8973,0.1864,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,22],[28,52,0.5385,0.91963,0.12846,0.85714,1.0,1.0,0.4286,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,8,0,20],[32,52,0.6154,0.90624,0.16605,0.85714,1.0,1.0,0.28571,1.0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,3,0,22],[36,52,0.6923,0.8125,0.24338,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,2,0,0,8,0,15],[40,52,0.7692,0.88839,0.15458,0.85714,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,7,0,18],[44,52,0.8462,0.875,0.23623,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,5,0,21],[48,52,0.9231,0.95536,0.0974,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,2,0,26],[52,52,1.0,0.8616,0.21275,0.82132,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,5,0,19]]},{"b":5,"e":0.85714,"k":"rising","v":0.61152,"x":0.84375,"p":[[0,44,0.0,0.69197,0.35013,0.42857,0.85714,1.0,0.0,1.0,4,13,0,4,0,1,0,0,0,0,0,5,0,0,1,0,0,4,0,0,4,0,13],[4,44,0.0909,0.67857,0.38132,0.42857,0.85714,1.0,0.0,1.0,6,15,1,6,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,2,0,15],[8,44,0.1818,0.61152,0.39334,0.24928,0.71429,1.0,0.0,1.0,7,11,0,7,0,1,0,0,1,0,0,2,0,0,3,0,0,3,0,0,4,0,11],[12,44,0.2727,0.75,0.35174,0.71429,1.0,1.0,0.0,1.0,4,17,0,4,0,1,0,0,1,0,0,0,0,0,1,0,0,6,0,0,2,0,17],[16,44,0.3636,0.78571,0.20825,0.71429,0.85714,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,8,0,0,6,0,11],[20,44,0.4545,0.66964,0.21261,0.53569,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,9,0,0,6,0,4],[24,44,0.5455,0.67409,0.20897,0.5713,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,3,0,0,4,0,0,6,0,0,9,0,0,6,0,4],[28,44,0.6364,0.71428,0.16751,0.57143,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,11,0,0,8,0,3],[32,44,0.7273,0.71872,0.22158,0.57143,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,4,0,0,1,0,0,5,0,0,8,0,0,8,0,6],[36,44,0.8182,0.82141,0.13832,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,16,0,6],[40,44,0.9091,0.79464,0.20184,0.71429,0.85714,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,7,0,0,9,0,10],[44,44,1.0,0.84375,0.15303,0.71429,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,12,0,11]]}]},{"i":"af24e6a3dfd828b6","q":"Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2 n, 2 n)$ consists of $4 n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0)$, $(2 n, 0),(2 n, 2 n),(0,2 n)$ into two regions with even areas is $$ \\frac{\\binom{4 n}{2 n}+\\binom{2 n}{n}}{2} $$","t":[{"b":0,"e":0.0,"k":"flat","v":0.06696,"x":0.09375,"p":[[0,5,0.0,0.09375,0.11071,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.06696,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.09366,0.10475,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.0758,"x":0.17411,"p":[[0,25,0.0,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.09821,0.14032,0.0,0.0,0.17857,0.0,0.42857,20,0,0,20,0,4,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.09375,0.15407,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,7,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,25,0.48,0.0758,0.107,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.17411,0.20433,0.0,0.14286,0.2857,0.0,1.0,11,1,0,11,0,12,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[20,25,0.8,0.09803,0.09732,0.0,0.14143,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.14732,0.1838,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[25,25,1.0,0.11607,0.18707,0.0,0.07143,0.14286,0.0,1.0,16,1,0,16,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"49a964080389c805","q":"Prove that, given any formation, each mobot may be colored in one of three colors - say, white, black, and blue - such that no two adjacent clumps of grass are mowed by different mobots of the same color. Two clumps of grass are adjacent if the distance between them is 1 . In your proof, you may use the Four-Color Theorem if you're familiar with it.","t":[{"b":2,"e":0.2857,"k":"flat","v":0.38387,"x":0.54004,"p":[[0,37,0.0,0.53122,0.23483,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,3,0,0,1,0,0,6,0,0,6,0,0,12,0,0,2,0,0],[4,37,0.1081,0.50442,0.2201,0.28571,0.571,0.71429,0.0,0.71429,2,0,0,2,0,1,0,0,6,0,0,5,0,0,5,0,0,13,0,0,0,0,0],[8,37,0.2162,0.42856,0.26244,0.24999,0.4998,0.71429,0.0,0.71429,5,0,0,5,0,3,0,0,5,0,0,3,0,0,6,0,0,10,0,0,0,0,0],[12,37,0.3243,0.42408,0.26117,0.24999,0.4286,0.71429,0.0,0.71429,6,0,0,6,0,2,0,0,2,0,0,9,0,0,3,0,0,10,0,0,0,0,0],[16,37,0.4324,0.45088,0.26752,0.2857,0.42857,0.71429,0.0,0.85714,4,0,0,4,0,3,0,0,6,0,0,4,0,0,2,0,0,12,0,0,1,0,0],[20,37,0.5405,0.54004,0.21954,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,3,0,0,4,0,0,6,0,0,14,0,0,1,0,0],[24,37,0.6486,0.4552,0.23816,0.28571,0.4998,0.60714,0.0,0.85714,4,0,0,4,0,1,0,0,5,0,0,6,0,0,8,0,0,7,0,0,1,0,0],[28,37,0.7568,0.38387,0.2536,0.14289,0.42857,0.57143,0.0,0.85714,6,0,0,6,0,3,0,0,5,0,0,5,0,0,8,0,0,4,0,0,1,0,0],[32,37,0.8649,0.39285,0.25753,0.14289,0.42859,0.57143,0.0,0.85714,6,0,0,6,0,3,0,0,4,0,0,6,0,0,7,0,0,5,0,0,1,0,0],[36,37,0.973,0.38838,0.27717,0.14286,0.42857,0.71429,0.0,0.85714,7,0,0,7,0,3,0,0,4,0,0,6,0,0,3,0,0,8,0,0,1,0,0],[37,37,1.0,0.44173,0.24832,0.2857,0.4286,0.71429,0.0,0.85714,3,0,0,3,0,4,0,0,5,0,0,6,0,0,4,0,0,9,0,0,1,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.54013,"x":0.56692,"p":[[0,6,0.0,0.54013,0.22226,0.42857,0.5712,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,4,0,0,5,0,0,8,0,0,8,0,0,4,0,0],[4,6,0.6667,0.56692,0.22441,0.39286,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,4,0,0,1,0,0,6,0,0,14,0,0,3,0,0],[6,6,1.0,0.56245,0.2141,0.42857,0.57143,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,3,0,0,5,0,0,8,0,0,11,0,0,3,0,0]]}]},{"i":"1f42e8e4515fa701","q":"Prove that the following inequality holds for all positive real numbers $x, y, z$ :\n\n$$\n\\frac{x^{3}}{y^{2}+z^{2}}+\\frac{y^{3}}{z^{2}+x^{2}}+\\frac{z^{3}}{x^{2}+y^{2}} \\geq \\frac{x+y+z}{2}\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.04911,"p":[[0,20,0.0,0.0267,0.08316,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.04911,0.12682,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.04018,"p":[[0,17,0.0,0.04018,0.11426,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4b970599d636ef46","q":"Show that, among 2048 integers, one can always find 1024 whose sum is divisible by 1024.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.14277,"p":[[0,9,0.0,0.14277,0.19233,0.0,0.0,0.1786,0.0,0.57143,17,0,0,17,0,7,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[4,9,0.4444,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,10,0.0,0.09375,0.19434,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,10,0.4,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"186afabb4ff17cc9","q":"\u064eA natural number  $n$  is given. Let  $f(x,y)$  be a polynomial of degree less than  $n$  such that for any positive integers  $x,y\\leq n, x+y \\leq n+1$  the equality  $f(x,y)=\\frac{x}{y}$  holds. Find  $f(0,0)$ .","t":[{"b":3,"e":0.71429,"k":"rising","v":0.13839,"x":0.60713,"p":[[0,15,0.0,0.16964,0.20025,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[4,15,0.2667,0.13839,0.16935,0.0,0.14286,0.17857,0.0,0.85714,13,0,0,13,0,11,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,15,0.5333,0.16518,0.1992,0.0,0.14286,0.2857,0.0,0.85714,12,0,0,12,0,11,0,0,6,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[12,15,0.8,0.54017,0.22512,0.28571,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,3,0,0,6,0,0,8,0,0,2,0,2],[15,15,1.0,0.60713,0.25754,0.28571,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,10,0,0,2,0,0,3,0,0,8,0,0,5,0,4]]},{"b":7,"e":0.0,"k":"falling","v":0.03571,"x":0.25,"p":[[0,58,0.0,0.25,0.25254,0.14286,0.14286,0.28571,0.0,1.0,5,1,0,5,0,15,0,0,7,0,0,1,0,0,0,0,0,1,0,0,2,0,1],[4,58,0.069,0.14286,0.13832,0.0,0.14286,0.17857,0.0,0.57143,11,0,0,11,0,13,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,58,0.1379,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,9,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,58,0.2069,0.06696,0.15966,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,58,0.2759,0.08482,0.14664,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,58,0.3448,0.20982,0.27196,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,6,0,0,7,0,0,0,0,0,1,0,0,2,0,0,1,0,1],[24,58,0.4138,0.15179,0.24727,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,8,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,0],[28,58,0.4828,0.15179,0.21998,0.0,0.14286,0.1786,0.0,1.0,15,1,0,15,0,9,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[32,58,0.5517,0.11159,0.18114,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,5,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[36,58,0.6207,0.15178,0.13803,0.0,0.14286,0.2857,0.0,0.57143,10,0,0,10,0,13,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,58,0.6897,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.08036,0.16728,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,58,0.8276,0.06696,0.07973,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c35a05d81c528634","q":"Let  $\\mathbb{N} = \\{1,2,3, \\ldots\\}$ . Determine if there exists a strictly increasing function  $f: \\mathbb{N} \\mapsto \\mathbb{N}$  with the following properties:\r\n\r\n(i)  $f(1) = 2$ ;\r\n\r\n(ii)  $f(f(n)) = f(n) + n, (n \\in \\mathbb{N})$ .","t":[{"b":1,"e":0.71429,"k":"rising","v":0.66964,"x":0.91071,"p":[[0,17,0.0,0.66964,0.39194,0.35714,0.85707,1.0,0.0,1.0,5,15,0,5,0,3,0,0,0,0,0,2,0,0,1,0,0,4,0,0,2,0,15],[4,17,0.2353,0.69419,0.39091,0.28571,1.0,1.0,0.0,1.0,3,17,0,3,1,3,0,0,2,0,0,2,0,0,0,0,0,1,0,0,3,0,17],[8,17,0.4706,0.88393,0.1357,0.71429,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,2,0,18],[12,17,0.7059,0.89286,0.13363,0.71429,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,2,0,19],[16,17,0.9412,0.85714,0.14286,0.71429,0.85714,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16],[17,17,1.0,0.91071,0.13243,0.71429,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,22]]},{"b":5,"e":0.21429,"k":"falling","v":0.27447,"x":0.77232,"p":[[0,17,0.0,0.77232,0.33666,0.64286,1.0,1.0,0.0,1.0,3,19,0,3,0,0,0,0,2,0,0,3,0,0,0,0,0,3,0,0,2,0,19],[4,17,0.2353,0.57143,0.44031,0.0,0.71429,1.0,0.0,1.0,10,14,0,10,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,14],[8,17,0.4706,0.63389,0.25986,0.42857,0.64286,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,2,0,0,4,0,0,7,0,0,7,0,0,3,0,6],[12,17,0.7059,0.27447,0.18051,0.14286,0.2143,0.42857,0.0,0.71429,2,0,0,2,1,13,0,0,5,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[16,17,0.9412,0.35267,0.19226,0.24999,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,5,0,0,6,0,0,12,0,0,5,0,0,0,0,0,1,0,0],[17,17,1.0,0.35265,0.20193,0.14286,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,11,0,0,2,0,0,11,0,0,5,0,0,1,0,0,1,0,0]]}]},{"i":"7b185effe56a2e59","q":"Let the set of all bijective functions taking positive integers to positive integers be  $\\mathcal B.$  Find all functions  $\\mathbf F:\\mathcal B\\to \\mathbb R$  such that   $$ (\\mathbf F(p)+\\mathbf F(q))^2=\\mathbf F(p \\circ p)+\\mathbf F(p\\circ q)+\\mathbf F(q\\circ p)+\\mathbf F(q\\circ q) $$   for all  $p,q \\in \\mathcal B.$  \n\n*Proposed by ckliao914*","t":[{"b":3,"e":0.14286,"k":"flat","v":0.17402,"x":0.32588,"p":[[0,31,0.0,0.29463,0.13801,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,9,0,0,16,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[4,31,0.129,0.30579,0.16676,0.14286,0.28571,0.42857,0.07143,0.71429,0,0,0,0,1,11,0,0,9,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[8,31,0.2581,0.32588,0.16839,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,9,0,0,12,0,0,7,0,0,1,0,0,3,0,0,0,0,0],[12,31,0.3871,0.31696,0.19145,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,9,0,0,16,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[16,31,0.5161,0.22321,0.08702,0.14286,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,13,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.24999,0.1129,0.14286,0.2857,0.28571,0.14286,0.571,0,0,0,0,0,14,0,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,31,0.7742,0.2142,0.07994,0.14286,0.14286,0.28571,0.14,0.4286,0,0,0,0,0,17,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.17402,0.06906,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,26,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.21428,0.08748,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,18,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.26786,"x":0.35712,"p":[[0,29,0.0,0.30356,0.13714,0.25,0.28571,0.32143,0.14286,0.71429,0,0,0,0,0,8,0,0,16,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[4,29,0.1379,0.35712,0.18895,0.2857,0.28571,0.4286,0.0,1.0,1,1,0,1,0,6,0,0,10,0,0,9,0,0,5,0,0,0,0,0,0,0,1],[8,29,0.2759,0.32589,0.1684,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,6,0,0,15,0,0,6,0,0,1,0,0,3,0,0,0,0,0],[12,29,0.4138,0.29911,0.10926,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,5,0,0,21,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[16,29,0.5517,0.26786,0.11709,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,9,0,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[20,29,0.6897,0.29463,0.12337,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,7,0,0,19,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[24,29,0.8276,0.3125,0.1357,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,6,0,0,19,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[28,29,0.9655,0.27231,0.10923,0.14286,0.28571,0.28571,0.14286,0.571,0,0,0,0,0,10,0,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[29,29,1.0,0.27222,0.09697,0.24999,0.28571,0.28571,0.14,0.571,0,0,0,0,0,8,0,0,20,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"0358ed5c822a9384","q":"Let  $x =\\sqrt{a}+\\sqrt{b}$ , where  $a$  and  $b$  are natural numbers,  $x$  is not an integer, and  $x < 1976$ . Prove that the fractional part of  $x$  exceeds  $10^{-19.76}$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.9241,"x":0.99554,"p":[[0,48,0.0,0.9375,0.10062,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,22],[4,48,0.0833,0.9241,0.10705,0.85714,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,19],[8,48,0.1667,0.93302,0.12365,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,7,0,22],[12,48,0.25,0.95089,0.07668,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22],[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],[20,48,0.4167,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[24,48,0.5,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,48,0.5833,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,48,0.6667,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[36,48,0.75,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[40,48,0.8333,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[44,48,0.9167,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[48,48,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":4,"e":0.85714,"k":"flat","v":0.84375,"x":0.93303,"p":[[0,46,0.0,0.93303,0.10706,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,21],[4,46,0.087,0.92186,0.10623,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,1,18],[8,46,0.1739,0.86161,0.07563,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,23,0,5],[12,46,0.2609,0.91071,0.07784,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,13],[16,46,0.3478,0.87053,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,5],[20,46,0.4348,0.84375,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,25,0,2],[24,46,0.5217,0.87054,0.07457,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,23,0,6],[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],[32,46,0.6957,0.875,0.06916,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,6],[36,46,0.7826,0.86604,0.06122,0.85711,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,26,0,4],[40,46,0.8696,0.8616,0.08364,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,21,0,6],[44,46,0.9565,0.875,0.05923,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,5],[46,46,1.0,0.87053,0.08268,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,21,0,7]]}]},{"i":"cea00e0864167a3a","q":"Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number  $1$ , an arc between a red and a blue point is assigned a number  $2$ , and an arc between a blue and a green point is assigned a number  $3$ . The arcs between two points of the same colour are assigned a number  $0$ . What is the greatest possible sum of all the numbers assigned to the arcs?","t":[{"b":5,"e":1.0,"k":"flat","v":0.95089,"x":1.0,"p":[[0,45,0.0,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[4,45,0.0889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,45,0.1778,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[12,45,0.2667,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,45,0.3556,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,45,0.4444,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,45,0.5333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[28,45,0.6222,0.98884,0.04414,1.0,1.0,1.0,0.7857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,30],[32,45,0.7111,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,45,0.8,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[40,45,0.8889,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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],[45,45,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":6,"e":1.0,"k":"flat","v":0.81919,"x":0.96875,"p":[[0,57,0.0,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[4,57,0.0702,0.96875,0.07772,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[8,57,0.1404,0.95982,0.08171,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,25],[12,57,0.2105,0.88392,0.11538,0.85708,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,13,0,13],[16,57,0.2807,0.81919,0.16365,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,5,1,0,10,0,10],[20,57,0.3509,0.89271,0.09475,0.85714,0.85714,1.0,0.71,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,16,0,12],[24,57,0.4211,0.91517,0.08645,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,15],[28,57,0.4912,0.87053,0.13175,0.85714,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,1,0,14,1,11],[32,57,0.5614,0.87724,0.0879,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,0,17,0,9],[36,57,0.6316,0.87053,0.07457,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,23,0,6],[40,57,0.7018,0.88838,0.09771,0.857,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,2,0,14,0,12],[44,57,0.7719,0.87499,0.08376,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,2,0,19,0,8],[48,57,0.8421,0.86828,0.09803,0.85711,0.85714,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,1,0,19,0,8],[52,57,0.9123,0.86829,0.08397,0.85714,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,0,20,0,7],[56,57,0.9825,0.87944,0.09527,0.85714,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,20,0,9],[57,57,1.0,0.84372,0.16018,0.83918,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,4,1,0,14,0,10]]}]},{"i":"7c3baaea4d341832","q":"Let  $n$  be a positive integer. An  $n\\times n$  board consisting of  $n^2$  cells, each being a unit square colored either black or white, is called *convex* if for every black colored cell, both the cell directly to the left of it and the cell directly above it are also colored black. We define the *beauty* of a board as the number of pairs of its cells  $(u,v)$  such that  $u$  is black,  $v$  is white, and  $u$  and  $v$  are in the same row or column. Determine the maximum possible beauty of a convex  $n\\times n$  board.\n\n*Proposed by Ivan Novak*","t":[{"b":4,"e":0.71429,"k":"flat","v":0.60267,"x":0.625,"p":[[0,22,0.0,0.61161,0.11425,0.57143,0.64286,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,0,16,0,0,0,0,0],[4,22,0.1818,0.61607,0.12595,0.57143,0.64286,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,10,0,0,15,0,0,1,0,0],[8,22,0.3636,0.62054,0.12682,0.57143,0.57143,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,13,0,0,2,0,0],[12,22,0.5455,0.60267,0.1461,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,10,0,0,8,0,0,12,0,0,1,0,1],[16,22,0.7273,0.62499,0.0928,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,15,0,0,0,0,0],[20,22,0.9091,0.60268,0.12745,0.53572,0.64286,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,7,0,0,8,0,0,16,0,0,0,0,0],[22,22,1.0,0.625,0.1171,0.57143,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,4,0,0,9,0,0,18,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"rising","v":0.51339,"x":0.82143,"p":[[0,32,0.0,0.57143,0.16751,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,4,0,0,8,0,0,15,0,0,0,0,0],[4,32,0.125,0.51339,0.20472,0.42857,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,3,0,0,2,0,0,7,0,0,9,0,0,9,0,0,1,0,0],[8,32,0.25,0.57588,0.145,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,7,0,0,8,0,0,14,0,0,0,0,0],[12,32,0.375,0.79464,0.11259,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,18,0,2],[16,32,0.5,0.76786,0.14617,0.71429,0.78571,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,11,0,0,13,0,3],[20,32,0.625,0.77679,0.11259,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,17,0,1],[24,32,0.75,0.82143,0.09449,0.71429,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,19,0,3],[28,32,0.875,0.78125,0.16746,0.71429,0.85714,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,8,0,0,15,0,4],[32,32,1.0,0.77232,0.11214,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,10,0,0,16,0,1]]}]},{"i":"d626cb0907293e0b","q":"Let  $n$  be a positive integer. Find the number of sequences  $a_0,a_1,a_2,\\dots,a_{2n}$  of integers in the range  $[0,n]$  such that for all integers  $0\\leq k\\leq n$  and all nonnegative integers  $m$ , there exists an integer  $k\\leq i\\leq 2k$  such that  $\\lfloor k/2^m\\rfloor=a_i.$ *Andrew Carratu*","t":[{"b":0,"e":0.57143,"k":"flat","v":0.36159,"x":0.47318,"p":[[0,26,0.0,0.375,0.17035,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,20,0,0,6,0,0,2,0,0,2,0,0,0,0,1],[4,26,0.1538,0.36159,0.19226,0.2857,0.28571,0.32143,0.14286,1.0,0,2,0,0,0,2,0,0,22,0,0,3,0,0,3,0,0,0,0,0,0,0,2],[8,26,0.3077,0.37498,0.19148,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,19,0,0,6,0,0,3,0,0,0,0,0,0,0,2],[12,26,0.4615,0.39732,0.19475,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,17,0,0,10,0,0,0,0,0,2,0,0,0,0,2],[16,26,0.6154,0.38384,0.1576,0.28571,0.28571,0.42857,0.14,0.85714,0,0,0,0,0,2,0,0,15,0,0,10,0,0,2,0,0,2,0,0,1,0,0],[20,26,0.7692,0.47318,0.2214,0.2857,0.42857,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,12,0,0,5,0,0,4,0,0,5,0,0,4,0,0],[24,26,0.9231,0.41518,0.2,0.28571,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,15,0,0,8,0,0,1,0,0,3,0,0,3,0,0],[26,26,1.0,0.42854,0.19883,0.28571,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,2,0,0,12,0,0,10,0,0,4,0,0,1,0,0,2,0,1]]},{"b":7,"e":0.28571,"k":"falling","v":0.29017,"x":0.44186,"p":[[0,39,0.0,0.44186,0.27756,0.2857,0.28571,0.57111,0.14,1.0,0,5,0,0,0,3,0,0,17,0,0,3,0,0,3,0,0,0,0,0,1,0,5],[4,39,0.1026,0.375,0.22799,0.2857,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,4,0,0,18,0,0,6,0,0,0,0,0,1,0,0,0,0,3],[8,39,0.2051,0.37053,0.19516,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,21,0,0,3,0,0,4,0,0,0,0,0,0,0,2],[12,39,0.3077,0.32143,0.11294,0.2857,0.28571,0.32143,0.14286,0.71429,0,0,0,0,0,3,0,0,21,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[16,39,0.4103,0.37052,0.23919,0.2857,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,6,0,0,17,0,0,3,0,0,2,0,0,1,0,0,0,0,3],[20,39,0.5128,0.31696,0.06902,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,26,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[24,39,0.6154,0.33928,0.16269,0.2857,0.28571,0.32143,0.14286,1.0,0,1,0,0,0,3,0,0,21,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[28,39,0.7179,0.33482,0.15405,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,2,0,0,24,0,0,3,0,0,1,0,0,0,0,0,2,0,0],[32,39,0.8205,0.35268,0.11837,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,20,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[36,39,0.9231,0.37491,0.18134,0.28571,0.28571,0.42857,0.14,0.85714,0,0,0,0,0,5,0,0,12,0,0,10,0,0,2,0,0,1,0,0,2,0,0],[39,39,1.0,0.29017,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"170c4e368b28fffc","q":"Find all functions  $f\\colon \\mathbb Z \\to \\mathbb Z$  for which\n\\[ f(x)+f(y)+xy \\quad \\text{divides} \\quad xf(x)-y^3 \\]\nfor all pairs of integers  $(x, y)$ . Here, we use the convention that  $a$  divides  $b$  if and only if there exists some integer  $c$  such that  $ac=b$ .\n\n*Dennis Chen and Andrew Wen*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.06696,"x":0.11831,"p":[[0,11,0.0,0.06696,0.11285,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.08929,0.11152,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.11831,0.1245,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,1,12,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.02679,"x":0.16964,"p":[[0,48,0.0,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.08929,0.13716,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,7,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.06696,0.12364,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.05348,0.12746,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.08481,0.1466,0.0,0.0,0.14286,0.0,0.571,22,0,0,22,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,48,0.5833,0.08929,0.12242,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.12277,0.22815,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,1,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[36,48,0.75,0.08027,0.12335,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.09375,0.11071,0.0,0.07143,0.14286,0.0,0.4286,16,0,0,16,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.16964,0.12595,0.14286,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,15,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.16071,0.13716,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,11,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fa6e776aabc77e63","q":"On the side  $AC$  of triangle  $ABC$  point  $D$  is chosen. The perpendicular bisector of segment  $BD$  intersects the circumcircle  $\\Omega$  of triangle  $ABC$  at  $P$ ,  $Q$ . Point  $E$  lies on the arc  $AC$  of circle  $\\Omega$ , that doesn't contain point  $B$ , such that  $\\angle ABD=\\angle CBE$ . \nProve that the orthocenter of the triangle  $PQE$  lies on the line  $AC$ *M. Zorka*","t":[{"b":0,"e":0.0,"k":"flat","v":0.00223,"x":0.01786,"p":[[0,12,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.00893,"p":[[0,8,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2d158e113d28dd6b","q":"Find all functions such that  $ f: \\mathbb{R}^\\plus{} \\rightarrow \\mathbb{R}^\\plus{}$  and  $ f(x\\plus{}f(y))\\equal{}yf(xy\\plus{}1)$  for every  $ x,y\\in \\mathbb{R}^\\plus{}$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,15,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,14,0.0,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,14,0.2857,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e6995a4cdb34afb5","q":"Mykolka the numismatist possesses  $241$  coins, each worth an integer number of turgiks. The total value of the coins is  $360$  turgiks. Is it necessarily true that the coins can be divided into three groups of equal total value?","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,18,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[12,18,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,34,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d3e95037a9981836","q":"Let  $\\triangle ABC$  and  $A'$ , $B'$ , $C'$  the symmetrics of vertex over opposite sides.The intersection of the  circumcircles of  $\\triangle ABB'$  and  $\\triangle ACC'$  is  $A_1$ . $B_1$  and  $C_1$  are defined similarly.Prove that lines  $AA_1$ , $BB_1$  and  $CC_1$  are concurent.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.08473,"x":0.13398,"p":[[0,30,0.0,0.10268,0.09606,0.0,0.14286,0.14286,0.0,0.4286,12,0,1,12,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.08473,0.07867,0.0,0.14286,0.14286,0.0,0.28571,14,0,3,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.13398,0.1235,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,19,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,30,0.6667,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.10688,0.08738,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.12036,0.08068,0.14,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"rising","v":0.12036,"x":0.33929,"p":[[0,28,0.0,0.12947,0.10926,0.0,0.14286,0.14286,0.0,0.4286,9,0,2,9,0,19,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.12036,0.11352,0.0,0.14286,0.14286,0.0,0.4286,11,0,1,11,0,17,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.28126,0.15765,0.14286,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,8,0,0,5,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.30358,0.14174,0.25,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,5,0,0,9,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.30362,0.15875,0.24999,0.42857,0.42857,0.0,0.43,5,0,1,5,0,3,0,0,7,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.25447,0.17029,0.14286,0.28571,0.42857,0.0,0.4286,7,0,0,7,0,6,0,0,6,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.30362,0.15051,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,10,0,0,7,0,0,13,0,0,0,0,0,1,0,0,0,0,0],[28,28,1.0,0.33929,0.13244,0.2857,0.42857,0.42857,0.0,0.4286,2,0,0,2,0,4,0,0,6,0,0,20,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e46e2eafe60cb317","q":"Consider the set  $A=\\{1,2,3\\ldots ,2^n\\}, n\\ge 2$ . Find the number of subsets  $B$  of  $A$  such that for any two elements of  $A$  whose sum is a power of  $2$  exactly one of them is in  $B$ .\n\n*Aleksandar Ivanov*","t":[{"b":4,"e":0.85714,"k":"flat","v":0.62944,"x":0.76332,"p":[[0,8,0.0,0.62944,0.20158,0.42859,0.57143,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,0,0,0,9,0,0,7,0,0,8,0,0,4,0,3],[4,8,0.5,0.76332,0.18081,0.71429,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,13,0,0,3,0,9],[8,8,1.0,0.7589,0.2096,0.57143,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,10,0,0,2,0,11]]},{"b":7,"e":0.71429,"k":"flat","v":0.51339,"x":0.74995,"p":[[0,30,0.0,0.73213,0.20125,0.67857,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,12,0,0,5,0,7],[4,30,0.1333,0.74995,0.19564,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,9,0,0,4,0,9],[8,30,0.2667,0.62944,0.20473,0.4286,0.57143,0.75,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,10,0,0,8,0,0,5,0,0,4,0,4],[12,30,0.4,0.52233,0.17717,0.42857,0.42859,0.60714,0.1429,1.0,0,2,0,0,0,1,0,0,0,0,0,20,0,0,3,0,0,6,0,0,0,0,2],[16,30,0.5333,0.55357,0.17034,0.42857,0.57141,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,13,0,0,8,0,0,7,0,0,0,0,2],[20,30,0.6667,0.51339,0.12807,0.42857,0.42859,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,18,0,0,7,0,0,5,0,0,1,0,0],[24,30,0.8,0.54459,0.12594,0.42857,0.571,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,13,0,0,9,0,0,9,0,0,0,0,0],[28,30,0.9333,0.57591,0.20036,0.42857,0.5005,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,14,0,0,5,0,0,6,0,0,2,0,3],[30,30,1.0,0.65625,0.1984,0.42857,0.71429,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,10,0,0,2,0,5]]}]},{"i":"a44b438c72043afc","q":"$a_1,a_2,\\dots ,a_n>0$  are positive real numbers such that  $\\sum_{i=1}^{n} \\frac{1}{a_i}=n$  prove that: $\\sum_{i<j} \\left(\\frac{a_i-a_j}{a_i+a_j}\\right)^2\\le\\frac{n^2}{2}\\left(1-\\frac{n}{\\sum_{i=1}^{n}a_i}\\right)$","t":[{"b":2,"e":0.14286,"k":"flat","v":0.12946,"x":0.14286,"p":[[0,11,0.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.14286,1e-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],[11,11,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0625,"x":0.16518,"p":[[0,30,0.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.16518,0.15612,0.14286,0.14286,0.14286,0.0,1.0,2,1,0,2,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,30,0.2667,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.12937,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.12938,0.04162,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4286a97cd417613a","q":"16. (MCD 3) Let $n \\in \\mathbb{N}, n \\geq 2$, and $A_{0}=\\left(a_{01}, a_{02}, \\ldots, a_{0 n}\\right)$ be any $n$-tuple of natural numbers such that $0 \\leq a_{0 i} \\leq i-1$, for $i=1, \\ldots, n$. The $n$-tuples $A_{1}=\\left(a_{11}, a_{12}, \\ldots, a_{1 n}\\right), A_{2}=\\left(a_{21}, a_{22}, \\ldots, a_{2 n}\\right), \\ldots$ are defined by $a_{i+1, j}=\\operatorname{Card}\\left\\{a_{i, l} \\mid 1 \\leq l \\leq j-1, a_{i, l} \\geq a_{i, j}\\right\\}, \\quad$ for $i \\in \\mathbb{N}$ and $j=1, \\ldots, n$. Prove that there exists $k \\in \\mathbb{N}$, such that $A_{k+2}=A_{k}$.","t":[{"b":3,"e":0.14286,"k":"falling","v":0.06022,"x":0.26339,"p":[[0,9,0.0,0.26325,0.29235,0.10714,0.14286,0.42857,0.0,0.85714,8,0,0,8,0,15,0,0,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0],[4,9,0.4444,0.26339,0.30745,0.0,0.14286,0.28571,0.0,0.85714,9,0,0,9,0,13,0,0,3,0,0,0,0,0,1,0,0,0,0,0,6,0,0],[8,9,0.8889,0.06022,0.06941,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,1,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.16947,"x":0.46875,"p":[[0,21,0.0,0.3258,0.33362,0.14286,0.14286,0.71429,0.0,1.0,6,1,0,6,0,14,0,0,3,0,0,0,0,0,0,0,0,2,0,0,6,0,1],[4,21,0.1905,0.44643,0.31894,0.14286,0.28571,0.75,0.0,1.0,2,1,0,2,0,10,0,0,5,0,0,1,0,0,2,0,0,4,0,0,7,0,1],[8,21,0.381,0.46875,0.37156,0.14286,0.42857,0.85714,0.0,1.0,5,3,0,5,0,9,0,0,1,0,0,3,0,0,0,0,0,2,0,0,9,0,3],[12,21,0.5714,0.33473,0.33242,0.105,0.14286,0.71429,0.0,0.85714,8,0,0,8,0,11,0,0,1,0,0,2,0,0,0,0,0,4,0,0,6,0,0],[16,21,0.7619,0.34812,0.3425,0.14214,0.14286,0.71429,0.0,1.0,7,1,0,7,0,12,0,0,1,0,0,2,0,0,0,0,0,3,0,0,6,0,1],[20,21,0.9524,0.17857,0.10101,0.14286,0.14286,0.2857,0.0,0.42857,3,0,0,3,0,20,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.16947,0.08334,0.14286,0.14286,0.1429,0.0,0.42857,2,0,0,2,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e567d31166c1d67e","q":"7. (GEO 3) Let $a, b, c$ be given integers $a>0, a c-b^{2}=P=P_{1} \\cdots P_{m}$ where $P_{1}, \\ldots, P_{m}$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which  $$ a x^{2}+2 b x y+c y^{2}=n $$  Prove that $M(n)$ is finite and $M(n)=M\\left(P^{k} \\cdot n\\right)$ for every integer $k \\geq 0$.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.44196,"x":0.58026,"p":[[0,6,0.0,0.58026,0.33881,0.25001,0.64286,0.85714,0.14,1.0,0,7,0,0,0,8,0,0,4,0,0,2,0,0,2,0,0,3,0,0,6,0,7],[4,6,0.6667,0.54009,0.33463,0.24999,0.5,0.85714,0.14,1.0,0,7,0,0,0,8,0,0,6,0,0,2,0,0,2,0,0,4,0,0,3,0,7],[6,6,1.0,0.44196,0.30589,0.14286,0.35714,0.71429,0.14286,1.0,0,1,0,0,0,14,0,0,2,0,0,2,0,0,3,0,0,4,0,0,6,0,1]]},{"b":5,"e":0.14286,"k":"volatile","v":0.14268,"x":0.64277,"p":[[0,7,0.0,0.64277,0.33895,0.28571,0.71429,1.0,0.14,1.0,0,12,0,0,0,5,0,0,5,0,0,3,0,0,1,0,0,4,0,0,2,0,12],[4,7,0.5714,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"77019e3bd9c85fa8","q":"10. A4 (GBR) The function $F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: For every $n \\geq 0$, (i) $F(4 n)=F(2 n)+F(n)$; (ii) $F(4 n+2)=F(4 n)+1$; (iii) $F(2 n+1)=F(2 n)+1$.  Prove that for each positive integer $m$, the number of integers $n$ with $0 \\leq n<2^{m}$ and $F(4 n)=F(3 n)$ is $F\\left(2^{m+1}\\right)$.","t":[{"b":2,"e":1.0,"k":"rising","v":0.4732,"x":0.89285,"p":[[0,28,0.0,0.62053,0.29366,0.53569,0.71429,0.71429,0.0,1.0,3,7,0,3,0,0,0,0,4,0,0,1,0,0,6,0,0,11,0,0,0,0,7],[4,28,0.1429,0.4732,0.33203,0.1429,0.28571,0.71429,0.0,1.0,2,6,0,2,0,7,0,0,8,0,0,0,0,0,5,0,0,3,0,0,1,0,6],[8,28,0.2857,0.625,0.31492,0.28571,0.71429,1.0,0.14286,1.0,0,9,0,0,0,5,0,0,4,0,0,3,0,0,2,0,0,7,0,0,2,0,9],[12,28,0.4286,0.59375,0.27225,0.28571,0.71429,0.71429,0.0,1.0,1,5,0,1,0,2,0,0,6,0,0,1,0,0,4,0,0,13,0,0,0,0,5],[16,28,0.5714,0.79464,0.24468,0.71429,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,1,0,0,3,0,0,2,0,0,8,0,0,1,0,16],[20,28,0.7143,0.86607,0.17835,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,2,0,19],[24,28,0.8571,0.78121,0.18557,0.57143,0.71429,1.0,0.4286,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,10,0,0,1,0,12],[28,28,1.0,0.89285,0.13363,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,5,0,18]]},{"b":3,"e":0.57143,"k":"flat","v":0.57589,"x":0.68748,"p":[[0,9,0.0,0.57589,0.26118,0.39286,0.57143,0.71429,0.0,1.0,1,3,0,1,0,2,0,0,5,0,0,3,0,0,7,0,0,7,0,0,4,0,3],[4,9,0.4444,0.68748,0.2911,0.5354,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,2,0,0,3,0,0,5,0,0,4,0,0,6,0,9],[8,9,0.8889,0.62496,0.13717,0.571,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,11,0,0,2,0,1],[9,9,1.0,0.62052,0.1048,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,14,0,0,13,0,0,1,0,0]]}]},{"i":"30b859feefc5c5b8","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  $$ P(x, y, z)=F\\left(x^{2}+y^{2}+z^{2}-x y z\\right) . $$  (Russia) Common remarks. The polynomial $x^{2}+y^{2}+z^{2}-x y z$ satisfies the condition (*), so every polynomial of the form $F\\left(x^{2}+y^{2}+z^{2}-x y z\\right)$ does satisfy (*). We will use without comment the fact that two polynomials have the same coefficients if and only if they are equal as functions.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.15178,"x":0.26786,"p":[[0,19,0.0,0.18303,0.12992,0.0,0.2857,0.28571,0.0,0.28571,10,0,0,10,0,3,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.26786,0.05922,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.25669,0.09081,0.2857,0.28571,0.28571,0.0,0.5,2,0,0,2,0,4,0,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.23661,0.09852,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,3,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.18749,0.16142,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,4,0,0,15,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[19,19,1.0,0.15178,0.13803,0.0,0.21428,0.28571,0.0,0.28571,14,0,0,14,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.2857,"k":"flat","v":0.125,"x":0.31696,"p":[[0,18,0.0,0.20527,0.12344,0.105,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,2,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.125,0.15872,0.0,0.0,0.28571,0.0,0.57143,19,0,0,19,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,18,0.4444,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.28571,1e-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],[16,18,0.8889,0.31696,0.10555,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,29,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[18,18,1.0,0.31695,0.08548,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"59d08923189c6845","q":"$BB_1$  is the angle bisector of  $\\triangle ABC$ , and  $I$  is its incenter. The perpendicular bisector of segment  $AC$  intersects the circumcircle of  $\\triangle AIC$  at  $D$  and  $E$ . Point  $F$  is on the segment  $B_1C$  such that  $AB_1=CF$ .Prove that the four points  $B, D, E$  and  $F$  are concyclic.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.03563,"p":[[0,33,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.01768,0.04678,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.04018,"p":[[0,19,0.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bb6430437ddd66d5","q":"A hunter and an invisible rabbit play on an infinite square grid, that is, where each cell has four neighbors: left, right, up, and down. First, the hunter colors each cell of the grid, but can only use a finite number of colors. The rabbit then chooses a cell on the grid, which will be its starting point. It then begins to move: each minute, it tells the hunter the color of the cell it is on, and then moves to one of the four adjacent cells. Of course, since it is invisible, the only information the hunter has access to are the colors the rabbit announces each minute just before moving.\nThe hunter wins if, after a finite time,\n$\\triangleright$ he can identify the cell the rabbit initially chose, or if\n$\\triangleright$ the rabbit returns to a cell it had previously visited.\nDoes the hunter have a winning strategy?","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,46,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,22,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"9e5111868a5e9b9f","q":"Decide whether or not there exists a nonconstant polynomial $Q(x)$ with integer coefficients with the following property: for every positive integer $n>2$, the numbers  $$ Q(0), Q(1), Q(2), \\ldots, Q(n-1) $$  produce at most $0.499 n$ distinct residues when taken modulo $n$.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,6,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,21,0.0,0.04464,0.15335,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,21,0.1905,0.04018,0.1439,0.0,0.0,0.0,0.0,0.71429,29,0,1,29,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,21,0.381,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"914f7dfe93a7f554","q":"A positive integer  $n$  is *friendly* if the difference of each pair of neighbouring digits of  $n$ , written in base  $10$ , is exactly  $1$ . *For example, 6787 is friendly, but 211 and 901 are not.*\n\nFind all odd natural numbers  $m$  for which there exists a friendly integer divisible by  $64m$ .","t":[{"b":4,"e":0.85714,"k":"flat","v":0.58022,"x":0.78122,"p":[[0,47,0.0,0.58022,0.27412,0.42857,0.71429,0.71429,0.0,0.85714,4,0,0,4,0,0,0,0,3,0,0,3,0,0,1,0,0,15,0,0,6,0,0],[4,47,0.0851,0.59375,0.23176,0.42857,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,5,0,0,4,0,0,0,0,0,16,0,0,5,0,0],[8,47,0.1702,0.66963,0.20025,0.71429,0.71429,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,3,0,0,1,0,0,18,0,0,6,0,1],[12,47,0.2553,0.75,0.17494,0.71429,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,13,0,0,12,0,3],[16,47,0.3404,0.70982,0.15765,0.71429,0.71429,0.75,0.14286,1.0,0,1,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,20,0,0,7,0,1],[20,47,0.4255,0.73212,0.17035,0.71429,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,4,0,0,1,0,0,12,0,0,12,0,2],[24,47,0.5106,0.75893,0.11538,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,14,0,0,13,0,1],[28,47,0.5957,0.77902,0.11203,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,14,1,0,11,0,3],[32,47,0.6809,0.70536,0.14258,0.71429,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,21,0,0,4,0,2],[36,47,0.766,0.70982,0.13592,0.71429,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,17,0,0,9,0,0],[40,47,0.8511,0.78122,0.13826,0.71429,0.78564,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,13,0,0,12,0,4],[44,47,0.9362,0.71428,0.15151,0.71429,0.71429,0.74996,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,17,0,0,5,0,3],[47,47,1.0,0.70088,0.16114,0.71429,0.71429,0.85704,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,15,0,0,9,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.60713,"x":0.75446,"p":[[0,43,0.0,0.61159,0.24284,0.42857,0.71429,0.75,0.0,1.0,1,3,0,1,0,0,0,0,4,0,0,7,0,0,3,0,0,9,0,0,5,0,3],[4,43,0.093,0.60713,0.22587,0.42857,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,3,0,0,5,0,0,3,0,0,12,0,0,7,0,0],[8,43,0.186,0.6875,0.2034,0.71429,0.71429,0.85704,0.0,1.0,1,2,0,1,0,0,0,0,1,0,0,4,0,0,1,0,0,16,0,0,7,0,2],[12,43,0.2791,0.73659,0.16794,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,14,0,0,8,0,4],[16,43,0.3721,0.70536,0.15947,0.71429,0.71429,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,22,0,0,4,0,2],[20,43,0.4651,0.71429,0.10101,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,22,0,0,6,0,0],[24,43,0.5581,0.74107,0.09062,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,21,0,0,9,0,0],[28,43,0.6512,0.71874,0.09093,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,25,0,0,5,0,0],[32,43,0.7442,0.75432,0.1143,0.71429,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,20,0,0,10,0,1],[36,43,0.8372,0.74553,0.05904,0.71429,0.71429,0.71429,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,7,0,0],[40,43,0.9302,0.75446,0.06422,0.71429,0.71429,0.85704,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,0,9,0,0],[43,43,1.0,0.74107,0.08328,0.71429,0.71429,0.75,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,23,0,0,8,0,0]]}]},{"i":"f98ce0fb2ce15a53","q":"Chip and Dale play the following game. Chip starts by splitting  $1001$  nuts between three piles, so Dale can see it. In response, Dale chooses some number  $N$  from  $1$  to  $1001$ . Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly  $N$  nuts in any one or more piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,6,0.0,0.08036,0.25738,0.0,0.0,0.0,0.0,1.0,29,2,1,29,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.30357,"p":[[0,54,0.0,0.30357,0.42671,0.0,0.0,0.71429,0.0,1.0,21,6,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,6],[4,54,0.0741,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,54,0.5185,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"2db95f3e7b672782","q":"4. (MON 1) ${ }^{\\mathrm{IMO} 5}$ Let $d$ be the sum of the lengths of all diagonals of a convex polygon of $n(n>3)$ vertices and let $p$ be its perimeter. Prove that  $$ \\frac{n-3}{2}<\\frac{d}{p}<\\frac{1}{2}\\left(\\left[\\frac{n}{2}\\right]\\left[\\frac{n+1}{2}\\right]-2\\right) . $$","t":[{"b":3,"e":0.0,"k":"flat","v":0.10268,"x":0.16964,"p":[[0,4,0.0,0.16964,0.24074,0.0,0.0,0.28571,0.0,0.71429,18,0,0,18,0,5,0,0,2,0,0,1,0,0,4,0,0,2,0,0,0,0,0],[4,4,1.0,0.10268,0.15663,0.0,0.0,0.14287,0.0,0.57143,20,0,0,20,0,5,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.10268,"x":0.14733,"p":[[0,6,0.0,0.14733,0.22442,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,3,0,0,3,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[4,6,0.6667,0.10268,0.17581,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,5,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[6,6,1.0,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bb3b09cb8a3ef139","q":"Do there exist strictly positive integers $a$ and $b$ such that $a^{n}+n^{b}$ and $b^{n}+n^{a}$ are coprime for all integers $n \\geqslant 0$?","t":[{"b":1,"e":0.14286,"k":"flat","v":0.125,"x":0.27232,"p":[[0,30,0.0,0.125,0.1171,0.0,0.14286,0.14286,0.0,0.42857,10,0,1,10,0,19,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.22759,0.15515,0.14286,0.14286,0.42857,0.0,0.71429,1,0,0,1,0,22,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[8,30,0.2667,0.23661,0.15407,0.14286,0.14286,0.42857,0.0,0.4286,3,0,0,3,0,17,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.22768,0.15093,0.14286,0.14286,0.42857,0.0,0.4286,3,0,0,3,0,18,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.27232,0.18336,0.14286,0.42857,0.42857,0.0,0.5714,6,0,0,6,0,9,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[20,30,0.6667,0.20982,0.12869,0.14286,0.14286,0.21429,0.0,0.42857,1,0,0,1,0,23,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.21402,0.14299,0.14286,0.14286,0.2857,0.0,0.71429,1,0,0,1,0,22,0,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[28,30,0.9333,0.22768,0.14223,0.14286,0.14286,0.42857,0.0,0.4286,2,0,0,2,0,19,0,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.20536,0.11258,0.14286,0.14286,0.17857,0.14286,0.42857,0,0,0,0,0,24,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.13384,"x":0.2767,"p":[[0,30,0.0,0.17848,0.14288,0.14286,0.14286,0.14286,0.0,0.4286,6,0,0,6,0,19,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.16072,0.11152,0.14286,0.14286,0.14286,0.0,0.42857,4,0,1,4,0,24,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.17857,0.11845,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,23,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.2009,0.12807,0.14286,0.14286,0.1786,0.0,0.4286,2,0,0,2,0,22,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.22312,0.13339,0.14286,0.14286,0.42857,0.0,0.42857,1,0,0,1,0,21,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.20536,0.11811,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,25,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.14286,0.11294,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,21,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.13384,0.07086,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.2767,0.14266,0.14286,0.14286,0.42857,0.14,0.42857,0,0,0,0,0,17,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d8aabf4b7fb71833","q":"A flea is initially at the point  $(0, 0)$  in the Cartesian plane. Then it makes  $n$  jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length  $1$ , the second of length  $2$ , the third of length  $4$ , and so on. The  $n^{th}$ -jump is of length  $2^{n-1}$ . Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the  $n$  jumps.","t":[{"b":3,"e":1.0,"k":"flat","v":0.65623,"x":0.86159,"p":[[0,31,0.0,0.86159,0.20357,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,9,0,17],[4,31,0.129,0.67409,0.343,0.42857,0.85714,1.0,0.0,1.0,3,9,0,3,0,3,0,0,1,0,0,2,0,0,2,0,0,3,0,0,9,0,9],[8,31,0.2581,0.74541,0.29629,0.67846,0.85714,1.0,0.0,1.0,2,9,0,2,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,0,13,0,9],[12,31,0.3871,0.72762,0.31211,0.571,0.85714,1.0,0.0,1.0,3,13,0,3,0,0,0,0,1,0,0,1,0,0,8,0,0,1,0,0,5,0,13],[16,31,0.5161,0.83482,0.30116,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,3,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,21],[20,31,0.6452,0.65623,0.3131,0.42857,0.78564,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,2,0,0,5,0,0,2,0,0,3,0,0,9,0,7],[24,31,0.7742,0.68744,0.33205,0.42857,0.857,1.0,0.0,1.0,3,10,0,3,0,1,0,0,2,0,0,3,0,0,4,0,0,0,0,0,9,0,10],[28,31,0.9032,0.77678,0.31122,0.71429,0.85714,1.0,0.0,1.0,2,15,0,2,0,2,0,0,1,0,0,0,0,0,1,0,0,5,0,0,6,0,15],[31,31,1.0,0.75443,0.30144,0.57143,0.85714,1.0,0.0,1.0,1,13,0,1,0,3,0,0,1,0,0,0,0,0,4,0,0,3,0,0,7,0,13]]},{"b":7,"e":1.0,"k":"rising","v":0.72321,"x":1.0,"p":[[0,26,0.0,0.78571,0.28121,0.85711,0.85714,1.0,0.0,1.0,1,11,1,1,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,14,0,11],[4,26,0.1538,0.72321,0.33491,0.42857,0.85714,1.0,0.0,1.0,3,11,1,3,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,0,11,0,11],[8,26,0.3077,0.81696,0.2237,0.85714,0.85714,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,2,0,0,14,0,11],[12,26,0.4615,0.97321,0.12596,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"ea84d93eaf8c3741","q":"A calculator can square a number or add  $1$  to it. It cannot add  $1$  two times in a row. By several operations it transformed a number  $x$  into a number  $S > x^n + 1$  ( $x, n,S$  are positive integers). Prove that  $S > x^n + x - 1$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"47cbb26c1ee152eb","q":"7. A7 (IRE) Let $a_{1}, a_{2}, \\ldots, a_{n}$ be positive real numbers, $n>1$. Denote by $g_{n}$ their geometric mean, and by $A_{1}, A_{2}, \\ldots, A_{n}$ the sequence of arithmetic means defined by $A_{k}=\\frac{a_{1}+a_{2}+\\cdots+a_{k}}{k}, k=1,2, \\ldots, n$. Let $G_{n}$ be the geometric mean of $A_{1}, A_{2}, \\ldots, A_{n}$. Prove the inequality  $$ n \\sqrt[n]{\\frac{G_{n}}{A_{n}}}+\\frac{g_{n}}{G_{n}} \\leq n+1 $$  and establish the cases of equality.","t":[{"b":2,"e":0.2857,"k":"rising","v":0.04911,"x":0.29911,"p":[[0,54,0.0,0.14732,0.14054,0.0,0.21428,0.28571,0.0,0.28571,15,0,0,15,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.1116,0.14166,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,2,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.13393,0.13803,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,5,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.11152,0.13234,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,6,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.2991,0.04164,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.29464,0.04971,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.29018,0.04351,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.29911,0.05486,0.2857,0.28571,0.28571,0.1429,0.4286,0,0,0,0,0,1,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.28125,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.29464,0.07087,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.2991,0.07457,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.28571,0.06186,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.2991,0.04164,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.0,"x":0.14286,"p":[[0,39,0.0,0.14286,0.13832,0.0,0.14286,0.28571,0.0,0.28571,15,0,0,15,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.10268,0.1394,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,0,2,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.11162,0.15459,0.0,0.0,0.28571,0.0,0.4286,20,0,0,20,0,2,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.11161,0.1461,0.0,0.0,0.28571,0.0,0.42857,20,0,0,20,0,0,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.11161,0.1461,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.08036,0.13333,0.0,0.0,0.17857,0.0,0.42857,23,0,0,23,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.10268,0.12992,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.06696,0.12364,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"648226cd52494943","q":"EST   Let $a, b, c$ be positive real numbers such that $\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}=a+b+c$. Prove that  $$ \\frac{1}{(2 a+b+c)^{2}}+\\frac{1}{(2 b+c+a)^{2}}+\\frac{1}{(2 c+a+b)^{2}} \\leq \\frac{3}{16} $$","t":[{"b":3,"e":0.28571,"k":"flat","v":0.22321,"x":0.375,"p":[[0,37,0.0,0.22321,0.22851,0.0,0.14286,0.42857,0.0,0.71429,11,0,0,11,0,8,0,0,4,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[4,37,0.1081,0.26786,0.1948,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,8,0,0,7,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[8,37,0.2162,0.29463,0.25488,0.0,0.28571,0.46418,0.0,0.71429,10,0,0,10,0,4,0,0,4,0,0,6,0,0,4,0,0,4,0,0,0,0,0],[12,37,0.3243,0.24107,0.23266,0.0,0.14286,0.42858,0.0,0.71429,12,0,0,12,0,5,0,0,3,0,0,6,0,0,5,0,0,1,0,0,0,0,0],[16,37,0.4324,0.30804,0.12428,0.14289,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,9,0,0,10,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[20,37,0.5405,0.28116,0.1101,0.14286,0.28571,0.42857,0.14,0.4286,0,0,0,0,0,10,0,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.375,0.09942,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,10,0,0,18,0,0,2,0,0,0,0,0,0,0,0],[28,37,0.7568,0.34821,0.13333,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,7,0,0,7,0,0,15,0,0,3,0,0,0,0,0,0,0,0],[32,37,0.8649,0.34375,0.11214,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,10,0,0,16,0,0,1,0,0,0,0,0,0,0,0],[36,37,0.973,0.29911,0.13533,0.14286,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,12,0,0,6,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[37,37,1.0,0.33928,0.12753,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,11,0,0,12,0,0,3,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.01786,"x":0.26786,"p":[[0,26,0.0,0.26786,0.20438,0.14286,0.2857,0.42857,0.0,0.71429,6,0,0,6,0,9,0,0,7,0,0,4,0,0,5,0,0,1,0,0,0,0,0],[4,26,0.1538,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.1875,0.20959,0.0,0.14286,0.32143,0.0,0.57143,15,0,0,15,0,4,0,0,5,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[12,26,0.4615,0.10268,0.17582,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,7,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,26,0.6154,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0a498e452752a2ec","q":"A coin is called a Cape Town coin if its value is $1 / n$ for some positive integer $n$. Given a collection of Cape Town coins of total value at most $99+\\frac{1}{2}$, prove that it is possible to split this collection into at most 100 groups each of total value at most 1. (Luxembourg)","t":[{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.07589,"p":[[0,25,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.07589,0.15966,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,25,0.32,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.02223,"x":0.07142,"p":[[0,9,0.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.07142,0.17126,0.0,0.0,0.0,0.0,0.857,25,0,0,25,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,9,0.8889,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5ce29f11f81fc12e","q":"An acute-angled triangle  $ABC$  has altitudes  $AD, BE$  and  $CF$ . Let  $Q$  be an interior point of the segment  $AD$ , and let the circumcircles of the triangles  $QDF$  and  $QDE$  meet the line  $BC$  again at points  $X$  and  $Y$  , respectively. Prove that  $BX = CY$  .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02677,"p":[[0,36,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.02677,0.10367,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,36,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,17,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"fb91ba864c08f7b3","q":"$n$  - some natural. We write on the board all such numbers  $d$ , that  $d\\leq 1000$  and  $d|n+k$  for some  $ 1\\leq k \\leq 1000$ . Let  $S(n)$  -sum of all written numbers. Prove , that  $S(n)<10^6$  and  $S(n)>10^6$  has infinitely many solutions.","t":[{"b":5,"e":0.42857,"k":"falling","v":0.20955,"x":0.51785,"p":[[0,61,0.0,0.45536,0.36323,0.14286,0.42857,0.85714,0.0,1.0,5,4,0,5,0,9,0,0,1,0,0,3,0,0,1,0,0,4,0,0,5,0,4],[4,61,0.0656,0.49542,0.36948,0.14286,0.42836,0.85714,0.0,1.0,5,5,0,5,0,5,0,0,6,0,0,0,0,0,3,0,0,1,0,0,7,0,5],[8,61,0.1311,0.45535,0.33396,0.24999,0.28571,0.75,0.0,1.0,4,4,0,4,0,4,0,0,10,0,0,1,0,0,2,0,0,3,0,0,4,0,4],[12,61,0.1967,0.51785,0.3549,0.14286,0.57143,0.85714,0.0,1.0,5,4,0,5,0,5,0,0,2,0,0,2,0,0,4,0,0,3,0,0,7,0,4],[16,61,0.2623,0.46429,0.34442,0.14286,0.42859,0.85714,0.0,1.0,7,2,0,7,0,2,0,0,5,0,0,3,0,0,4,0,0,1,0,0,8,0,2],[20,61,0.3279,0.36157,0.33688,0.0,0.28571,0.57143,0.0,1.0,9,2,0,9,0,4,0,0,7,0,0,1,0,0,4,0,0,0,0,0,5,0,2],[24,61,0.3934,0.38838,0.33355,0.14286,0.28571,0.57143,0.0,1.0,6,4,0,6,0,7,0,0,5,0,0,3,0,0,4,0,0,1,0,0,2,0,4],[28,61,0.459,0.20955,0.21432,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,10,0,0,6,0,0,1,0,0,3,0,0,2,0,0,0,0,0],[32,61,0.5246,0.29018,0.26842,0.0,0.2857,0.4286,0.0,1.0,10,1,1,10,0,3,0,0,8,0,0,4,0,0,4,0,0,1,0,0,1,0,1],[36,61,0.5902,0.38383,0.32236,0.14286,0.28571,0.60714,0.0,1.0,5,3,0,5,0,7,0,0,8,0,0,2,0,0,2,0,0,2,0,0,3,0,3],[40,61,0.6557,0.33036,0.25364,0.14286,0.2857,0.4286,0.0,0.85714,5,0,0,5,0,6,0,0,10,0,0,4,0,0,2,0,0,2,0,0,3,0,0],[44,61,0.7213,0.40179,0.33964,0.14286,0.28571,0.57143,0.0,1.0,7,4,0,7,0,4,0,0,7,0,0,2,0,0,5,0,0,0,0,0,3,0,4],[48,61,0.7869,0.44195,0.37004,0.14286,0.42835,0.85704,0.0,1.0,7,4,1,7,0,7,0,0,2,0,0,0,0,0,5,0,0,2,0,0,5,0,4],[52,61,0.8525,0.40169,0.34528,0.105,0.28571,0.71429,0.0,1.0,8,2,0,8,0,6,0,0,3,0,0,0,0,0,5,0,0,4,0,0,4,0,2],[56,61,0.918,0.46426,0.3234,0.24999,0.42859,0.75,0.0,1.0,5,3,0,5,0,3,0,0,6,0,0,3,0,0,6,0,0,1,0,0,5,0,3],[60,61,0.9836,0.36607,0.33108,0.14286,0.2857,0.57143,0.0,1.0,6,3,0,6,0,9,0,0,5,0,0,1,0,0,4,0,0,1,0,0,3,0,3],[61,61,1.0,0.25446,0.24415,0.10714,0.14288,0.42857,0.0,1.0,8,1,0,8,0,9,0,0,6,0,0,5,0,0,2,0,0,0,0,0,1,0,1]]},{"b":6,"e":0.57143,"k":"rising","v":0.47766,"x":0.71872,"p":[[0,9,0.0,0.47766,0.35465,0.14286,0.42857,0.85714,0.0,1.0,6,4,0,6,0,4,0,0,4,0,0,3,0,0,2,0,0,4,0,0,5,0,4],[4,9,0.4444,0.71872,0.15767,0.57143,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,8,0,0,10,0,0,10,0,2],[8,9,0.8889,0.71425,0.13834,0.57143,0.71429,0.85714,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,8,0,0,9,0,2],[9,9,1.0,0.70089,0.13054,0.57143,0.71429,0.75,0.5714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,11,0,0,6,0,2]]}]},{"i":"b383a1c39643e297","q":"A prime number  $p > 2$  and  $x,y \\in \\left\\{ 1,2,\\ldots, \\frac{p-1}{2} \\right\\}$  are given. Prove that if  $x\\left( p-x\\right)y\\left( p-y\\right)$  is a perfect square, then  $x = y$ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.2679,"x":0.39732,"p":[[0,8,0.0,0.27231,0.22687,0.10714,0.2857,0.42857,0.0,1.0,8,1,0,8,0,4,0,0,11,0,0,4,0,0,4,0,0,0,0,0,0,0,1],[4,8,0.5,0.2679,0.13249,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,6,0,0,16,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[8,8,1.0,0.39732,0.12233,0.28571,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,13,0,0,3,0,0,2,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.15628,"x":0.28571,"p":[[0,18,0.0,0.27232,0.1488,0.25,0.28571,0.28571,0.0,0.71429,4,0,1,4,0,4,0,0,17,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[4,18,0.2222,0.27231,0.18334,0.14286,0.28571,0.28571,0.0,0.85714,5,0,0,5,0,5,0,0,15,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[8,18,0.4444,0.28571,0.15972,0.2857,0.28571,0.32143,0.0,0.71429,5,0,0,5,0,1,0,0,18,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[12,18,0.6667,0.28122,0.14048,0.2857,0.28571,0.28571,0.0,0.571,4,0,0,4,0,2,0,0,19,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[16,18,0.8889,0.25886,0.20641,0.0,0.28571,0.42857,0.0,0.71429,9,0,1,9,0,3,0,0,11,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[18,18,1.0,0.15628,0.20003,0.0,0.07143,0.2857,0.0,0.71429,16,0,0,16,0,7,0,0,2,0,0,5,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"a33a084608554553","q":"Denote by $\\mathbb{N}$ the positive integers. Let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a function such that, for any $w, x, y, z \\in \\mathbb{N}$,\n\n$$\nf(f(f(z))) f(w x f(y f(z)))=z^{2} f(x f(y)) f(w)\n$$\n\nShow that $f(n!) \\geq n$ ! for every positive integer $n$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.08929,"x":0.28114,"p":[[0,16,0.0,0.16509,0.29905,0.0,0.0,0.14286,0.0,1.0,20,3,0,20,0,5,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[4,16,0.25,0.28114,0.36333,0.0,0.14143,0.57111,0.0,1.0,15,4,0,15,0,5,0,0,3,0,0,0,0,0,2,0,0,2,0,0,1,0,4],[8,16,0.5,0.26339,0.34646,0.0,0.07143,0.42858,0.0,1.0,16,4,0,16,0,4,0,0,1,0,0,4,0,0,2,0,0,1,0,0,0,0,4],[12,16,0.75,0.10715,0.23958,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[16,16,1.0,0.08929,0.16269,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,7,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.26339,"p":[[0,28,0.0,0.15178,0.30916,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,2],[4,28,0.1429,0.26339,0.37134,0.0,0.0,0.35714,0.0,1.0,17,4,0,17,0,4,0,0,3,0,0,0,0,0,1,0,0,1,0,0,2,0,4],[8,28,0.2857,0.16061,0.29395,0.0,0.0,0.21214,0.0,1.0,23,2,0,23,0,1,0,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,2],[12,28,0.4286,0.05357,0.10565,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2dbc248ce501151d","q":"$I,\\Omega$  are the incenter and the circumcircle of triangle  $ABC$ , respectively, and the tangents of  $B,C$  to  $\\Omega$  intersect at  $L$ . Assume that  $P\\neq C$  is a point on  $\\Omega$  such that  $CI,AP$ , and the circle with center  $L$  and radius  $LC$  are concurrent. Let the foot from  $I$  to  $AB$  be  $F$ , the midpoint of  $BC$  be  $M$ ,  $X$  is a point on  $\\Omega$  s.t.  $AI,BC,PX$  are concurrent. Prove that the lines  $AI,AX,MF$  form an isosceles triangle.\n\n*Proposed by ckliao914*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,19,0.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01777,0.04701,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,7,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"82fb4080627ff253","q":"$ABCDEF$  is a cyclic hexagon with circumcenter  $O$ , and  $AD, BE, CF$  are concurrent at  $X$ .  $P$  is a point on the plane. The circumenter of  $PAB$  is  $O_{AB}$ . Define  $O_{BC}, O_{CD}$ ,  $O_{DE}, O_{EF}, O_{FA}$  similarly. Prove that  $O_{AB} O_{DE}, O_{BC}O_{EF}, O_{CD}O_{FA}$ ,  $OX$  are concurrent.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,10,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,9,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4c646342a7233a59","q":"A triangle  $ABC$  is given. A circle  $\\gamma$  centered at  $A$  meets segments  $AB$  and  $AC$ . The common chord of  $\\gamma$  and the circumcircle of  $ABC$  meets  $AB$  and  $AC$  at  $X$  and  $Y$ , respectively. The segments  $CX$  and  $BY$  meet  $\\gamma$  at point  $S$  and  $T$ , respectively. The circumcircles of triangles  $ACT$  and  $BAS$  meet at points  $A$  and  $P$ . Prove that  $CX, BY$  and  $AP$  concur.","t":[{"b":1,"e":0.0,"k":"falling","v":0.13393,"x":0.37052,"p":[[0,21,0.0,0.37052,0.21974,0.24999,0.28571,0.57111,0.0,0.85714,1,0,0,1,0,7,0,0,12,0,0,3,0,0,3,0,0,5,0,0,1,0,0],[4,21,0.1905,0.17411,0.17029,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,5,0,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[8,21,0.381,0.13393,0.19541,0.0,0.0,0.14286,0.0,0.85714,17,0,0,17,0,8,0,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[12,21,0.5714,0.14728,0.17298,0.0,0.14286,0.14287,0.0,0.571,13,0,0,13,0,12,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[16,21,0.7619,0.17856,0.202,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,10,0,0,5,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[20,21,0.9524,0.22768,0.19186,0.14286,0.14286,0.32143,0.0,0.71429,7,0,0,7,0,11,0,0,6,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[21,21,1.0,0.18746,0.21845,0.0,0.14286,0.32142,0.0,0.71429,13,0,0,13,0,10,0,0,1,0,0,3,0,0,4,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.12714,"x":0.31247,"p":[[0,7,0.0,0.31247,0.24595,0.14286,0.2857,0.4642,0.0,1.0,5,1,0,5,0,7,0,0,11,0,0,1,0,0,4,0,0,3,0,0,0,0,1],[4,7,0.5714,0.16965,0.16536,0.0,0.14288,0.28571,0.0,0.71429,10,0,0,10,0,11,0,0,9,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[7,7,1.0,0.12714,0.12592,0.0,0.14143,0.2857,0.0,0.28571,14,0,0,14,1,6,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3bd65ae46063b32b","q":"$a_n,b_n,c_n$  are three sequences of positive integers satisfying   $$ \\prod_{d|n}a_d=2^n-1,\\prod_{d|n}b_d=\\frac{3^n-1}{2},\\prod_{d|n}c_d=\\gcd(2^n-1,\\frac{3^n-1}{2}) $$   for all  $n\\in \\mathbb{N}$ . Prove that  $\\gcd(a_n,b_n)|c_n$  for all  $n\\in \\mathbb{N}$","t":[{"b":0,"e":0.2857,"k":"flat","v":0.20982,"x":0.29456,"p":[[0,13,0.0,0.20982,0.2082,0.14286,0.14286,0.28571,0.0,1.0,5,1,0,5,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,13,0.3077,0.29456,0.13343,0.14289,0.28571,0.42857,0.14,0.57143,0,0,0,0,0,10,0,0,13,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[8,13,0.6154,0.24553,0.06423,0.14289,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.25446,0.07771,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.26785,0.06916,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,3,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.25893,"p":[[0,10,0.0,0.15177,0.14695,0.0,0.14286,0.14286,0.0,0.571,10,0,0,10,0,15,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,10,0.4,0.25893,0.22428,0.14286,0.14286,0.28571,0.0,1.0,3,1,0,3,0,15,0,0,9,0,0,1,0,0,0,0,0,3,0,0,0,0,1],[8,10,0.8,0.04911,0.06546,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,2,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.00893,0.02961,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"37596ad31d1c4c1b","q":"Denote by  $\\left(ABC\\right)$  the circumcircle of a triangle  $ABC$ .\r\nLet  $ABC$  be an isosceles right-angled triangle with  $AB=AC=1$  and  $\\measuredangle CAB=90^{\\circ}$ . Let  $D$  be the midpoint of the side  $BC$ , and let  $E$  and  $F$  be two points on the side  $BC$ .\r\nLet  $M$  be the point of intersection of the circles  $\\left(ADE\\right)$  and  $\\left(ABF\\right)$  (apart from  $A$ ).\r\nLet  $N$  be the point of intersection of the line  $AF$  and the circle  $\\left(ACE\\right)$  (apart from  $A$ ).\r\nLet  $P$  be the point of intersection of the line  $AD$  and the circle  $\\left(AMN\\right)$ .\r\nFind the length of  $AP$ .","t":[{"b":1,"e":0.71429,"k":"rising","v":0.12946,"x":0.35714,"p":[[0,5,0.0,0.16518,0.19597,0.0,0.0,0.42857,0.0,0.5714,17,0,0,17,0,3,0,0,3,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[4,5,0.8,0.12946,0.19019,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,1,0,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[5,5,1.0,0.35714,0.11294,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,21,0,0,7,0,0,3,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"rising","v":0.16071,"x":0.38393,"p":[[0,35,0.0,0.16071,0.18123,0.0,0.0,0.28571,0.0,0.42857,17,0,0,17,0,1,0,0,7,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.20534,0.21108,0.0,0.14288,0.42857,0.0,0.71429,14,0,0,14,0,3,0,0,5,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[8,35,0.2286,0.17411,0.18118,0.0,0.14285,0.28571,0.0,0.4286,16,0,0,16,0,0,0,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.19643,0.1948,0.0,0.21428,0.42857,0.0,0.4286,15,0,0,15,0,1,0,0,5,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.16964,0.21261,0.0,0.0,0.42857,0.0,0.57143,19,0,1,19,0,0,0,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[20,35,0.5714,0.25891,0.20956,0.0,0.28571,0.42857,0.0,0.57143,11,0,0,11,0,2,0,0,4,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[24,35,0.6857,0.20981,0.22009,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,3,0,0,1,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[28,35,0.8,0.21875,0.22011,0.0,0.14286,0.42857,0.0,0.57143,14,0,0,14,0,3,0,0,3,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[32,35,0.9143,0.28115,0.19397,0.14214,0.42857,0.42857,0.0,0.57143,7,0,0,7,0,7,0,0,0,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[35,35,1.0,0.38393,0.14914,0.28571,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,5,0,0,3,0,0,17,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"318c11a66eec9ea1","q":"Define a sequence of integers by  $a_0=1$  , and  $a_n=\\sum_{k=0}^{n-1} \\binom{n}{k}a_k$   ,  $n \\geq 1$  . Let  $m$  be a positive integer , let  $p$  be a prime , and let  $q$  and  $r$  be non-negative integers . Prove that :  $$ a_{p^mq+r} \\equiv a_{p^{m-1}q+r} \\pmod{p^m} $$","t":[{"b":0,"e":0.14286,"k":"flat","v":0.00893,"x":0.07143,"p":[[0,37,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.09812,"p":[[0,7,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.09812,0.06616,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3e99ad95a1e32070","q":"An integer $n \\geqslant 2$ is written on the board. Each day, someone chooses $p$, a prime divisor of the integer $n$ written on the board, erases it, and writes $n+\\frac{n}{p}$ in its place. Show that $p=3$ is chosen infinitely many times.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.38384,"x":0.80801,"p":[[0,19,0.0,0.80801,0.2541,0.67857,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,4,0,0,1,0,0,3,0,0,3,0,0,4,0,17],[4,19,0.2105,0.79909,0.27401,0.71429,0.85714,1.0,0.0,1.0,1,14,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,0,9,0,14],[8,19,0.4211,0.72321,0.32525,0.42857,0.85714,1.0,0.0,1.0,1,14,0,1,0,2,0,0,4,0,0,3,0,0,0,0,0,3,0,0,5,0,14],[12,19,0.6316,0.42409,0.30822,0.14289,0.35714,0.71429,0.0,1.0,3,3,0,3,0,7,0,0,6,0,0,6,0,0,1,0,0,3,0,0,3,0,3],[16,19,0.8421,0.58483,0.31815,0.28571,0.4286,1.0,0.0,1.0,1,10,0,1,0,1,0,0,7,0,0,10,0,0,0,0,0,2,0,0,1,0,10],[19,19,1.0,0.38384,0.23817,0.2857,0.28571,0.42857,0.0,1.0,1,3,0,1,0,4,0,0,14,0,0,8,0,0,1,0,0,1,0,0,0,0,3]]},{"b":4,"e":0.14286,"k":"flat","v":0.72766,"x":0.87946,"p":[[0,21,0.0,0.87946,0.2055,0.85714,1.0,1.0,0.0,1.0,1,17,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,11,0,17],[4,21,0.1905,0.83915,0.21061,0.71429,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,3,0,0,7,0,16],[8,21,0.381,0.84819,0.20809,0.82143,0.92857,1.0,0.28571,1.0,0,16,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,8,0,16],[12,21,0.5714,0.87498,0.16659,0.85711,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,9,0,16],[16,21,0.7619,0.86158,0.22157,0.857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,8,0,18],[20,21,0.9524,0.72766,0.23788,0.5354,0.78571,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,3,0,0,5,0,0,2,0,0,6,0,0,8,0,8],[21,21,1.0,0.76784,0.18122,0.71429,0.78564,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,9,0,0,9,0,7]]}]},{"i":"6e90f9b8e744ad76","q":". Let $n \\geqslant 2$ be an integer and let $x_{1}, \\ldots, x_{n}$ be positive real numbers such that $x_{1} x_{2} \\cdots x_{n}=1$. Show that\n\n$$\n\\frac{1}{n-1+x_{1}}+\\frac{1}{n-1+x_{2}}+\\cdots+\\frac{1}{n-1+x_{n}} \\leqslant 1\n$$","t":[{"b":6,"e":1.0,"k":"volatile","v":0.21429,"x":1.0,"p":[[0,13,0.0,0.21429,0.3977,0.0,0.0,0.03571,0.0,1.0,24,6,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[4,13,0.3077,0.21429,0.39609,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,5],[8,13,0.6154,0.53125,0.47814,0.0,0.71429,1.0,0.0,1.0,14,15,0,14,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,15],[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],[13,13,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":7,"e":0.0,"k":"volatile","v":0.03571,"x":0.625,"p":[[0,23,0.0,0.1875,0.3719,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[4,23,0.1739,0.23661,0.41282,0.0,0.0,0.17857,0.0,1.0,24,6,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,6],[8,23,0.3478,0.17411,0.36897,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[12,23,0.5217,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[16,23,0.6957,0.44196,0.47697,0.0,0.0,1.0,0.0,1.0,17,11,0,17,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,11],[20,23,0.8696,0.625,0.4598,0.0,0.92857,1.0,0.0,1.0,11,16,0,11,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,16],[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]]}]},{"i":"a79ce28de9b6ebb5","q":"10. C5 (SWE) At a round table are 1994 girls, playing a game with a deck of $n$ cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when and only when each girl is holding at most one card. (a) Prove that if $n \\geq 1994$, then the game cannot end. (b) Prove that if $n<1994$, then the game must end.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.23659,"x":0.42862,"p":[[0,9,0.0,0.29018,0.21275,0.0,0.42857,0.42857,0.0,0.71429,10,0,0,10,0,1,0,0,2,0,0,17,0,0,1,0,0,1,0,0,0,0,0],[4,9,0.4444,0.23659,0.20703,0.0,0.2857,0.42857,0.0,0.571,13,0,0,13,0,1,0,0,3,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[8,9,0.8889,0.42857,1e-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],[9,9,1.0,0.42862,0.00025,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.06241,"x":0.3125,"p":[[0,18,0.0,0.24554,0.20589,0.0,0.42857,0.42857,0.0,0.42857,13,0,0,13,0,0,0,0,2,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.3125,0.21852,0.0,0.42857,0.42857,0.0,0.71429,9,0,0,9,0,1,0,0,2,0,0,17,0,0,1,0,0,2,0,0,0,0,0],[8,18,0.4444,0.23659,0.20703,0.0,0.28571,0.42857,0.0,0.571,13,0,0,13,0,1,0,0,3,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[12,18,0.6667,0.06241,0.12334,0.0,0.0,0.035,0.0,0.4286,24,0,0,24,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.06696,0.12364,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7df4a734940c6594","q":"A plane is cut into unit squares, which are then colored in  $n$  colors. A polygon  $P$  is created from  $n$  unit squares that are connected by their sides. It is known that any cell polygon created by  $P$  with translation, covers  $n$  unit squares in different colors. Prove that the plane can be covered with copies of  $P$  so that each cell is covered exactly once.","t":[{"b":2,"e":1.0,"k":"volatile","v":0.60266,"x":1.0,"p":[[0,5,0.0,0.60266,0.45839,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,17],[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":3,"e":0.0,"k":"volatile","v":0.00893,"x":0.55348,"p":[[0,11,0.0,0.45535,0.47034,0.0,0.14286,1.0,0.0,1.0,14,12,0,14,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,12],[4,11,0.3636,0.55348,0.45007,0.0,0.71429,1.0,0.0,1.0,10,14,0,10,0,3,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,14],[8,11,0.7273,0.52232,0.43829,0.0,0.57143,1.0,0.0,1.0,11,12,0,11,0,0,0,0,3,0,0,2,0,0,0,0,0,3,0,0,1,0,12],[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]]}]},{"i":"f48b5b2d498114fe","q":"20. A6 (IRN) Let $A$ be a nonempty set of positive integers. Suppose that there are positive integers $b_{1}, \\ldots, b_{n}$ and $c_{1}, \\ldots, c_{n}$ such that (i) for each $i$ the set $b_{i} A+c_{i}=\\left\\{b_{i} a+c_{i} \\mid a \\in A\\right\\}$ is a subset of $A$, and (ii) the sets $b_{i} A+c_{i}$ and $b_{j} A+c_{j}$ are disjoint whenever $i \\neq j$.  Prove that  $$ \\frac{1}{b_{1}}+\\cdots+\\frac{1}{b_{n}} \\leq 1 $$","t":[{"b":0,"e":0.85714,"k":"flat","v":0.63838,"x":0.82589,"p":[[0,27,0.0,0.68302,0.22512,0.57142,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,4,0,0,3,0,0,9,0,0,11,0,2],[4,27,0.1481,0.70533,0.20807,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,1,0,0,2,0,0,2,0,0,5,0,0,7,0,0,13,0,2],[8,27,0.2963,0.72766,0.16115,0.67857,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,11,0,0,11,0,2],[12,27,0.4444,0.66068,0.15872,0.57143,0.71429,0.857,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,8,0,0,9,0,0],[16,27,0.5926,0.63838,0.22012,0.57143,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,0,0,0,2,0,0,2,0,0,3,0,0,17,0,0,6,0,0],[20,27,0.7407,0.80803,0.12168,0.85711,0.85714,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,0,24,0,1],[24,27,0.8889,0.82142,0.07142,0.857,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,25,0,0],[27,27,1.0,0.82589,0.15458,0.85714,0.85714,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,1]]},{"b":4,"e":0.71429,"k":"rising","v":0.44642,"x":0.86607,"p":[[0,20,0.0,0.70089,0.23517,0.67857,0.78564,0.85714,0.0,1.0,1,1,0,1,0,2,0,0,0,0,0,2,0,0,3,0,0,8,0,0,15,0,1],[4,20,0.2,0.67407,0.24284,0.57143,0.71429,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,0,0,0,2,0,0,5,0,0,9,0,0,12,0,1],[8,20,0.4,0.44642,0.29178,0.14286,0.42857,0.71429,0.0,0.85714,4,0,0,4,0,5,0,0,5,0,0,3,0,0,4,0,0,6,0,0,5,0,0],[12,20,0.6,0.45534,0.25363,0.28571,0.42857,0.71429,0.0,0.85714,2,0,0,2,0,5,0,0,6,0,0,4,0,0,4,0,0,9,0,0,2,0,0],[16,20,0.8,0.85712,0.10718,0.85714,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,20,0,7],[20,20,1.0,0.86607,0.11811,0.85714,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,18,0,9]]}]},{"i":"6443faaa8df540fb","q":"14. (ROM) Prove that a convex pentagon (a five-sided polygon) $A B C D E$ with equal sides and for which the interior angles satisfy the condition $\\angle A \\geq \\angle B \\geq \\angle C \\geq \\angle D \\geq \\angle E$ is a regular pentagon.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.04464,"p":[[0,18,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.03572,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,8,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d36685d2bd96f9e8","q":"A convex quadrilateral  $ABCD$  with  $AC \\neq BD$  is inscribed in a circle with center  $O$ . Let  $E$  be the intersection of diagonals  $AC$  and  $BD$ . If  $P$  is a point inside  $ABCD$  such that  $\\angle PAB+\\angle PCB=\\angle PBC+\\angle PDC=90^\\circ$ , prove that  $O$ ,  $P$  and  $E$  are collinear.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.22767,"p":[[0,68,0.0,0.14732,0.24085,0.0,0.0,0.28571,0.0,0.857,21,0,0,21,0,2,0,0,3,0,0,2,0,0,2,0,0,1,0,0,1,0,0],[4,68,0.0588,0.14731,0.26117,0.0,0.0,0.1786,0.0,1.0,22,1,0,22,0,2,0,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,1],[8,68,0.1176,0.16516,0.30115,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,2,0,0,0,0,0,0,0,0,3,0,0,2,0,0,1,0,1],[12,68,0.1765,0.12054,0.27919,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[16,68,0.2353,0.16964,0.30606,0.0,0.0,0.1786,0.0,1.0,23,1,0,23,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,1],[20,68,0.2941,0.13393,0.22286,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,5,0,0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[24,68,0.3529,0.10268,0.26058,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[28,68,0.4118,0.22767,0.2985,0.0,0.0,0.42857,0.0,0.85714,17,0,0,17,0,3,0,0,3,0,0,2,0,0,1,0,0,4,0,0,2,0,0],[32,68,0.4706,0.03571,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,68,0.5294,0.04018,0.15663,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[40,68,0.5882,0.09375,0.21609,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[44,68,0.6471,0.05357,0.12242,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,68,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,68,0.7647,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,68,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,68,0.8824,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,68,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.85714,"k":"rising","v":0.08929,"x":0.84372,"p":[[0,53,0.0,0.23658,0.33993,0.0,0.0,0.46418,0.0,1.0,20,1,0,20,0,0,0,0,3,0,0,1,0,0,1,0,0,3,0,0,3,0,1],[4,53,0.0755,0.08929,0.23077,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[8,53,0.1509,0.26344,0.30954,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,2,0,0,3,0,0,7,0,0,0,0,0,1,0,0,3,0,1],[12,53,0.2264,0.20535,0.33107,0.0,0.0,0.28571,0.0,1.0,20,2,0,20,0,2,0,0,4,0,0,0,0,0,0,0,0,2,0,0,2,0,2],[16,53,0.3019,0.16963,0.26349,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,4,0,0,2,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[20,53,0.3774,0.1607,0.26182,0.0,0.0,0.2857,0.0,1.0,21,1,0,21,0,1,0,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,1],[24,53,0.4528,0.14723,0.24609,0.0,0.0,0.1786,0.0,1.0,19,1,0,19,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,1],[28,53,0.5283,0.12491,0.23348,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,7,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[32,53,0.6038,0.12496,0.25697,0.0,0.0,0.035,0.0,0.85714,24,0,0,24,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0],[36,53,0.6792,0.53569,0.35354,0.14286,0.64264,0.85714,0.0,1.0,4,5,0,4,0,5,0,0,3,0,0,3,0,0,1,0,0,5,0,0,6,0,5],[40,53,0.7547,0.84372,0.1651,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,5,0,0,11,0,12],[44,53,0.8302,0.81695,0.18294,0.71429,0.85714,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,14,0,9],[48,53,0.9057,0.80798,0.17358,0.71429,0.85707,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,6,0,0,11,0,9],[52,53,0.9811,0.8169,0.14396,0.71429,0.857,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,14,0,7],[53,53,1.0,0.8214,0.15152,0.71429,0.85707,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,7,0,0,17,0,6]]}]},{"i":"0cea95e35f2176b2","q":"10. (ITA 1) ${ }^{\\mathrm{IMO5}}$ Let $V$ be a finite subset of Euclidean space consisting of points $(x, y, z)$ with integer coordinates. Let $S_{1}, S_{2}, S_{3}$ be the projections of $V$ onto the $y z, x z, x y$ planes, respectively. Prove that  $$ |V|^{2} \\leq\\left|S_{1}\\right|\\left|S_{2}\\right|\\left|S_{3}\\right| $$  ( $|X|$ denotes the number of elements of $X$ ).","t":[{"b":1,"e":0.14286,"k":"flat","v":0.26785,"x":0.36158,"p":[[0,15,0.0,0.35713,0.18209,0.2857,0.42857,0.4286,0.0,0.71429,3,0,0,3,0,3,0,0,9,0,0,11,0,0,4,0,0,2,0,0,0,0,0],[4,15,0.2667,0.26785,0.19802,0.14286,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,10,0,0,11,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[8,15,0.5333,0.33036,0.1448,0.28571,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,4,0,0,20,0,0,4,0,0,3,0,0,0,0,0,1,0,0],[12,15,0.8,0.36158,0.13821,0.28571,0.28571,0.571,0.14286,0.57143,0,0,0,0,0,2,0,0,20,0,0,1,0,0,9,0,0,0,0,0,0,0,0],[15,15,1.0,0.32589,0.15663,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,5,0,0,20,0,0,3,0,0,2,0,0,1,0,0,1,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.27677,"x":0.4375,"p":[[0,11,0.0,0.30357,0.22232,0.14286,0.28571,0.42857,0.0,0.71429,7,0,1,7,0,4,0,0,8,0,0,7,0,0,3,0,0,3,0,0,0,0,0],[4,11,0.3636,0.27677,0.20181,0.14286,0.28571,0.42857,0.0,0.85714,5,0,1,5,0,9,0,0,7,0,0,7,0,0,3,0,0,0,0,0,1,0,0],[8,11,0.7273,0.3125,0.25862,0.10714,0.28571,0.42857,0.0,1.0,8,2,3,8,0,3,0,0,7,0,0,9,0,0,3,0,0,0,0,0,0,0,2],[11,11,1.0,0.4375,0.24206,0.2857,0.28571,0.57143,0.14286,1.0,0,3,0,0,0,4,0,0,13,0,0,3,0,0,7,0,0,2,0,0,0,0,3]]}]},{"i":"61170fc7037c0396","q":"Determine all integers  $n \\geq 2$  such that there exists a permutation  $x_0, x_1, \\ldots, x_{n - 1}$  of the numbers  $0, 1, \\ldots, n - 1$  with the property that the  $n$  numbers  $$ x_0, \\hspace{0.3cm} x_0 + x_1, \\hspace{0.3cm} \\ldots, \\hspace{0.3cm} x_0 + x_1 + \\ldots  + x_{n - 1} $$  are pairwise distinct modulo  $n$ .","t":[{"b":4,"e":0.0,"k":"falling","v":0.15616,"x":0.74107,"p":[[0,30,0.0,0.74107,0.27765,0.42857,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,13,0,0,2,0,0,0,0,0,0,0,17],[4,30,0.1333,0.59821,0.26351,0.42857,0.42857,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,1,0,0,21,0,0,0,0,0,0,0,0,1,0,9],[8,30,0.2667,0.59821,0.26351,0.42857,0.42857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,19,0,0,2,0,0,1,0,0,0,0,9],[12,30,0.4,0.45536,0.16536,0.42857,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,2],[16,30,0.5333,0.4107,0.22516,0.42857,0.42857,0.42857,0.0,1.0,2,2,0,2,0,5,0,0,0,0,0,21,0,0,1,0,0,0,0,0,1,0,2],[20,30,0.6667,0.29911,0.16506,0.14286,0.35714,0.42857,0.0,0.71429,2,0,0,2,0,11,0,0,3,0,0,15,0,0,0,0,0,1,0,0,0,0,0],[24,30,0.8,0.20081,0.13771,0.14286,0.14286,0.28571,0.0,0.42857,4,0,0,4,0,18,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.16955,0.14034,0.105,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,15,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.15616,0.13534,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.571,"k":"falling","v":0.45535,"x":0.67411,"p":[[0,12,0.0,0.67411,0.27254,0.42857,0.50001,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,3,0,0,0,0,0,0,0,13],[4,12,0.3333,0.63839,0.26483,0.42857,0.42859,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,18,0,0,3,0,0,0,0,0,0,0,11],[8,12,0.6667,0.59819,0.23808,0.42857,0.42857,0.67857,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,18,0,0,6,0,0,0,0,0,0,0,8],[12,12,1.0,0.45535,0.05573,0.42857,0.42857,0.42858,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"02867d1be4fc3043","q":"Circle  $k$  and its diameter  $AB$  are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on  $AB$  and two other vertices on  $k.$","t":[{"b":0,"e":0.0,"k":"rising","v":0.20067,"x":0.4195,"p":[[0,41,0.0,0.20067,0.21057,0.0,0.14286,0.42857,0.0,0.71,11,0,0,11,0,11,0,0,1,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[4,41,0.0976,0.27231,0.22687,0.14286,0.14286,0.32143,0.0,0.71429,2,0,0,2,0,19,0,0,3,0,0,1,0,0,2,0,0,5,0,0,0,0,0],[8,41,0.1951,0.35712,0.24221,0.14286,0.28571,0.57141,0.0,0.857,3,0,0,3,0,8,0,0,8,0,0,3,0,0,4,0,0,5,0,0,1,0,0],[12,41,0.2927,0.32589,0.24284,0.14286,0.21428,0.42858,0.0,0.85714,2,0,0,2,0,14,0,0,3,0,0,6,0,0,2,0,0,3,0,0,2,0,0],[16,41,0.3902,0.38836,0.2348,0.14286,0.42857,0.57143,0.0,0.857,4,0,0,4,0,6,0,0,1,0,0,10,0,0,7,0,0,3,0,0,1,0,0],[20,41,0.4878,0.29006,0.24613,0.14286,0.14286,0.57111,0.0,0.71429,5,0,0,5,0,14,0,0,2,0,0,1,0,0,6,0,0,4,0,0,0,0,0],[24,41,0.5854,0.3571,0.26482,0.14286,0.42857,0.57141,0.0,0.857,6,0,0,6,0,8,0,0,0,0,0,6,0,0,7,0,0,4,0,0,1,0,0],[28,41,0.6829,0.29462,0.22567,0.14286,0.21428,0.4286,0.0,0.71429,6,0,0,6,0,10,0,0,1,0,0,8,0,0,5,0,0,2,0,0,0,0,0],[32,41,0.7805,0.35253,0.22561,0.14286,0.35714,0.57141,0.0,0.71429,3,0,0,3,0,9,0,0,4,0,0,6,0,0,6,0,0,4,0,0,0,0,0],[36,41,0.878,0.31249,0.22141,0.14286,0.2857,0.4286,0.0,0.71429,3,0,0,3,0,12,0,0,4,0,0,6,0,0,3,0,0,4,0,0,0,0,0],[40,41,0.9756,0.37503,0.21354,0.14286,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,7,0,0,2,0,0,10,0,0,7,0,0,3,0,0,0,0,0],[41,41,1.0,0.4195,0.2341,0.24999,0.4286,0.57143,0.0,0.85714,3,0,0,3,0,5,0,0,4,0,0,5,0,0,10,0,0,4,0,0,1,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.20092,"p":[[0,5,0.0,0.20092,0.22831,0.0,0.14286,0.42857,0.0,0.85714,12,0,1,12,0,10,0,0,1,0,0,6,0,0,1,0,0,1,0,0,1,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8b81a80e6fe907bb","q":"$\\boxed{\\text{G5}}$  The incircle of a triangle  $ABC$  touches its sides  $BC$ , $CA$ , $AB$  at the points  $A_1$ , $B_1$ , $C_1$ .Let the projections of the orthocenter  $H_1$  of the triangle  $A_{1}B_{1}C_{1}$  to the lines  $AA_1$  and  $BC$  be  $P$  and  $Q$ ,respectively. Show that  $PQ$  bisects the line segment  $B_{1}C_{1}$","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,10,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"68ed99719bb9ca68","q":"14. (IRE 1) The circle inscribed in a triangle $A B C$ touches the sides $B C, C A, A B$ in $D, E, F$, respectively, and $X, Y, Z$ are the midpoints of $E F, F D, D E$, respectively. Prove that the centers of the inscribed circle and of the circles around $X Y Z$ and $A B C$ are collinear.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.125,"p":[[0,25,0.0,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,25,0.48,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"falling","v":0.0,"x":0.21875,"p":[[0,13,0.0,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[4,13,0.3077,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[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]]}]},{"i":"19a10ea057a572b7","q":"1. (BUL 1) Let $A B C$ be a triangle with bisectors $A A_{1}, B B_{1}, C C_{1}\\left(A_{1} \\in\\right.$ $B C$, etc.) and $M$ their common point. Consider the triangles $M B_{1} A$, $M C_{1} A, M C_{1} B, M A_{1} B, M A_{1} C, M B_{1} C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $A B=$ $B C=C A$.","t":[{"b":5,"e":0.2857,"k":"rising","v":0.17857,"x":0.37945,"p":[[0,8,0.0,0.17857,0.19561,0.0,0.14286,0.2857,0.0,0.85714,13,0,0,13,0,5,0,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,8,0.5,0.19643,0.18472,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,6,0,0,10,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[8,8,1.0,0.37945,0.21311,0.25,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,5,0,0,6,0,0,8,0,0,6,0,0,4,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.01786,"x":0.27668,"p":[[0,27,0.0,0.27668,0.22001,0.14214,0.2857,0.42857,0.0,0.71429,7,0,0,7,0,7,0,0,7,0,0,6,0,0,2,0,0,3,0,0,0,0,0],[4,27,0.1481,0.15624,0.19016,0.0,0.14286,0.2857,0.0,0.71429,15,0,0,15,0,7,0,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[8,27,0.2963,0.14286,0.19885,0.0,0.0,0.2857,0.0,0.71429,18,0,0,18,0,3,0,0,8,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[12,27,0.4444,0.08035,0.15943,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,5,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,27,0.5926,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.08473,0.11209,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4345fd9d239462ba","q":"Anthony writes the  $(n+1)^2$  distinct positive integer divisors of  $10^n$ , each once, on a whiteboard. On a move, he may choose any two distinct numbers  $a$  and  $b$  on the board, erase them both, and write  $\\gcd(a, b)$  twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.\n\n*Proposed by Andrew Wen*","t":[{"b":1,"e":0.14286,"k":"volatile","v":0.20089,"x":0.91071,"p":[[0,5,0.0,0.83482,0.34646,1.0,1.0,1.0,0.0,1.0,4,25,0,4,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,25],[4,5,0.8,0.91071,0.24679,1.0,1.0,1.0,0.0,1.0,2,26,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[5,5,1.0,0.20089,0.21975,0.0,0.14286,0.42857,0.0,0.71429,13,0,8,13,0,7,0,0,3,0,0,6,0,0,1,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.78124,"x":0.83929,"p":[[0,7,0.0,0.81249,0.34708,0.82143,1.0,1.0,0.0,1.0,4,23,0,4,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,23],[4,7,0.5714,0.83929,0.21943,0.71429,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,7,0,0,4,0,17],[7,7,1.0,0.78124,0.22301,0.67857,0.85714,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0,7,0,0,5,0,12]]}]},{"i":"b84d73f977d431c4","q":"6. N6 (BLR) Prove that for every real number $M$ there exists an infinite arithmetic progression such that: (i) each term is a positive integer and the common difference is not divisible by 10 ; (ii) the sum of the digits of each term (in decimal representation) exceeds $M$.","t":[{"b":6,"e":0.14286,"k":"flat","v":0.11589,"x":0.20089,"p":[[0,7,0.0,0.11589,0.10968,0.0,0.14286,0.14286,0.0,0.57143,10,0,1,10,0,20,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.20089,0.27862,0.0,0.14286,0.14286,0.0,1.0,10,3,0,10,0,16,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[7,7,1.0,0.16063,0.07786,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.08027,"x":0.20088,"p":[[0,39,0.0,0.1517,0.23129,0.0,0.14286,0.14286,0.0,1.0,11,2,0,11,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,39,0.1026,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,1,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.08473,0.07866,0.0,0.14286,0.14286,0.0,0.2857,14,0,1,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.28571,10,0,1,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.10491,0.07573,0.0,0.14286,0.14286,0.0,0.28571,10,0,1,10,0,20,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.11152,0.06898,0.105,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.14286,0.20516,0.0,0.14286,0.14286,0.0,1.0,14,1,1,14,0,12,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[28,39,0.7179,0.20088,0.18507,0.0,0.14286,0.2857,0.0,0.57143,9,0,1,9,0,11,0,0,6,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[32,39,0.8205,0.16964,0.16536,0.0,0.14286,0.17857,0.0,0.57143,9,0,2,9,0,15,0,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[36,39,0.9231,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.10714,0.07986,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bf66553a0ee9d7ba","q":"Determine all infinite sets  $A$  of positive integers with the following propety:\nIf  $a,b \\in A$  and  $a \\ge b$  then  $\\left\\lfloor \\frac{a}{b} \\right\\rfloor \\in A$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,52,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,52,0.0769,0.00893,0.0346,0.0,0.0,0.0,0.0,0.143,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,52,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,52,0.2308,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,52,0.3077,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,52,0.3846,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,52,0.5385,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,52,0.6154,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,11,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5d25f836bf6a5253","q":"An  $n \\times m$  matrix is nice if it contains every integer from  $1$  to  $mn$  exactly once and  $1$  is the only entry which is the smallest both in its row and in its column. Prove that the number of  $n \\times m$  nice matrices is  $(nm)!n!m!/(n+m-1)!$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.02232,"x":0.08929,"p":[[0,17,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,13,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"2c03224043ebf67a","q":"Consider acute  $\\triangle ABC$  with altitudes  $AA_1, BB_1$  and  $CC_1$  ( $A_1 \\in BC,B_1 \\in  AC,C_1 \\in  AB$ ). A point  $C' $  on the extension of  $B_1A_1$  beyond  $A_1$  is such that  $A_1C' = B_1C_1$ . Analogously, a point  $B'$  on the extension of A $_1C_1$  beyond  $C_1$  is such that  $C_1B'  = A_1B_1$  and a point  $A' $  on the extension of  $C_1B_1$  beyond  $B_1$  is such that  $B_1A'  = C_1A_1$ . Denote by  $A'', B'', C''$  the symmetric points of  $A' , B' , C'$   with respect to  $BC, CA$  and  $AB$  respectively. Prove that if  $R, R'$   and R'' are circumradiii of  $\\triangle ABC, \\triangle A'B'C'$  and  $\\triangle A''B''C''$ , then  $R, R'$   and  $R'' $  are sidelengths of a triangle with area equals one half of the area of  $\\triangle ABC$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.02232,"x":0.11161,"p":[[0,7,0.0,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.09813,"x":0.12948,"p":[[0,16,0.0,0.09813,0.06616,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.12054,0.06298,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.10706,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.12948,0.04164,0.14286,0.14286,0.14286,0.0,0.143,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.12938,0.04162,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9c996d27ae646e59","q":"8. (GBR) ${ }^{\\mathrm{IMO} 2}$ Four different points $A, B, C, D$ are chosen on a circle $\\Gamma$ such that the triangle $B C D$ is not right-angled. Prove that: (a) The perpendicular bisectors of $A B$ and $A C$ meet the line $A D$ at certain points $W$ and $V$, respectively, and that the lines $C V$ and $B W$ meet at a certain point $T$. (b) The length of one of the line segments $A D, B T$, and $C T$ is the sum of the lengths of the other two. Original formulation. In triangle $A B C$ the angle at $A$ is the smallest. A line through $A$ meets the circumcircle again at the point $U$ lying on the $\\operatorname{arc} B C$ opposite to $A$. The perpendicular bisectors of $C A$ and $A B$ meet $A U$ at $V$ and $W$, respectively, and the lines $C V, B W$ meet at $T$. Show that $A U=T B+T C$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.02223,"x":0.30357,"p":[[0,5,0.0,0.30357,0.36725,0.0,0.14286,0.50002,0.0,1.0,11,4,0,11,0,10,0,0,2,0,0,1,0,0,0,0,0,1,0,0,3,0,4],[4,5,0.8,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.71429,"k":"rising","v":0.19195,"x":0.89731,"p":[[0,17,0.0,0.19195,0.26631,0.0,0.14286,0.1786,0.0,1.0,13,1,0,13,0,11,0,0,3,0,0,1,0,0,1,0,0,0,0,0,2,0,1],[4,17,0.2353,0.35705,0.42412,0.0,0.14143,0.85704,0.0,1.0,15,6,0,15,0,5,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,6],[8,17,0.4706,0.85708,0.1288,0.857,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,24,0,6],[12,17,0.7059,0.86159,0.12622,0.82132,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,13,0,11],[16,17,0.9412,0.89079,0.11145,0.85714,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,18,1,11],[17,17,1.0,0.89731,0.0735,0.85714,0.85714,1.0,0.71429,1.0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,21,0,10]]}]},{"i":"1a472ddcfd99f8d2","q":"$a$  and  $b$  are natural numbers such that  $b > a > 1$ , and  $a$  does not divide  $b$ .  The sequence of natural numbers  $\\{b_n\\}_{n=1}^\\infty$  satisfies  $b_{n + 1} \\geq 2b_n \\forall n \\in \\mathbb{N}$ .  Does there exist a sequence  $\\{a_n\\}_{n=1}^\\infty$  of natural numbers such that for all  $n \\in \\mathbb{N}$ ,  $a_{n + 1} - a_n \\in \\{a, b\\}$ , and for all  $m, l \\in \\mathbb{N}$  ( $m$  may be equal to  $l$ ),  $a_m + a_l \\not\\in \\{b_n\\}_{n=1}^\\infty$ ?","t":[{"b":0,"e":0.71429,"k":"volatile","v":0.22759,"x":0.69642,"p":[[0,9,0.0,0.22759,0.18855,0.14286,0.14286,0.28571,0.0,0.85714,4,0,0,4,0,15,0,0,9,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[4,9,0.4444,0.27661,0.16742,0.14286,0.28571,0.32143,0.0,0.71429,1,0,0,1,0,13,0,0,10,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[8,9,0.8889,0.26774,0.16658,0.14286,0.2143,0.32143,0.0,0.71429,1,0,0,1,0,15,0,0,8,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[9,9,1.0,0.69642,0.14173,0.71429,0.71429,0.74996,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,18,0,0,8,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.14705,"x":0.24553,"p":[[0,16,0.0,0.24553,0.12992,0.14286,0.28571,0.28571,0.0,0.57143,3,0,1,3,0,9,0,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[4,16,0.25,0.18299,0.11956,0.14286,0.14286,0.28571,0.0,0.57,5,0,0,5,0,15,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,16,0.5,0.15626,0.08268,0.14286,0.14286,0.14287,0.0,0.42857,3,0,0,3,0,24,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.15161,0.07089,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.14705,0.06669,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3147a2268a821369","q":"7. G1 (FRA) Let $B$ be a point on a circle $S_{1}$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_{1}$. Let $C$ be a point not on $S_{1}$ such that the line segment $A C$ meets $S_{1}$ at two distinct points. Let $S_{2}$ be the circle touching $A C$ at $C$ and touching $S_{1}$ at a point $D$ on the opposite side of $A C$ from $B$. Prove that the circumcenter of triangle $B C D$ lies on the circumcircle of triangle $A B C$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.28565,"x":0.51782,"p":[[0,24,0.0,0.40622,0.30743,0.10714,0.42857,0.57143,0.0,1.0,8,2,0,8,0,2,0,0,2,0,0,8,0,0,5,0,0,3,0,0,2,0,2],[4,24,0.1667,0.28565,0.22859,0.10714,0.28571,0.4286,0.0,0.71429,8,0,1,8,0,7,0,0,2,0,0,9,0,0,4,0,0,2,0,0,0,0,0],[8,24,0.3333,0.41515,0.282,0.1429,0.42859,0.60714,0.0,1.0,5,1,0,5,0,5,0,0,2,0,0,8,0,0,4,0,0,5,0,0,2,0,1],[12,24,0.5,0.51782,0.21942,0.42857,0.57121,0.57143,0.0,1.0,1,2,0,1,0,2,0,0,3,0,0,7,0,0,14,0,0,1,0,0,2,0,2],[16,24,0.6667,0.38389,0.25859,0.14286,0.42857,0.57143,0.0,1.0,7,1,0,7,0,2,0,0,3,0,0,7,0,0,10,0,0,2,0,0,0,0,1],[20,24,0.8333,0.46423,0.14721,0.42857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,0,0,0,16,0,0,11,0,0,2,0,0,0,0,0],[24,24,1.0,0.48211,0.18469,0.42857,0.57121,0.57143,0.0,0.71429,2,0,0,2,0,2,0,0,1,0,0,8,0,0,15,0,0,4,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.37945,"x":0.71864,"p":[[0,27,0.0,0.42853,0.23143,0.28571,0.42857,0.57111,0.0,1.0,3,1,0,3,0,3,0,0,4,0,0,11,0,0,5,0,0,5,0,0,0,0,1],[4,27,0.1481,0.41053,0.2221,0.2857,0.42859,0.57141,0.0,0.71429,4,0,0,4,0,2,0,0,6,0,0,7,0,0,8,0,0,5,0,0,0,0,0],[8,27,0.2963,0.37945,0.25154,0.14286,0.42857,0.57143,0.0,0.71429,5,0,1,5,0,6,0,0,3,0,0,5,0,0,7,0,0,6,0,0,0,0,0],[12,27,0.4444,0.46868,0.27945,0.24999,0.571,0.71429,0.0,1.0,5,1,0,5,0,3,0,0,2,0,0,3,0,0,9,0,0,8,0,0,1,0,1],[16,27,0.5926,0.44193,0.24315,0.2857,0.42857,0.57143,0.0,1.0,3,1,0,3,0,4,0,0,2,0,0,10,0,0,7,0,0,4,0,0,1,0,1],[20,27,0.7407,0.71864,0.18407,0.67836,0.71429,0.857,0.14,1.0,0,5,0,0,0,1,0,0,0,0,0,2,0,0,5,0,0,15,0,0,4,0,5],[24,27,0.8889,0.55341,0.13741,0.571,0.57143,0.57143,0.14,0.71429,0,0,0,0,0,2,0,0,1,0,0,2,0,0,21,0,0,6,0,0,0,0,0],[27,27,1.0,0.64283,0.15153,0.57143,0.71429,0.71429,0.14286,0.857,0,0,0,0,0,2,0,0,0,0,0,1,0,0,7,0,0,21,0,0,1,0,0]]}]},{"i":"d6001098fd6ebe70","q":"$n \\geq 4$ players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players bad if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let $w_{i}$ and $\\ell_{i}$ be respectively the number of wins and losses of the $i$ th player. Prove that  $$ \\sum_{i=1}^{n}\\left(w_{i}-\\ell_{i}\\right)^{3} \\geq 0 $$  (South Korea)","t":[{"b":1,"e":0.57143,"k":"rising","v":0.30357,"x":0.94643,"p":[[0,29,0.0,0.30357,0.33834,0.0,0.14286,0.42857,0.0,1.0,11,4,0,11,0,7,0,0,2,0,0,5,0,0,2,0,0,0,0,0,1,0,4],[4,29,0.1379,0.56249,0.41333,0.14286,0.71429,1.0,0.0,1.0,7,10,2,7,0,4,0,0,1,0,0,2,0,0,2,0,0,0,0,0,6,0,10],[8,29,0.2759,0.74107,0.35792,0.60714,0.92857,1.0,0.0,1.0,2,16,0,2,0,4,0,0,2,0,0,0,0,0,0,0,0,2,0,0,6,0,16],[12,29,0.4138,0.60714,0.39609,0.14286,0.78571,1.0,0.0,1.0,3,13,0,3,0,7,0,0,1,0,0,3,0,0,1,0,0,1,0,0,3,0,13],[16,29,0.5517,0.91964,0.15126,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,23],[20,29,0.6897,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[24,29,0.8276,0.93749,0.15545,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,26],[28,29,0.9655,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[29,29,1.0,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22]]},{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.52232,"p":[[0,21,0.0,0.52232,0.40972,0.14286,0.50001,1.0,0.0,1.0,7,10,0,7,0,5,0,0,1,0,0,3,0,0,2,0,0,1,0,0,3,0,10],[4,21,0.1905,0.41063,0.3843,0.0,0.28571,0.85714,0.0,1.0,10,4,0,10,0,4,0,0,3,0,0,3,0,0,1,0,0,1,0,0,6,0,4],[8,21,0.381,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,21,0.5714,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c80e92ebe26f9993","q":"A sphere with center on the plane of the face  $ABC$  of a tetrahedron  $SABC$  passes through  $A$ ,  $B$  and  $C$ , and meets the edges  $SA$ ,  $SB$ ,  $SC$  again at  $A_1$ ,  $B_1$ ,  $C_1$ , respectively.  The planes through  $A_1$ ,  $B_1$ ,  $C_1$  tangent to the sphere meet at  $O$ .  Prove that  $O$  is the circumcenter of the tetrahedron  $SA_1B_1C_1$ .","t":[{"b":0,"e":0.14286,"k":"rising","v":0.08482,"x":0.28125,"p":[[0,6,0.0,0.11161,0.13709,0.0,0.0,0.28571,0.0,0.28571,19,0,0,19,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.28125,0.04351,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.14286,"p":[[0,17,0.0,0.11161,0.1461,0.0,0.0,0.2857,0.0,0.4286,19,0,0,19,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.14286,0.14725,0.0,0.07143,0.28571,0.0,0.42857,16,0,2,16,0,1,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0ea29435c394a04f","q":"An acute angled triangle  $\\mathcal{T}$  is inscribed in circle  $\\Omega$ .Denote by  $\\Gamma$  the nine-point circle of  $\\mathcal{T}$ .A circle  $\\omega$  passes through two of the vertices of  $\\mathcal{T}$ , and centre of  $\\Omega$ .Prove that the common external tangents of  $\\Gamma$  and  $\\omega$  meet on the external bisector of the angle at third vertex of  $\\mathcal{T}$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00438,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,19,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"39ce611e8764a2fe","q":"9. C3 (RUS) Define a $k$-clique to be a set of $k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3 -cliques has at least one person in common, and there are no 5 -cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.","t":[{"b":1,"e":0.857,"k":"flat","v":0.02232,"x":0.16518,"p":[[0,15,0.0,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.12053,0.23176,0.0,0.0,0.17857,0.0,1.0,23,1,1,23,0,1,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[12,15,0.8,0.12946,0.19019,0.0,0.0,0.2857,0.0,0.71429,19,0,0,19,0,4,0,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[15,15,1.0,0.16518,0.21162,0.0,0.14286,0.2857,0.0,1.0,15,1,0,15,0,5,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,1]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,26,0.0,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63d4b8e3c61775ee","q":"15. (PRK 2) Does there exist a set $M$ with the following properties? (i) The set $M$ consists of 1992 natural numbers. (ii) Every element in $M$ and the sum of any number of elements have the form $m^{k}(m, k \\in \\mathbb{N}, k \\geq 2)$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[4,20,0.2,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2670526e3b018339","q":"Define a function  $g: \\mathbb{N} \\mapsto \\mathbb{N}$  by the following rule:\r\n(a)  $g$  is nondecrasing\r\n(b) for each  $n$ ,  $g(n)$  i sthe number of times  $n$  appears in the range of  $g$ ,\r\n\r\nProve that  $g(1) = 1$  and  $g(n+1) = 1 + g( n +1 - g(g(n)))$  for all  $n \\in \\mathbb{N}$","t":[{"b":0,"e":0.571,"k":"flat","v":0.5535,"x":0.81249,"p":[[0,15,0.0,0.79461,0.20808,0.57143,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,5,0,0,6,0,12],[4,15,0.2667,0.80801,0.27344,0.57143,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,2,0,0,0,0,20],[8,15,0.5333,0.5535,0.32093,0.25,0.57143,0.75,0.0,1.0,3,6,3,3,0,5,0,0,1,0,0,0,0,0,12,0,0,3,0,0,2,0,6],[12,15,0.8,0.70085,0.30382,0.571,0.78571,1.0,0.0,1.0,2,11,1,2,0,1,0,0,2,0,0,1,0,0,8,0,0,2,0,0,5,0,11],[15,15,1.0,0.81249,0.20653,0.57143,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,2,0,16]]},{"b":6,"e":1.0,"k":"flat","v":0.82141,"x":0.9375,"p":[[0,12,0.0,0.82141,0.20826,0.71429,0.85714,1.0,0.14286,1.0,0,12,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,5,0,0,10,0,12],[4,12,0.3333,0.82588,0.22514,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,1,0,0,6,0,0,5,0,0,1,0,18],[8,12,0.6667,0.9375,0.12846,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,25],[12,12,1.0,0.92411,0.15966,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0,25]]}]},{"i":"dd54e5cf7e4a5a67","q":"In every cell of a square table is a number. The sum of the largest two numbers in each row\nis  $a$  and the sum of the largest two numbers in each column is b. Prove that  $a = b$ .","t":[{"b":1,"e":0.14,"k":"flat","v":0.08929,"x":0.15625,"p":[[0,21,0.0,0.15625,0.12555,0.10714,0.14286,0.17868,0.0,0.42857,8,0,1,8,0,16,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.12938,0.12036,0.0,0.14286,0.1786,0.0,0.4286,12,0,0,12,0,12,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.10696,0.10095,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.09821,0.12595,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,7,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.11153,0.06898,0.105,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.13822,0.05628,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.10705,"x":0.1875,"p":[[0,20,0.0,0.11607,0.1729,0.0,0.07143,0.14286,0.0,0.85714,16,0,0,16,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,20,0.2,0.13839,0.12619,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,16,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.17857,0.14725,0.0,0.21428,0.28571,0.0,0.4286,11,0,0,11,0,5,0,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.1875,0.16146,0.0,0.28571,0.28571,0.0,0.4286,13,0,0,13,0,0,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.13821,0.14933,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,15,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,20,1.0,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"35c700d2197bd054","q":"$(FRA 4)$  A right-angled triangle  $OAB$  has its right angle at the point  $B.$  An arbitrary circle with center on the line  $OB$  is tangent to the line  $OA.$  Let  $AT$  be the tangent to the circle different from  $OA$  ( $T$  is the point of tangency). Prove that the median from  $B$  of the triangle  $OAB$  intersects  $AT$  at a point  $M$  such that  $MB = MT.$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.10703,"p":[[0,17,0.0,0.06249,0.12334,0.0,0.0,0.14286,0.0,0.571,23,0,1,23,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.10703,0.16742,0.0,0.0,0.14286,0.0,0.571,19,0,1,19,0,8,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,17,0.4706,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.03125,"x":0.12057,"p":[[0,25,0.0,0.12054,0.16016,0.0,0.0,0.1429,0.0,0.57143,17,0,0,17,0,8,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,25,0.16,0.08035,0.14253,0.0,0.0,0.14286,0.0,0.571,22,0,0,22,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,25,0.32,0.12057,0.20242,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,3,0,0,0,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[12,25,0.48,0.04009,0.08907,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.0758,0.07966,0.0,0.07,0.14286,0.0,0.2857,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.04447,0.06596,0.0,0.0,0.14071,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b110b4b1e670a2b0","q":"Circles  ${\\omega_1}$  ,  ${\\omega_2}$  are externally tangent at point M and tangent internally with circle  ${\\omega_3}$  at points  ${K}$  and  $L$  respectively. Let  ${A}$  and   ${B}$  be the points that their common tangent at point  ${M}$  of circles  ${\\omega_1}$  and  ${\\omega_2}$  intersect with circle  ${\\omega_3.}$  Prove that if  ${\\angle KAB=\\angle LAB}$  then the segment  ${AB}$  is diameter of circle  ${\\omega_3.}$ Theoklitos Paragyiou (Cyprus)","t":[{"b":1,"e":0.0,"k":"flat","v":0.02232,"x":0.16964,"p":[[0,20,0.0,0.13393,0.32915,0.0,0.0,0.0,0.0,1.0,26,4,1,26,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,20,0.2,0.16964,0.3597,0.0,0.0,0.03571,0.0,1.0,24,5,1,24,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,20,0.4,0.09822,0.24856,0.0,0.0,0.03571,0.0,1.0,24,2,1,24,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,20,0.6,0.08482,0.18162,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,20,0.8,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.15625,"p":[[0,18,0.0,0.15625,0.34323,0.0,0.0,0.0,0.0,1.0,25,4,2,25,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4],[4,18,0.2222,0.04455,0.1765,0.0,0.0,0.0,0.0,1.0,28,1,2,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,18,0.4444,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[18,18,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"211d153b3e99ba87","q":"3. I 3 (SWE 3) ${ }^{\\mathrm{IMO}}$ Let $P(x)$ be a polynomial with integer coefficients. If $n(P)$ is the number of (distinct) integers $k$ such that $P^{2}(k)=1$, prove that  $$ n(P)-\\operatorname{deg}(P) \\leq 2 $$  where $\\operatorname{deg}(P)$ denotes the degree of the polynomial $P$.","t":[{"b":0,"e":0.42857,"k":"falling","v":0.4241,"x":0.66518,"p":[[0,6,0.0,0.66518,0.28928,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,2,0,0,3,0,0,8,0,0,1,0,0,5,0,0,3,0,10],[4,6,0.6667,0.56691,0.30195,0.28571,0.71429,0.71429,0.0,1.0,4,2,1,4,0,2,0,0,3,0,0,0,0,0,5,0,0,11,0,0,5,0,2],[6,6,1.0,0.4241,0.1906,0.39286,0.42857,0.4286,0.0,0.85714,2,0,0,2,0,2,0,0,4,0,0,17,0,0,2,0,0,4,0,0,1,0,0]]},{"b":3,"e":0.2857,"k":"falling","v":0.33035,"x":0.69639,"p":[[0,12,0.0,0.69639,0.30463,0.571,0.78564,1.0,0.0,1.0,2,9,0,2,0,2,0,0,1,0,0,2,0,0,4,0,0,5,0,0,7,0,9],[4,12,0.3333,0.5,0.31943,0.2857,0.42859,0.71429,0.0,1.0,5,4,0,5,0,2,0,0,3,0,0,7,0,0,2,0,0,7,0,0,2,0,4],[8,12,0.6667,0.40619,0.16402,0.28571,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,5,0,0,7,0,0,10,0,0,8,0,0,2,0,0,0,0,0],[12,12,1.0,0.33035,0.19702,0.14286,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,10,0,0,8,0,0,8,0,0,2,0,0,2,0,0,1,0,0]]}]},{"i":"3eaf49602a6f9d86","q":"28. (GBR 5) The sequence $\\left\\{a_{n}\\right\\}$ of integers is defined by $a_{1}=2, a_{2}=7$, and  $$ -\\frac{1}{2}<a_{n+1}-\\frac{a_{n}^{2}}{a_{n-1}} \\leq \\frac{1}{2}, \\quad \\text { for } n \\geq 2 $$  Prove that $a_{n}$ is odd for all $n>1$.","t":[{"b":1,"e":0.0,"k":"falling","v":0.02232,"x":0.31473,"p":[[0,19,0.0,0.31473,0.39523,0.0,0.10714,0.60714,0.0,1.0,15,6,0,15,1,3,0,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,6],[4,19,0.2105,0.26339,0.36265,0.0,0.14286,0.28571,0.0,1.0,14,5,0,14,0,8,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,5],[8,19,0.4211,0.04018,0.08918,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.08929,"x":0.42409,"p":[[0,17,0.0,0.42409,0.44389,0.0,0.21429,1.0,0.0,1.0,14,10,0,14,0,2,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,10],[4,17,0.2353,0.35714,0.33311,0.0,0.28571,0.57143,0.0,1.0,9,3,0,9,0,5,0,0,4,0,0,4,0,0,3,0,0,2,0,0,2,0,3],[8,17,0.4706,0.08929,0.15047,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,6,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,17,0.7059,0.10268,0.1394,0.0,0.0,0.17857,0.0,0.4286,19,0,0,19,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.12938,0.16505,0.0,0.14143,0.14287,0.0,0.71429,15,0,0,15,0,10,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[17,17,1.0,0.09598,0.13554,0.0,0.0,0.2857,0.0,0.42857,20,0,0,20,1,2,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3bd2ad7744e24a36","q":"13. (POL 5) Given two congruent triangles $A_{1} A_{2} A_{3}$ and $B_{1} B_{2} B_{3}\\left(A_{i} A_{k}=\\right.$ $B_{i} B_{k}$ ), prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.","t":[{"b":2,"e":1.0,"k":"rising","v":0.55356,"x":0.89732,"p":[[0,15,0.0,0.55356,0.31693,0.28571,0.57143,0.85714,0.0,1.0,3,3,2,3,0,2,0,0,7,0,0,1,0,0,4,0,0,4,0,0,8,0,3],[4,15,0.2667,0.70533,0.25739,0.57143,0.71429,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,4,0,0,1,0,0,4,0,0,8,0,0,7,0,7],[8,15,0.5333,0.79015,0.14723,0.71429,0.85714,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,17,0,4],[12,15,0.8,0.89732,0.08917,0.85714,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,20,0,11],[15,15,1.0,0.88838,0.05906,0.85714,0.85714,0.85714,0.857,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,7]]},{"b":7,"e":0.85714,"k":"rising","v":0.56247,"x":0.90186,"p":[[0,15,0.0,0.60698,0.29229,0.53539,0.71429,0.85714,0.0,1.0,2,3,0,2,0,4,0,0,1,0,0,1,0,0,6,0,0,8,0,0,7,0,3],[4,15,0.2667,0.60712,0.26963,0.42857,0.71414,0.85714,0.0,1.0,2,2,0,2,0,0,0,0,5,0,0,4,0,0,4,0,0,6,0,0,9,0,2],[8,15,0.5333,0.56247,0.3272,0.39286,0.57143,0.85714,0.0,1.0,6,3,0,6,0,0,0,0,2,0,0,2,0,0,8,0,0,3,0,0,8,0,3],[12,15,0.8,0.89286,0.06186,0.85714,0.85714,0.89286,0.8571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,8],[15,15,1.0,0.90186,0.17653,0.85714,0.93,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,16]]}]},{"i":"2fdb6ecad9f37806","q":"A point  $P$  lies on side  $AB$  of a convex quadrilateral  $ABCD$ . Let  $\\omega$  be the inscribed circumference of triangle  $CPD$  and  $I$  the centre of  $\\omega$ . It is known that  $\\omega$  is tangent to the inscribed circumferences of triangles  $APD$  and  $BPC$  at points  $K$  and  $L$  respectively. Let  $E$  be the point where the lines  $AC$  and  $BD$  intersect, and  $F$  the point where the lines  $AK$  and  $BL$  intersect. Prove that the points  $E, I, F$  are collinear.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,19,0.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,59,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,59,0.1356,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,59,0.2034,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,59,0.2712,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,59,0.339,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,59,0.4068,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,59,0.4746,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,59,0.5424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,59,0.6102,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,59,0.8136,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,59,0.8814,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"d9e4d5b1b2087bbf","q":"A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circle, the sum of the unit outgoing vectors along the edges of the graph is zero. Prove that the total length of the web is equal to the number of its vertices on the circle.","t":[{"b":2,"e":0.0,"k":"falling","v":0.00893,"x":0.61607,"p":[[0,55,0.0,0.61607,0.43659,0.0,0.85714,1.0,0.0,1.0,9,16,0,9,0,1,0,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,16],[4,55,0.0727,0.47768,0.46649,0.0,0.42857,1.0,0.0,1.0,14,11,0,14,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,11],[8,55,0.1455,0.55804,0.45647,0.0,0.71429,1.0,0.0,1.0,12,14,0,12,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,0,1,0,14],[12,55,0.2182,0.46875,0.4808,0.0,0.2143,1.0,0.0,1.0,16,13,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,13],[16,55,0.2909,0.4866,0.4895,0.0,0.35714,1.0,0.0,1.0,16,14,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,14],[20,55,0.3636,0.29911,0.44372,0.0,0.0,1.0,0.0,1.0,21,9,0,21,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[24,55,0.4364,0.51339,0.47897,0.0,0.57144,1.0,0.0,1.0,14,15,0,14,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,15],[28,55,0.5091,0.27678,0.44311,0.0,0.0,0.89275,0.0,1.0,23,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,8],[32,55,0.5818,0.16071,0.35129,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[36,55,0.6545,0.05804,0.22548,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[40,55,0.7273,0.09375,0.24643,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,55,0.8,0.09375,0.27804,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[48,55,0.8727,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.0133,0.05465,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"volatile","v":0.0,"x":0.53571,"p":[[0,12,0.0,0.53571,0.4711,0.0,0.71429,1.0,0.0,1.0,13,14,0,13,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,14],[4,12,0.3333,0.52231,0.46237,0.0,0.64264,1.0,0.0,1.0,12,14,0,12,0,2,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,14],[8,12,0.6667,0.52232,0.46375,0.0,0.71429,1.0,0.0,1.0,13,13,0,13,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,13],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e55bfaef70a5e7cb","q":"An equilateral triangle is divided into 9000000 congruent equilateral triangles by lines parallel to its sides. Each vertex of the small triangles is coloured in one of three colours. Prove that there exist three points of the same colour being the vertices of a triangle with its sides parallel to the sides of the original triangle.","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.00893,"p":[[0,11,0.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,16,0.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"6995a8ffda8c7b3c","q":"5. (HUN 1) ${ }^{\\mathrm{IMO}}$ Let $a, b, c, d, e$ be real numbers. Prove that the expression  $$ \\begin{gathered} (a-b)(a-c)(a-d)(a-e)+(b-a)(b-c)(b-d)(b-e)+(c-a)(c-b)(c-d)(c-e) \\\\ +(d-a)(d-b)(d-c)(d-e)+(e-a)(e-b)(e-c)(e-d) \\end{gathered} $$  is nonnegative.","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.40625,"p":[[0,33,0.0,0.34375,0.47496,0.0,0.0,1.0,0.0,1.0,21,11,1,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[4,33,0.1212,0.33482,0.45402,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[8,33,0.2424,0.40625,0.45332,0.0,0.21428,1.0,0.0,1.0,15,11,0,15,0,1,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,11],[12,33,0.3636,0.32589,0.43336,0.0,0.0,1.0,0.0,1.0,18,9,0,18,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[16,33,0.4848,0.22768,0.41011,0.0,0.0,0.14286,0.0,1.0,23,7,0,23,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[20,33,0.6061,0.10268,0.29284,0.0,0.0,0.0,0.0,1.0,28,3,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[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":2,"e":0.0,"k":"falling","v":0.0,"x":0.33482,"p":[[0,11,0.0,0.33482,0.45402,0.0,0.0,1.0,0.0,1.0,19,10,1,19,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[4,11,0.3636,0.32589,0.42743,0.0,0.0,0.89286,0.0,1.0,18,8,0,18,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,8],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0545dff47db87bc4","q":"Determine the maximum integer  $ n $  such that for each positive integer  $ k \\le \\frac{n}{2} $  there are two positive divisors of  $ n $  with difference  $ k $ .","t":[{"b":1,"e":0.1429,"k":"falling","v":0.05786,"x":0.27232,"p":[[0,106,0.0,0.27232,0.32607,0.14286,0.14286,0.1429,0.0,1.0,6,3,0,6,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,3],[4,106,0.0377,0.19188,0.2893,0.0,0.14286,0.14286,0.0,1.0,13,3,0,13,0,13,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[8,106,0.0755,0.20972,0.28344,0.0,0.14286,0.14286,0.0,1.0,10,2,0,10,0,16,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,2],[12,106,0.1132,0.17839,0.28349,0.0,0.0,0.14287,0.0,0.85714,17,0,0,17,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0,4,0,0],[16,106,0.1509,0.08473,0.07866,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,106,0.1887,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,106,0.2264,0.10714,0.12877,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,19,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,106,0.2642,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,106,0.3019,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,106,0.3396,0.09367,0.07663,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,106,0.3774,0.08036,0.08702,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,106,0.4151,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,106,0.4528,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,106,0.4906,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,106,0.5283,0.05786,0.06995,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,106,0.566,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,106,0.6038,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,106,0.6415,0.07589,0.07973,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,106,0.6792,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,106,0.717,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,106,0.7547,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,106,0.7925,0.09375,0.08458,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,106,0.8302,0.07134,0.07978,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,106,0.8679,0.0667,0.07101,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,106,0.9057,0.09366,0.07662,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,106,0.9434,0.08929,0.07784,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,106,0.9811,0.09813,0.08323,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[106,106,1.0,0.07589,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.09357,"x":0.16964,"p":[[0,25,0.0,0.16964,0.23538,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,18,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[4,25,0.16,0.14732,0.18723,0.0,0.14286,0.14286,0.0,0.85714,11,0,0,11,0,16,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[8,25,0.32,0.09357,0.07657,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.13393,0.08703,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.14732,0.0977,0.14286,0.14286,0.17857,0.0,0.28571,7,0,0,7,0,17,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.1183,0.09211,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,17,0,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.15616,0.09008,0.14286,0.14286,0.1786,0.0,0.28571,5,0,0,5,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.10714,0.09449,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5af4144746c0ba97","q":"A triangle and a circle are in the same plane. Show that the area of the intersection of the triangle and the circle is at most one third of the area of the triangle plus one half of the area of the circle.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,26,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,13,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"e53a098e679b2397","q":"11. G2 (CAN) ${ }^{\\mathrm{IMO} 2}$ Let $P$ be a point inside $\\triangle A B C$ such that  $$ \\angle A P B-\\angle C=\\angle A P C-\\angle B . $$  Let $D, E$ be the incenters of $\\triangle A P B, \\triangle A P C$ respectively. Show that $A P, B D$ and $C E$ meet in a point.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.19196,"x":0.2767,"p":[[0,16,0.0,0.23214,0.20124,0.0,0.2857,0.28571,0.0,1.0,9,1,1,9,0,3,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[4,16,0.25,0.19196,0.14555,0.0,0.2857,0.28571,0.0,0.42857,10,0,1,10,0,4,0,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.25446,0.16263,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,3,0,0,13,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[12,16,0.75,0.2767,0.10689,0.2857,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,4,0,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.25893,0.12078,0.2857,0.2857,0.28571,0.0,0.4286,4,0,0,4,0,3,0,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.16071,"p":[[0,10,0.0,0.16071,0.15465,0.0,0.14288,0.28571,0.0,0.4286,14,0,1,14,0,3,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b27665a4af0d4554","q":"An equilateral pentagon $A M N P Q$ is inscribed in triangle $A B C$ such that $M \\in \\overline{A B}$, $Q \\in \\overline{A C}$, and $N, P \\in \\overline{B C}$. Let $S$ be the intersection of $\\overline{M N}$ and $\\overline{P Q}$. Denote by $\\ell$ the angle bisector of $\\angle M S Q$. Prove that $\\overline{O I}$ is parallel to $\\ell$, where $O$ is the circumcenter of triangle $A B C$, and $I$ is the incenter of triangle $A B C$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,23,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01777,"p":[[0,9,0.0,0.01777,0.0777,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"653bbb829101059b","q":"EST   Let $f$ be a non-constant function from the set of positive integers into the set of positive integers, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a, b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.11607,"x":0.16518,"p":[[0,49,0.0,0.12054,0.17896,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,49,0.0816,0.12474,0.1056,0.0,0.14286,0.14287,0.0,0.4286,10,0,0,10,0,17,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.16518,0.16793,0.14286,0.14286,0.14286,0.0,1.0,5,1,0,5,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,49,0.2449,0.13366,0.087,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.11607,0.06621,0.14286,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.13821,0.07562,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.12937,0.06544,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.12054,0.08828,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.11608,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.14259,0.07986,0.14214,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,22,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.12929,0.04159,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.12054,"x":0.1383,"p":[[0,23,0.0,0.12947,0.1488,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,23,0.1739,0.12937,0.05484,0.14286,0.14286,0.14286,0.0,0.2857,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.13822,0.12103,0.14214,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,24,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,23,0.5217,0.12054,0.08073,0.14286,0.14286,0.14286,0.0,0.42857,7,0,1,7,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.1383,0.0435,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e665796f6827737d","q":"18. (USA 2) Let $[x]$ denote the greatest integer less than or equal to $x$. Pick any $x_{1}$ in $[0,1)$ and define the sequence $x_{1}, x_{2}, x_{3}, \\ldots$ by $x_{n+1}=0$ if $x_{n}=0$ and $x_{n+1}=1 / x_{n}-\\left[1 / x_{n}\\right]$ otherwise. Prove that  $$ x_{1}+x_{2}+\\cdots+x_{n}<\\frac{F_{1}}{F_{2}}+\\frac{F_{2}}{F_{3}}+\\cdots+\\frac{F_{n}}{F_{n+1}} $$  where $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$ for $n \\geq 1$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.05348,"x":0.20527,"p":[[0,25,0.0,0.20527,0.18539,0.105,0.14286,0.32143,0.0,0.71429,8,0,0,8,0,13,0,0,3,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[4,25,0.16,0.13384,0.2111,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,15,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,25,0.32,0.15625,0.19678,0.0,0.14286,0.14286,0.0,1.0,10,1,0,10,0,17,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[12,25,0.48,0.14732,0.21572,0.0,0.14286,0.14286,0.0,0.85714,14,0,0,14,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[16,25,0.64,0.08473,0.08639,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.05348,0.07774,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.08482,0.10012,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.12723,"x":0.32143,"p":[[0,33,0.0,0.22767,0.25217,0.10714,0.14286,0.28571,0.0,1.0,8,1,0,8,0,13,0,0,6,0,0,1,0,0,0,0,0,2,0,0,1,0,1],[4,33,0.1212,0.20973,0.23688,0.0,0.14286,0.28571,0.0,0.85714,11,0,0,11,0,9,0,0,7,0,0,1,0,0,0,0,0,3,0,0,1,0,0],[8,33,0.2424,0.19643,0.19804,0.10714,0.14286,0.2857,0.0,0.71429,8,0,0,8,0,15,0,0,4,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[12,33,0.3636,0.1875,0.21852,0.0,0.14286,0.2857,0.0,1.0,10,1,0,10,0,13,0,0,4,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[16,33,0.4848,0.16518,0.19269,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,11,0,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[20,33,0.6061,0.12723,0.20496,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,7,0,1,3,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[24,33,0.7273,0.32143,0.24744,0.14286,0.2857,0.42857,0.0,1.0,3,2,0,3,0,10,0,0,8,0,0,5,0,0,3,0,0,1,0,0,0,0,2],[28,33,0.8485,0.23215,0.17034,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,18,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[32,33,0.9697,0.22322,0.10062,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,18,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.1875,0.10972,0.14286,0.14286,0.2857,0.0,0.4286,3,0,0,3,0,19,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fa622792f5259302","q":"An acute triangle is a triangle that has all angles less than  $90^{\\circ}$  ( $90^{\\circ}$  is a Right Angle). Let  $ABC$  be an acute triangle with altitudes  $AD$ ,  $BE$ , and  $CF$  meeting at  $H$ . The circle passing through points  $D$ ,  $E$ , and  $F$  meets  $AD$ ,  $BE$ , and  $CF$  again at  $X$ ,  $Y$ , and  $Z$  respectively. Prove the following inequality:   $$ \\frac{AH}{DX}+\\frac{BH}{EY}+\\frac{CH}{FZ} \\geq 3. $$","t":[{"b":2,"e":0.28571,"k":"flat","v":0.35266,"x":0.52676,"p":[[0,14,0.0,0.52676,0.30813,0.28571,0.42859,0.75,0.0,1.0,1,6,0,1,0,3,0,0,10,0,0,3,0,0,3,0,0,4,0,0,2,0,6],[4,14,0.2857,0.40174,0.21554,0.28571,0.28571,0.571,0.14286,0.85714,0,0,0,0,0,6,0,0,12,0,0,4,0,0,4,0,0,4,0,0,2,0,0],[8,14,0.5714,0.35266,0.15559,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,4,0,0,17,0,0,6,0,0,2,0,0,3,0,0,0,0,0],[12,14,0.8571,0.39274,0.23153,0.2857,0.28571,0.4642,0.14,1.0,0,2,0,0,0,5,0,0,16,0,0,3,0,0,2,0,0,4,0,0,0,0,2],[14,14,1.0,0.39727,0.22509,0.2857,0.28571,0.571,0.14286,1.0,0,1,0,0,0,6,0,0,13,0,0,3,0,0,6,0,0,1,0,0,2,0,1]]},{"b":7,"e":0.85714,"k":"flat","v":0.37054,"x":0.56701,"p":[[0,29,0.0,0.51783,0.32684,0.28571,0.571,0.71429,0.0,1.0,3,7,0,3,0,3,0,0,7,0,0,2,0,0,6,0,0,4,0,0,0,0,7],[4,29,0.1379,0.56701,0.27773,0.39286,0.50071,0.85714,0.14286,1.0,0,4,0,0,0,4,0,0,4,0,0,8,0,0,2,0,0,5,0,0,5,0,4],[8,29,0.2759,0.5625,0.29,0.28571,0.42859,0.85704,0.14286,1.0,0,6,0,0,0,1,0,0,12,0,0,4,0,0,1,0,0,5,0,0,3,0,6],[12,29,0.4138,0.45089,0.21461,0.28571,0.42857,0.60714,0.1429,1.0,0,1,0,0,0,1,0,0,14,0,0,8,0,0,1,0,0,5,0,0,2,0,1],[16,29,0.5517,0.38838,0.12991,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,12,0,0,1,0,0,3,0,0,0,0,0],[20,29,0.6897,0.43748,0.16339,0.28571,0.42857,0.571,0.143,0.71429,0,0,0,0,0,1,0,0,11,0,0,11,0,0,3,0,0,6,0,0,0,0,0],[24,29,0.8276,0.41515,0.19349,0.28571,0.35714,0.57111,0.14286,0.71429,0,0,0,0,0,4,0,0,12,0,0,6,0,0,3,0,0,7,0,0,0,0,0],[28,29,0.9655,0.37054,0.14222,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,22,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[29,29,1.0,0.38839,0.12993,0.28571,0.35714,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,16,0,0,12,0,0,1,0,0,3,0,0,0,0,0]]}]},{"i":"07872c5a129f0002","q":"25. (HUN 3) Prove the identity  $$ (z+a)^{n}=z^{n}+a \\sum_{k=1}^{n}\\binom{n}{k}(a-k b)^{k-1}(z+k b)^{n-k} . $$","t":[{"b":3,"e":0.0,"k":"flat","v":0.22322,"x":0.27232,"p":[[0,9,0.0,0.24554,0.33737,0.0,0.07143,0.42857,0.0,1.0,16,3,0,16,0,5,0,0,2,0,0,3,0,0,0,0,0,2,0,0,1,0,3],[4,9,0.4444,0.27232,0.20316,0.10714,0.28571,0.42857,0.0,0.57143,8,0,0,8,0,5,0,0,6,0,0,8,0,0,5,0,0,0,0,0,0,0,0],[8,9,0.8889,0.22322,0.16728,0.14286,0.1429,0.42857,0.0,0.57143,7,0,0,7,0,10,0,0,6,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[9,9,1.0,0.2366,0.1411,0.14286,0.2857,0.28571,0.0,0.57143,5,0,0,5,0,7,0,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.06696,"x":0.32134,"p":[[0,22,0.0,0.32134,0.34997,0.0,0.14286,0.60714,0.0,1.0,13,3,0,13,0,4,0,0,2,0,0,3,0,0,2,0,0,4,0,0,1,0,3],[4,22,0.1818,0.27222,0.3261,0.0,0.14286,0.57111,0.0,1.0,12,2,0,12,0,9,0,0,2,0,0,0,0,0,2,0,0,4,0,0,1,0,2],[8,22,0.3636,0.17848,0.22305,0.0,0.14286,0.1786,0.0,1.0,12,1,0,12,0,12,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,1],[12,22,0.5455,0.30353,0.3169,0.0,0.14286,0.571,0.0,1.0,10,2,0,10,0,7,0,0,5,0,0,0,0,0,5,0,0,1,0,0,2,0,2],[16,22,0.7273,0.26339,0.26513,0.0,0.14286,0.46429,0.0,0.71429,11,0,0,11,0,7,0,0,3,0,0,3,0,0,3,0,0,5,0,0,0,0,0],[20,22,0.9091,0.13822,0.12103,0.0,0.14286,0.14287,0.0,0.4286,10,0,0,10,0,15,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.06696,0.09438,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0956a4f46527ab12","q":"A circle of diameter  $AB$  is given. There are points   $C$  and  $ D$  on this circle, on different sides of the diameter  such that holds   $AC <BC$  or  $AC<AD$ . The point  $P$  lies on the segment  $BC$  and   $\\angle CAP = \\angle ABC$ . The perpendicular from the point   $C$  to the line  $AB$  intersects the direction  $BD$  at the point  $Q$ . Lines  $PQ$  and  $AD$  intersect at point  $R$ , and the lines  $PQ$  and  $CD$  intersect at point  $T$ . If   $AR=RQ$ , prove that the lines  $AT$  and  $PQ$  are perpendicular.","t":[{"b":4,"e":0.0,"k":"flat","v":0.00438,"x":0.00893,"p":[[0,6,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,62,0.0,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,62,0.0645,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,62,0.129,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,62,0.1935,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,62,0.2581,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,62,0.3226,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,62,0.3871,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,62,0.4516,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,62,0.5161,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,62,0.5806,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,62,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"dd052dc15f9abc1c","q":"$ABC$  is a triangle and  $\\omega$  its incircle. Let  $P,Q,R$  be the intersections with  $\\omega$  and the sides  $BC,CA,AB$  respectively.  $AP$  cuts  $\\omega$  in  $P$  and  $X$ .  $BX,CX$  cut  $\\omega$  in  $M,N$  respectively. Show that  $MR,NQ,AP$  are parallel or concurrent.","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.01339,"p":[[0,11,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,11,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"rising","v":0.00893,"x":0.28126,"p":[[0,27,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.11174,0.10577,0.0,0.14286,0.14286,0.0,0.29,13,0,0,13,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.28126,0.05629,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,0,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.26339,0.08827,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c6dd326aede9e737","q":"An isosceles trapezoid  $ABCD$ , inscribed in  $\\omega$ , satisfies  $AB=CD, AD<BC, AD<CD$ .\nA circle with center  $D$  and passing  $A$  hits  $BD, CD, \\omega$  at  $E, F, P(\\not= A)$ , respectively.\nLet  $AP \\cap EF = Q$ , and  $\\omega$  meet  $CQ$  and the circumcircle of  $\\triangle BEQ$  at  $R(\\not= C), S(\\not= B)$ , respectively. \nProve that  $\\angle BER= \\angle FSC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,65,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,65,0.1846,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,65,0.2462,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,65,0.5538,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,65,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,65,0.7385,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,50,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.04446,0.06595,0.0,0.0,0.14071,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"620123a2317f3eff","q":"56. (USS 3) In a group of interpreters each one speaks one or several foreign languages; 24 of them speak Japanese, 24 Malay, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malay, and exactly 12 speak Farsi.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.04018,"x":0.16955,"p":[[0,28,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0759,0.07974,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0959,0.10349,0.0,0.14143,0.14286,0.0,0.35714,15,0,0,15,0,13,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.16509,0.10781,0.14286,0.14286,0.1786,0.0,0.42857,5,0,0,5,0,19,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.16955,0.06625,0.14286,0.14286,0.14287,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.14732,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04902,"x":0.14286,"p":[[0,32,0.0,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.04902,0.11626,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,32,0.5,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.14277,0.12372,0.105,0.14286,0.1429,0.0,0.57143,8,0,0,8,0,19,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,32,0.875,0.14286,0.07143,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.13831,0.07563,0.14286,0.14286,0.1429,0.0,0.28571,5,0,0,5,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ac5979768414d88c","q":"$100$  red points divide a blue circle into  $100$  arcs such that their lengths are all positive integers from  $1$  to  $100$  in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.","t":[{"b":2,"e":0.0,"k":"flat","v":0.06697,"x":0.11831,"p":[[0,15,0.0,0.06697,0.06902,0.0,0.03571,0.14286,0.0,0.1429,16,0,1,16,2,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.11607,0.08328,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.11831,0.05817,0.14286,0.14286,0.14286,0.0,0.21429,6,0,0,6,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.05357,"x":0.07804,"p":[[0,4,0.0,0.07804,0.0699,0.0,0.14143,0.14286,0.0,0.1429,14,0,0,14,1,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9903b4cb4fc32147","q":"A and B play tennis. The player to first win at least four points and at least two more than the other player wins. We know that A gets a point each time with probability  $p\\le \\frac12$ , independent of the game so far. Prove that the probability that A wins is at most  $2p^2$ .","t":[{"b":2,"e":0.42857,"k":"rising","v":0.33482,"x":0.52231,"p":[[0,8,0.0,0.33482,0.25657,0.0,0.42857,0.46429,0.0,1.0,9,1,0,9,0,2,0,0,2,0,0,11,0,0,6,0,0,1,0,0,0,0,1],[4,8,0.5,0.46875,0.1439,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,18,0,0,10,0,0,0,0,0,0,0,1],[8,8,1.0,0.52231,0.06784,0.42857,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,11,0,0,21,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"rising","v":0.32589,"x":0.50447,"p":[[0,14,0.0,0.32589,0.1996,0.14286,0.42857,0.42857,0.0,0.71429,6,0,0,6,0,4,0,0,2,0,0,16,0,0,3,0,0,1,0,0,0,0,0],[4,14,0.2857,0.43304,0.23551,0.42857,0.42857,0.42857,0.0,1.0,3,2,0,3,0,3,0,0,0,0,0,19,0,0,3,0,0,1,0,0,1,0,2],[8,14,0.5714,0.48661,0.07016,0.42857,0.42857,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0],[12,14,0.8571,0.48659,0.07014,0.42857,0.42857,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0],[14,14,1.0,0.50447,0.07129,0.42857,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0]]}]},{"i":"1905d3dd85b9ddb7","q":"A rectangle  $ABCD$  has side lengths  $AB = m$ ,  $AD = n$ , with  $m$  and  $n$  relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points  $A_1 = A, A_2, A_3, \\ldots , A_k = C$ . Show that \\[ A_1A_2 - A_2A_3 + A_3A_4 - \\cdots + A_{k-1}A_k = {\\sqrt{m^2+n^2}\\over mn}.  \\]","t":[{"b":0,"e":0.2857,"k":"flat","v":0.29018,"x":0.37499,"p":[[0,8,0.0,0.31696,0.21349,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,10,0,0,10,0,0,4,0,0,2,0,0,3,0,0,1,0,0],[4,8,0.5,0.37499,0.2335,0.14286,0.28571,0.57111,0.14286,1.0,0,1,0,0,0,10,0,0,9,0,0,4,0,0,4,0,0,3,0,0,1,0,1],[8,8,1.0,0.29018,0.12103,0.2857,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,5,0,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.21867,"x":0.28571,"p":[[0,21,0.0,0.28571,0.19885,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,8,0,0,11,0,0,5,0,0,2,0,0,1,0,0,1,0,0],[4,21,0.1905,0.27679,0.26711,0.10714,0.14288,0.42857,0.0,0.85714,8,0,0,8,0,10,0,0,3,0,0,5,0,0,2,0,0,1,0,0,3,0,0],[8,21,0.381,0.23652,0.16219,0.14286,0.28571,0.28571,0.0,0.71429,6,0,0,6,0,7,0,0,13,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[12,21,0.5714,0.21875,0.13356,0.14286,0.21428,0.28571,0.0,0.57143,4,0,0,4,0,12,0,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,21,0.7619,0.22304,0.12857,0.14286,0.14286,0.28571,0.0,0.71429,1,0,0,1,0,17,0,0,11,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[20,21,0.9524,0.21867,0.10711,0.14286,0.14288,0.28571,0.0,0.42857,1,0,0,1,0,17,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.25445,0.18806,0.14286,0.2143,0.28571,0.0,1.0,2,1,0,2,0,14,0,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,1]]}]},{"i":"1d8bdf24a05ae2cf","q":"A set $A$ of integers is called sum-full if $A \\subseteq A+A$, i.e. each element $a \\in A$ is the sum of some pair of (not necessarily different) elements $b, c \\in A$. A set $A$ of integers is said to be zero-sum-free if 0 is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$.\n\nDoes there exist a sum-full zero-sum-free set of integers?\n\nOrigin. Romania (Dan Schwarz).\nRemark. The original formulation of this problem had a weaker definition of zero-sum-free that did not require all nonzero integers to be sums of finite nonempty subsets of $A$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,8,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,27,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0331b4c12208317e","q":"A sequence of positive integers $\\left(a_{n}\\right)_{n \\geq 1}$ is of Fibonacci type if it satisfies the recursive relation $a_{n+2}=a_{n+1}+a_{n}$ for all $n \\geq 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences?","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.09822,"p":[[0,35,0.0,0.0625,0.12846,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.09822,0.18364,0.0,0.0,0.14286,0.0,0.71429,22,0,1,22,0,5,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[8,35,0.2286,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,35,0.5714,0.03125,0.08553,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,27,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.04911,0.13175,0.0,0.0,0.0,0.0,0.57143,27,0,1,27,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,27,0.2963,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"93452f3c321bb424","q":"2. (CAN) Let $R_{1}, R_{2}, \\ldots$ be the family of finite sequences of positive integers defined by the following rules: $R_{1}=(1)$, and if $R_{n-1}=\\left(x_{1}, \\ldots, x_{s}\\right)$, then  $$ R_{n}=\\left(1,2, \\ldots, x_{1}, 1,2, \\ldots, x_{2}, \\ldots, 1,2, \\ldots, x_{s}, n\\right) $$  For example, $R_{2}=(1,2), R_{3}=(1,1,2,3), R_{4}=(1,1,1,2,1,2,3,4)$. Prove that if $n>1$, then the $k$ th term from the left in $R_{n}$ is equal to 1 if and only if the $k$ th term from the right in $R_{n}$ is different from 1.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.16508,"x":0.2632,"p":[[0,12,0.0,0.20759,0.13987,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,15,0,0,9,0,1,1,0,0,2,0,0,0,0,0,0,0,0],[4,12,0.3333,0.16508,0.11901,0.14286,0.14286,0.2857,0.0,0.571,6,0,0,6,0,17,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,12,0.6667,0.19197,0.15815,0.14286,0.14286,0.2857,0.0,0.85714,4,0,0,4,0,19,0,0,6,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[12,12,1.0,0.2632,0.16803,0.14286,0.2857,0.32143,0.0,0.71429,3,0,0,3,0,11,0,0,10,0,0,5,0,0,2,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"flat","v":0.16938,"x":0.2299,"p":[[0,12,0.0,0.18308,0.10863,0.14286,0.14286,0.2857,0.0,0.43,3,0,0,3,0,20,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.1808,0.11005,0.14286,0.14286,0.2857,0.0,0.42857,5,0,0,5,0,15,0,0,10,0,1,1,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.16938,0.12601,0.14,0.14286,0.2857,0.0,0.57143,6,0,0,6,0,17,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,12,1.0,0.2299,0.1414,0.14286,0.24999,0.28571,0.0,0.57143,3,0,0,3,0,12,0,1,13,0,0,0,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"aded24193c1cbea7","q":"A finite family of finite sets  $F$  is given, satisfying two conditions:\n(i) if  $A, B \\in F$ , then  $A \\cup B \\in F$ ;\n(ii) if  $A \\in F$ , then the number of elements  $| A |$  is not a multiple of  $3$ .\nProve that you can specify at most two elements so that every set of the family  $F$  contains at least one of them.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,31,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,3,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"be6a4f024a6183ba","q":"7. (IND 4) Circles $G, G_{1}, G_{2}$ are three circles related to each other as follows: Circles $G_{1}$ and $G_{2}$ are externally tangent to one another at a point $W$ and both these circles are internally tangent to the circle $G$. Points $A, B, C$ are located on the circle $G$ as follows: Line $B C$ is a direct common tangent to the pair of circles $G_{1}$ and $G_{2}$, and line $W A$ is the transverse common tangent at $W$ to $G_{1}$ and $G_{2}$, with $W$ and $A$ lying on the same side of the line $B C$. Prove that $W$ is the incenter of the triangle $A B C$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"080551915f545629","q":"2. A2 (FRA) ${ }^{\\mathrm{IMO} 1}$ Let $m$ and $n$ be positive integers. The set $A=\\left\\{a_{1}, a_{2}, \\ldots\\right.$, $\\left.a_{m}\\right\\}$ is a subset of $\\{1,2, \\ldots, n\\}$. Whenever $a_{i}+a_{j} \\leq n, 1 \\leq i \\leq j \\leq m$, $a_{i}+a_{j}$ also belongs to $A$. Prove that  $$ \\frac{a_{1}+a_{2}+\\cdots+a_{m}}{m} \\geq \\frac{n+1}{2} . $$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.05804,"x":0.16964,"p":[[0,9,0.0,0.16518,0.15198,0.14286,0.14286,0.14287,0.0,0.71429,7,0,0,7,0,19,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,9,0.4444,0.16964,0.10972,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,24,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.85714,"k":"volatile","v":0.11152,"x":0.57587,"p":[[0,10,0.0,0.12054,0.1017,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,21,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.19197,0.18073,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,24,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[8,10,0.8,0.11152,0.12743,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,20,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[10,10,1.0,0.57587,0.2004,0.42857,0.64286,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,1,0,0,9,0,0,4,0,0,13,0,0,3,0,0]]}]},{"i":"85d869213327810a","q":"4. Let $A B$ be a diameter of a circle $\\gamma$. A point $C$ different from $A$ and $B$ is on the circle $\\gamma$. Let $D$ be the projection of the point $C$ onto the line $A B$. Consider three other circles $\\gamma_{1}, \\gamma_{2}$, and $\\gamma_{3}$ with the common tangent $A B: \\gamma_{1}$ inscribed in the triangle $A B C$, and $\\gamma_{2}$ and $\\gamma_{3}$ tangent to both (the segment) $C D$ and $\\gamma$. Prove that $\\gamma_{1}, \\gamma_{2}$, and $\\gamma_{3}$ have two common tangents.","t":[{"b":4,"e":0.57143,"k":"volatile","v":0.30344,"x":0.81695,"p":[[0,3,0.0,0.30344,0.28961,0.0,0.21428,0.57143,0.0,1.0,11,1,1,11,0,5,0,0,2,0,0,2,0,0,10,0,0,0,0,0,1,0,1],[3,3,1.0,0.81695,0.19962,0.57143,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,3,0,16]]},{"b":7,"e":0.0,"k":"falling","v":0.01786,"x":0.3347,"p":[[0,38,0.0,0.3347,0.2937,0.14214,0.14286,0.57143,0.0,1.0,7,2,1,7,0,10,0,0,0,0,0,2,0,0,10,0,0,1,0,0,0,0,2],[4,38,0.1053,0.31695,0.2806,0.0,0.28571,0.57143,0.0,1.0,10,1,0,10,0,6,0,0,0,0,0,2,0,0,13,0,0,0,0,0,0,0,1],[8,38,0.2105,0.30353,0.29175,0.0,0.14288,0.57141,0.0,1.0,9,2,0,9,0,8,0,0,3,0,0,0,0,0,10,0,0,0,0,0,0,0,2],[12,38,0.3158,0.08036,0.17105,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[16,38,0.4211,0.12936,0.20931,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,6,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[20,38,0.5263,0.03572,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,1,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.03571,0.07986,0.0,0.0,0.0,0.0,0.2857,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"06bdf35f0db13bd9","q":"A game is played on a 2001 x 2001 board as follows. The first player's piece is the policeman, the second player's piece is the robber. Each piece can move one square south, one square east or one square northwest. In addition, the policeman (but not the robber) can move from the bottom right to the top left square in a single move. The policeman starts in the central square, and the robber starts one square diagonally northeast of the policeman. If the policeman moves onto the same square as the robber, then the robber is captured and the first player wins. However, the robber may move onto the same square as the policeman without being captured (and play continues). Show that the robber can avoid capture for at least 10000 moves, but that the policeman can ultimately capture the robber.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.00446,"x":0.0757,"p":[[0,15,0.0,0.0757,0.12349,0.0,0.0,0.14286,0.0,0.571,20,0,1,20,0,9,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.03563,0.0713,0.0,0.0,0.0,0.0,0.2857,25,0,1,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.02224,0.05168,0.0,0.0,0.0,0.0,0.143,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0133,0.05465,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.0625,"p":[[0,32,0.0,0.03554,0.07962,0.0,0.0,0.0,0.0,0.28571,26,0,2,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.42857,22,0,2,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.06241,0.10055,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,1,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,32,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c5d0b12503450032","q":"5. (ROM 1) Let $D$ be the interior of the circle $C$ and let $A \\in C$. Show that the function $f: D \\rightarrow \\mathbb{R}, f(M)=\\frac{|M A|}{\\left|M M^{\\prime}\\right|}$, where $M^{\\prime}=(A M \\cap C$, is strictly convex; i.e., $f(P)<\\frac{f\\left(M_{1}\\right)+f\\left(M_{2}\\right)}{2}, \\forall M_{1}, M_{2} \\in D, M_{1} \\neq M_{2}$, where $P$ is the midpoint of the segment $M_{1} M_{2}$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.12491,"p":[[0,15,0.0,0.12491,0.16268,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,5,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,15,0.2667,0.10705,0.12875,0.0,0.0,0.28571,0.0,0.28571,18,0,0,18,0,4,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.08472,"x":0.12946,"p":[[0,29,0.0,0.12946,0.1488,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,5,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,29,0.1379,0.10268,0.13474,0.0,0.0,0.2857,0.0,0.42857,19,0,0,19,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.12946,0.13997,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,4,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.08929,0.12753,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,13,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,29,0.5517,0.08472,0.13287,0.0,0.0,0.14286,0.0,0.571,19,0,0,19,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,29,0.6897,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.2857,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.10268,0.12492,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,16,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,29,0.9655,0.12501,0.13716,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,20,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[29,29,1.0,0.10714,0.15152,0.0,0.14286,0.14286,0.0,0.85714,13,0,0,13,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"3b2c8c2253023277","q":"A regular  $2004$ -sided polygon is given, with all of its diagonals drawn. After some sides and diagonals are removed, every vertex has at most five segments coming out of it. Prove that one can color the vertices with two colors such that at least  $\\frac{3}{5}$  of the remaining segments have ends with different colors.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,37,0.0,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.09375,0.10479,0.0,0.14286,0.14286,0.0,0.4286,15,0,1,15,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.01777,0.05904,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.08473,"x":0.21866,"p":[[0,12,0.0,0.09821,0.10972,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.08473,0.07866,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.10937,0.09276,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,1,16,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.21866,0.07138,0.14286,0.2857,0.28571,0.14,0.28571,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2aca373d66a01cba","q":"An open necklace can contain rubies, emeralds and sapphires. At every step we can perform any of the following operations:\n$\\left(1^{\\circ}\\right)$ We can replace two consecutive rubies with an emerald and a sapphire, where the emerald is on the left of the sapphire.\n$\\left(2^{\\circ}\\right)$ We can replace three consecutive emeralds with a sapphire and a ruby, where the sapphire is on the left of the ruby.\n(3 ) If we find two consecutive sapphires then we can remove them.\n(4 ) If we find consecutively and in this order a ruby, an emerald, and a sapphire, then we can remove them.\n\nFurthermore we can also reverse all of the above operations. For example, by reversing $\\left(3^{\\circ}\\right)$ we can put two consecutive sapphires on any position we wish.\nInitially the necklace has one sapphire (and no other precious stones). Decide, with proof, whether there is a finite sequence of steps such that at the end of this sequence the necklace contains one emerald (and no other precious stones).\nRemark. A necklace is open if its precious stones are on a line from left to right. We are not allowed to move a precious stone from the rightmost position to the leftmost as we would be able to do if the necklace was closed.\n(Cyprus)","t":[{"b":1,"e":1.0,"k":"rising","v":0.37499,"x":0.95534,"p":[[0,59,0.0,0.37499,0.37414,0.0,0.28571,0.60682,0.0,1.0,12,6,1,12,0,0,0,0,6,0,0,5,0,0,1,0,0,1,0,0,1,0,6],[4,59,0.0678,0.45982,0.41762,0.0,0.28571,1.0,0.0,1.0,10,9,0,10,0,2,0,0,6,0,0,0,0,0,1,0,0,2,0,0,2,0,9],[8,59,0.1356,0.53794,0.38506,0.28571,0.50001,1.0,0.0,1.0,7,11,1,7,0,0,0,0,4,0,1,4,0,0,4,0,0,1,0,0,0,0,11],[12,59,0.2034,0.49998,0.3677,0.21427,0.57143,0.75,0.0,1.0,8,7,1,8,0,0,0,0,5,0,0,1,0,0,6,0,0,4,0,0,1,0,7],[16,59,0.2712,0.80802,0.25409,0.57143,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,3,0,0,2,0,0,5,0,0,1,0,0,3,0,18],[20,59,0.339,0.74106,0.31631,0.57143,0.85714,1.0,0.0,1.0,3,14,0,3,0,1,0,0,0,0,0,0,0,0,7,0,0,3,0,0,4,0,14],[24,59,0.4068,0.70536,0.28333,0.57143,0.57143,1.0,0.0,1.0,2,13,0,2,0,0,0,0,1,0,0,0,0,0,15,0,0,1,0,0,0,0,13],[28,59,0.4746,0.95089,0.12682,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,27],[32,59,0.5424,0.91518,0.15916,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,2,0,24],[36,59,0.6102,0.94196,0.13296,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,26],[40,59,0.678,0.87944,0.22902,0.85714,1.0,1.0,0.0,1.0,1,23,1,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,2,0,23],[44,59,0.7458,0.88393,0.17655,0.71429,1.0,1.0,0.42857,1.0,0,21,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,0,2,0,21],[48,59,0.8136,0.92857,0.12877,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,4,0,23],[52,59,0.8814,0.91964,0.2111,1.0,1.0,1.0,0.0,1.0,1,27,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,27],[56,59,0.9492,0.90625,0.14555,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,4,0,21],[59,59,1.0,0.95534,0.10379,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,3,0,26]]},{"b":5,"e":0.57143,"k":"rising","v":0.45981,"x":0.67854,"p":[[0,8,0.0,0.45982,0.45279,0.0,0.35714,1.0,0.0,1.0,14,12,0,14,0,0,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,12],[4,8,0.5,0.45981,0.34206,0.2857,0.42857,0.60714,0.0,1.0,6,7,0,6,0,1,0,0,7,0,0,6,0,0,4,0,0,1,0,0,0,0,7],[8,8,1.0,0.67854,0.15154,0.57143,0.57143,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,19,0,0,6,0,0,3,0,4]]}]},{"i":"f33491d9f6b86e9f","q":"3. (CZS 6) ${ }^{\\mathrm{IMO}}$ Prove that the sum of an odd number of unit vectors passing through the same point $O$ and lying in the same half-plane whose border passes through $O$ has length greater than or equal to 1 .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02678,"p":[[0,30,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.02678,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03572,"p":[[0,33,0.0,0.03124,0.11136,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,33,0.1212,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,33,0.4848,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.03572,0.10102,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"7dd2cf773053ee9a","q":"**5.** Prove that neither the closed nor the open interval can be decomposed into finitely many mutually disjoint proper subsets which are all congruent by translation. **(St. 2)**","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.06696,"p":[[0,7,0.0,0.06696,0.1712,0.0,0.0,0.0,0.0,0.857,25,0,0,25,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,7,0.5714,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.06241,"p":[[0,5,0.0,0.01786,0.04726,0.0,0.0,0.0,0.0,0.143,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.06241,0.08695,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4f9281c700e83da5","q":"19. $\\mathbf{G} 4 \\mathbf{( P O L})^{\\mathrm{IMO}}$ In a convex quadrilateral $A B C D$ the diagonal $B D$ does not bisect the angles $A B C$ and $C D A$. The point $P$ lies inside $A B C D$ and satisfies  $$ \\angle P B C=\\angle D B A \\quad \\text { and } \\quad \\angle P D C=\\angle B D A . $$  Prove that $A B C D$ is a cyclic quadrilateral if and only if $A P=C P$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.05358,"p":[[0,18,0.0,0.03563,0.0713,0.0,0.0,0.0,0.0,0.2857,25,0,1,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.04465,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,1,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.05358,0.09943,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03563,"p":[[0,3,0.0,0.03563,0.0713,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7c6987d9d9a51686","q":"Call a subset  $ S$  of  $ \\{1,2,\\dots,n\\}$  *mediocre* if it has the following property: Whenever  $ a$  and  $ b$  are elements of  $ S$  whose average is an integer, that average is also an element of  $ S.$  Let  $ A(n)$  be the number of mediocre subsets of  $ \\{1,2,\\dots,n\\}.$  [For instance, every subset of  $ \\{1,2,3\\}$  except  $ \\{1,3\\}$  is mediocre, so  $ A(3)\\equal{}7.$ ] Find all positive integers  $ n$  such that  $ A(n\\plus{}2)\\minus{}2A(n\\plus{}1)\\plus{}A(n)\\equal{}1.$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,16,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,2,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"de02e165ee3e289a","q":"Assign each edge in a graph an arbitrary nonnegative weight. Is it always possible to assign each vertex of the graph a nonnegative weight, so that the sum of the weights of vertices equals the sum of the weights of edges, and each edge's weight is at most the difference between the weights of its endpoints?\n\n*Proposed by Evan Chang*","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.10268,"p":[[0,28,0.0,0.09812,0.16532,0.0,0.0,0.14286,0.0,0.857,18,0,0,18,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,28,0.1429,0.09821,0.09061,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.10268,0.10248,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.07143,0.10101,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.08036,"x":0.14286,"p":[[0,7,0.0,0.09357,0.12162,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.14286,0.11845,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,0,13,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a4eeb0f06c0004d9","q":"$2n-1$  distinct positive real numbers with sum  $S $  are given. Prove that there are at least  $\\binom {2n-2}{n-1}$  different ways to choose  $n $  numbers among them such that their sum is at least  $\\frac {S}{2}$ .\n\n*Proposed by Amirhossein Gorzi*","t":[{"b":4,"e":0.0,"k":"flat","v":0.02232,"x":0.03125,"p":[[0,14,0.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,2,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02679,0.10375,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,2,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,2,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[14,14,1.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,31,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.02232,0.08073,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.04018,0.1439,0.0,0.0,0.0,0.0,0.71429,29,0,8,29,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,31,0.7742,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"198940b810426b3b","q":"A positive integer $n$ is downhill if its decimal representation $\\overline{a_{k} a_{k-1} \\ldots a_{0}}$ satisfies $a_{k} \\geq a_{k-1} \\geq \\ldots \\geq a_{0}$. A real-coefficient polynomial $P$ is integer-valued if $P(n)$ is an integer for all integer $n$, and downhill-integervalued if $P(n)$ is an integer for all downhill positive integers $n$. Is it true that every downhill-integer-valued polynomial is also integer-valued?","t":[{"b":4,"e":0.0,"k":"flat","v":0.00447,"x":0.05804,"p":[[0,23,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.05804,0.19186,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[8,23,0.3478,0.04018,0.1394,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,23,0.5217,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,15,0.0,0.04018,0.16457,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,15,0.2667,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"971a05c8d9e65369","q":"5. (FRA 1) Given $\\triangle A B C$ with no side equal to another side, let $G, K$, and $H$ be its centroid, incenter, and orthocenter, respectively. Prove that $\\angle G K H>90^{\\circ}$.","t":[{"b":0,"e":0.85714,"k":"volatile","v":0.32141,"x":0.84821,"p":[[0,16,0.0,0.5,0.33881,0.14286,0.5,0.85714,0.0,0.85714,3,0,0,3,0,8,0,0,3,0,0,2,0,0,1,0,0,2,0,0,13,0,0],[4,16,0.25,0.39277,0.30937,0.14286,0.28571,0.75,0.0,0.85714,3,0,0,3,0,9,0,0,9,0,0,1,0,0,0,0,0,2,0,0,8,0,0],[8,16,0.5,0.32141,0.26962,0.14286,0.2857,0.28571,0.0,0.85714,3,0,0,3,0,11,0,0,11,0,0,0,0,0,1,0,0,1,0,0,5,0,0],[12,16,0.75,0.84375,0.04164,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0],[16,16,1.0,0.84821,0.03458,0.85714,0.85714,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.27223,"x":0.45981,"p":[[0,7,0.0,0.42848,0.32935,0.14286,0.28571,0.75,0.0,0.85714,5,0,0,5,0,7,0,0,5,0,0,2,0,0,0,0,0,5,0,0,8,0,0],[4,7,0.5714,0.45981,0.3108,0.2857,0.28571,0.85714,0.0,0.85714,2,0,0,2,0,5,0,0,12,0,0,1,0,0,0,0,0,1,0,0,11,0,0],[7,7,1.0,0.27223,0.04192,0.2857,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"44161f940f85a10f","q":"26. C6 (POL) Let $n$ be an even positive integer. Show that there is a permutation $x_{1}, x_{2}, \\ldots, x_{n}$ of $1,2, \\ldots, n$ such that for every $1 \\leq i \\leq n$ the number $x_{i+1}$ is one of $2 x_{i}, 2 x_{i}-1,2 x_{i}-n, 2 x_{i}-n-1$ (where we take $x_{n+1}=x_{1}$ ).","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,4,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.07141,"p":[[0,22,0.0,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.07141,0.202,0.0,0.0,0.0,0.0,1.0,26,1,1,26,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[8,22,0.3636,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.04455,0.15327,0.0,0.0,0.0,0.0,0.857,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"db555178518e8043","q":"A natural number is written on the blackboard. Whenever number  $ x$  is written, one can write any of the numbers  $ 2x \\plus{} 1$  and  $ \\frac {x}{x \\plus{} 2}$ . At some moment the number  $ 2008$  appears on the blackboard. Show that it was there from the very beginning.","t":[{"b":6,"e":0.28571,"k":"rising","v":0.08928,"x":0.48214,"p":[[0,43,0.0,0.08928,0.18123,0.0,0.0,0.14286,0.0,0.85714,23,0,1,23,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,43,0.093,0.32589,0.1931,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,11,0,0,10,0,0,6,0,0,3,0,0,0,0,0,2,0,0],[8,43,0.186,0.41506,0.21545,0.28571,0.42857,0.57143,0.14,1.0,0,1,0,0,0,7,0,0,7,0,0,7,0,0,6,0,0,4,0,0,0,0,1],[12,43,0.2791,0.39284,0.17127,0.28571,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,5,0,0,10,0,0,8,0,0,6,0,0,3,0,0,0,0,0],[16,43,0.3721,0.375,0.21053,0.24999,0.28571,0.46429,0.14286,1.0,0,1,0,0,0,8,0,0,10,0,0,6,0,0,4,0,0,3,0,0,0,0,1],[20,43,0.4651,0.36604,0.20494,0.24999,0.28571,0.42858,0.14286,0.85714,0,0,0,0,0,8,0,0,11,0,0,6,0,0,3,0,0,2,0,0,2,0,0],[24,43,0.5581,0.47768,0.21902,0.28571,0.42859,0.60714,0.14286,0.85714,0,0,0,0,0,4,0,0,6,0,0,9,0,0,5,0,0,4,0,0,4,0,0],[28,43,0.6512,0.48214,0.25692,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,5,0,0,7,0,0,8,0,0,3,0,0,2,0,0,6,0,1],[32,43,0.7442,0.35265,0.17487,0.2857,0.28571,0.4642,0.14286,0.85714,0,0,0,0,0,7,0,0,13,0,0,4,0,0,7,0,0,0,0,0,1,0,0],[36,43,0.8372,0.41517,0.23786,0.2857,0.35714,0.57143,0.14286,1.0,0,2,0,0,0,7,0,0,9,0,0,5,0,0,7,0,0,1,0,0,1,0,2],[40,43,0.9302,0.41071,0.15872,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,10,0,0,10,0,0,6,0,0,3,0,0,0,0,0],[43,43,1.0,0.38831,0.17228,0.28571,0.42857,0.42857,0.14,1.0,0,1,0,0,0,5,0,0,8,0,0,13,0,0,5,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.71429,"k":"volatile","v":0.05804,"x":0.66964,"p":[[0,6,0.0,0.05804,0.1551,0.0,0.0,0.0,0.0,0.71429,27,0,3,27,0,1,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,6,0.6667,0.15625,0.28203,0.0,0.0,0.2857,0.0,1.0,22,2,0,22,0,1,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[6,6,1.0,0.66964,0.2126,0.57143,0.71429,0.85704,0.2857,1.0,0,4,0,0,0,0,0,0,4,0,0,3,0,0,5,0,0,11,0,0,5,0,4]]}]},{"i":"1fdd53b6f8bbd497","q":"A lottery ticket has  $50$  cells into which one must put a permutation of  $1, 2, 3, ... , 50$ . Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize?","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.20536,"p":[[0,32,0.0,0.20536,0.32131,0.0,0.0,0.2857,0.0,1.0,19,1,10,19,0,4,0,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,1],[4,32,0.125,0.11607,0.23538,0.0,0.0,0.14286,0.0,1.0,23,1,9,23,0,3,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[8,32,0.25,0.12947,0.27976,0.0,0.0,0.03571,0.0,1.0,24,1,9,24,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,1],[12,32,0.375,0.05357,0.17405,0.0,0.0,0.0,0.0,0.71429,28,0,11,28,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[16,32,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,12,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.29018,"x":0.76784,"p":[[0,20,0.0,0.29018,0.35801,0.0,0.07143,0.71429,0.0,1.0,16,2,9,16,0,4,0,0,0,0,0,2,0,0,1,0,0,5,0,0,2,0,2],[4,20,0.2,0.73209,0.22519,0.57132,0.78564,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,7,0,0,10,0,6],[8,20,0.4,0.75887,0.19708,0.57132,0.85714,0.85714,0.28571,1.0,0,6,0,0,0,0,0,0,1,0,0,3,0,0,6,0,0,3,0,0,13,0,6],[12,20,0.6,0.71872,0.20041,0.67856,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,10,0,0,11,0,3],[16,20,0.8,0.76784,0.17405,0.71429,0.78564,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,10,0,0,10,0,6],[20,20,1.0,0.71873,0.20973,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,10,0,0,8,0,5]]}]},{"i":"9c35732be416ccf2","q":"Let  $I$  be the incenter of triangle  $ABC$ ;  $H_B, H_C$  the orthocenters of triangles  $ACI$  and  $ABI$  respectively;  $K$  the touching point of the incircle with the side  $BC$ . Prove that  $H_B, H_C$  and K are collinear.\n*Proposed by M.Plotnikov*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,32,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,32,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02009,"p":[[0,26,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.02009,0.05426,0.0,0.0,0.0,0.0,0.21429,28,0,0,28,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"430d297e4c80b854","q":"A knight moves on a two-dimensional grid. From any square, it can move 2 units in one axisparallel direction, then move 1 unit in an orthogonal direction, the way a regular knight moves in a game of chess. The knight starts at the origin. As it moves, it keeps track of a number $t$, which is initially 0 . When the knight lands at the point $(a, b)$, the number is changed from $x$ to $a x+b$.\n\nShow that, for any integers $a$ and $b$, it is possible for the knight to land at the points $(1, a)$ and $(-1, a)$ with $t$ equal to $b$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.01786,"x":0.11152,"p":[[0,5,0.0,0.11152,0.24931,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[4,5,0.8,0.04911,0.16213,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[5,5,1.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.08929,"p":[[0,12,0.0,0.08929,0.25191,0.0,0.0,0.0,0.0,1.0,27,2,1,27,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c9819602ad218bfd","q":"Does there exist a positive integer $N$ which is divisible by at least 2024 distinct primes and whose positive divisors $1=d_{1}<d_{2}<\\ldots<d_{k}=N$ are such that the number\n\n$$\n\\frac{d_{2}}{d_{1}}+\\frac{d_{3}}{d_{2}}+\\ldots+\\frac{d_{k}}{d_{k-1}}\n$$\n\nis an integer?","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.37946,"p":[[0,53,0.0,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,53,0.0755,0.1875,0.38038,0.0,0.0,0.0,0.0,1.0,25,5,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[8,53,0.1509,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[12,53,0.2264,0.37946,0.46922,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,10],[16,53,0.3019,0.32366,0.41954,0.0,0.0,0.85714,0.0,1.0,17,5,0,17,1,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,5,0,5],[20,53,0.3774,0.18304,0.38172,0.0,0.0,0.0,0.0,1.0,26,5,1,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,5],[24,53,0.4528,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,1,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[28,53,0.5283,0.125,0.31894,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[32,53,0.6038,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,53,0.6792,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,53,0.7547,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,53,0.8302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,53,0.9057,0.02009,0.07832,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[53,53,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.15625,"p":[[0,16,0.0,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[4,16,0.25,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,16,0.5,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"27e3fe35abc43618","q":"15. G3 (IND) Let $A B C$ be a triangle and let $P$ be a point in its interior. Denote by $D, E, F$ the feet of the perpendiculars from $P$ to the lines $B C$, $C A$, and $A B$, respectively. Suppose that  $$ A P^{2}+P D^{2}=B P^{2}+P E^{2}=C P^{2}+P F^{2} $$  Denote by $I_{A}, I_{B}, I_{C}$ the excenters of the triangle $A B C$. Prove that $P$ is the circumcenter of the triangle $I_{A} I_{B} I_{C}$.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,11,0.0,0.03571,0.14286,0.0,0.0,0.0,0.0,0.71429,30,0,1,30,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.00446,"x":0.03125,"p":[[0,23,0.0,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,23,0.1739,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6e24b0ad5cf2e37f","q":"Consider a triangle  $ABC$  and a point  $D$  on its side  $\\overline{AB}$ . Let  $I$  be a point inside  $\\triangle ABC$  on the angle bisector of  $ACB$ . The second intersections of lines  $AI$  and  $CI$  with circle  $ACD$  are  $P$  and  $Q$ , respectively. Similarly, the second intersection of lines  $BI$  and  $CI$  with circle  $BCD$  are  $R$  and  $S$ , respectively. Show that if  $P\\neq Q$  and  $R\\neq S$ , then lines  $AB$ ,  $PQ$  and  $RS$  pass through a point or are parallel.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,23,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7f0845ab134e41fe","q":"Alice and Bob are playing a game. First, Alice chooses a partition $\\mathcal{C}$ of the positive integers into a (not necessarily finite) set of sets, such that each positive integer is in exactly one of the sets in $\\mathcal{C}$. Then Bob does the following operation a finite number of times.\nChoose a set $S \\in \\mathcal{C}$ not previously chosen, and let $D$ be the set of all positive integers dividing at least one element in $S$. Then add the set $D \\backslash S$ (possibly the empty set) to $\\mathcal{C}$.\nBob wins if there are two equal sets in $\\mathcal{C}$ after he has done all his moves, otherwise, Alice wins. Determine which player has a winning strategy.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,12,0.0,0.04464,0.12078,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,38,0.0,0.04018,0.12993,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,38,0.1053,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,38,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"afc85737ad43f6f5","q":"6. (POL 2) ${ }^{\\mathrm{IMO} 2}$ Does there exist a finite set $M$ of points in space, not all in the same plane, such that for each two points $A, B \\in M$ there exist two other points $C, D \\in M$ such that lines $A B$ and $C D$ are parallel?","t":[{"b":1,"e":0.0,"k":"falling","v":0.0,"x":0.50446,"p":[[0,45,0.0,0.47321,0.47707,0.0,0.35714,1.0,0.0,1.0,16,11,2,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,11],[4,45,0.0889,0.50446,0.4615,0.0,0.78564,1.0,0.0,1.0,13,10,0,13,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,10],[8,45,0.1778,0.34375,0.46683,0.0,0.0,1.0,0.0,1.0,20,10,1,20,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,10],[12,45,0.2667,0.44643,0.47749,0.0,0.0,1.0,0.0,1.0,17,10,1,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,10],[16,45,0.3556,0.08927,0.24154,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[20,45,0.4444,0.25,0.42408,0.0,0.0,0.32143,0.0,1.0,23,7,0,23,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7],[24,45,0.5333,0.14732,0.34345,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,3],[28,45,0.6222,0.10713,0.29012,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[32,45,0.7111,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,45,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[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":2,"e":0.0,"k":"falling","v":0.0,"x":0.58929,"p":[[0,33,0.0,0.41518,0.46478,0.0,0.0,0.89286,0.0,1.0,17,8,0,17,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,8],[4,33,0.1212,0.58929,0.44571,0.0,0.85714,1.0,0.0,1.0,10,12,1,10,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,12],[8,33,0.2424,0.42409,0.44675,0.0,0.14286,0.89286,0.0,1.0,14,8,0,14,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,8],[12,33,0.3636,0.44196,0.4634,0.0,0.14285,1.0,0.0,1.0,16,10,0,16,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,10],[16,33,0.4848,0.35714,0.44175,0.0,0.0,0.85714,0.0,1.0,18,7,0,18,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,7],[20,33,0.6061,0.26339,0.42274,0.0,0.0,0.85714,0.0,1.0,23,5,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,5],[24,33,0.7273,0.10714,0.28571,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,1],[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],[32,33,0.9697,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[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]]}]},{"i":"dc4c48035c895f0d","q":"Dima has written number  $ 1/80!,\\,1/81!,\\,\\dots,1/99!$  on  $ 20$  infinite pieces of papers as decimal fractions (the following is written on the last piece:  $ \\frac {1}{99!} \\equal{} 0{,}{00\\dots 00}10715\\dots$ , 155 0-s before 1).  Sasha wants to cut a fragment of  $ N$  consecutive digits from one of pieces without the comma. For which maximal  $ N$  he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? \r\n\r\n*A. Golovanov*","t":[{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.01786,"p":[[0,3,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.04018,"p":[[0,13,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1acfc1987e0967af","q":"A number of 17 workers stand in a row. Every contiguous group of at least 2 workers is a brigade. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker's number of assignments is divisible by 4. Prove that the number of such ways to assign the leaders is divisible by 17 .\n\nRussia, Mikhail Antipov","t":[{"b":3,"e":0.14286,"k":"falling","v":0.07134,"x":0.56695,"p":[[0,18,0.0,0.40179,0.3719,0.14286,0.28571,0.75,0.0,1.0,6,7,0,6,0,8,0,0,6,0,0,2,0,0,1,0,0,1,0,0,1,0,7],[4,18,0.2222,0.56695,0.36506,0.28571,0.49979,1.0,0.0,1.0,5,10,0,5,0,0,0,0,6,0,0,5,0,0,2,0,0,2,0,0,2,0,10],[8,18,0.4444,0.33929,0.31084,0.14286,0.28571,0.46431,0.0,1.0,7,3,0,7,0,7,0,0,6,0,0,4,0,0,2,0,0,2,0,0,1,0,3],[12,18,0.6667,0.07134,0.1184,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.09804,0.11532,0.0,0.07,0.14286,0.0,0.4286,16,0,0,16,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.10714,0.12372,0.0,0.14286,0.14286,0.0,0.4286,15,0,0,15,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.39732,"p":[[0,62,0.0,0.36161,0.29011,0.14286,0.28571,0.60714,0.0,1.0,7,2,0,7,0,2,0,0,11,0,0,3,0,0,1,0,0,6,0,0,0,0,2],[4,62,0.0645,0.35268,0.37284,0.0,0.28571,0.53571,0.0,1.0,12,5,0,12,0,1,0,0,8,0,0,3,0,0,0,0,0,0,0,0,3,0,5],[8,62,0.129,0.39732,0.30037,0.14286,0.42857,0.46431,0.0,1.0,5,4,0,5,0,5,0,0,4,0,0,10,0,0,2,0,0,2,0,0,0,0,4],[12,62,0.1935,0.26339,0.35013,0.0,0.14286,0.42857,0.0,1.0,15,5,0,15,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,5],[16,62,0.2581,0.05795,0.15505,0.0,0.0,0.035,0.0,0.85714,24,0,0,24,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,62,0.3226,0.0982,0.23263,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[24,62,0.3871,0.04464,0.13092,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,62,0.4516,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,62,0.5161,0.0625,0.17835,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[36,62,0.5806,0.04018,0.12993,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[40,62,0.6452,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,62,0.7742,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,62,0.8387,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,62,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,62,0.9677,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[62,62,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"aa09683b241055e4","q":"Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer  $ n $  there exists a triangle of area  $ n $  having its vertices of the chosen colour.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,41,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,69,0.0,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,69,0.058,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,69,0.1159,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.01787,0.0692,0.0,0.0,0.0,0.0,0.286,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,69,0.2319,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[48,69,0.6957,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,69,0.8116,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[69,69,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"10c760898f26496b","q":"9. (IND 4) (a) Show that the set $\\mathbb{Q}^{+}$of all positive rational numbers can be partitioned into three disjoint subsets $A, B, C$ satisfying the following conditions:  $$ B A=B, \\quad B^{2}=C, \\quad B C=A, $$  where $H K$ stands for the set $\\{h k \\mid h \\in H, k \\in K\\}$ for any two subsets $H, K$ of $\\mathbb{Q}^{+}$and $H^{2}$ stands for $H H$. (b) Show that all positive rational cubes are in $A$ for such a partition of $\\mathbb{Q}^{+}$. (c) Find such a partition $\\mathbb{Q}^{+}=A \\cup B \\cup C$ with the property that for no positive integer $n \\leq 34$ are both $n$ and $n+1$ in $A$; that is,  $$ \\min \\{n \\in \\mathbb{N} \\mid n \\in A, n+1 \\in A\\}>34 $$","t":[{"b":2,"e":0.85714,"k":"falling","v":0.57127,"x":0.84375,"p":[[0,21,0.0,0.84375,0.15714,0.85714,0.85714,0.85714,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,3],[4,21,0.1905,0.81696,0.1197,0.85711,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,24,0,2],[8,21,0.381,0.83481,0.13418,0.85714,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,20,0,6],[12,21,0.5714,0.59372,0.17896,0.42857,0.57141,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,13,0,0,9,0,0,4,0,0,4,0,2],[16,21,0.7619,0.62501,0.18118,0.42857,0.57143,0.85704,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,10,0,0,6,0,0,6,0,0,9,0,0],[20,21,0.9524,0.66075,0.18118,0.53575,0.57143,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,9,0,0,3,0,0,11,0,1],[21,21,1.0,0.57127,0.16354,0.42857,0.5712,0.60607,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,14,0,0,10,0,0,3,0,0,4,0,1]]},{"b":5,"e":1.0,"k":"flat","v":0.80357,"x":0.875,"p":[[0,29,0.0,0.83928,0.16269,0.85714,0.85714,0.85714,0.0,1.0,1,4,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,4],[4,29,0.1379,0.85713,0.11297,0.85714,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,23,0,6],[8,29,0.2759,0.80357,0.18814,0.85714,0.85714,0.85714,0.0,1.0,1,3,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,22,0,3],[12,29,0.4138,0.86607,0.04971,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,3],[16,29,0.5517,0.84821,0.06121,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,26,0,2],[20,29,0.6897,0.85713,0.07143,0.85713,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,4],[24,29,0.8276,0.8616,0.06667,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,25,0,4],[28,29,0.9655,0.875,0.06916,0.85714,0.85714,0.85714,0.71429,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,24,0,6],[29,29,1.0,0.86159,0.05629,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,27,0,3]]}]},{"i":"67f6ea0eea0f257c","q":"6. A6 (VIE 1) ${ }^{\\text {IMO6 }}$ Let $S$ be a square with sides of length 100 and let $L$ be a path within $S$ that does not meet itself and that is composed of linear segments $A_{0} A_{1}, A_{1} A_{2}, \\ldots, A_{n-1} A_{n}$ with $A_{0} \\neq A_{n}$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $\\frac{1}{2}$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than 1 and the length of that part of $L$ that lies between $X$ and $Y$ is not smaller than 198.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.06688,"p":[[0,22,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.05803,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.06688,0.10085,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,16,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.04464,0.10374,0.0,0.0,0.0,0.0,0.4286,26,0,2,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,3,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"58ac19dce5b58275","q":"Determine all natural integers  $n$  for which there is no triplet  $(a, b, c)$  of natural numbers such that:  $$ n = \\frac{a \\cdot  \\,\\,lcm(b, c) + b  \\cdot  lcm \\,\\,(c, a) + c  \\cdot  lcm \\,\\, (a, b)}{lcm \\,\\,(a, b, c)} $$","t":[{"b":3,"e":0.42857,"k":"flat","v":0.40624,"x":0.66069,"p":[[0,27,0.0,0.41509,0.23528,0.2857,0.42857,0.46431,0.0,1.0,1,2,0,1,0,4,0,0,10,0,0,9,0,0,3,0,0,2,0,0,1,0,2],[4,27,0.1481,0.66069,0.30671,0.42857,0.71429,1.0,0.0,1.0,1,9,0,1,0,2,0,0,3,0,0,6,0,0,2,0,0,3,0,0,6,0,9],[8,27,0.2963,0.50445,0.28568,0.28571,0.49979,0.85714,0.0,1.0,2,1,0,2,0,4,0,0,5,0,0,5,0,0,6,0,0,1,0,0,8,0,1],[12,27,0.4444,0.40624,0.27224,0.2857,0.28571,0.57143,0.0,1.0,3,1,1,3,0,4,0,0,11,0,0,4,0,0,3,0,0,2,0,0,4,0,1],[16,27,0.5926,0.50001,0.24997,0.28571,0.42857,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,12,0,0,6,0,0,3,0,0,3,0,0,6,0,1],[20,27,0.7407,0.58036,0.24727,0.39286,0.50001,0.85714,0.1429,0.85714,0,0,0,0,0,1,0,0,7,0,0,8,0,0,1,0,0,3,0,0,12,0,0],[24,27,0.8889,0.55802,0.21234,0.42857,0.571,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,4,0,0,9,0,0,6,0,0,6,0,0,6,0,0],[27,27,1.0,0.48212,0.2335,0.28571,0.42857,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,15,0,0,6,0,0,2,0,0,3,0,0,5,0,1]]},{"b":4,"e":0.71429,"k":"flat","v":0.36157,"x":0.5,"p":[[0,64,0.0,0.40177,0.24597,0.2857,0.42857,0.4642,0.0,1.0,3,1,0,3,0,3,0,0,9,0,0,9,0,0,2,0,0,3,0,0,2,0,1],[4,64,0.0625,0.36157,0.26957,0.24999,0.28571,0.4286,0.0,1.0,5,2,1,5,0,3,0,0,11,0,0,6,0,0,3,0,0,0,0,0,2,0,2],[8,64,0.125,0.46427,0.16364,0.39293,0.42859,0.57111,0.14286,0.85714,0,0,0,0,0,1,0,0,7,0,0,14,0,0,4,0,0,5,0,0,1,0,0],[12,64,0.1875,0.46429,0.17494,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,5,0,0,16,0,0,3,0,0,4,0,0,2,0,0],[16,64,0.25,0.48215,0.17405,0.42857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,1,0,0,3,0,0,21,0,0,0,0,0,5,0,0,1,0,1],[20,64,0.3125,0.47768,0.18765,0.42857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,19,0,0,0,0,0,4,0,0,2,0,1],[24,64,0.375,0.43304,0.14054,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,8,0,0,16,0,0,3,0,0,4,0,0,0,0,0],[28,64,0.4375,0.45089,0.14773,0.42857,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,22,0,0,1,0,0,2,0,0,2,0,0],[32,64,0.5,0.41963,0.15541,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,1,0,0,12,0,0,12,0,0,2,0,0,5,0,0,0,0,0],[36,64,0.5625,0.5,0.19232,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,17,0,0,3,0,0,3,0,0,3,0,1],[40,64,0.625,0.42857,0.15971,0.39286,0.42857,0.4286,0.14286,0.857,0,0,0,0,0,3,0,0,5,0,0,18,0,0,2,0,0,3,0,0,1,0,0],[44,64,0.6875,0.45087,0.14771,0.42857,0.42857,0.4642,0.14286,0.857,0,0,0,0,0,1,0,0,6,0,0,17,0,0,4,0,0,3,0,0,1,0,0],[48,64,0.75,0.48214,0.14616,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,16,0,0,7,0,0,2,0,0,2,0,0],[52,64,0.8125,0.45535,0.17654,0.42857,0.42857,0.4286,0.14286,1.0,0,1,0,0,0,1,0,0,6,0,0,19,0,0,2,0,0,1,0,0,2,0,1],[56,64,0.875,0.41515,0.15711,0.28571,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,1,0,0,8,0,0,15,0,0,5,0,0,1,0,0,1,0,0],[60,64,0.9375,0.39732,0.18118,0.2857,0.42857,0.42857,0.0,0.85714,1,0,0,1,0,3,0,0,8,0,0,15,0,0,2,0,0,1,0,0,2,0,0],[64,64,1.0,0.46426,0.15969,0.42857,0.42857,0.571,0.14286,0.857,0,0,0,0,0,1,0,0,6,0,0,16,0,0,3,0,0,5,0,0,1,0,0]]}]},{"i":"76330b5fda93b81c","q":"4. (GDR 1) Let $n_{1}, n_{2}$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_{1}$ and $M_{2}$ consisting of $2 n_{1}$ and $2 n_{2}$ points, respectively, and such that no three points of the union $M_{1} \\cup M_{2}$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ( $g$ not being included in either) contains exactly half of the points of $M_{1}$ and exactly half of the points of $M_{2}$.","t":[{"b":0,"e":1.0,"k":"rising","v":0.80357,"x":0.99554,"p":[[0,23,0.0,0.80357,0.30462,0.67857,1.0,1.0,0.0,1.0,2,20,2,2,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,2,0,20],[4,23,0.1739,0.91518,0.16698,0.85714,1.0,1.0,0.28571,1.0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,4,0,23],[8,23,0.3478,0.93304,0.19556,1.0,1.0,1.0,0.1429,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,27],[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],[16,23,0.6957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[23,23,1.0,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25]]},{"b":7,"e":1.0,"k":"flat","v":0.85714,"x":1.0,"p":[[0,18,0.0,0.85714,0.27433,0.85714,1.0,1.0,0.0,1.0,1,22,1,1,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,22],[4,18,0.2222,0.93304,0.18552,1.0,1.0,1.0,0.0,1.0,1,25,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,25],[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],[12,18,0.6667,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,18,0.8889,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"fb315cdf9e45ead4","q":"$N$  teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order  $\\mathcal{A} = (a_1 \\ge a_2 \\ge \\cdots \\ge a_N )$  is known, as well as which team obtained which score. Find the number of sequences  $\\mathcal{A}$  such that the outcome of all matches is uniquely determined by this information.\n\n*Proposed by Dominic Yeo, United Kingdom.*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0067,"x":0.14733,"p":[[0,10,0.0,0.10714,0.18898,0.0,0.0,0.14286,0.0,1.0,17,1,1,17,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,10,0.4,0.14733,0.26841,0.0,0.0,0.14286,0.0,1.0,20,2,3,20,0,6,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[8,10,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.0982,"x":0.16071,"p":[[0,9,0.0,0.16071,0.27605,0.0,0.0,0.14286,0.0,1.0,17,2,1,17,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[4,9,0.4444,0.11607,0.20652,0.0,0.0,0.14286,0.0,1.0,18,1,3,18,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[8,9,0.8889,0.0982,0.16532,0.0,0.0,0.14286,0.0,0.57143,21,0,2,21,0,5,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[9,9,1.0,0.09821,0.14032,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,10,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"48caa5e5a9437b5b","q":"A polynomial $f(x)$ with real coefficients is called generating, if for each polynomial $\\varphi(x)$ with real coefficients there exist a positive integer $k$ and polynomials $g_{1}(x), \\ldots, g_{k}(x)$ with real coefficients such that\n\n$$\n\\varphi(x)=f\\left(g_{1}(x)\\right)+\\cdots+f\\left(g_{k}(x)\\right) .\n$$\n\nFind all generating polynomials.","t":[{"b":2,"e":0.2857,"k":"flat","v":0.22759,"x":0.35701,"p":[[0,36,0.0,0.35701,0.16373,0.2857,0.28571,0.4286,0.0,0.71429,1,0,0,1,0,3,0,0,16,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[4,36,0.1111,0.33928,0.17405,0.2857,0.28571,0.4286,0.0,0.71429,2,0,1,2,0,3,0,0,17,0,0,3,0,0,5,0,0,2,0,0,0,0,0],[8,36,0.2222,0.30355,0.18812,0.14286,0.2857,0.32143,0.0,0.85714,3,0,1,3,0,6,0,0,15,0,0,2,0,0,5,0,0,0,0,0,1,0,0],[12,36,0.3333,0.35267,0.11283,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,19,0,0,9,0,0,2,0,0,1,0,0,0,0,0],[16,36,0.4444,0.25893,0.09062,0.2857,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,1,0,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.27678,0.03458,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.29017,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.29016,0.06661,0.2857,0.28571,0.28571,0.14286,0.571,0,0,0,0,0,2,0,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,36,0.8889,0.28572,5e-05,0.2857,0.28571,0.28571,0.2857,0.286,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.22759,0.1002,0.14286,0.2857,0.28571,0.0,0.28571,4,0,1,4,0,5,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.32138,"x":0.41514,"p":[[0,7,0.0,0.41514,0.21827,0.2857,0.42857,0.57143,0.0,1.0,3,1,3,3,0,2,0,0,8,0,0,5,0,0,12,0,0,1,0,0,0,0,1],[4,7,0.5714,0.32138,0.19242,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,4,0,0,16,0,0,4,0,0,1,0,0,4,0,0,0,0,0],[7,7,1.0,0.37495,0.1665,0.2857,0.42857,0.4642,0.0,0.71429,2,0,0,2,0,2,0,0,11,0,0,9,0,0,7,0,0,1,0,0,0,0,0]]}]},{"i":"df4382a5cd43c345","q":"Each student chooses  $1$  math problem and  $1$  physics problem among  $20$  math problems and  $11$  physics problems. No same pair of problem is selected by two students. And at least one of the problems selected by any student is selected by at most one other student. At most how many students are there?","t":[{"b":3,"e":1.0,"k":"flat","v":0.74104,"x":0.93303,"p":[[0,31,0.0,0.80802,0.18768,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,5,0,0,5,0,13],[4,31,0.129,0.76338,0.21903,0.67857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,1,0,0,1,0,0,5,0,0,9,0,0,5,0,10],[8,31,0.2581,0.79909,0.18511,0.67857,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,6,0,0,7,0,11],[12,31,0.3871,0.76339,0.24119,0.57143,0.71429,1.0,0.1429,1.0,0,13,0,0,0,1,0,0,1,0,0,3,0,0,4,0,0,8,0,0,2,0,13],[16,31,0.5161,0.79015,0.23956,0.57143,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,3,0,0,4,0,15],[20,31,0.6452,0.78568,0.20206,0.57143,0.78564,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,7,0,0,4,0,12],[24,31,0.7742,0.74104,0.25113,0.57143,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,2,0,0,4,0,0,9,0,0,3,0,11],[28,31,0.9032,0.78123,0.22584,0.71429,0.78564,1.0,0.0,1.0,1,11,0,1,0,0,0,0,1,0,0,0,0,0,4,0,0,10,0,0,5,0,11],[31,31,1.0,0.93303,0.11285,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22]]},{"b":4,"e":0.85714,"k":"flat","v":0.683,"x":0.84372,"p":[[0,9,0.0,0.76784,0.25443,0.67857,0.85707,1.0,0.0,1.0,1,12,0,1,0,0,0,0,1,0,0,4,0,0,2,0,0,6,0,0,6,0,12],[4,9,0.4444,0.7991,0.2693,0.71429,0.85714,1.0,0.0,1.0,1,15,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,0,6,0,15],[8,9,0.8889,0.683,0.25188,0.57132,0.71429,0.85714,0.0,1.0,1,6,0,1,0,0,0,0,4,0,0,1,0,0,6,0,0,8,0,0,6,0,6],[9,9,1.0,0.84372,0.1689,0.71429,0.85714,1.0,0.4286,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,7,0,14]]}]},{"i":"e63c14ac92a419e5","q":"3. (NET 3) ${ }^{\\mathrm{IMO} 3}$ The weight $w(p)$ of a polynomial $p, p(x)=\\sum_{i=0}^{n} a_{i} x^{i}$, with integer coefficients $a_{i}$ is defined as the number of its odd coefficients. For $i=0,1,2, \\ldots$, let $q_{i}(x)=(1+x)^{i}$. Prove that for any finite sequence $0 \\leq i_{1}<i_{2}<\\cdots<i_{n}$, the inequality  $$ w\\left(q_{i_{1}}+\\cdots+q_{i_{n}}\\right) \\geq w\\left(q_{i_{1}}\\right) $$  holds.","t":[{"b":0,"e":0.0,"k":"flat","v":0.03116,"x":0.04911,"p":[[0,6,0.0,0.03116,0.05889,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.1429,"k":"flat","v":0.03125,"x":0.07588,"p":[[0,4,0.0,0.07588,0.15557,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,4,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7d103da3846d1dea","q":"Consider a triangle  $ABC$  with  $AB < AC$  and let  $H$  and  $O$  be its orthocenter and circumcenter, respectively. A line starting from  $B$  cuts the lines  $AO$  and  $AH$  at  $M$  and  $M'$  so that  $M'$  is the midpoint of  $BM$ . Another line starting from  $C$  cuts the lines  $AH$  and  $AO$  at  $N$  and  $N'$  so that  $N'$  is the midpoint of  $CN$ . Prove that  $M, M', N, N'$  are on the same circle.","t":[{"b":1,"e":0.1429,"k":"flat","v":0.14286,"x":0.22321,"p":[[0,10,0.0,0.14286,0.14286,0.0,0.14286,0.28571,0.0,0.42857,15,0,5,15,0,3,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.14286,0.12877,0.0,0.14286,0.28571,0.0,0.28571,13,0,4,13,0,6,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.20982,0.11836,0.14286,0.2857,0.28571,0.0,0.28571,7,0,1,7,0,3,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.22321,0.10677,0.14286,0.2857,0.28571,0.0,0.28571,5,0,1,5,0,4,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.2857,"k":"flat","v":0.16072,"x":0.29018,"p":[[0,21,0.0,0.16072,0.13243,0.0,0.21435,0.28571,0.0,0.28571,12,0,2,12,0,4,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.20089,0.14664,0.0,0.2857,0.28571,0.0,0.57143,9,0,2,9,0,4,0,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,21,0.381,0.16518,0.13882,0.0,0.2143,0.28571,0.0,0.42857,12,0,3,12,0,4,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.2857,0.20823,0.2857,0.28571,0.28571,0.0,0.71429,7,0,0,7,0,0,0,0,20,0,0,0,0,0,1,0,0,4,0,0,0,0,0],[16,21,0.7619,0.29018,0.10999,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,2,0,0,27,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,21,0.9524,0.28571,1e-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],[21,21,1.0,0.29017,0.06667,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"2bfbe284bf7cb8b9","q":"8. G2 (JAP) A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point in its interior, and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.","t":[{"b":4,"e":0.2857,"k":"flat","v":0.12472,"x":0.4375,"p":[[0,38,0.0,0.12472,0.15865,0.0,0.07,0.1786,0.0,0.57143,16,0,0,16,0,8,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,38,0.1053,0.16054,0.23893,0.0,0.14,0.17868,0.0,1.0,15,1,1,15,0,9,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[8,38,0.2105,0.2767,0.34061,0.0,0.14293,0.28571,0.0,1.0,12,5,1,12,0,5,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[12,38,0.3158,0.22991,0.31172,0.0,0.14288,0.2857,0.0,1.0,14,3,2,14,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,1,3],[16,38,0.4211,0.4375,0.42248,0.0,0.28571,1.0,0.0,1.0,11,10,0,11,0,1,0,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,10],[20,38,0.5263,0.23214,0.3567,0.0,0.0,0.28571,0.0,1.0,18,5,0,18,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,5],[24,38,0.6316,0.13169,0.21583,0.0,0.0,0.16075,0.0,0.857,19,0,0,19,0,5,0,1,4,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[28,38,0.7368,0.35692,0.20531,0.24999,0.28571,0.57143,0.0,0.57143,4,0,0,4,0,4,0,0,9,0,0,2,0,0,13,0,0,0,0,0,0,0,0],[32,38,0.8421,0.27229,0.22963,0.0,0.2857,0.571,0.0,0.57143,11,0,6,11,0,1,0,0,9,0,0,2,0,0,9,0,0,0,0,0,0,0,0],[36,38,0.9474,0.35713,0.19231,0.2857,0.28571,0.57143,0.0,0.57143,4,0,1,4,0,1,0,0,14,0,0,1,0,0,12,0,0,0,0,0,0,0,0],[38,38,1.0,0.21416,0.21721,0.0,0.21428,0.42857,0.0,0.57143,14,0,0,14,0,2,0,0,7,0,0,4,0,0,5,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.12027,"p":[[0,15,0.0,0.10232,0.13464,0.0,0.0,0.14287,0.0,0.4286,18,0,0,18,0,7,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.08009,0.10046,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.12027,0.17533,0.0,0.0,0.1429,0.0,0.57143,18,0,2,18,0,7,0,0,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[12,15,0.8,0.08482,0.18851,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[15,15,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c006bbccfababc0c","q":"$ABCDEF$  is a convex hexagon, such that  $|\\angle FAB| = |\\angle BCD| =|\\angle DEF|$  and   $|AB| =|BC|,$   $|CD| = |DE|$ ,  $|EF| = |FA|$ . Prove that the lines  $AD$ ,  $BE$  and  $CF$  are concurrent.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,28,0.0,0.04911,0.11633,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,29,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.02232,0.08073,0.0,0.0,0.0,0.0,0.4286,29,0,1,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.08036,0.16342,0.0,0.0,0.03571,0.0,0.71429,24,0,1,24,0,2,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,29,0.4138,0.08036,0.17835,0.0,0.0,0.0,0.0,0.71429,25,0,1,25,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,29,0.5517,0.05357,0.14617,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"2f84957ed3978ad6","q":"13. (ROM) Let $A_{0}, A_{1}, \\ldots, A_{n}$ be points in a plane such that (i) $A_{0} A_{1} \\leq \\frac{1}{2} A_{1} A_{2} \\leq \\cdots \\leq \\frac{1}{2^{n-1}} A_{n-1} A_{n}$ and (ii) $0<\\measuredangle A_{0} A_{1} A_{2}<\\measuredangle A_{1} A_{2} A_{3}<\\cdots<\\measuredangle A_{n-2} A_{n-1} A_{n}<180^{\\circ}$, where all these angles have the same orientation. Prove that the segments $A_{k} A_{k+1}, A_{m} A_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \\leq k \\leq$ $m-2<n-2$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"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],[4,18,0.2222,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,10,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,10,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"3c52a29996c2d1ca","q":"At the entrance to a cave is a rotating round table. On top of the table are  $n$  identical barrels, evenly spaced along its circumference. Inside each barrel is a herring either with its head up or its head down. In a move, Ali Baba chooses from  $1$  to  $n$  of the barrels and turns them upside down. Then the table spins around. When it stops, it is impossible to tell which barrels have been turned over. The cave will open if the heads of the herrings in all  $n$  barrels are up or are all down. Determine all values of  $n$  for which Ali Baba can open the cave in a fi\fnite number of moves.\n\n*(11 points)*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,8,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d3702f97064fac9b","q":"Circles  $C_1$  and  $C_2$  intersect at different points  $A$  and  $B$ . The straight lines tangents to  $C_1$  that pass through  $A$  and  $B$  intersect at  $T$ . Let  $M$  be a point on  $C_1$  that is out of  $C_2$ . The  $MT$  line intersects  $C_1$  at  $C$  again, the  $MA$  line intersects again to  $C_2$  in  $K$  and the line  $AC$  intersects again to the circumference  $C_2$  in  $L$ . Prove that the  $MC$  line passes through the midpoint of the  $KL$  segment.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"1564258ddf854365","q":"32. (POL 2) In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted.","t":[{"b":2,"e":0.571,"k":"rising","v":0.32589,"x":0.60272,"p":[[0,9,0.0,0.32589,0.38172,0.0,0.21428,0.71429,0.0,1.0,15,5,3,15,0,1,0,0,6,0,0,0,0,0,0,0,0,5,0,0,0,0,5],[4,9,0.4444,0.49107,0.38289,0.24999,0.35714,1.0,0.0,1.0,7,9,1,7,0,1,0,0,8,0,0,2,0,0,1,0,0,4,0,0,0,0,9],[8,9,0.8889,0.60268,0.16262,0.42857,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,18,0,0,0,0,1],[9,9,1.0,0.60272,0.18113,0.42857,0.71429,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,11,0,0,1,0,0,16,0,0,0,0,2]]},{"b":4,"e":0.0,"k":"volatile","v":0.03125,"x":0.57139,"p":[[0,10,0.0,0.21427,0.32536,0.0,0.0,0.32143,0.0,1.0,20,3,2,20,0,0,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,3],[4,10,0.4,0.57139,0.26726,0.39286,0.64286,0.71429,0.0,1.0,2,4,0,2,0,1,0,0,5,0,0,3,0,0,5,0,0,12,0,0,0,0,4],[8,10,0.8,0.21427,0.32731,0.0,0.0,0.4286,0.0,1.0,21,2,0,21,0,0,0,0,2,0,0,2,0,0,1,0,0,4,0,0,0,0,2],[10,10,1.0,0.03125,0.09933,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e48e3d07fa5b50b4","q":"14. (IND 2) A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.","t":[{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.01339,"p":[[0,8,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,74,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,74,0.0541,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,1,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,74,0.1081,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,74,0.1622,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,74,0.2162,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,74,0.2703,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,1,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,74,0.3243,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,74,0.3784,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,74,0.4324,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,74,0.4865,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,74,0.5405,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,74,0.5946,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,74,0.6486,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,74,0.7027,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,74,0.7568,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,74,0.8108,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,74,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,74,0.9189,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,74,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[74,74,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4ffd07676f34e6fa","q":"A real number  $a$  is given. The sequence  $n_{1}< n_{2}< n_{3}< ...$  consists of all the positive integral  $n$  such that  $\\{na\\}< \\frac{1}{10}$ . Prove that there are at most three different numbers among the numbers  $n_{2}-n_{1}$ ,  $n_{3}-n_{2}$ ,  $n_{4}-n_{3}$ ,  $\\ldots$ .\n\n*A corollary of a theorem from ergodic theory*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.13375,"x":0.1517,"p":[[0,16,0.0,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,3,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.1517,0.04973,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.14286,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.14286,1e-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],[16,16,1.0,0.1426,0.00084,0.14286,0.14286,0.14286,0.14,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.10714,"x":0.14706,"p":[[0,19,0.0,0.14706,0.06669,0.14286,0.14286,0.14286,0.0,0.42857,2,0,2,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.12938,0.04162,0.14286,0.14286,0.14286,0.0,0.143,3,0,3,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,6,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.1425,0.00095,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.143,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"228fe5f6523ad2ae","q":"8. (FRA 3) Let $u_{1}, u_{2}, \\ldots, u_{m}$ be $m$ vectors in the plane, each of length less than or equal to 1 , which add up to zero. Show that one can rearrange $u_{1}, u_{2}, \\ldots, u_{m}$ as a sequence $v_{1}, v_{2}, \\ldots, v_{m}$ such that each partial sum $v_{1}, v_{1}+v_{2}, v_{1}+v_{2}+v_{3}, \\ldots, v_{1}+v_{2}+\\cdots+v_{m}$ has length less than or equal to $\\sqrt{5}$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.04902,"x":0.09813,"p":[[0,18,0.0,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.0892,0.0691,0.0,0.14286,0.14286,0.0,0.1429,12,0,1,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.09813,0.07518,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,1,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00447,"x":0.04911,"p":[[0,6,0.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"69392c8764e78a9c","q":"A rectangular building consists of  $30$  square rooms situated like the cells of a  $2 \\times 15$  board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,6,0.0,0.06696,0.15966,0.0,0.0,0.0,0.0,0.57143,27,0,1,27,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,43,0.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"78923cfa6344d5b8","q":"A point $P$ lies in the interior of the triangle $A B C$. The lines $A P, B P$, and $C P$ intersect $B C, C A$, and $A B$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $A B D E, B C E F, C A F D, A E P F, B F P D$, and $C D P E$ are concyclic, then all six are concyclic.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.04447,"x":0.12036,"p":[[0,12,0.0,0.09143,0.11873,0.0,0.0,0.14286,0.0,0.42857,17,0,7,17,0,11,0,1,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.10259,0.14824,0.0,0.07,0.14286,0.0,0.71429,16,0,5,16,0,13,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,12,0.6667,0.12036,0.10776,0.0,0.14286,0.1429,0.0,0.4286,11,0,3,11,0,16,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.04447,0.06596,0.0,0.0,0.14071,0.0,0.1429,22,0,2,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.08037,"x":0.24995,"p":[[0,13,0.0,0.10248,0.14813,0.0,0.0,0.14286,0.0,0.571,17,0,3,17,0,11,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,13,0.3077,0.10258,0.12484,0.0,0.14286,0.14286,0.0,0.571,14,0,6,14,0,16,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,13,0.6154,0.08037,0.08703,0.0,0.07143,0.14286,0.0,0.28571,16,0,5,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.19198,0.21902,0.0,0.1429,0.2857,0.0,0.71429,13,0,2,13,0,7,0,0,7,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[13,13,1.0,0.24995,0.25759,0.0,0.1429,0.42857,0.0,1.0,9,1,0,9,0,10,0,0,4,0,0,4,0,0,1,0,0,3,0,0,0,0,1]]}]},{"i":"6f1b478e17682e92","q":"Let  $ABC$  be an isosceles triangle with  $AB = AC$ ,  $P$  be the midpoint of the minor arc  $AB$  of its circumcircle, and  $Q$  be the midpoint of  $AC$ . A circumcircle of triangle  $APQ$  centered at  $O$  meets  $AB$  for the second\ntime at point  $K$ . Prove that lines  $PO$  and  $KQ$  meet on the bisector of angle  $ABC$ .","t":[{"b":4,"e":0.28571,"k":"flat","v":0.17857,"x":0.39731,"p":[[0,63,0.0,0.2321,0.23346,0.0,0.21428,0.42857,0.0,0.71429,13,0,0,13,0,3,0,0,6,0,0,5,0,0,3,0,0,2,0,0,0,0,0],[4,63,0.0635,0.17857,0.15567,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,7,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,63,0.127,0.2008,0.2471,0.0,0.14286,0.2857,0.0,0.85714,11,0,0,11,0,12,0,0,3,0,0,3,0,0,0,0,0,0,0,0,3,0,0],[12,63,0.1905,0.21876,0.22011,0.0,0.14288,0.28579,0.0,0.85714,11,0,1,11,0,6,0,0,8,0,0,4,0,0,1,0,0,1,0,0,1,0,0],[16,63,0.254,0.29018,0.22442,0.14286,0.28571,0.42857,0.0,1.0,5,1,1,5,0,9,0,0,6,0,0,8,0,0,2,0,0,1,0,0,0,0,1],[20,63,0.3175,0.24097,0.20343,0.0,0.2857,0.42857,0.0,0.71429,9,0,1,9,0,6,0,0,8,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[24,63,0.381,0.28561,0.24487,0.14286,0.2143,0.42857,0.0,1.0,5,1,3,5,0,11,0,0,6,0,0,5,0,0,2,0,0,1,0,0,1,0,1],[28,63,0.4444,0.23652,0.20709,0.0,0.14288,0.42857,0.0,0.71429,9,0,2,9,0,8,0,0,4,0,0,9,0,0,0,0,0,2,0,0,0,0,0],[32,63,0.5079,0.28124,0.20354,0.14286,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,5,0,0,8,0,0,7,0,0,4,0,0,1,0,0,0,0,0],[36,63,0.5714,0.36159,0.22298,0.14286,0.35714,0.42858,0.0,0.85714,3,0,0,3,0,6,0,0,7,0,0,9,0,0,2,0,0,4,0,0,1,0,0],[40,63,0.6349,0.39731,0.23071,0.24999,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,6,0,0,7,0,0,6,0,0,6,0,0,3,0,0,2,0,0],[44,63,0.6984,0.29464,0.20182,0.14286,0.28571,0.42857,0.0,0.857,5,0,0,5,0,6,0,0,9,0,0,9,0,0,1,0,0,1,0,0,1,0,0],[48,63,0.7619,0.32589,0.2321,0.14286,0.28571,0.42857,0.0,1.0,4,1,0,4,0,7,0,0,7,0,0,10,0,0,1,0,0,1,0,0,1,0,1],[52,63,0.8254,0.33927,0.18812,0.14286,0.28571,0.42858,0.0,0.71429,1,0,1,1,0,9,0,0,8,0,0,8,0,0,3,0,0,3,0,0,0,0,0],[56,63,0.8889,0.25893,0.19045,0.14286,0.2857,0.42857,0.0,0.71429,6,0,0,6,0,9,0,0,6,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[60,63,0.9524,0.29911,0.21535,0.14286,0.28571,0.42857,0.0,0.85714,6,0,0,6,0,5,0,0,7,0,0,12,0,0,0,0,0,0,0,0,2,0,0],[63,63,1.0,0.23659,0.17714,0.14286,0.2857,0.28571,0.0,0.71429,6,0,2,6,0,7,0,0,16,0,0,0,0,0,1,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.14286,"x":0.34821,"p":[[0,56,0.0,0.14286,0.14725,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,4,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.30357,0.17405,0.2857,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,3,0,0,15,0,0,7,0,0,1,0,0,2,0,0,0,0,0],[8,56,0.1429,0.30803,0.08073,0.2857,0.28571,0.32143,0.14286,0.42857,0,0,0,0,0,3,0,0,21,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[12,56,0.2143,0.28125,0.10999,0.24999,0.28571,0.32143,0.0,0.42857,1,0,0,1,0,7,0,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.30134,0.10675,0.2857,0.28571,0.32143,0.0,0.57143,1,0,0,1,0,3,0,1,19,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[20,56,0.3571,0.34821,0.10062,0.28571,0.35714,0.42857,0.14286,0.5714,0,0,0,0,0,3,0,0,13,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[24,56,0.4286,0.30803,0.10779,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,7,0,0,13,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.27669,0.12855,0.14286,0.28571,0.32143,0.0,0.57143,2,0,0,2,0,7,0,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[32,56,0.5714,0.29464,0.07936,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,4,0,0,22,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[36,56,0.6429,0.32142,0.07986,0.2857,0.28571,0.42857,0.14286,0.42857,0,0,0,0,0,2,0,0,20,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.3125,0.0974,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,2,0,0,19,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.32589,0.08918,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,17,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.32143,0.10102,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,5,0,0,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.3125,0.09062,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,4,0,0,18,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.27678,0.10062,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,5,0,0,22,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"684c0b66c3cc6433","q":"Consider the sequence  $(a_n)_{n\\in \\mathbb{N}}$  with  $a_0=a_1=a_2=a_3=1$  and  $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$ . Prove that all the terms of this sequence are integer numbers.","t":[{"b":4,"e":0.0,"k":"flat","v":0.08036,"x":0.17411,"p":[[0,46,0.0,0.13839,0.16554,0.0,0.14286,0.2857,0.0,0.71429,15,0,0,15,0,7,0,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,46,0.087,0.16516,0.18932,0.0,0.14286,0.17857,0.0,0.85714,11,0,0,11,0,13,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[8,46,0.1739,0.14286,0.15152,0.0,0.14286,0.1786,0.0,0.57143,12,0,0,12,0,12,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,46,0.2609,0.08927,0.13712,0.0,0.0,0.14286,0.0,0.571,19,0,0,19,0,9,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,46,0.3478,0.11152,0.14165,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,14,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,46,0.4348,0.10705,0.14723,0.0,0.07,0.14286,0.0,0.71429,16,0,0,16,0,11,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,46,0.5217,0.13839,0.19719,0.0,0.14286,0.14286,0.0,0.85714,15,0,0,15,0,10,0,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[28,46,0.6087,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.09375,0.12682,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,11,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,46,0.7826,0.125,0.12752,0.0,0.14286,0.14286,0.0,0.57143,12,0,0,12,0,14,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,46,0.8696,0.1026,0.09603,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.17411,0.09268,0.14286,0.14286,0.28571,0.0,0.28571,4,0,0,4,0,17,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04455,"x":0.16964,"p":[[0,17,0.0,0.13839,0.12619,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,15,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.08929,0.1171,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.10491,0.10099,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,1,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.16964,0.18013,0.0,0.14286,0.2857,0.0,0.71429,12,0,1,12,0,10,0,0,4,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[16,17,0.9412,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.06688,0.07966,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce0bbe2dab5a44b4","q":"A PUMaC grader is grading the submissions of forty students  $s_1, s_2, ..., s_{40}$  for the individual finals round, which has three problems. After grading a problem of student  $s_i$ , the grader either: $\\bullet$   grades another problem of the same student, or $\\bullet$  grades the same problem of the student  $s_{i-1}$  or  $s_{i+1}$  (if  $i > 1$  and  $i < 40$ , respectively).\nHe grades each problem exactly once, starting with the first problem of  $s_1$  and ending with the third problem of  $s_{40}$ . Let  $N$  be the number of different orders the grader may grade the students\u2019 problems in this way. Find the remainder when  $N$  is divided by  $100$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.14732,"x":0.17857,"p":[[0,26,0.0,0.15616,0.07459,0.14286,0.14286,0.14286,0.0,0.28571,3,0,1,3,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.16964,0.10972,0.14286,0.14286,0.14286,0.0,0.71429,1,0,1,1,0,27,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,26,0.3077,0.16946,0.16539,0.14286,0.14286,0.14286,0.0,1.0,4,1,3,4,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,26,0.4615,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.42857,1,0,1,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.15179,0.07087,0.14286,0.14286,0.14286,0.0,0.42857,2,0,2,2,0,27,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.17857,0.06186,0.14286,0.14286,0.1786,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.16062,0.04728,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.13821,"x":0.20527,"p":[[0,26,0.0,0.20527,0.17477,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,24,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[4,26,0.1538,0.16964,0.06622,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,27,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.14732,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,2,0,2,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.16072,0.06916,0.14286,0.14286,0.14286,0.0,0.42857,1,0,1,1,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,2,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3616ece36ef55dd5","q":"18. (GBR) The altitudes through the vertices $A, B, C$ of an acute-angled triangle $A B C$ meet the opposite sides at $D, E, F$, respectively. The line through $D$ parallel to $E F$ meets the lines $A C$ and $A B$ at $Q$ and $R$, respectively. The line $E F$ meets $B C$ at $P$. Prove that the circumcircle of the triangle $P Q R$ passes through the midpoint of $B C$.","t":[{"b":5,"e":0.0,"k":"rising","v":0.15615,"x":0.34819,"p":[[0,13,0.0,0.15615,0.23515,0.0,0.14143,0.14287,0.0,1.0,15,1,3,15,0,11,0,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,1],[4,13,0.3077,0.25,0.30093,0.0,0.14286,0.42857,0.0,1.0,12,1,5,12,0,9,0,0,1,0,0,5,0,0,0,0,0,1,0,0,3,0,1],[8,13,0.6154,0.34819,0.33298,0.14286,0.2143,0.571,0.0,1.0,6,4,3,6,0,10,0,0,5,0,0,2,0,0,3,0,0,0,0,0,2,0,4]]},{"b":6,"e":1.0,"k":"rising","v":0.15625,"x":0.77229,"p":[[0,20,0.0,0.16518,0.24513,0.0,0.14286,0.17857,0.0,1.0,14,2,2,14,0,10,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,20,0.2,0.16506,0.25278,0.0,0.07,0.14286,0.0,1.0,16,1,5,16,0,9,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[8,20,0.4,0.15625,0.20935,0.0,0.14286,0.14286,0.0,1.0,11,1,0,11,0,16,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,20,0.6,0.20963,0.26485,0.0,0.14286,0.17857,0.0,1.0,10,2,2,10,0,14,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,2],[16,20,0.8,0.77229,0.22548,0.71429,0.857,1.0,0.0,1.0,1,9,0,1,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,9,0,9],[20,20,1.0,0.6964,0.27606,0.42859,0.71429,1.0,0.14286,1.0,0,9,0,0,0,2,0,0,3,0,0,4,0,0,3,0,0,5,0,0,6,0,9]]}]},{"i":"8b20c4275a7f1250","q":"A set  $X$  consisting of  $n$  positive integers is called  $\\textit{good}$  if the following condition holds:\nFor any two different subsets of  $X$ , say  $A$  and  $B$ , the number  $s(A) - s(B)$  is not divisible by  $2^n$ .\n(Here, for a set  $A$ ,  $s(A)$  denotes the sum of the elements of  $A$ )\nGiven  $n$ , find the number of good sets of size  $n$ , all of whose elements is strictly less than  $2^n$ .","t":[{"b":3,"e":0.85714,"k":"rising","v":0.66061,"x":0.94196,"p":[[0,25,0.0,0.71429,0.25505,0.57143,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,0,0,0,14,0,0,1,0,0,3,0,11],[4,25,0.16,0.66964,0.28669,0.57143,0.57143,1.0,0.0,1.0,1,11,1,1,0,2,0,0,1,0,0,2,0,0,13,0,0,1,0,0,1,0,11],[8,25,0.32,0.66964,0.28446,0.57143,0.57143,1.0,0.0,1.0,1,10,0,1,0,3,0,0,0,0,0,0,0,0,15,0,0,1,0,0,2,0,10],[12,25,0.48,0.66061,0.27853,0.57143,0.64286,0.89286,0.14,1.0,0,8,0,0,0,5,0,0,0,0,0,0,0,0,11,0,0,5,0,0,3,0,8],[16,25,0.64,0.92411,0.14279,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,5,0,23],[20,25,0.8,0.91964,0.18536,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,24],[24,25,0.96,0.94196,0.07016,0.85714,1.0,1.0,0.85714,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,0,19],[25,25,1.0,0.91071,0.1171,0.85714,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,12,0,17]]},{"b":4,"e":1.0,"k":"flat","v":0.69196,"x":0.83929,"p":[[0,11,0.0,0.78125,0.2696,0.57143,0.92857,1.0,0.0,1.0,1,16,1,1,0,1,0,0,0,0,0,2,0,0,6,0,0,4,0,0,2,0,16],[4,11,0.3636,0.69196,0.25781,0.57143,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,1,0,0,15,0,0,1,0,0,1,0,11],[8,11,0.7273,0.83929,0.17035,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,7,0,14],[11,11,1.0,0.80357,0.17405,0.57143,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,5,0,0,7,0,11]]}]},{"i":"590bc5f21f02dc62","q":"Determine with proof whether there is a subset $X \\subseteq \\mathbb{Z}$ with the following property: for any $n \\in \\mathbb{Z}$, there is exactly one solution to $a+2 b=n$, with $a, b \\in X$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.03571,"p":[[0,29,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,2,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.00884,"x":0.0625,"p":[[0,17,0.0,0.0625,0.14698,0.0,0.0,0.03571,0.0,0.71429,24,0,2,24,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,17,0.2353,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7f1c0b07766d48e5","q":"Each positive integer number is coloured red or blue. A function $f$ from the set of positive integer numbers into itself has the following two properties:\n(a) if $x \\leq y$, then $f(x) \\leq f(y)$; and\n(b) if $x, y$ and $z$ are all (not necessarily distinct) positive integer numbers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$.\n\nProve that there exists a positive number $a$ such that $f(x) \\leq a x$ for all positive integer numbers $x$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00447,"x":0.10268,"p":[[0,37,0.0,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.07589,0.13825,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.06697,0.12869,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.09374,0.15403,0.0,0.0,0.14286,0.0,0.571,21,0,0,21,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,37,0.6486,0.10268,0.15146,0.0,0.0,0.21429,0.0,0.57143,20,0,0,20,0,3,0,2,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,37,0.7568,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.03125,0.08553,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.02232,"x":0.0759,"p":[[0,19,0.0,0.0759,0.12871,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.04018,0.08918,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.04008,0.11411,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,19,0.8421,0.02455,0.06341,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6121884cd67f3fb8","q":"22. (ROM 4) Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.02679,"p":[[0,14,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0267,0.05558,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.04464,"p":[[0,22,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"af5b6ac0f088a42d","q":"A positive integer \\( r \\) is given, find the largest real number \\( C \\) such that there exists a geometric sequence  $\\{ a_n \\}_{n\\ge 1}$   with common ratio \\( r \\) satisfying  $$ \\| a_n \\| \\ge C $$  for all positive integers \\( n \\). Here,  $\\|  x \\|$  denotes the distance from the real number \\( x \\) to the nearest integer.","t":[{"b":2,"e":0.71429,"k":"falling","v":0.48657,"x":0.75,"p":[[0,18,0.0,0.75,0.27433,0.57143,0.78571,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,0,0,0,2,0,0,6,0,0,5,0,0,2,0,14],[4,18,0.2222,0.59822,0.15335,0.42857,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,8,0,0,10,0,0,2,0,1],[8,18,0.4444,0.51339,0.10012,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,17,0,0,11,0,0,4,0,0,0,0,0],[12,18,0.6667,0.48657,0.15913,0.42857,0.4998,0.57143,0.0,0.85714,2,0,0,2,0,0,0,0,0,0,0,14,0,0,14,0,0,1,0,0,1,0,0],[16,18,0.8889,0.5535,0.12241,0.42857,0.57121,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,16,0,0,4,0,0,0,0,1],[18,18,1.0,0.53124,0.0817,0.42857,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,11,0,0,19,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.85714,"k":"flat","v":0.68746,"x":0.9241,"p":[[0,19,0.0,0.78571,0.30723,0.67857,1.0,1.0,0.0,1.0,2,17,2,2,0,1,0,0,0,0,0,4,0,0,1,0,0,2,0,0,5,0,17],[4,19,0.2105,0.79463,0.20185,0.57143,0.78571,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,6,0,0,2,0,14],[8,19,0.4211,0.76784,0.29828,0.57132,0.92857,1.0,0.0,1.0,1,16,1,1,0,2,0,0,0,0,0,4,0,0,3,0,0,2,0,0,4,0,16],[12,19,0.6316,0.9241,0.12364,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,22],[16,19,0.8421,0.79017,0.2575,0.57143,0.92857,1.0,0.0,1.0,1,16,1,1,0,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,3,0,16],[19,19,1.0,0.68746,0.20343,0.5713,0.57143,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,2,0,0,4,0,7]]}]},{"i":"9bfc51024b22476c","q":"**5.** In triangle  $ABC$ , let  $r_A$  be the line that passes through the midpoint of  $BC$  and is perpendicular to the internal bisector of  $\\angle{BAC}$ . Define  $r_B$  and  $r_C$  similarly. Let  $H$  and  $I$  be the orthocenter and incenter of  $ABC$ , respectively. Suppose that the three lines  $r_A$ ,  $r_B$ ,  $r_C$  define a triangle. Prove that the circumcenter of this triangle is the midpoint of  $HI$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.04465,"x":0.21865,"p":[[0,36,0.0,0.16509,0.19598,0.0,0.14286,0.17857,0.0,0.71429,13,0,1,13,0,11,0,0,2,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[4,36,0.1111,0.16062,0.19481,0.0,0.14286,0.14286,0.0,0.85714,12,0,0,12,0,13,0,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[8,36,0.2222,0.14732,0.13592,0.0,0.14286,0.2857,0.0,0.4286,11,0,1,11,0,12,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.21865,0.2474,0.105,0.14286,0.2857,0.0,1.0,8,2,0,8,0,14,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[16,36,0.4444,0.17848,0.22017,0.0,0.14286,0.1429,0.0,1.0,10,1,0,10,0,15,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[20,36,0.5556,0.17411,0.22227,0.0,0.14286,0.1429,0.0,1.0,11,1,0,11,0,14,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[24,36,0.6667,0.15625,0.1488,0.10714,0.14286,0.14287,0.0,0.71429,8,0,0,8,0,18,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[28,36,0.7778,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.09795,0.06604,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.00893,"x":0.23652,"p":[[0,32,0.0,0.19179,0.21316,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,9,0,0,4,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[4,32,0.125,0.21429,0.18558,0.10714,0.14286,0.42857,0.0,0.71429,8,0,0,8,0,12,0,0,2,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[8,32,0.25,0.23652,0.23857,0.0,0.14286,0.42857,0.0,0.85714,11,0,0,11,0,6,0,0,6,0,0,5,0,0,1,0,0,2,0,0,1,0,0],[12,32,0.375,0.18284,0.18639,0.0,0.14286,0.2857,0.0,0.71429,10,0,0,10,0,13,0,0,2,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[16,32,0.5,0.08929,0.20124,0.0,0.0,0.14286,0.0,0.85714,23,0,3,23,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[20,32,0.625,0.12053,0.18593,0.0,0.0,0.17857,0.0,0.714,20,0,2,20,0,4,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[24,32,0.75,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.03125,0.08552,0.0,0.0,0.0,0.0,0.42857,27,0,1,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9352e109f8328dc9","q":"7. G1 (ARM) Let $A B C$ be a triangle and $M$ an interior point. Prove that  $$ \\min \\{M A, M B, M C\\}+M A+M B+M C<A B+A C+B C . $$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.00446,"x":0.07589,"p":[[0,28,0.0,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.04018,"x":0.05134,"p":[[0,4,0.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.05134,0.06737,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,1,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ab77eeb7c733ac2f","q":"Circles  $ \\Gamma_1$  and  $ \\Gamma_2$  are internally tangent to circle  $ \\Gamma$  at  $ P$  and  $ Q$ , respectively. Let  $ P_1$  and  $ Q_1$  are on  $ \\Gamma_1$  and  $ \\Gamma_2$  respectively such that  $ P_1Q_1$  is the common tangent of  $ P_1$  and  $ Q_1$ . Assume that  $ \\Gamma_1$  and  $ \\Gamma_2$  intersect at  $ R$  and  $ R_1$ . Define  $ O_1,O_2,O_3$  as the intersection of  $ PQ$  and  $ P_1Q_1$ , the intersection of  $ PR$  and  $ P_1R_1$ , and the intersection  $ QR$  and  $ Q_1R_1$ . Prove that the points  $ O_1,O_2,O_3$  are collinear.\r\n*Rudi Adha Prihandoko, Bandung*","t":[{"b":1,"e":0.0,"k":"flat","v":0.04464,"x":0.24104,"p":[[0,19,0.0,0.08927,0.17031,0.0,0.0,0.14286,0.0,0.57143,23,0,1,23,0,4,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,19,0.2105,0.13839,0.2382,0.0,0.0,0.1786,0.0,0.85714,22,0,0,22,0,2,0,0,2,0,0,1,0,0,4,0,0,0,0,0,1,0,0],[8,19,0.4211,0.24104,0.26104,0.0,0.14286,0.571,0.0,0.85714,14,0,0,14,0,4,0,0,3,0,0,2,0,0,8,0,0,0,0,0,1,0,0],[12,19,0.6316,0.10266,0.21788,0.0,0.0,0.03571,0.0,0.857,24,0,0,24,0,2,0,0,3,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[16,19,0.8421,0.1741,0.23071,0.0,0.0,0.28571,0.0,0.857,18,0,0,18,0,1,0,0,7,0,0,2,0,0,3,0,0,0,0,0,1,0,0],[19,19,1.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.05357,"x":0.16969,"p":[[0,12,0.0,0.12945,0.22117,0.0,0.0,0.2857,0.0,0.85714,22,0,0,22,0,1,0,0,4,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[4,12,0.3333,0.09821,0.20958,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[8,12,0.6667,0.16969,0.24603,0.0,0.0,0.42895,0.0,0.57143,21,0,0,21,0,1,0,0,0,0,0,3,0,0,7,0,0,0,0,0,0,0,0],[12,12,1.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fa6f083ab0be055e","q":"5. N5 (ARM) Let $n, k$ be positive integers such that $n$ is not divisible by 3 and $k \\geq n$. Prove that there exists a positive integer $m$ that is divisible by $n$ and the sum of whose digits in decimal representation is $k$.","t":[{"b":1,"e":0.2857,"k":"volatile","v":0.23228,"x":0.55354,"p":[[0,7,0.0,0.4598,0.36374,0.10714,0.42859,0.71429,0.0,1.0,8,6,0,8,0,2,0,0,4,0,0,3,0,0,4,0,0,4,0,0,1,0,6],[4,7,0.5714,0.55354,0.37923,0.14286,0.57143,1.0,0.0,1.0,5,10,0,5,0,5,0,0,1,0,0,2,0,0,4,0,0,5,0,0,0,0,10],[7,7,1.0,0.23228,0.13249,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,0,15,0,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.1429,"k":"falling","v":0.14277,"x":0.51777,"p":[[0,37,0.0,0.51777,0.35857,0.24999,0.5,0.85714,0.0,1.0,5,7,0,5,0,3,0,0,5,0,0,3,0,0,2,0,0,5,0,0,2,0,7],[4,37,0.1081,0.36607,0.35524,0.0,0.28571,0.60714,0.0,1.0,10,4,0,10,0,5,0,0,2,0,0,5,0,0,2,0,0,2,0,0,2,0,4],[8,37,0.2162,0.45982,0.37412,0.14286,0.28571,0.89286,0.0,1.0,5,8,0,5,0,6,0,0,6,0,0,3,0,0,1,0,0,2,0,0,1,0,8],[12,37,0.3243,0.43304,0.35801,0.14286,0.28571,0.75,0.0,1.0,4,6,0,4,0,9,0,0,5,0,0,2,0,0,2,0,0,2,0,0,2,0,6],[16,37,0.4324,0.33473,0.28713,0.14286,0.28571,0.4286,0.0,1.0,6,3,0,6,0,7,0,0,6,0,0,6,0,0,3,0,0,1,0,0,0,0,3],[20,37,0.5405,0.37497,0.34763,0.10714,0.28571,0.57143,0.0,1.0,8,5,0,8,0,4,0,0,9,0,0,0,0,0,4,0,0,1,0,0,1,0,5],[24,37,0.6486,0.31695,0.31487,0.0,0.2857,0.57111,0.0,1.0,10,3,0,10,0,5,0,0,5,0,0,3,0,0,4,0,0,2,0,0,0,0,3],[28,37,0.7568,0.19643,0.25939,0.0,0.14286,0.28571,0.0,1.0,13,2,0,13,0,9,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[32,37,0.8649,0.31694,0.2493,0.0,0.28571,0.57111,0.0,0.71429,9,0,0,9,0,3,0,0,5,0,0,5,0,0,7,0,0,3,0,0,0,0,0],[36,37,0.973,0.20088,0.23379,0.0,0.14286,0.28571,0.0,1.0,12,1,0,12,0,8,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[37,37,1.0,0.14277,0.15567,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,9,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"405b09a668a6ef81","q":"17. (NET 2) ${ }^{\\mathrm{IMO} 6}$ Let $n$ be an integer greater than 1 . In a circular arrangement of $n$ lamps $L_{0}, \\ldots, L_{n-1}$, each one of that can be either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $S_{t e p_{0}}, S t e p_{1}, \\ldots$. If $L_{j-1}(j$ is taken $\\bmod n)$ is ON, then $S_{t e p}^{j}$ changes the status of $L_{j}$ (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If $L_{j-1}$ is OFF, then $S_{t e p}^{j}$ does not change anything at all. Show that: (a) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again. (b) If $n$ has the form $2^{k}$, then all lamps are ON after $n^{2}-1$ steps. (c) If $n$ has the form $2^{k}+1$, then all lamps are ON after $n^{2}-n+1$ steps.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.03571,"x":0.11152,"p":[[0,5,0.0,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.12491,"x":0.15616,"p":[[0,11,0.0,0.12491,0.07782,0.14286,0.14286,0.14286,0.0,0.42857,6,0,1,6,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.15616,0.0827,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,27,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.1292,0.04156,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"04f2c34d9e4f4886","q":"A convex pentagon  $ABCDE$  satisfies that the sidelengths and  $AC,AD\\leq \\sqrt 3.$  Let us choose  $2011$  distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least  $403$  points out of the  $2011$  given points.\n{Edited}\n{I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}","t":[{"b":1,"e":0.14286,"k":"flat","v":0.1384,"x":0.18303,"p":[[0,3,0.0,0.18303,0.16065,0.0,0.14286,0.28571,0.0,0.57143,11,0,8,11,0,6,0,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[3,3,1.0,0.1384,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.10268,"x":0.26339,"p":[[0,20,0.0,0.23652,0.13182,0.14286,0.2857,0.28571,0.0,0.42857,4,0,2,4,0,9,0,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.26339,0.19269,0.14286,0.2857,0.32143,0.0,0.85714,5,0,2,5,0,8,0,0,11,0,0,6,0,0,0,0,0,1,0,0,1,0,0],[8,20,0.4,0.18741,0.14916,0.0,0.14286,0.28571,0.0,0.42857,9,0,7,9,0,9,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.16071,0.10564,0.14286,0.14286,0.28571,0.0,0.28571,7,0,1,7,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ba2dc70fc8a6d135","q":"A regular  $2012$ -gon is inscribed in a circle. Find the maximal  $k$  such that we can choose  $k$  vertices from given  $2012$  and construct a convex  $k$ -gon without parallel sides.","t":[{"b":1,"e":0.42857,"k":"rising","v":0.27223,"x":0.57143,"p":[[0,33,0.0,0.27223,0.29748,0.0,0.14286,0.42857,0.0,1.0,10,1,4,10,0,10,0,0,2,0,0,3,0,0,0,0,0,5,0,0,1,0,1],[4,33,0.1212,0.40625,0.37983,0.14286,0.21429,0.64286,0.0,1.0,7,7,0,7,0,9,0,0,2,0,0,1,0,0,5,0,0,0,0,0,1,0,7],[8,33,0.2424,0.41516,0.34875,0.14286,0.35714,0.71429,0.0,1.0,6,4,0,6,0,9,0,0,1,0,0,2,0,0,3,0,0,6,0,0,1,0,4],[12,33,0.3636,0.49554,0.28344,0.28571,0.50001,0.71429,0.0,1.0,2,1,0,2,0,4,0,0,7,0,0,3,0,0,3,0,0,7,0,0,5,0,1],[16,33,0.4848,0.54017,0.30874,0.28571,0.57143,0.85714,0.0,1.0,1,4,0,1,0,5,0,0,7,0,0,1,0,0,4,0,0,5,0,0,5,0,4],[20,33,0.6061,0.44179,0.33965,0.14286,0.28571,0.71429,0.0,1.0,3,5,0,3,0,9,0,0,5,0,0,2,0,0,3,0,0,3,0,0,2,0,5],[24,33,0.7273,0.57143,0.30514,0.28571,0.71429,0.85714,0.0,1.0,2,2,0,2,0,3,0,0,5,0,0,4,0,0,1,0,0,5,0,0,10,0,2],[28,33,0.8485,0.55356,0.32878,0.14286,0.64286,0.75,0.0,1.0,3,5,0,3,0,6,0,0,0,0,0,3,0,0,4,0,0,8,0,0,3,0,5],[32,33,0.9697,0.45087,0.30744,0.24999,0.49979,0.57143,0.0,1.0,5,2,0,5,0,3,0,0,6,0,0,2,0,0,9,0,0,0,0,0,5,0,2],[33,33,1.0,0.52679,0.2911,0.28571,0.57143,0.75,0.0,1.0,2,1,0,2,0,5,0,0,3,0,0,4,0,0,4,0,0,6,0,0,7,0,1]]},{"b":2,"e":0.14286,"k":"falling","v":0.08482,"x":0.28125,"p":[[0,21,0.0,0.28125,0.29555,0.10714,0.14286,0.46429,0.0,1.0,8,2,4,8,0,12,0,0,2,0,0,2,0,0,4,0,0,1,0,0,1,0,2],[4,21,0.1905,0.28107,0.30204,0.0,0.14286,0.42857,0.0,1.0,11,1,3,11,0,7,0,0,3,0,0,4,0,0,1,0,0,3,0,0,2,0,1],[8,21,0.381,0.10714,0.07986,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.13393,0.07936,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,22,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.08483,0.07017,0.0,0.14286,0.14286,0.0,0.143,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.10714,0.07986,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.08482,0.07873,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1bd524c0f518e986","q":"Find all all positive integers x,y,and z satisfying the equation  $x^3=3^y7^z+8$","t":[{"b":5,"e":0.85714,"k":"flat","v":0.79017,"x":0.94196,"p":[[0,32,0.0,0.79017,0.23416,0.67857,0.85714,1.0,0.28571,1.0,0,14,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,6,0,0,4,0,14],[4,32,0.125,0.83479,0.1602,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,8,0,12],[8,32,0.25,0.87052,0.1689,0.85714,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,9,0,16],[12,32,0.375,0.90177,0.11542,0.85714,0.92857,1.0,0.571,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,16],[16,32,0.5,0.91963,0.1126,0.85714,1.0,1.0,0.714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,20],[20,32,0.625,0.8616,0.12103,0.71429,0.85714,1.0,0.57143,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,12,0,11],[24,32,0.75,0.94196,0.10012,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,23],[28,32,0.875,0.93303,0.09439,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,20],[32,32,1.0,0.90179,0.10374,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,15]]},{"b":7,"e":0.85714,"k":"flat","v":0.68746,"x":0.83034,"p":[[0,18,0.0,0.83034,0.15338,0.71429,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,11,0,10],[4,18,0.2222,0.74997,0.18901,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,0,0,0,1,0,0,5,0,0,13,0,0,5,0,7],[8,18,0.4444,0.80803,0.1365,0.71429,0.85707,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,11,0,7],[12,18,0.6667,0.79018,0.18205,0.67857,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,7,0,0,7,0,10],[16,18,0.8889,0.77231,0.17075,0.71429,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,16,0,0,6,0,7],[18,18,1.0,0.68746,0.19046,0.57132,0.71429,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,8,0,0,10,0,2]]}]},{"i":"3944f27d5e23b0e1","q":"A convex hexagon  $ABCDEF$  is given such that  $AB||DE$ ,  $BC||EF$ ,  $CD||FA$ . The point  $M, N, K$  are common points of the lines  $BD$  and  $AE$ ,  $AC$  and  $DF$ ,  $CE$  and  $BF$  respectively. Prove that perpendiculars drawn from  $M, N, K$  to lines  $AB, CD, EF$  respectively concurrent.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,31,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,18,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"32b0f71ab3d07ed0","q":"At a dinner party there are $N$ hosts and $N$ guests, seated around a circular table, where $N \\geq 4$. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Prove that no matter how the 2 N people are seated at the dinner party, at least $N$ pairs of guests will chat with one another.","t":[{"b":1,"e":0.0,"k":"flat","v":0.13839,"x":0.25893,"p":[[0,22,0.0,0.15625,0.12037,0.14286,0.14286,0.1786,0.0,0.57143,7,0,2,7,0,17,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,22,0.1818,0.13839,0.15765,0.0,0.14286,0.14286,0.0,0.71429,12,0,2,12,0,14,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[8,22,0.3636,0.22759,0.23656,0.14286,0.14286,0.28571,0.0,1.0,6,2,3,6,0,14,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[12,22,0.5455,0.19643,0.21053,0.0,0.14286,0.2857,0.0,1.0,9,1,1,9,0,13,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[16,22,0.7273,0.21875,0.13825,0.14286,0.14288,0.28571,0.0,0.57143,4,0,0,4,0,13,0,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,22,0.9091,0.25893,0.16917,0.14286,0.2857,0.42857,0.0,0.71429,4,0,0,4,0,10,0,0,9,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[22,22,1.0,0.21875,0.09438,0.14286,0.21428,0.2857,0.0,0.42857,1,0,0,1,0,15,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.16509,"x":0.21849,"p":[[0,4,0.0,0.21849,0.19238,0.0,0.14288,0.42857,0.0,0.71429,9,0,1,9,0,9,0,0,5,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[4,4,1.0,0.16509,0.09525,0.14286,0.14286,0.1786,0.0,0.42857,4,0,0,4,0,20,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e56ba03573781f82","q":"Do there exist  $2011$  positive integers  $a_1 < a_2 < \\ldots < a_{2011}$  such that  $\\gcd(a_i,a_j) = a_j - a_i$  for any  $i$ ,  $j$  such that  $1 \\le i < j \\le 2011$ ?","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.01339,"p":[[0,13,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,5,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00893,"x":0.03125,"p":[[0,23,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,5,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,6,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,10,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"11cb4165e2e6a165","q":"Determine the polygons with  $n$  sides  $(n \\ge 4)$ , 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. **Note:** 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":1,"e":0.42857,"k":"flat","v":0.16072,"x":0.33036,"p":[[0,33,0.0,0.19179,0.16608,0.0,0.14286,0.32143,0.0,0.42857,10,0,4,10,0,9,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.18293,0.15662,0.105,0.14286,0.2857,0.0,0.57143,8,0,1,8,0,13,0,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[8,33,0.2424,0.16072,0.1171,0.14286,0.14286,0.28571,0.0,0.4286,7,0,0,7,0,16,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.30803,0.08073,0.2857,0.28571,0.32143,0.14286,0.4286,0,0,0,0,0,3,0,0,21,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.30348,0.09295,0.2857,0.28571,0.42857,0.14,0.4286,0,0,0,0,0,5,0,0,18,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.27232,0.10326,0.24999,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,7,0,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.30808,0.08834,0.2857,0.28571,0.42857,0.14286,0.43,0,0,0,0,0,4,0,0,19,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.33036,0.10374,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,5,0,0,12,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.2633,0.09534,0.14286,0.28571,0.28571,0.14,0.4286,0,0,0,0,0,10,0,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.31696,0.08552,0.2857,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,19,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14,"k":"falling","v":0.02232,"x":0.21429,"p":[[0,20,0.0,0.21429,0.16367,0.10714,0.21428,0.28571,0.0,0.57143,8,0,3,8,0,8,0,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,20,0.2,0.14732,0.13592,0.0,0.14286,0.1786,0.0,0.57143,10,0,2,10,0,14,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.08027,0.10672,0.0,0.0,0.14286,0.0,0.4286,18,0,6,18,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.2857,22,0,1,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"39a24981cb2204b8","q":"Consider a circular necklace with  $\\displaystyle{2013}$  beads. Each bead can be paintes either green or white. A painting of the necklace is called  *good* if among any  $\\displaystyle{21}$  successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.**Note.** Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings.\n\n*Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.*","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.08036,"p":[[0,19,0.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,3,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,4,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.08036,0.11258,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.03125,"x":0.05357,"p":[[0,6,0.0,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,25,0,5,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,4,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b4e9ddfc2c19d21b","q":"Circles  $\\Omega $  and  $\\omega $  are tangent at a point  $P$  ( $\\omega $  lies inside  $\\Omega $ ). A chord  $AB$  of  $\\Omega $  is tangent to  $\\omega $  at   $C;$  the line  $PC$  meets  $\\Omega $  again at  $Q.$  Chords  $QR$  and  $QS$  of  $ \\Omega $  are tangent to  $\\omega .$  Let  $I,X,$  and  $Y$  be the incenters of the triangles  $APB,$   $ARB,$  and  $ASB,$  respectively. Prove that  $\\angle PXI+\\angle PYI=90^{\\circ }.$","t":[{"b":2,"e":0.14286,"k":"flat","v":0.02679,"x":0.14286,"p":[[0,17,0.0,0.07589,0.10091,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0892,0.11148,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.14286,1e-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],[17,17,1.0,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.12036,"p":[[0,21,0.0,0.12036,0.10166,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,15,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6c2a6d8916957247","q":"Alexander has chosen a natural number  $N>1$  and has written down in a line,and in increasing order,all his positive divisors  $d_1<d_2<\\ldots <d_s$  (where  $d_1=1$  and  $d_s=N$ ).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these  $s-1$  numbers (the greatest common divisors) is equal to  $N-2$ .Find all possible values of  $N$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,57,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,57,0.0702,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,57,0.1404,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,57,0.2105,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,57,0.7018,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.0,"k":"flat","v":0.00446,"x":0.08485,"p":[[0,44,0.0,0.08485,0.18166,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,4,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[4,44,0.0909,0.04465,0.10972,0.0,0.0,0.0,0.0,0.5714,25,0,0,25,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,44,0.1818,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.0134,0.04165,0.0,0.0,0.0,0.0,0.143,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0bed5efe505c805b","q":"11. G5 (NZL) ${ }^{\\mathrm{IMO} 5}$ Let $A B C D E F$ be a convex hexagon with $A B=B C=$ $C D, D E=E F=F A$, and $\\measuredangle B C D=\\measuredangle E F A=\\pi / 3$ (that is, $60^{\\circ}$ ). Let $G$ and $H$ be two points interior to the hexagon such that angles $A G B$ and $D H E$ are both $2 \\pi / 3$ (that is, $120^{\\circ}$ ). Prove that $A G+G B+G H+D H+$ $H E \\geq C F$.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.56696,"x":0.82147,"p":[[0,13,0.0,0.56696,0.27078,0.28571,0.71429,0.85714,0.0,0.85714,1,0,1,1,0,2,0,0,8,0,0,2,0,0,2,0,0,7,0,0,10,0,0],[4,13,0.3077,0.78125,0.20511,0.71429,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,1,0,0,2,0,0,1,0,0,6,0,0,15,0,6],[8,13,0.6154,0.82147,0.13819,0.85714,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,23,0,3],[12,13,0.9231,0.80802,0.11632,0.857,0.85714,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,26,0,0],[13,13,1.0,0.80803,0.12168,0.71429,0.85714,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,21,0,2]]},{"b":2,"e":0.85714,"k":"rising","v":0.62053,"x":0.80802,"p":[[0,23,0.0,0.62053,0.32656,0.28571,0.78564,0.85714,0.0,1.0,3,2,3,3,0,4,0,0,2,0,0,0,0,0,2,0,0,5,0,0,14,0,2],[4,23,0.1739,0.70089,0.24836,0.67857,0.85714,0.85714,0.14286,1.0,0,1,0,0,0,3,0,0,2,0,0,2,0,0,1,0,0,5,0,0,18,0,1],[8,23,0.3478,0.6875,0.26591,0.71429,0.78564,0.85714,0.0,1.0,3,1,2,3,0,0,0,0,1,0,0,2,0,0,1,0,0,9,0,0,15,0,1],[12,23,0.5217,0.77678,0.13333,0.71429,0.85714,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,9,0,0,20,0,0],[16,23,0.6957,0.72765,0.17261,0.71429,0.78564,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,10,0,0,16,0,0],[20,23,0.8696,0.76337,0.16603,0.71429,0.85714,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,8,0,0,20,0,0],[23,23,1.0,0.80802,0.09851,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,22,0,1]]}]},{"i":"668e1578aca3261e","q":"38. (ROM 4) Let $m_{j}>0$ for $j=1,2, \\ldots, n$ and $a_{1} \\leq \\cdots \\leq a_{n}<b_{1} \\leq \\cdots \\leq$ $b_{n}<c_{1} \\leq \\cdots \\leq c_{n}$ be real numbers. Prove:  $$ \\left[\\sum_{j=1}^{n} m_{j}\\left(a_{j}+b_{j}+c_{j}\\right)\\right]^{2}>3\\left(\\sum_{j=1}^{n} m_{j}\\right)\\left[\\sum_{j=1}^{n} m_{j}\\left(a_{j} b_{j}+b_{j} c_{j}+c_{j} a_{j}\\right)\\right] . $$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,13,0.0,0.04911,0.19759,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,16,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"0dd2358e4bb727db","q":"21. $\\mathbf{C 1}(\\mathbf{C O L})^{\\mathrm{IMO} 1}$ Let $n$ be a positive integer. Each point $(x, y)$ in the plane, where $x$ and $y$ are nonnegative integers with $x+y \\leq n$, is colored red or blue, subject to the following condition: If a point $(x, y)$ is red, then so are all points $\\left(x^{\\prime}, y^{\\prime}\\right)$ with $x^{\\prime} \\leq x$ and $y^{\\prime} \\leq y$. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$-coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.11599,"x":0.16036,"p":[[0,28,0.0,0.11599,0.06619,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.15161,0.04975,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.14733,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.15599,0.04173,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.16036,0.04739,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.15617,0.04166,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.15599,"p":[[0,11,0.0,0.15599,0.16117,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,11,0.3636,0.12046,0.1017,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,18,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"e6eafc76bb9b3cc5","q":"A point  $H$  lies on the side  $AB$  of regular polygon  $ABCDE$ . A circle with center  $H$  and radius  $HE$  meets the segments  $DE$  and  $CD$  at points  $G$  and  $F$  respectively. It is known that  $DG=AH$ . Prove that  $CF=AH$ .","t":[{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.03125,"p":[[0,25,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.03125,"p":[[0,26,0.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,2,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"960b192d1ea66d6a","q":"13. (IND) In town $A$, there are $n$ girls and $n$ boys, and each girl knows each boy. In town $B$, there are $n$ girls $g_{1}, g_{2}, \\ldots, g_{n}$ and $2 n-1$ boys $b_{1}, b_{2}, \\ldots$, $b_{2 n-1}$. The girl $g_{i}, i=1,2, \\ldots, n$, knows the boys $b_{1}, b_{2}, \\ldots, b_{2 i-1}$, and no others. For all $r=1,2, \\ldots, n$, denote by $A(r), B(r)$ the number of different ways in which $r$ girls from town $A$, respectively town $B$, can dance with $r$ boys from their own town, forming $r$ pairs, each girl with a boy she knows. Prove that $A(r)=B(r)$ for each $r=1,2, \\ldots, n$.","t":[{"b":2,"e":0.14286,"k":"volatile","v":0.39062,"x":0.79464,"p":[[0,12,0.0,0.66963,0.37702,0.2857,1.0,1.0,0.14286,1.0,0,17,0,0,0,7,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,17],[4,12,0.3333,0.79464,0.33108,0.57143,1.0,1.0,0.14286,1.0,0,22,0,0,0,4,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,22],[8,12,0.6667,0.68078,0.2907,0.48214,0.57143,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,4,0,0,2,0,1,8,0,0,0,0,0,4,0,11],[12,12,1.0,0.39062,0.15565,0.28571,0.28571,0.57143,0.14286,0.643,0,0,0,0,0,3,0,0,15,0,0,2,0,0,11,1,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.40625,"x":0.95981,"p":[[0,28,0.0,0.60937,0.38713,0.24999,0.67856,1.0,0.14286,1.0,0,15,0,0,0,8,0,0,7,0,0,0,0,0,1,0,0,0,1,0,0,0,15],[4,28,0.1429,0.72322,0.36058,0.28571,1.0,1.0,0.14286,1.0,0,19,0,0,0,4,0,0,7,0,0,0,0,0,0,0,0,1,0,0,1,0,19],[8,28,0.2857,0.61598,0.33599,0.28571,0.64286,1.0,0.14,1.0,0,11,0,0,0,4,0,0,8,0,0,2,0,0,2,0,0,3,0,0,2,0,11],[12,28,0.4286,0.52661,0.36517,0.2857,0.28571,1.0,0.14,1.0,0,11,0,0,0,7,0,0,11,0,0,2,0,0,0,0,0,0,0,0,1,0,11],[16,28,0.5714,0.48897,0.32309,0.2857,0.28571,0.85704,0.14286,1.0,0,7,0,0,0,4,0,0,16,0,0,1,0,0,1,0,0,0,1,0,2,0,7],[20,28,0.7143,0.40625,0.29257,0.2857,0.28571,0.46429,0.14286,1.0,0,5,0,0,0,7,0,0,16,0,0,1,0,0,2,0,0,0,0,0,1,0,5],[24,28,0.8571,0.49553,0.36941,0.14286,0.28571,1.0,0.14286,1.0,0,9,0,0,0,10,0,0,10,0,0,0,0,0,0,0,0,0,0,0,3,0,9],[28,28,1.0,0.95981,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]}]},{"i":"4d31acb22ee9d877","q":"2. A2 (YUG 1) Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that  $$ \\operatorname{area}\\left(K \\cap Q_{i}\\right)=\\frac{1}{4} \\text { area } K(i=1,2,3,4,), $$  where the $Q_{i}$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than 4.","t":[{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.03125,"p":[[0,22,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,3,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,2,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.00893,"x":0.04009,"p":[[0,28,0.0,0.04009,0.07337,0.0,0.0,0.035,0.0,0.28571,24,0,1,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,3,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,2,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,1,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,2,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.02223,0.07225,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"174bfaa40dc8c9b2","q":"5-Two circles  $ S_1$  and  $ S_2$  with equal radius and intersecting at two points are given in the plane.A line  $ l$  intersects  $ S_1$  at  $ B,D$  and  $ S_2$  at  $ A,C$ (the order of the points on the line are as follows: $ A,B,C,D$ ).Two circles  $ W_1$  and  $ W_2$  are drawn such that both of them are tangent externally at  $ S_1$  and internally at  $ S_2$  and also tangent to  $ l$  at both sides.Suppose  $ W_1$  and  $ W_2$  are tangent.Then PROVE  $ AB \\equal{} CD$ .","t":[{"b":5,"e":0.14286,"k":"flat","v":0.03125,"x":0.09821,"p":[[0,7,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.09821,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.00438,"x":0.08473,"p":[[0,10,0.0,0.04911,0.13651,0.0,0.0,0.0,0.0,0.71429,26,0,1,26,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,10,0.4,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.08473,0.07867,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"01db0ec53f542d26","q":"5. (ROM) Let $A B C D$ be a regular tetrahedron and $M, N$ distinct points in the planes $A B C$ and $A D C$ respectively. Show that the segments $M N, B N, M D$ are the sides of a triangle.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,8,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,14,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"42e5c696001ca091","q":"A square with side  $2008$  is broken into regions that are all squares with side  $1$ . In every region, either  $0$  or  $1$  is written, and the number of  $1$ 's and  $0$ 's is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than  $\\frac{1}{2^{10^6}}$ .","t":[{"b":1,"e":0.14286,"k":"rising","v":0.02679,"x":0.27232,"p":[[0,46,0.0,0.07143,0.12877,0.0,0.0,0.14286,0.0,0.42857,23,0,11,23,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.12054,0.20238,0.0,0.0,0.28571,0.0,0.85714,21,0,10,21,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[8,46,0.1739,0.18304,0.18638,0.0,0.14286,0.28571,0.0,0.71429,12,0,6,12,0,8,0,0,5,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[12,46,0.2609,0.08929,0.17035,0.0,0.0,0.14286,0.0,0.71429,22,0,1,22,0,6,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[16,46,0.3478,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.42857,22,0,1,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.2232,0.20803,0.0,0.14286,0.32143,0.0,0.71429,9,0,0,9,0,10,0,0,5,0,0,3,0,0,4,0,0,1,0,0,0,0,0],[32,46,0.6957,0.22312,0.12852,0.14286,0.2857,0.28571,0.0,0.42857,4,0,0,4,0,11,0,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.21429,0.13832,0.14286,0.14286,0.28571,0.0,0.4286,5,0,0,5,0,12,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.27232,0.20935,0.14286,0.28571,0.42857,0.0,0.85714,4,0,0,4,0,11,0,0,8,0,0,6,0,0,1,0,0,0,0,0,2,0,0],[44,46,0.9565,0.25893,0.20341,0.14286,0.28571,0.32143,0.0,0.85714,7,0,0,7,0,6,0,0,11,0,0,4,0,0,3,0,0,0,0,0,1,0,0],[46,46,1.0,0.25893,0.16536,0.14286,0.28571,0.32143,0.0,0.57143,5,0,0,5,0,7,0,0,12,0,0,5,0,0,3,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.08036,"x":0.15624,"p":[[0,14,0.0,0.08036,0.15126,0.0,0.0,0.14286,0.0,0.57143,23,0,15,23,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,14,0.2857,0.15624,0.23784,0.0,0.0,0.28571,0.0,0.85714,20,0,8,20,0,2,0,0,4,0,0,1,0,0,4,0,0,0,0,0,1,0,0]]}]},{"i":"c71f68e7f6788374","q":"Anna and Bob play the following game. In the beginning, Bob writes down the numbers  $1, 2, ... , 2022$  on a piece of paper, such that half of the numbers are on the left and half on the right. Furthermore, we assume that the  $1011$  numbers on both sides are written in some order.\nAfter Bob does this, Anna has the opportunity to swap the positions of the two numbers lying on different sides of the paper if they have different parity. Anna wins if, after finitely many moves, all odd numbers end up on the left, in increasing order, and all even ones end up on the right, in increasing order. Can Bob write down a arrangement of numbers for which Anna cannot win?\nFor example, Bob could write down numbers in the following way:   $$ 4, 2, 5, 7, 9, ... , 2021\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,,\\, 3, 1, 6, 8, 10, ... , 2022 $$   Then Anna could swap the numbers  $1, 4$  and then swap  $2, 3$  to win. However, if Anna swapped\nthe pairs  $3, 4$  and  $1, 2$ , the resulting numbers on the left and on the right would not be in increasing order, and hence Anna would not win.","t":[{"b":0,"e":0.0,"k":"flat","v":0.03138,"x":0.15625,"p":[[0,12,0.0,0.03138,0.13262,0.0,0.0,0.0,0.0,0.71429,30,0,1,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,12,0.3333,0.05804,0.16698,0.0,0.0,0.0,0.0,0.71429,28,0,2,28,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,12,0.6667,0.15625,0.25841,0.0,0.0,0.28579,0.0,0.85714,22,0,0,22,0,1,0,0,2,0,0,1,0,0,4,0,0,1,0,0,1,0,0],[12,12,1.0,0.10268,0.18293,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,3,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.09813,"p":[[0,8,0.0,0.09813,0.25363,0.0,0.0,0.0,0.0,1.0,27,1,1,27,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,1],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0da3a434db3203fb","q":"Consider a regular $n$-gon with $n>3$, and call a line acceptable if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons.\n(a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having 3 or 4 sides.\n(b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have 3 or 4 sides.","t":[{"b":3,"e":0.0,"k":"rising","v":0.43749,"x":0.70981,"p":[[0,9,0.0,0.43749,0.37276,0.0,0.57121,0.71429,0.0,1.0,11,3,8,11,0,2,0,0,1,0,0,1,0,0,3,0,0,8,0,0,3,0,3],[4,9,0.4444,0.51338,0.309,0.28571,0.57143,0.71429,0.0,0.85714,7,0,7,7,0,0,0,0,2,0,0,2,0,0,6,0,0,9,0,0,6,0,0],[8,9,0.8889,0.70981,0.21865,0.57143,0.71429,0.85714,0.0,1.0,2,1,2,2,0,0,0,0,0,0,0,0,0,0,7,0,0,8,0,0,14,0,1]]},{"b":4,"e":0.71429,"k":"volatile","v":0.29015,"x":0.68301,"p":[[0,9,0.0,0.43749,0.3213,0.10714,0.4998,0.71429,0.0,1.0,8,1,8,8,0,2,0,0,2,0,0,4,0,0,7,0,0,3,0,0,5,0,1],[4,9,0.4444,0.29015,0.33402,0.0,0.14286,0.57143,0.0,1.0,15,1,12,15,0,3,0,0,2,0,0,1,0,0,5,0,0,2,0,0,3,0,1],[8,9,0.8889,0.68301,0.198,0.571,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,6,0,0,11,0,2],[9,9,1.0,0.6428,0.1557,0.57143,0.57143,0.71429,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,0,0,0,16,0,0,10,0,0,5,0,0]]}]},{"i":"b3b25b5633be37b0","q":"**interesting function** $S$  is a set with  $n$  elements and  $P(S)$  is the set of all subsets of  $S$  and $f : P(S) \\rightarrow \\mathbb N$ is a function with these properties:\nfor every subset  $A$  of  $S$  we have  $f(A)=f(S-A)$ .\nfor every two subsets of  $S$  like  $A$  and  $B$  we have  $max(f(A),f(B))\\ge f(A\\cup B)$ prove that number of natural numbers like  $x$  such that there exists  $A\\subseteq  S$  and  $f(A)=x$  is less than  $n$ .\n\ntime allowed for this question was 1 hours and 30 minutes.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,5,0.0,0.03125,0.09268,0.0,0.0,0.0,0.0,0.4286,28,0,1,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,29,0.0,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"71339ea50b9882b4","q":"Alice has arranged 200 boxes in her living room. Each box contains a paper on which she has written a non-zero natural number; the 200 numbers are not necessarily distinct. Each minute, and as long as it is possible, Alice performs an action of the following form: she chooses three boxes, containing integers $a, b$, and $c$ such that $a+b=c$, and also chooses an arbitrary integer $k \\geqslant 2$; she then replaces the integer $c$ with the integer $k \\times c$. If she can no longer perform such an action, she stops definitively.\nProve that, regardless of the initial situation and Alice's choices, she will be forced to stop at some point.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,40,0.0,0.08036,0.24984,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,40,0.1,0.06679,0.17115,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[8,40,0.2,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,40,0.9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.12052,"p":[[0,23,0.0,0.07142,0.19229,0.0,0.0,0.0,0.0,0.85714,27,0,1,27,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,23,0.1739,0.12052,0.27687,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,4,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[8,23,0.3478,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.04911,0.12169,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e1abe83141f326bc","q":"Determine all real values of  $A$  for which there exist distinct complex numbers  $x_1$ ,  $x_2$  such that the following three equations hold:\n\\begin{align*}x_1(x_1+1)&=Ax_2(x_2+1)&=Ax_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\\end{align*}","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,38,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7a0de545eea2fd4a","q":"Consider a sequence of positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ such that for $k \\geqslant 2$ we have\n\n$$\na_{k+1}=\\frac{a_{k}+a_{k-1}}{2015^{i}}\n$$\n\nwhere $2015^{i}$ is the maximal power of 2015 that divides $a_{k}+a_{k-1}$. Prove that if this sequence is periodic then its period is divisible by 3 .","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.36161,"p":[[0,15,0.0,0.26339,0.4317,0.0,0.0,0.57145,0.0,1.0,23,8,0,23,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[4,15,0.2667,0.27232,0.43793,0.0,0.0,0.78571,0.0,1.0,23,8,1,23,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[8,15,0.5333,0.36161,0.47243,0.0,0.0,1.0,0.0,1.0,20,11,0,20,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11],[12,15,0.8,0.28125,0.44961,0.0,0.0,1.0,0.0,1.0,23,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.4375,"p":[[0,31,0.0,0.3125,0.45518,0.0,0.0,1.0,0.0,1.0,21,9,2,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,9],[4,31,0.129,0.4375,0.49608,0.0,0.0,1.0,0.0,1.0,18,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14],[8,31,0.2581,0.27232,0.43793,0.0,0.0,0.78571,0.0,1.0,23,8,1,23,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8],[12,31,0.3871,0.20089,0.391,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[16,31,0.5161,0.15179,0.35344,0.0,0.0,0.0,0.0,1.0,27,4,1,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[20,31,0.6452,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,31,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"84dedc02e6fb6e75","q":"$a_0 = a_1 = 1$  and  ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$  for all positive integers n.\r\n\r\nprove that  $a_n$  is one integer for all positive integers n.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,17,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[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],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,31,0.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,31,0.129,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"ab4e393ab3f552eb","q":"A triangle  $ABC$  is given. Let  $M$  be the midpoint of the side  $AC$  of the triangle and  $Z$  the image of point  $B$  along the line  $BM$ . The circle with center  $M$  and radius  $MB$  intersects the lines  $BA$  and  $BC$  at the points  $E$  and  $G$  respectively. Let  $H$  be the point of intersection of  $EG$  with the line  $AC$ , and  $K$  the point of intersection of  $HZ$  with the line  $EB$ . The perpendicular from point  $K$  to the line  $BH$   intersects the lines  $BZ$  and  $BH$  at the points  $L$  and  $N$ , respectively.\nIf  $P$  is the second point of intersection of the circumscribed circles of the triangles  $KZL$  and  $BLN$ , prove that, the lines  $BZ, KN$  and  $HP$  intersect at a common point.","t":[{"b":4,"e":0.71429,"k":"rising","v":0.36604,"x":0.68748,"p":[[0,13,0.0,0.52231,0.3285,0.2857,0.42857,0.85714,0.0,1.0,2,2,0,2,0,4,0,0,9,0,0,3,0,0,0,0,0,0,0,0,12,0,2],[4,13,0.3077,0.45089,0.33713,0.25,0.35714,0.75,0.0,1.0,6,3,0,6,0,2,0,0,8,0,0,4,0,0,0,0,0,4,0,0,5,0,3],[8,13,0.6154,0.36604,0.16339,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,22,0,0,3,0,0,4,0,0,0,0,0,2,0,0],[12,13,0.9231,0.58478,0.24317,0.28571,0.5712,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,10,0,0,3,0,0,5,0,0,2,0,0,12,0,0],[13,13,1.0,0.68748,0.25615,0.42859,0.85714,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,7,0,0,3,0,0,1,0,0,2,0,0,16,0,3]]},{"b":6,"e":0.85714,"k":"rising","v":0.40625,"x":0.78121,"p":[[0,29,0.0,0.48214,0.31693,0.2857,0.28571,0.85714,0.14286,1.0,0,4,0,0,0,5,0,0,15,0,0,1,0,0,0,0,0,0,0,0,7,0,4],[4,29,0.1379,0.54015,0.3108,0.28571,0.35714,0.85714,0.14286,1.0,0,6,0,0,0,3,0,0,13,0,0,1,0,0,2,0,0,3,0,0,4,0,6],[8,29,0.2759,0.48661,0.28316,0.28571,0.28571,0.71429,0.14286,1.0,0,3,0,0,0,3,0,0,15,0,0,2,0,0,0,0,0,5,0,0,4,0,3],[12,29,0.4138,0.48213,0.33645,0.28571,0.28571,0.85714,0.0,1.0,2,7,0,2,0,2,0,0,16,0,0,1,0,0,1,0,0,0,0,0,3,0,7],[16,29,0.5517,0.57141,0.31339,0.2857,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,12,0,0,3,0,0,1,0,0,2,0,0,5,0,7],[20,29,0.6897,0.40625,0.24772,0.2857,0.28571,0.60714,0.0,0.85714,1,0,0,1,0,3,0,0,18,0,0,1,0,0,1,0,0,3,0,0,5,0,0],[24,29,0.8276,0.6696,0.2299,0.42857,0.857,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,3,0,0,3,0,0,3,0,0,17,0,0],[28,29,0.9655,0.76338,0.20706,0.8214,0.85714,0.85714,0.2857,1.0,0,2,0,0,0,0,0,0,4,0,0,1,0,0,1,0,0,2,0,0,22,0,2],[29,29,1.0,0.78121,0.1287,0.71429,0.85707,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,5,0,0,22,0,0]]}]},{"i":"7910d1ec41e82558","q":"A sequence $a_{1}, a_{2}, \\ldots$ of positive integers is defined recursively by $a_{1}=2$ and  $$ a_{n+1}=a_{n}^{n+1}-1 \\quad \\text { for } n \\geq 1 $$  Prove that for every odd prime $p$ and integer $k$, some term of the sequence is divisible by $p^{k}$.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.04464,"x":0.19188,"p":[[0,15,0.0,0.09821,0.12595,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.15179,0.13333,0.0,0.14286,0.28571,0.0,0.42857,12,0,2,12,0,7,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.19188,0.12685,0.14286,0.1429,0.28571,0.0,0.4286,6,0,0,6,0,12,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.15178,"p":[[0,24,0.0,0.14286,0.17128,0.0,0.0,0.28571,0.0,0.4286,17,0,5,17,0,4,0,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.12054,0.12428,0.0,0.14286,0.1786,0.0,0.4286,14,0,4,14,0,10,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.15178,0.13333,0.0,0.14286,0.2857,0.0,0.42857,11,0,2,11,0,10,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.10272,0.11983,0.0,0.07143,0.14287,0.0,0.43,16,0,6,16,0,10,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,1,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dc04a2345acf132f","q":"13. G7 (ARM) The point $M$ inside the convex quadrilateral $A B C D$ is such that $M A=M C, \\angle A M B=\\angle M A D+\\angle M C D, \\angle C M D=\\angle M C B+\\angle M A B$. Prove that $A B \\cdot C M=B C \\cdot M D$ and $B M \\cdot A D=M A \\cdot C D$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.03116,"x":0.20955,"p":[[0,18,0.0,0.07134,0.13358,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.09375,0.14555,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,10,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,18,0.4444,0.08027,0.10672,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.03116,0.08541,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.06242,0.08695,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.20955,0.12249,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,1,16,0,0,9,0,1,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.13829,"p":[[0,32,0.0,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.0959,0.15108,0.0,0.0,0.1429,0.0,0.71429,19,0,0,19,0,7,0,1,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,32,0.25,0.07375,0.11797,0.0,0.0,0.14287,0.0,0.36,22,0,0,22,0,4,0,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.08018,0.11249,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.13829,0.17669,0.0,0.0,0.2857,0.0,0.571,18,0,0,18,0,3,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,32,0.625,0.10714,0.12372,0.0,0.0,0.2857,0.0,0.28571,17,0,0,17,0,6,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.10928,0.12996,0.0,0.0,0.2857,0.0,0.28571,18,0,0,18,0,3,0,1,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.08246,0.13253,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,1,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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":"4e52314139fc5c9a","q":"A circle is divided into congruent arcs by 432 points. The points are colored in four colors such that some 108 points are colored red, some 108 points are colored green, some 108 points are colored blue, and the remaining 108 points are colored yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,17,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,43,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,4,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,43,0.7442,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"853dbc627eb87970","q":"14. (JAP 2) In the coordinate plane a rectangle with vertices $(0,0),(m, 0)$, $(0, n),(m, n)$ is given where both $m$ and $n$ are odd integers. The rectangle is partitioned into triangles in such a way that (i) each triangle in the partition has at least one side (to be called a \"good\" side) that lies on a line of the form $x=j$ or $y=k$, where $j$ and $k$ are integers, and the altitude on this side has length 1 ; (ii) each \"bad\" side (i.e., a side of any triangle in the partition that is not a \"good\" one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.01786,"p":[[0,5,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.02679,"x":0.09366,"p":[[0,7,0.0,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,3,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02679,0.09061,0.0,0.0,0.0,0.0,0.42857,29,0,2,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.09366,0.15404,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,3,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5ede5361bea609ea","q":"7. (FIN 1) Show that any two points lying inside a regular $n$-gon $E$ can be joined by two circular arcs lying inside $E$ and meeting at an angle of at least $\\left(1-\\frac{2}{n}\\right) \\pi$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,15,0.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.04018,"p":[[0,16,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d651694b4b81ad82","q":"20. (USS 3) ${ }^{\\text {IMO6 }}$ Let $f(x)=x^{2}+x+p, p \\in \\mathbb{N}$. Prove that if the numbers $f(0), f(1), \\ldots, f([\\sqrt{p / 3}])$ are primes, then all the numbers $f(0), f(1), \\ldots$, $f(p-2)$ are primes.","t":[{"b":0,"e":0.28571,"k":"volatile","v":0.3125,"x":0.61158,"p":[[0,10,0.0,0.3125,0.35613,0.0,0.14286,0.50002,0.0,1.0,10,3,4,10,0,10,0,0,2,0,0,2,0,0,0,0,0,1,0,0,4,0,3],[4,10,0.4,0.61158,0.32584,0.28571,0.71429,0.85714,0.0,1.0,1,7,0,1,0,5,0,0,4,0,0,1,0,0,4,0,0,4,0,0,6,0,7],[8,10,0.8,0.33026,0.33968,0.14214,0.14286,0.60714,0.0,1.0,7,3,4,7,0,11,0,0,5,0,0,0,0,0,1,0,0,2,0,0,3,0,3],[10,10,1.0,0.53113,0.17212,0.42857,0.571,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,7,0,0,14,0,0,4,0,0,1,0,1]]},{"b":3,"e":0.71429,"k":"rising","v":0.37932,"x":0.83033,"p":[[0,25,0.0,0.37932,0.36713,0.0,0.28571,0.71429,0.0,1.0,10,5,1,10,0,4,0,0,3,0,0,6,0,0,0,0,0,2,0,0,2,0,5],[4,25,0.16,0.48212,0.36551,0.14286,0.4998,0.85714,0.0,1.0,6,5,1,6,0,5,0,0,3,0,0,2,0,0,4,0,0,2,0,0,5,0,5],[8,25,0.32,0.63837,0.35532,0.25,0.78564,1.0,0.0,1.0,2,9,0,2,0,6,0,0,1,0,0,2,0,0,1,0,0,4,0,0,7,0,9],[12,25,0.48,0.55802,0.32799,0.24999,0.571,0.85704,0.0,1.0,2,6,1,2,0,6,0,0,1,0,0,5,0,0,4,0,0,4,0,0,4,0,6],[16,25,0.64,0.77677,0.28333,0.71421,0.85714,1.0,0.14286,1.0,0,14,0,0,0,3,0,0,1,0,0,3,0,0,0,0,0,4,0,0,7,0,14],[20,25,0.8,0.73658,0.15201,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,13,0,0,5,0,5],[24,25,0.96,0.83033,0.14917,0.71429,0.85714,1.0,0.571,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,12,0,0,5,0,12],[25,25,1.0,0.82128,0.15579,0.71429,0.85714,1.0,0.571,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,9,0,0,7,0,11]]}]},{"i":"7f245cdfd43ed3fe","q":"$O$  is a point inside triangle  $ABC$  such that  $OA=OB+OC$ . Suppose  $B',C'$  be midpoints of arcs  $\\overarc{AOC}$  and  $AOB$ . Prove that circumcircles  $COC'$  and  $BOB'$  are tangent to each other.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,12,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.01786,"p":[[0,12,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"72f5bce3dd1bc398","q":"11. $(\\mathbf{G B R})^{\\mathrm{IMO} 2}$ Let $a_{1}, a_{2}, a_{3}, \\ldots$ be any infinite increasing sequence of positive integers. (For every integer $i>0, a_{i+1}>a_{i}$.) Prove that there are infinitely many $m$ for which positive integers $x, y, h, k$ can be found such that $0<h<k<m$ and $a_{m}=x a_{h}+y a_{k}$.","t":[{"b":2,"e":0.2857,"k":"falling","v":0.15169,"x":0.45982,"p":[[0,17,0.0,0.45982,0.41762,0.0,0.42857,1.0,0.0,1.0,9,11,1,9,0,4,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,11],[4,17,0.2353,0.41954,0.36938,0.14286,0.2857,0.85714,0.0,1.0,4,7,0,4,0,11,0,0,3,0,0,4,0,0,1,0,0,0,0,0,2,0,7],[8,17,0.4706,0.19188,0.24645,0.0,0.14286,0.1429,0.0,1.0,10,2,0,10,0,15,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[12,17,0.7059,0.30803,0.36614,0.0,0.14286,0.42857,0.0,1.0,11,6,0,11,0,9,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,6],[16,17,0.9412,0.18751,0.18363,0.0,0.14286,0.32142,0.0,0.71429,10,0,0,10,0,12,0,0,2,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[17,17,1.0,0.15169,0.16725,0.0,0.14286,0.17868,0.0,0.571,13,0,0,13,0,11,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.20528,"x":0.54911,"p":[[0,16,0.0,0.40606,0.39638,0.14,0.2143,1.0,0.0,1.0,7,9,0,7,0,9,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,9],[4,16,0.25,0.54911,0.41666,0.14286,0.42859,1.0,0.0,1.0,5,14,0,5,0,6,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,14],[8,16,0.5,0.29901,0.34511,0.0,0.14286,0.42858,0.0,1.0,10,5,0,10,0,10,0,0,1,0,0,4,0,0,2,0,0,0,0,0,0,0,5],[12,16,0.75,0.22304,0.21117,0.105,0.14286,0.42857,0.0,1.0,8,1,0,8,0,11,0,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[16,16,1.0,0.20528,0.19867,0.14286,0.14286,0.2857,0.0,1.0,6,1,0,6,0,17,0,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"e7b1811a33a6555f","q":"17. (SWE 1) Let $\\alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $n$. Prove that: (a) the inequality $\\alpha\\left(n^{2}\\right) \\leq \\frac{1}{2} \\alpha(n)(\\alpha(n)+1)$ holds; (b) the above inequality is an equality for infinitely many positive integers; (c) there exists a sequence $\\left(n_{i}\\right)_{1}^{\\infty}$ such that $\\alpha\\left(n_{i}^{2}\\right) / \\alpha\\left(n_{i}\\right) \\rightarrow 0$ as $i \\rightarrow \\infty$.  Alternative parts: Prove that there exists a sequence $\\left(n_{i}\\right)_{1}^{\\infty}$ such that $\\alpha\\left(n_{i}^{2}\\right) / \\alpha\\left(n_{i}\\right)$ tends to (d) $\\infty$; (e) an arbitrary real number $\\gamma \\in(0,1)$; (f) an arbitrary real number $\\gamma \\geq 0$.","t":[{"b":5,"e":0.42857,"k":"flat","v":0.33929,"x":0.40625,"p":[[0,42,0.0,0.39286,0.16752,0.42857,0.42857,0.42857,0.0,0.85714,4,0,4,4,0,0,0,0,0,0,0,26,0,0,1,0,0,0,0,0,1,0,0],[4,42,0.0952,0.36161,0.14719,0.28571,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,13,0,0,15,0,0,0,0,0,0,0,0,0,0,1],[8,42,0.1905,0.35714,0.08748,0.28571,0.42857,0.42857,0.14286,0.42857,0,0,0,0,0,2,0,0,12,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.39286,0.10102,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,10,0,0,17,0,0,4,0,0,0,0,0,0,0,0],[16,42,0.381,0.34821,0.08703,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,17,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[20,42,0.4762,0.33929,0.10565,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,16,0,0,11,0,0,2,0,0,0,0,0,0,0,0],[24,42,0.5714,0.37054,0.09354,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,10,0,0,19,0,0,1,0,0,0,0,0,0,0,0],[28,42,0.6667,0.40625,0.0724,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,23,0,0,2,0,0,0,0,0,0,0,0],[32,42,0.7619,0.38839,0.08918,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,9,0,0,20,0,0,2,0,0,0,0,0,0,0,0],[36,42,0.8571,0.38839,0.0735,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,10,0,0,21,0,0,1,0,0,0,0,0,0,0,0],[40,42,0.9524,0.375,0.07784,0.28571,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,13,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[42,42,1.0,0.36161,0.07974,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"falling","v":0.12054,"x":0.4375,"p":[[0,26,0.0,0.43308,0.07563,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,26,0,0,2,0,0,1,0,0,0,0,0],[4,26,0.1538,0.4375,0.10677,0.42857,0.42857,0.42857,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,1],[8,26,0.3077,0.33482,0.18074,0.39286,0.42857,0.42857,0.0,0.57143,7,0,7,7,0,0,0,0,1,0,0,23,0,0,1,0,0,0,0,0,0,0,0],[12,26,0.4615,0.25893,0.21559,0.0,0.42857,0.42857,0.0,0.57143,13,0,12,13,0,0,0,0,0,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[16,26,0.6154,0.21875,0.22011,0.0,0.21429,0.42857,0.0,0.57143,16,0,16,16,0,0,0,0,0,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[20,26,0.7692,0.12054,0.19269,0.0,0.0,0.42857,0.0,0.42857,23,0,23,23,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e391d7ba25402bbb","q":"$ABC$  is a triangle such that  $AC<AB<BC$  and  $D$  is a point on side  $AB$  satisfying  $AC=AD$ .\nThe circumcircle of  $ABC$  meets with the bisector of angle  $A$  again at  $E$  and meets with  $CD$  again at  $F$ . $K$  is an intersection point of  $BC$  and  $DE$ . Prove that  $CK=AC$  is a necessary and sufficient condition for  $DK \\cdot EF = AC \\cdot DF$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.14268,"x":0.15624,"p":[[0,4,0.0,0.15624,0.19349,0.0,0.14286,0.1786,0.0,0.71429,14,0,0,14,0,10,0,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[4,4,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.12491,"p":[[0,17,0.0,0.04911,0.11071,0.0,0.0,0.03571,0.0,0.57143,24,0,1,24,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.12491,0.18471,0.0,0.0,0.14287,0.0,0.71429,18,0,1,18,0,7,0,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[8,17,0.4706,0.07589,0.13825,0.0,0.0,0.14286,0.0,0.57143,22,0,1,22,0,6,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,17,0.7059,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"f04105be7b02f460","q":"$n\\ge 3$  guests met at a party. Some of them know each other but there is no quartet of different guests  $a, b, c, d$  such that in pairs  $\\lbrace a, b \\rbrace, \\lbrace b, c \\rbrace, \\lbrace c, d \\rbrace, \\lbrace d, a \\rbrace$  guests know each other but in pairs  $\\lbrace a, c \\rbrace, \\lbrace b, d \\rbrace$  guests don't know each other. We say a nonempty set of guests  $X$  is an *ingroup*, when guests from  $X$  know each other pairwise and there are no guests not from  $X$  knowing all guests from  $X$ . Prove that there are at most  $\\frac{n(n-1)}{2}$  different ingroups at that party.","t":[{"b":6,"e":0.0,"k":"flat","v":0.02232,"x":0.05348,"p":[[0,17,0.0,0.03339,0.06898,0.0,0.0,0.0,0.0,0.2857,25,0,4,25,1,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.05348,0.13237,0.0,0.0,0.035,0.0,0.71429,24,0,1,24,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,17,0.9412,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.03125,"x":0.06696,"p":[[0,14,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0625,0.08702,0.0,0.0,0.14286,0.0,0.28571,20,0,2,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.06696,0.07973,0.0,0.0,0.14286,0.0,0.2857,18,0,2,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4ab91d912d16e189","q":"A quadrilateral  $ABCD$  without parallel sidelines is circumscribed around a circle centered at  $I$ . Let  $K, L, M$  and  $N$  be the midpoints of  $AB, BC, CD$  and  $DA$  respectively. It is known that  $AB \\cdot CD = 4IK \\cdot IM$ . Prove that  $BC \\cdot AD = 4IL \\cdot IN$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,32,0.0,0.03125,0.12234,0.0,0.0,0.0,0.0,0.57143,30,0,1,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,32,0.125,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,32,0.625,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[32,32,1.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,18,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,1,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"feb4976ea5d45043","q":"A circle  $\\Gamma$  is tangent to the side  $BC$  of a triangle  $ABC$  at  $X$  and tangent to the side  $AC$  at  $Y$ . A point  $P$  is taken on the side  $AB$ . Let  $XP$  and  $YP$  intersect  $\\Gamma$  at  $K$  and  $L$  for the second time,  $AK$  and  $BL$  intersect  $\\Gamma$  at  $R$  and  $S$  for the second time. Prove that  $XR$  and  $YS$  intersect on  $AB$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,30,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,20,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"86679533fd605331","q":"$E$  is the intersection point of the diagonals of the cyclic quadrilateral,  $ABCD, F$  is the intersection point of the lines  $AB$  and  $CD, M$  is the midpoint of the side  $AB$ , and  $N$  is the midpoint of the side  $CD$ . The circles circumscribed around the triangles  $ABE$  and  $ACN$  intersect for the second time at point  $K$ . Prove that the points  $F, K, M$  and  $N$  lie on one circle.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,46,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.01339,"x":0.14286,"p":[[0,17,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.14241,0.00104,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f769ea780989e6e0","q":"9. (FRA 4) For all integers $n, n \\geq 0$, there exist uniquely determined integers $a_{n}, b_{n}, c_{n}$ such that  $$ (1+4 \\sqrt[3]{2}-4 \\sqrt[3]{4})^{n}=a_{n}+b_{n} \\sqrt[3]{2}+c_{n} \\sqrt[3]{4} $$  Prove that $c_{n}=0$ implies $n=0$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,23,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,14,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"cf36c4da65f79ea7","q":"Define the function  $ g(\\cdot): \\mathbb{Z} \\to \\{0,1\\}$  such that  $ g(n) \\equal{} 0$  if  $ n < 0$ , and  $ g(n) \\equal{} 1$  otherwise. Define the function  $ f(\\cdot): \\mathbb{Z} \\to \\mathbb{Z}$  such that  $ f(n) \\equal{} n \\minus{} 1024g(n \\minus{} 1024)$  for all  $ n \\in \\mathbb{Z}$ . Define also the sequence of integers  $ \\{a_i\\}_{i \\in \\mathbb{N}}$  such that  $ a_0 \\equal{} 1$  e  $ a_{n \\plus{} 1} \\equal{} 2f(a_n) \\plus{} \\ell$ , where  $ \\ell \\equal{} 0$  if  $ \\displaystyle \\prod_{i \\equal{} 0}^n{\\left(2f(a_n) \\plus{} 1 \\minus{} a_i\\right)} \\equal{} 0$ , and  $ \\ell \\equal{} 1$  otherwise. How many distinct elements are in the set  $ S: \\equal{} \\{a_0,a_1,\\ldots,a_{2009}\\}$ ?\r\n*(Paolo Leonetti)*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.02679,"x":0.26786,"p":[[0,36,0.0,0.25447,0.15866,0.14286,0.28571,0.32143,0.0,0.57143,5,0,3,5,0,7,0,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[4,36,0.1111,0.14277,0.18211,0.0,0.0,0.28571,0.0,0.57143,17,0,1,17,0,5,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[8,36,0.2222,0.08928,0.13243,0.0,0.0,0.14286,0.0,0.5714,19,0,1,19,0,8,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,36,0.3333,0.06696,0.11285,0.0,0.0,0.14286,0.0,0.42857,22,0,3,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.02679,0.05577,0.0,0.0,0.0,0.0,0.143,26,0,2,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.08482,0.14665,0.0,0.0,0.14286,0.0,0.4286,22,0,5,22,0,5,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.05795,0.10005,0.0,0.0,0.14071,0.0,0.28571,23,0,3,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.17856,0.15969,0.0,0.14286,0.28571,0.0,0.571,10,0,5,10,0,10,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[32,36,0.8889,0.23214,0.13243,0.14286,0.2857,0.28571,0.0,0.57143,3,0,2,3,0,12,0,0,12,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[36,36,1.0,0.26786,0.15047,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,10,0,0,8,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"flat","v":0.28572,"x":0.40178,"p":[[0,6,0.0,0.28572,0.16752,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,10,0,0,6,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[4,6,0.6667,0.38393,0.15746,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,4,0,0,7,0,0,13,0,0,6,0,0,1,0,0,0,0,0],[6,6,1.0,0.40178,0.19703,0.28571,0.42857,0.57143,0.0,0.857,1,0,0,1,0,6,0,0,6,0,0,7,0,0,10,0,0,1,0,0,1,0,0]]}]},{"i":"10b162215e5d84e8","q":"Arrange arbitrarily  $1,2,\\ldots ,25$  on a circumference. We consider all  $25$  sums obtained by adding  $5$  consecutive numbers. If the number of distinct residues of those sums modulo  $5$  is  $d$    $(0\\le d\\le 5)$ ,find all possible values of  $d$ .","t":[{"b":1,"e":0.2857,"k":"flat","v":0.13839,"x":0.21866,"p":[[0,32,0.0,0.13839,0.14054,0.0,0.14286,0.14286,0.0,0.57143,11,0,4,11,0,15,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,32,0.125,0.16071,0.12753,0.10714,0.14286,0.28571,0.0,0.57143,8,0,5,8,0,14,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,32,0.25,0.17857,0.21129,0.0,0.14286,0.2857,0.0,1.0,10,1,5,10,0,13,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,32,0.375,0.21428,0.19885,0.14286,0.14286,0.28571,0.0,1.0,6,1,1,6,0,14,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[16,32,0.5,0.18303,0.18638,0.14286,0.14286,0.17857,0.0,1.0,6,1,4,6,0,18,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[20,32,0.625,0.21866,0.10711,0.14286,0.14286,0.28571,0.14,0.57143,0,0,0,0,0,19,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,32,0.75,0.20535,0.07936,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,19,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.19196,0.07667,0.14286,0.14286,0.2857,0.0,0.28571,1,0,1,1,0,19,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.1875,0.06621,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.12491,"x":0.19196,"p":[[0,28,0.0,0.19196,0.18073,0.14286,0.14286,0.2857,0.0,1.0,6,1,2,6,0,15,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,28,0.1429,0.17857,0.13363,0.14286,0.14286,0.17857,0.0,0.71429,4,0,0,4,0,20,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,28,0.2857,0.14732,0.17308,0.0,0.14286,0.14286,0.0,0.57143,13,0,3,13,0,12,0,0,3,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,28,0.4286,0.1517,0.15542,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,15,0,0,4,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[16,28,0.5714,0.16509,0.11357,0.14286,0.14286,0.17857,0.0,0.57143,5,0,0,5,0,19,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,28,0.7143,0.12491,0.09277,0.0,0.14286,0.14286,0.0,0.28571,9,0,1,9,0,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.15616,0.05488,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.14723,0.04352,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"47978f0820ef87cf","q":"$ AA_{3}$  and  $ BB_{3}$  are altitudes of acute-angled  $ \\triangle ABC$ . Points  $ A_{1}$  and  $ B_{1}$  are second points of intersection lines  $ AA_{3}$  and  $ BB_{3}$  with circumcircle of  $ \\triangle ABC$  respectively.  $ A_{2}$  and  $ B_{2}$  are points on  $ BC$  and  $ AC$  respectively.  $ A_{1}A_{2}\\parallel AC$ ,  $ B_{1}B_{2}\\parallel BC$ . Point  $ M$  is midpoint of  $ A_{2}B_{2}$ .  $ \\angle BCA \\equal{} x$ . Find  $ \\angle A_{3}MB_{3}$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.04911,"x":0.12946,"p":[[0,20,0.0,0.07366,0.12305,0.0,0.0,0.14286,0.0,0.35714,23,0,1,23,0,2,0,0,6,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.06695,0.11699,0.0,0.0,0.07143,0.0,0.357,22,0,3,22,3,1,0,1,3,0,2,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.04911,0.08458,0.0,0.0,0.07143,0.0,0.2857,20,0,5,20,8,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.08706,0.099,0.0,0.03571,0.14287,0.0,0.28571,16,0,1,16,2,8,0,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.12946,0.09007,0.07132,0.14286,0.16075,0.0,0.28571,7,0,0,7,4,13,0,4,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.10044,0.11689,0.0,0.03571,0.16075,0.0,0.357,16,0,0,16,2,6,0,2,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.08705,"p":[[0,33,0.0,0.08031,0.12209,0.0,0.0,0.14286,0.0,0.357,21,0,1,21,1,3,0,0,6,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.02009,0.05712,0.0,0.0,0.0,0.0,0.2857,27,0,4,27,3,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.08705,0.12967,0.0,0.0,0.2857,0.0,0.28571,22,0,1,22,0,0,0,1,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.01785,0.05646,0.0,0.0,0.0,0.0,0.28571,28,0,6,28,2,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.01116,0.05085,0.0,0.0,0.0,0.0,0.2857,30,0,10,30,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,7,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.00665,0.02068,0.0,0.0,0.0,0.0,0.07143,29,0,5,29,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[33,33,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8cd60b05ed8a182b","q":"Find all prime numbers  $p,q,r$  for  $p|2q+5$ ,  $q|2r+5$  and  $r|2p+5$ .","t":[{"b":4,"e":0.71429,"k":"rising","v":0.69195,"x":0.9375,"p":[[0,15,0.0,0.69195,0.2345,0.42859,0.71429,0.85714,0.28571,1.0,0,7,0,0,0,0,0,0,3,0,0,6,0,0,3,0,0,8,0,0,5,0,7],[4,15,0.2667,0.75443,0.20899,0.57142,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,9,0,0,2,0,11],[8,15,0.5333,0.75446,0.22933,0.67857,0.85707,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,7,0,0,8,0,9],[12,15,0.8,0.70089,0.25092,0.42857,0.71429,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,4,0,0,6,0,0,1,0,0,7,0,0,6,0,8],[15,15,1.0,0.9375,0.08703,0.85714,1.0,1.0,0.71429,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,20]]},{"b":5,"e":1.0,"k":"flat","v":0.62043,"x":0.71874,"p":[[0,17,0.0,0.62043,0.23603,0.42857,0.71429,0.75,0.14,1.0,0,4,0,0,0,1,0,0,3,0,0,9,0,0,2,0,0,9,0,0,4,0,4],[4,17,0.2353,0.67397,0.24044,0.42857,0.71429,0.85714,0.14,1.0,0,6,0,0,0,1,0,0,1,0,0,9,0,0,2,0,0,7,0,0,6,0,6],[8,17,0.4706,0.70981,0.2612,0.5354,0.78571,0.89286,0.0,1.0,1,8,1,1,0,0,0,0,2,0,0,5,0,0,4,0,0,4,0,0,8,0,8],[12,17,0.7059,0.63837,0.22584,0.42857,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,11,0,0,2,0,0,10,0,0,1,0,6],[16,17,0.9412,0.71874,0.17673,0.71429,0.71429,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,15,0,0,6,0,4],[17,17,1.0,0.69639,0.17408,0.57132,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,8,0,0,8,0,3]]}]},{"i":"8ed239337608b6a3","q":"6. (CUB) Let $P(z)$ and $Q(z)$ be complex-variable polynomials, with degree not less than 1. Let  $$ P_{k}=\\{z \\in \\mathbb{C} \\mid P(z)=k\\}, \\quad Q_{k}=\\{z \\in \\mathbb{C} \\mid Q(z)=k\\} $$  Let also $P_{0}=Q_{0}$ and $P_{1}=Q_{1}$. Prove that $P(z) \\equiv Q(z)$.","t":[{"b":1,"e":1.0,"k":"rising","v":0.20982,"x":1.0,"p":[[0,32,0.0,0.29464,0.35524,0.0,0.21428,0.32143,0.0,1.0,13,5,0,13,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[4,32,0.125,0.32588,0.38337,0.0,0.14288,0.60714,0.0,1.0,14,5,0,14,0,3,0,0,5,0,0,0,0,0,2,0,0,1,0,0,2,0,5],[8,32,0.25,0.29909,0.35777,0.0,0.21428,0.46418,0.0,1.0,14,5,0,14,0,2,0,0,7,0,0,1,0,0,2,0,0,1,0,0,0,0,5],[12,32,0.375,0.20982,0.32338,0.0,0.0,0.2857,0.0,1.0,17,4,0,17,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[16,32,0.5,0.29455,0.36062,0.0,0.14286,0.42857,0.0,1.0,13,5,0,13,0,5,0,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,5],[20,32,0.625,0.45536,0.39357,0.14286,0.28571,1.0,0.0,1.0,7,10,0,7,0,3,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,10],[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],[28,32,0.875,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[32,32,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.34375,"p":[[0,35,0.0,0.30803,0.37476,0.0,0.21428,0.35714,0.0,1.0,14,6,0,14,0,2,0,0,8,0,0,0,0,0,1,0,0,1,0,0,0,0,6],[4,35,0.1143,0.30356,0.34208,0.0,0.21428,0.42858,0.0,1.0,11,5,0,11,0,5,0,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,5],[8,35,0.2286,0.34375,0.38937,0.0,0.2857,0.53571,0.0,1.0,12,7,1,12,0,3,0,0,8,0,0,1,0,0,0,0,0,0,0,0,1,0,7],[12,35,0.3429,0.19195,0.29364,0.0,0.07143,0.2857,0.0,1.0,16,3,1,16,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[16,35,0.4571,0.25447,0.36897,0.0,0.0,0.32143,0.0,1.0,17,5,1,17,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[20,35,0.5714,0.13393,0.20497,0.0,0.0,0.28571,0.0,0.85714,20,0,0,20,0,1,0,0,8,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[24,35,0.6857,0.12491,0.19149,0.0,0.0,0.17857,0.0,0.85714,18,0,0,18,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[28,35,0.8,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63bc21b899481af6","q":"Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer $n$ there exists a triangle of area $n$ having its vertices of the chosen colour.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"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],[4,61,0.0656,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,61,0.1311,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,61,0.1967,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,61,0.2623,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,61,0.3279,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,61,0.3934,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,61,0.459,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,61,0.5246,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,61,0.5902,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,61,0.6557,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,61,0.7213,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,61,0.7869,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,61,0.8525,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,61,0.918,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,61,0.9836,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,30,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b4cfea519499cbb","q":"Consider 6 points in the plane, arranged such that the triangle formed by any three of them has three sides of distinct lengths. Show that there exists a triangle for which the shortest side is also the longest side of another triangle.","t":[{"b":5,"e":0.0,"k":"flat","v":0.05786,"x":0.17392,"p":[[0,20,0.0,0.12938,0.11494,0.0,0.14286,0.14286,0.0,0.42857,9,0,3,9,0,20,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.12938,0.12036,0.0,0.14286,0.14286,0.0,0.57143,10,0,5,10,0,17,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.17392,0.1504,0.14214,0.14286,0.14287,0.0,0.57143,6,0,4,6,0,20,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,20,0.6,0.08027,0.08696,0.0,0.07,0.14286,0.0,0.28571,16,0,6,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.05795,0.07864,0.0,0.0,0.14286,0.0,0.28571,20,0,1,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.05786,0.06995,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14,"k":"flat","v":0.13839,"x":0.26786,"p":[[0,15,0.0,0.15179,0.13804,0.10714,0.14286,0.14286,0.0,0.57143,8,0,5,8,0,19,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.26786,0.18814,0.14286,0.14286,0.42857,0.0,0.57143,4,0,1,4,0,13,0,0,3,0,0,7,0,0,5,0,0,0,0,0,0,0,0],[8,15,0.5333,0.18285,0.13476,0.14286,0.14286,0.14286,0.0,0.57143,3,0,1,3,0,23,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[12,15,0.8,0.15626,0.15714,0.0,0.14286,0.14287,0.0,0.57143,10,0,0,10,0,16,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[15,15,1.0,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"aef08e51a34a40dd","q":"2. (USA 3) At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_{1}, C_{2}, \\ldots, C_{k}$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \\geq 4$, then $k \\geq 2 n$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.14286,"p":[[0,22,0.0,0.14286,0.17496,0.0,0.07143,0.28571,0.0,0.71429,16,0,3,16,0,5,0,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,22,0.1818,0.125,0.17768,0.0,0.0,0.14287,0.0,0.71429,17,0,4,17,0,8,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,22,0.3636,0.07143,0.13363,0.0,0.0,0.14286,0.0,0.5714,23,0,1,23,0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,22,0.5455,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,5,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,5,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03571,"x":0.13393,"p":[[0,33,0.0,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,23,0,1,23,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.13393,0.20497,0.0,0.0,0.2857,0.0,0.85714,19,0,7,19,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[8,33,0.2424,0.11161,0.18809,0.0,0.0,0.17857,0.0,0.71429,22,0,4,22,0,2,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[12,33,0.3636,0.10267,0.17213,0.0,0.0,0.14286,0.0,0.857,19,0,6,19,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,33,0.4848,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,4,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.07589,0.15966,0.0,0.0,0.0,0.0,0.57143,25,0,2,25,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[24,33,0.7273,0.03571,0.10101,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2d9cdc6d06dd1a43","q":"A crazy physicist discovered a new kind of particle which he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.  (i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.  (ii) At any moment, he may double the whole family of imons in his lab by creating a copy $I^{\\prime}$ of each imon $I$. During this procedure, the two copies $I^{\\prime}$ and $J^{\\prime}$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I^{\\prime}$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment.  Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.  (Japan)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.19196,"x":0.22321,"p":[[0,20,0.0,0.20536,0.07936,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,19,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.19196,0.07668,0.14286,0.14286,0.28571,0.0,0.28571,1,0,1,1,0,19,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.22321,0.07936,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,15,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.22303,0.07954,0.14286,0.2857,0.28571,0.14,0.42857,0,0,0,0,0,15,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.20518,0.0795,0.14286,0.14288,0.28571,0.0,0.28571,1,0,0,1,0,16,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.2008,0.07023,0.14286,0.14286,0.2857,0.14,0.28571,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.20071,"x":0.22768,"p":[[0,7,0.0,0.20071,0.09365,0.14286,0.14286,0.28571,0.0,0.42857,2,0,1,2,0,16,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.22768,0.07873,0.14286,0.2857,0.28571,0.14286,0.42857,0,0,0,0,0,14,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.21866,0.07138,0.14286,0.2857,0.28571,0.14,0.28571,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"97d6ca3edfa23105","q":"$100$  distinct natural numbers  $a_1, a_2, a_3, \\ldots, a_{100}$  are written on the board. Then, under each number  $a_i$ , someone wrote a number  $b_i$ , such that  $b_i$  is the sum of  $a_i$  and the greatest common factor of the other  $99$  numbers. What is the least possible number of distinct natural numbers that can be among  $b_1, b_2, b_3, \\ldots, b_{100}$ ?","t":[{"b":0,"e":0.14286,"k":"flat","v":0.06701,"x":0.23214,"p":[[0,40,0.0,0.06701,0.12377,0.0,0.0,0.14286,0.0,0.43,23,0,3,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.17411,0.15458,0.0,0.14286,0.2857,0.0,0.57143,10,0,0,10,0,10,0,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,40,0.2,0.20982,0.10705,0.14286,0.21428,0.28571,0.0,0.4286,3,0,0,3,0,13,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.19866,0.12466,0.14286,0.14286,0.28571,0.0,0.42857,4,0,0,4,1,14,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.20974,0.1129,0.14286,0.14288,0.28571,0.0,0.42857,2,0,0,2,0,17,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.17848,0.08752,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,21,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.18304,0.11971,0.14286,0.14286,0.2857,0.0,0.4286,4,0,0,4,0,19,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.18304,0.09606,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,21,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.16518,0.09523,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.23214,0.09943,0.14286,0.21428,0.28571,0.14286,0.4286,0,0,0,0,0,16,0,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.20527,0.13338,0.14286,0.14286,0.2857,0.0,0.57143,3,0,0,3,0,18,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.07589,"x":0.14723,"p":[[0,16,0.0,0.07589,0.10092,0.0,0.0,0.14286,0.0,0.42857,18,0,4,18,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.13839,0.19393,0.0,0.14286,0.14286,0.0,0.85714,15,0,0,15,0,11,0,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0],[8,16,0.5,0.1383,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,1,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.14723,0.04352,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2df7942f520ddbea","q":"19. (ROM 2) Let $a, b, n$ be positive integers, $b>1$ and $b^{n}-1 \\mid a$. Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.11589,"x":0.17857,"p":[[0,28,0.0,0.14268,0.11294,0.14,0.14286,0.14286,0.0,0.57143,7,0,3,7,0,20,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,28,0.1429,0.13831,0.10402,0.105,0.14286,0.1429,0.0,0.42857,8,0,5,8,0,18,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.11589,0.07517,0.105,0.14286,0.14286,0.0,0.28571,8,0,3,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.17857,0.07143,0.14286,0.14286,0.14287,0.14286,0.42857,0,0,0,0,0,25,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.17411,0.12745,0.14286,0.14286,0.1786,0.0,0.71429,4,0,1,4,0,20,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,28,0.7143,0.16518,0.08073,0.14286,0.14286,0.1429,0.0,0.4286,2,0,0,2,0,24,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.13384,0.07086,0.14286,0.14286,0.14286,0.0,0.4286,4,0,1,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.16063,0.07786,0.14286,0.14286,0.1429,0.0,0.28571,3,0,2,3,0,22,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"rising","v":0.11161,"x":0.30347,"p":[[0,11,0.0,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,7,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.15152,0.06125,0.14286,0.14286,0.14286,0.0,0.28571,2,0,1,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.17393,0.09274,0.14286,0.14286,0.2857,0.0,0.42857,3,0,1,3,0,20,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.30347,0.11719,0.24999,0.28571,0.42857,0.14,0.571,0,0,0,0,0,8,0,0,13,0,0,10,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"396c0129f1993d33","q":"16. (KOR 2) Prove that $N=\\frac{5^{125}-1}{5^{25}-1}$ is a composite number.","t":[{"b":1,"e":0.0,"k":"falling","v":0.04464,"x":0.22321,"p":[[0,13,0.0,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,8,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[4,13,0.3077,0.22321,0.41178,0.0,0.0,0.03571,0.0,1.0,24,7,11,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[8,13,0.6154,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,12,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[12,13,0.9231,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,2,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[13,13,1.0,0.04464,0.18708,0.0,0.0,0.0,0.0,1.0,30,1,5,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]},{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.36161,"p":[[0,51,0.0,0.19196,0.38896,0.0,0.0,0.0,0.0,1.0,25,6,10,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[4,51,0.0784,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,8,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[8,51,0.1569,0.36161,0.46564,0.0,0.0,1.0,0.0,1.0,18,11,5,18,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11],[12,51,0.2353,0.1875,0.3719,0.0,0.0,0.0,0.0,1.0,25,4,6,25,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,4],[16,51,0.3137,0.13393,0.33108,0.0,0.0,0.0,0.0,1.0,27,4,7,27,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[20,51,0.3922,0.12946,0.32996,0.0,0.0,0.0,0.0,1.0,27,4,5,27,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[24,51,0.4706,0.16071,0.36202,0.0,0.0,0.0,0.0,1.0,26,5,11,26,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[28,51,0.549,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,14,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[32,51,0.6275,0.125,0.33072,0.0,0.0,0.0,0.0,1.0,28,4,9,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[36,51,0.7059,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,11,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,51,0.7843,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,10,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[44,51,0.8627,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c790f31a8918c9db","q":"$5000$  movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the fi\frst kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie that no one else in this group had watched. Prove that the chairman can always split the participants into exactly 100 groups. (A group consisting of one person is allowed; in this case this person submits a report).","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,6,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,6,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,11,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"78a75b562e077409","q":"27. C4 (FIN) Determine whether or not there exist two disjoint infinite sets $\\mathcal{A}$ and $\\mathcal{B}$ of points in the plane satisfying the following conditions: (i) No three points in $\\mathcal{A} \\cup \\mathcal{B}$ are collinear, and the distance between any two points in $\\mathcal{A} \\cup \\mathcal{B}$ is at least 1. (ii) There is a point of $\\mathcal{A}$ in any triangle whose vertices are in $\\mathcal{B}$, and there is a point of $\\mathcal{B}$ in any triangle whose vertices are in $\\mathcal{A}$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,8,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,2,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,34,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,5,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,1,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1bf6295e1c6ea2ea","q":"Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying\n\n$$\nP(x)^{10}+P(x)^{9}=Q(x)^{21}+Q(x)^{20} .\n$$\n\n## Ilya Bogdanov, Russia","t":[{"b":1,"e":0.0,"k":"flat","v":0.00893,"x":0.12491,"p":[[0,26,0.0,0.12491,0.07782,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.11598,0.09737,0.0,0.14286,0.14286,0.0,0.28571,11,0,4,11,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.28571,12,0,6,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.11607,0.08328,0.0,0.14286,0.14286,0.0,0.28571,9,0,1,9,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.28571,15,0,3,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.02679,"x":0.13839,"p":[[0,14,0.0,0.11607,0.08328,0.0,0.14286,0.14286,0.0,0.28571,9,0,3,9,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.11598,0.14478,0.0,0.14286,0.14286,0.0,0.71429,13,0,2,13,0,16,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,14,0.5714,0.13393,0.11811,0.0,0.14286,0.17857,0.0,0.42857,11,0,2,11,0,13,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.13839,0.15355,0.0,0.14286,0.14286,0.0,0.85714,9,0,1,9,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[14,14,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce3d0887c335f85c","q":"12. (HUN 3) At $n$ distinct points of a circular race course there are $n$ cars ready to start. Each car moves at a constant speed and covers the circle in an hour. On hearing the initial signal, each of them selects a direction and starts moving immediately. If two cars meet, both of them change directions and go on without loss of speed. Show that at a certain moment each car will be at its starting point.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.72766,"x":0.83482,"p":[[0,10,0.0,0.72766,0.29312,0.57143,0.71429,1.0,0.0,1.0,3,12,3,3,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,2,0,12],[4,10,0.4,0.78125,0.18205,0.57143,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,9,0,0,3,0,11],[8,10,0.8,0.83482,0.16016,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,11,0,0,3,0,14],[10,10,1.0,0.7857,0.2143,0.71429,0.71429,1.0,0.0,1.0,1,12,1,1,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,0,1,0,12]]},{"b":7,"e":0.42857,"k":"flat","v":0.59817,"x":0.59817,"p":[[0,5,0.0,0.59817,0.30397,0.42857,0.57143,0.75,0.0,1.0,4,7,4,4,0,0,0,0,1,0,0,5,0,0,8,0,0,6,0,0,1,0,7]]}]},{"i":"a0f57c5f00672c5f","q":"Annie has a permutation  $(a_1, a_2, \\dots ,a_{2019})$  of  $S=\\{1,2,\\dots,2019\\}$ , and Yannick wants to guess her permutation. With each guess Yannick gives Annie an  $n$ -tuple  $(y_1, y_2, \\dots, y_{2019})$  of integers in  $S$ , and then Annie gives the number of indices  $i\\in S$  such that  $a_i=y_i$ . \n\n(a) Show that Yannick can always guess Annie's permutation with at most  $1200000$  guesses. \n(b) Show that Yannick can always guess Annie's permutation with at most  $24000$  guesses.\n\n*Yannick Yao*","t":[{"b":1,"e":0.28571,"k":"rising","v":0.15179,"x":0.68303,"p":[[0,36,0.0,0.15624,0.21534,0.0,0.0,0.28571,0.0,0.857,18,0,5,18,0,2,0,0,8,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[4,36,0.1111,0.19643,0.28738,0.0,0.0,0.28571,0.0,1.0,18,1,4,18,0,2,0,0,6,0,0,2,0,0,0,0,0,1,0,0,2,0,1],[8,36,0.2222,0.24553,0.26301,0.0,0.2857,0.42857,0.0,0.85714,14,0,5,14,0,1,0,0,8,0,0,3,0,0,2,0,0,3,0,0,1,0,0],[12,36,0.3333,0.17848,0.2113,0.0,0.14143,0.28571,0.0,0.71429,15,0,10,15,0,4,0,0,8,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[16,36,0.4444,0.15179,0.19541,0.0,0.0,0.28571,0.0,0.57143,17,0,8,17,0,5,0,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[20,36,0.5556,0.32588,0.31182,0.10714,0.2857,0.57143,0.0,1.0,8,3,2,8,0,7,0,0,6,0,0,1,0,0,6,0,0,0,0,0,1,0,3],[24,36,0.6667,0.40179,0.32031,0.24999,0.28571,0.57143,0.0,1.0,7,3,5,7,0,1,0,0,9,0,0,6,0,0,2,0,0,0,0,0,4,0,3],[28,36,0.7778,0.44642,0.36377,0.0,0.42859,0.75,0.0,1.0,9,5,6,9,0,1,0,0,4,0,0,4,0,0,4,0,0,2,0,0,3,0,5],[32,36,0.8889,0.68303,0.39242,0.28571,0.85714,1.0,0.0,1.0,6,12,4,6,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,10,0,12],[36,36,1.0,0.46871,0.23208,0.42857,0.571,0.57143,0.0,0.85714,4,0,0,4,0,1,0,0,2,0,0,8,0,0,11,0,0,4,0,0,2,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.08928,"x":0.23661,"p":[[0,22,0.0,0.23661,0.23585,0.0,0.28571,0.28571,0.0,0.85714,12,0,1,12,0,2,0,0,11,0,0,3,0,0,1,0,0,2,0,0,1,0,0],[4,22,0.1818,0.12054,0.13415,0.0,0.0,0.28571,0.0,0.28571,17,0,1,17,0,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.08928,0.12752,0.0,0.0,0.2857,0.0,0.28571,21,0,1,21,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.125,0.13716,0.0,0.0,0.2857,0.0,0.28571,17,0,1,17,0,2,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.09821,0.1357,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"07ac19af626e5cc4","q":"Find all functions  $f:\\mathbb{N}\\to\\mathbb{N}$  such that for all  $x,y\\in\\mathbb{N}$ :  $$ 0\\le y+f(x)-f^{f(y)}(x)\\le1 $$  that here  $$ f^n(x)=\\underbrace{f(f(\\ldots(f}_{n}(x))\\ldots) $$","t":[{"b":3,"e":0.2857,"k":"flat","v":0.17411,"x":0.2857,"p":[[0,40,0.0,0.26786,0.1915,0.10714,0.28571,0.42857,0.0,0.57143,8,0,0,8,0,4,0,0,7,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[4,40,0.1,0.21428,0.22304,0.0,0.14286,0.42857,0.0,0.85714,13,0,0,13,0,4,0,0,6,0,0,6,0,0,2,0,0,0,0,0,1,0,0],[8,40,0.2,0.26339,0.18249,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,1,0,0,9,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[12,40,0.3,0.17411,0.20433,0.0,0.0,0.42857,0.0,0.57143,17,0,0,17,0,2,0,0,4,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[16,40,0.4,0.20982,0.19228,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,3,0,0,10,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[20,40,0.5,0.20536,0.20806,0.0,0.21428,0.42857,0.0,0.71429,14,0,0,14,0,2,0,0,7,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[24,40,0.6,0.2857,0.20823,0.10714,0.28571,0.4286,0.0,0.57143,8,0,0,8,0,3,0,0,9,0,0,5,0,0,7,0,0,0,0,0,0,0,0],[28,40,0.7,0.17411,0.19144,0.0,0.14286,0.28571,0.0,0.57143,15,0,0,15,0,4,0,0,6,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[32,40,0.8,0.25893,0.18707,0.0,0.28571,0.42857,0.0,0.57143,9,0,0,9,0,2,0,0,9,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[36,40,0.9,0.17857,0.16366,0.0,0.21428,0.28571,0.0,0.4286,13,0,0,13,0,3,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.24554,0.16065,0.14286,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,3,0,0,16,0,0,4,0,0,2,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.20535,"x":0.32588,"p":[[0,29,0.0,0.20535,0.17473,0.0,0.2857,0.28571,0.0,0.57143,12,0,0,12,0,1,0,0,13,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[4,29,0.1379,0.32588,0.20895,0.14286,0.28571,0.42857,0.0,0.85714,5,0,0,5,0,4,0,0,8,0,0,10,0,0,3,0,0,1,0,0,1,0,0],[8,29,0.2759,0.22321,0.17474,0.0,0.28571,0.42857,0.0,0.4286,10,0,0,10,0,4,0,0,8,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.26786,0.20438,0.0,0.28571,0.42857,0.0,0.71429,9,0,0,9,0,3,0,0,7,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[16,29,0.5517,0.23661,0.22192,0.0,0.2857,0.42857,0.0,0.71429,13,0,0,13,0,0,0,0,9,0,0,7,0,0,1,0,0,2,0,0,0,0,0],[20,29,0.6897,0.29021,0.24351,0.0,0.28571,0.42857,0.0,0.85714,10,0,0,10,0,2,0,0,5,0,0,11,0,0,2,0,0,0,0,0,2,0,0],[24,29,0.8276,0.27679,0.17835,0.10714,0.28571,0.42857,0.0,0.57143,8,0,0,8,0,1,0,0,9,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[28,29,0.9655,0.25446,0.19475,0.10714,0.28571,0.32143,0.0,0.85714,8,0,0,8,0,3,0,0,13,0,0,6,0,0,1,0,0,0,0,0,1,0,0],[29,29,1.0,0.22322,0.17474,0.0,0.28571,0.42857,0.0,0.4286,11,0,0,11,0,1,0,0,11,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"452bf11e18a2db46","q":"Find all  $f: R \\longrightarrow R$  such that\r\n\r\n\\[f(xy+f(x))=xf(y)+f(x)\\]\r\n\r\nfor every pair of real numbers  $x,y$ .","t":[{"b":1,"e":0.57143,"k":"flat","v":0.37944,"x":0.49995,"p":[[0,6,0.0,0.49995,0.24998,0.28571,0.4998,0.60714,0.0,1.0,2,2,0,2,0,0,0,0,9,0,0,5,0,0,8,0,0,3,0,0,3,0,2],[4,6,0.6667,0.37944,0.12165,0.28571,0.28571,0.4642,0.2857,0.57143,0,0,0,0,0,0,0,0,19,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[6,6,1.0,0.43748,0.14697,0.28571,0.42859,0.57143,0.1429,0.71429,0,0,0,0,0,1,0,0,12,0,0,4,0,0,14,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.31694,"x":0.56696,"p":[[0,39,0.0,0.42409,0.20665,0.28571,0.42857,0.57143,0.0,1.0,2,1,0,2,0,0,0,0,13,0,0,5,0,0,8,0,0,3,0,0,0,0,1],[4,39,0.1026,0.4107,0.27374,0.28571,0.28571,0.71429,0.0,1.0,6,1,1,6,0,0,0,0,11,0,0,1,0,0,5,0,0,8,0,0,0,0,1],[8,39,0.2051,0.33925,0.23347,0.24999,0.28571,0.571,0.0,0.85714,6,0,0,6,0,2,0,0,12,0,0,2,0,0,7,0,0,2,0,0,1,0,0],[12,39,0.3077,0.31694,0.25184,0.0,0.28571,0.571,0.0,0.71429,10,0,0,10,0,0,0,0,8,0,0,5,0,0,5,0,0,4,0,0,0,0,0],[16,39,0.4103,0.40174,0.14475,0.28571,0.28571,0.571,0.2857,0.71429,0,0,0,0,0,0,0,0,18,0,0,4,0,0,8,0,0,2,0,0,0,0,0],[20,39,0.5128,0.56696,0.20666,0.28571,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,1,0,0,3,0,0,17,0,0,0,0,1],[24,39,0.6154,0.44643,0.21943,0.28571,0.28571,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,18,0,0,4,0,0,2,0,0,6,0,0,0,0,2],[28,39,0.7179,0.41963,0.16727,0.28571,0.28571,0.57111,0.2857,0.71429,0,0,0,0,0,0,0,0,17,0,0,6,0,0,3,0,0,6,0,0,0,0,0],[32,39,0.8205,0.34373,0.12297,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,25,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[36,39,0.9231,0.33928,0.10565,0.28571,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[39,39,1.0,0.35268,0.12869,0.28571,0.28571,0.32143,0.2857,0.71429,0,0,0,0,0,0,0,0,24,0,0,3,0,0,3,0,0,2,0,0,0,0,0]]}]},{"i":"9730170f0148d1df","q":"A chess tournament took place between  $2n+1$  players.  Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly  $k$  games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than  $n-\\sqrt{2k}$  and no more than  $n+\\sqrt{2k}$  games.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.11607,"x":0.43304,"p":[[0,35,0.0,0.43304,0.33405,0.14286,0.35714,0.71429,0.0,1.0,4,3,2,4,0,10,0,0,2,0,0,1,0,0,3,0,0,7,0,0,2,0,3],[4,35,0.1143,0.26775,0.27377,0.0,0.14286,0.46418,0.0,0.85714,9,0,5,9,0,11,0,0,2,0,0,2,0,0,2,0,0,5,0,0,1,0,0],[8,35,0.2286,0.23661,0.2615,0.14286,0.14286,0.17857,0.0,1.0,6,1,3,6,0,18,0,0,2,0,0,1,0,0,0,0,0,3,0,0,1,0,1],[12,35,0.3429,0.24554,0.24284,0.14286,0.14286,0.17857,0.0,1.0,3,1,1,3,0,21,0,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,1],[16,35,0.4571,0.20089,0.2392,0.10714,0.14286,0.14286,0.0,1.0,8,1,4,8,0,17,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,1],[20,35,0.5714,0.13838,0.21863,0.0,0.14286,0.14286,0.0,1.0,14,1,2,14,0,15,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[24,35,0.6857,0.11607,0.1729,0.0,0.14286,0.14286,0.0,1.0,12,1,2,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,35,0.8,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.2857,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.4286,"k":"flat","v":0.14286,"x":0.52677,"p":[[0,18,0.0,0.34372,0.33664,0.14286,0.14286,0.60714,0.0,1.0,7,3,5,7,0,12,0,0,1,0,0,0,0,0,4,0,0,4,0,0,1,0,3],[4,18,0.2222,0.29902,0.26817,0.14286,0.14286,0.57143,0.0,1.0,5,1,4,5,0,14,0,0,2,0,0,2,0,0,4,0,0,4,0,0,0,0,1],[8,18,0.4444,0.2008,0.25472,0.0,0.14286,0.1786,0.0,0.85714,12,0,5,12,0,12,0,0,2,0,0,1,0,0,0,0,0,4,0,0,1,0,0],[12,18,0.6667,0.14286,0.25754,0.0,0.0,0.14286,0.0,1.0,18,1,8,18,0,10,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1],[16,18,0.8889,0.52677,0.24073,0.42857,0.4998,0.71429,0.14286,1.0,0,3,0,0,0,4,0,0,3,0,0,9,0,0,6,0,0,6,0,0,1,0,3],[18,18,1.0,0.36158,0.14715,0.2857,0.35714,0.42857,0.14286,0.57143,0,0,0,0,0,6,0,0,10,0,0,9,0,0,7,0,0,0,0,0,0,0,0]]}]},{"i":"f16b37471ef2522f","q":"Can every positive rational number $q$ be written as\n\n$$\n\\frac{a^{2021}+b^{2023}}{c^{2022}+d^{2024}}\n$$\n\nwhere $a, b, c, d$ are all positive integers?\nProposed by United Kingdom","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,66,0.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.06696,0.11837,0.0,0.0,0.03571,0.0,0.28571,24,0,2,24,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,66,0.1212,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,66,0.1818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,66,0.2424,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,66,0.303,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,66,0.3636,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,66,0.4242,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,66,0.4848,0.03571,0.11845,0.0,0.0,0.0,0.0,0.57143,29,0,3,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,66,0.5455,0.03125,0.07771,0.0,0.0,0.0,0.0,0.2857,27,0,1,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,66,0.6061,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,66,0.6667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,66,0.7273,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[56,66,0.8485,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,66,0.9091,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,58,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,58,0.1379,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,58,0.7586,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d5555a3ab92a8a0a","q":"10.6  Let the insphere of a pyramid  $SABC$  touch the faces  $SAB, SBC, SCA$  at  $D, E, F$  respectively. Find all the possible values of the sum of the angles  $SDA, SEB, SFC$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.12947,"x":0.15625,"p":[[0,26,0.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,2,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.14286,1e-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],[12,26,0.4615,0.14286,1e-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],[16,26,0.6154,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.12946,"x":0.14286,"p":[[0,21,0.0,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,1,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.14286,1e-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],[12,21,0.5714,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,1,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2a0970993e7a8ae6","q":"11.4 Let  $AA_1$  and  $BB_1$  are altitudes of an acute non-isosceles triangle  $ABC$ ,  $A'$  is a midpoint of  $BC$  and  $B'$  is a midpoint of  $AC$ . A segement  $A_1B_1$  intersects  $A'B'$  at point  $C'$ . Prove that  $CC'\\perp HO$ , where  $H$  is a orthocenter and  $O$  is a circumcenter of  $ABC$ . \r\n(*L. Emel'yanov*)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,17,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.00446,"x":0.02679,"p":[[0,17,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ac4b9f78246dbb64","q":"A bagel is a loop of  $2a+2b+4$  unit squares which can be obtained by cutting a concentric  $a\\times b$  hole out of an  $(a +2)\\times (b+2)$  rectangle, for some positive integers a and b. (The side of length a of the hole is parallel to the side of length  $a+2$  of the rectangle.)\nConsider an infinite grid of unit square cells. For each even integer  $n \\ge 8$ , a bakery of order  $n$  is a finite set of cells  $ S$  such that, for every  $n$ -cell bagel  $B$  in the grid, there exists a congruent copy of  $B$  all of whose cells are in  $S$ . (The copy can be translated and rotated.) We denote by  $f(n)$  the smallest possible number of cells in a bakery of order  $ n$ .\nFind a real number  $\\alpha$  such that, for all sufficiently large even integers  $n \\ge 8$ , we have   $$ \\frac{1}{100}<\\frac{f (n)}{n^ {\\alpha}}<100 $$  *Proposed by Nikolai Beluhov*","t":[{"b":4,"e":0.0,"k":"flat","v":0.02679,"x":0.04464,"p":[[0,11,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.42857,24,0,4,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02679,"x":0.08482,"p":[[0,13,0.0,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,1,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,13,0.3077,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5f60cc27d27a3b1a","q":"$ABCD$  is a rhombus. Take points  $E$ ,  $F$ ,  $G$ ,  $H$  on sides  $AB$ ,  $BC$ ,  $CD$ ,  $DA$  respectively so that  $EF$  and  $GH$  are tangent to the incircle of  $ABCD$ . Show that  $EH$  and  $FG$  are parallel.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,29,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c429f0139e27f367","q":"Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \\(0\\le n\\le 52\\), and cuts the deck into a pile with the top \\(n\\) cards and a pile with the remaining \\(52-n\\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck,     the cards that were in the deck of size \\(n\\)     appear in order, as do the cards that were in the deck of size \\(52-n\\).)\n\nAlice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee?\n\n*Proposed by Espen Slettnes*","t":[{"b":1,"e":0.0,"k":"flat","v":0.05795,"x":0.0758,"p":[[0,6,0.0,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,5,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.06241,0.15943,0.0,0.0,0.035,0.0,0.85714,24,0,11,24,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[6,6,1.0,0.05795,0.07863,0.0,0.0,0.14286,0.0,0.28571,20,0,2,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.05804,"x":0.11599,"p":[[0,31,0.0,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.42857,22,0,8,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.08483,0.13767,0.0,0.0,0.14286,0.0,0.71429,18,0,3,18,0,12,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,31,0.2581,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,4,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,2,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.08482,0.09354,0.0,0.14286,0.14286,0.0,0.42857,15,0,2,15,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.08464,0.07002,0.0,0.14286,0.14286,0.0,0.14286,13,0,1,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,2,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.11599,0.06618,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,2,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"77b771acbbf32de8","q":"22. (UKR) (a) Do there exist functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ and $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that  $$ f(g(x))=x^{2} \\quad \\text { and } \\quad g(f(x))=x^{3} \\quad \\text { for all } x \\in \\mathbb{R} ? $$  (b) Do there exist functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ and $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that  $$ f(g(x))=x^{2} \\quad \\text { and } \\quad g(f(x))=x^{4} \\quad \\text { for all } x \\in \\mathbb{R} ? $$","t":[{"b":0,"e":0.42857,"k":"flat","v":0.33473,"x":0.53125,"p":[[0,6,0.0,0.53125,0.42142,0.0,0.57143,1.0,0.0,1.0,11,10,11,11,0,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,3,0,10],[4,6,0.6667,0.33473,0.22457,0.14286,0.35714,0.42895,0.0,0.71429,5,0,5,5,0,6,0,0,5,0,0,9,0,0,3,0,0,4,0,0,0,0,0],[6,6,1.0,0.44201,0.20316,0.28571,0.42859,0.57143,0.0,1.0,2,1,2,2,0,1,0,0,7,0,0,10,0,0,8,0,0,3,0,0,0,0,1]]},{"b":5,"e":0.0,"k":"flat","v":0.60268,"x":0.66518,"p":[[0,9,0.0,0.66518,0.35644,0.53571,0.71429,1.0,0.0,1.0,5,11,5,5,0,1,0,0,1,0,0,1,0,0,2,0,0,8,0,0,3,0,11],[4,9,0.4444,0.60268,0.36375,0.39285,0.71429,0.89286,0.0,1.0,5,8,5,5,0,2,0,0,1,0,0,5,0,0,2,0,0,2,0,0,7,0,8],[8,9,0.8889,0.61147,0.32185,0.28571,0.64071,0.89286,0.0,1.0,2,8,2,2,0,1,0,0,6,0,0,5,0,0,2,0,0,3,0,0,5,0,8]]}]},{"i":"c92e884e4d5d074e","q":"A non-isosceles triangle  $A_{1}A_{2}A_{3}$  has sides  $a_{1}$ ,  $a_{2}$ ,  $a_{3}$  with the side  $a_{i}$  lying opposite to the vertex  $A_{i}$ . Let  $M_{i}$  be the midpoint of the side  $a_{i}$ , and let  $T_{i}$  be the point where the inscribed circle of triangle  $A_{1}A_{2}A_{3}$  touches the side  $a_{i}$ . Denote by  $S_{i}$  the reflection of the point  $T_{i}$  in the interior angle bisector of the angle  $A_{i}$ . Prove that the lines  $M_{1}S_{1}$ ,  $M_{2}S_{2}$  and  $M_{3}S_{3}$  are concurrent.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.0,"x":0.03571,"p":[[0,51,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,5,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,51,0.0784,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,11,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,51,0.1569,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,3,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,51,0.2353,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,11,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,51,0.3137,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,51,0.3922,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,6,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,51,0.4706,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,3,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,51,0.549,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,4,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,51,0.6275,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,51,0.7059,0.02232,0.0724,0.0,0.0,0.0,0.0,0.2857,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,51,0.7843,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,51,0.8627,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.02679,0.07524,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01339,"x":0.02232,"p":[[0,5,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,7,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f6278d3b701389ca","q":"Daniel chooses a positive integer  $n$  and tells Ana. With this information, Ana chooses a positive integer  $k$  and tells Daniel. Daniel draws  $n$  circles on a piece of paper and chooses  $k$  different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the  $k$  points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of  $k$  that allows Ana to achieve her goal regardless of how Daniel chose the  $n$  circumferences and the  $k$  points.","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.04911,"p":[[0,38,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,6,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.04911,0.13175,0.0,0.0,0.0,0.0,0.57143,27,0,10,27,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,38,0.2105,0.02911,0.07586,0.0,0.0,0.0,0.0,0.36,27,0,2,27,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,6,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,30,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,7,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"351e685a24189bad","q":"5. Given $n$ points in the plane such that no three of them are collinear, prove that one can find at least $\\binom{n-3}{2}$ convex quadrilaterals with their vertices at these points.","t":[{"b":0,"e":0.1429,"k":"flat","v":0.10268,"x":0.22321,"p":[[0,15,0.0,0.14277,0.13363,0.0,0.14286,0.1786,0.0,0.42857,11,0,3,11,0,13,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.22321,0.16342,0.14286,0.14286,0.42857,0.0,0.42857,7,0,1,7,0,10,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.13393,0.12339,0.0,0.14286,0.14287,0.0,0.42857,11,0,3,11,0,14,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.14723,0.09771,0.14286,0.14286,0.14286,0.0,0.4286,5,0,2,5,0,23,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.07134,"x":0.13384,"p":[[0,12,0.0,0.09821,0.10374,0.0,0.14286,0.14286,0.0,0.42857,14,0,2,14,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.10705,0.10711,0.0,0.14286,0.14286,0.0,0.42857,12,0,2,12,0,18,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.07134,0.10708,0.0,0.0,0.14286,0.0,0.42857,20,0,3,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"227a2274202c9e3b","q":"A  $100 \\times 100$  table is given. For each  $k, 1 \\le k \\le 100$ , the  $k$ -th row of the table contains the numbers  $1,2,\\dotsc,k$  in increasing order (from left to right) but not necessarily in consecutive cells; the remaining  $100-k$  cells are filled with zeroes. Prove that there exist two columns such that the sum of the numbers in one of the columns is at least  $19$  times as large as the sum of the numbers in the other column.","t":[{"b":0,"e":0.0,"k":"flat","v":0.16518,"x":0.23214,"p":[[0,8,0.0,0.16518,0.25028,0.0,0.0,0.28571,0.0,1.0,18,1,18,18,0,3,0,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[4,8,0.5,0.23214,0.26184,0.0,0.2857,0.28571,0.0,1.0,12,2,12,12,0,3,0,0,12,0,0,2,0,0,0,0,0,1,0,0,0,0,2]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.24552,"p":[[0,9,0.0,0.24552,0.25562,0.0,0.28571,0.28571,0.0,1.0,11,2,11,11,0,2,0,0,14,0,0,1,0,0,2,0,0,0,0,0,0,0,2],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,31,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7d520f7d31a02f53","q":"**Problem 2**\nConsider the infinite chessboard whose rows and columns are indexed by positive integers. Is it\npossible to put a single positive rational number into each cell of the chessboard so that each positive rational\nnumber appears exactly once and the sum of every row and of every column is finite?","t":[{"b":1,"e":0.14,"k":"flat","v":0.16963,"x":0.29911,"p":[[0,25,0.0,0.29017,0.29119,0.0,0.2857,0.42857,0.0,1.0,11,2,6,11,0,4,0,0,4,0,0,7,0,0,3,0,0,0,0,0,1,0,2],[4,25,0.16,0.21875,0.25501,0.0,0.14286,0.42857,0.0,1.0,13,1,5,13,0,6,0,0,4,0,0,6,0,0,1,0,0,0,0,0,1,0,1],[8,25,0.32,0.20089,0.21975,0.0,0.14286,0.42857,0.0,0.71429,12,0,4,12,0,10,0,0,1,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[12,25,0.48,0.29911,0.29957,0.0,0.14286,0.4643,0.0,1.0,10,2,3,10,0,7,0,0,2,0,0,5,0,0,3,0,0,3,0,0,0,0,2],[16,25,0.64,0.29464,0.24468,0.14286,0.21428,0.42858,0.0,1.0,6,1,1,6,0,10,0,0,3,0,0,6,0,0,5,0,0,1,0,0,0,0,1],[20,25,0.8,0.20509,0.17483,0.14,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,15,0,0,6,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[24,25,0.96,0.16963,0.14476,0.14286,0.14286,0.14287,0.0,0.571,7,0,0,7,0,18,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[25,25,1.0,0.23205,0.1741,0.14286,0.14286,0.42857,0.0,0.57143,5,0,0,5,0,14,0,0,4,0,0,6,0,0,3,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0758,"x":0.3259,"p":[[0,16,0.0,0.20536,0.26471,0.0,0.14286,0.2857,0.0,1.0,14,1,9,14,0,7,0,0,4,0,0,3,0,0,1,0,0,1,0,0,1,0,1],[4,16,0.25,0.32587,0.35575,0.0,0.14288,0.5711,0.0,1.0,12,5,3,12,0,5,0,0,2,0,0,3,0,0,5,0,0,0,0,0,0,0,5],[8,16,0.5,0.27232,0.25595,0.0,0.21428,0.42857,0.0,1.0,9,1,4,9,0,7,0,0,4,0,0,7,0,0,3,0,0,0,0,0,1,0,1],[12,16,0.75,0.3259,0.36287,0.0,0.14286,0.50002,0.0,1.0,12,5,2,12,0,5,0,0,3,0,0,4,0,0,0,0,0,3,0,0,0,0,5],[16,16,1.0,0.0758,0.0873,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b6137066bce023ad","q":"Find all monic nonconstant polynomials  $P$  with integer coefficients for which there exist positive integers  $a$  and  $m$  such that for all positive integers  $n\\equiv a\\pmod m$ ,  $P(n)$  is nonzero and   $$ 2022\\cdot\\frac{(n+1)^{n+1} - n^n}{P(n)} $$   is an integer. \n\n*Jaedon Whyte, Luke Robitaille, and Pitchayut Saengrungkongka*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,6,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,13,0.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"7599a1de89a195c1","q":"20. (CZS 1) Given $n(n \\geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^{\\circ}$, prove that these points can be denoted by $A_{1}, A_{2}, \\ldots, A_{n}$ in such a way that for each $i, j, k, 1 \\leq i<j<k \\leq n$, angle $A_{i} A_{j} A_{k}$ is greater than or equal to $120^{\\circ}$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03116,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.03116,0.0589,0.0,0.0,0.0,0.0,0.1429,25,0,3,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,9,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b774674d66e106db","q":"A sequence of real numbers $a_{1}, a_{2}, \\ldots$ satisfies the relation  $$ a_{n}=-\\max _{i+j=n}\\left(a_{i}+a_{j}\\right) \\quad \\text { for all } n>2017 $$  Prove that this sequence is bounded, i.e., there is a constant $M$ such that $\\left|a_{n}\\right| \\leqslant M$ for all positive integers $n$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,28,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,17,0.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,2,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"87ebd9580bc53881","q":"$2 n$ distinct tokens are placed at the vertices of a regular $2 n$-gon, with one token placed at each vertex. A move consists of choosing an edge of the $2 n$-gon and interchanging the two tokens at the endpoints of that edge. Suppose that after a finite number of moves, every pair of tokens have been interchanged exactly once. Prove that some edge has never been chosen.\n(Russia) Alexander Gribalko","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,34,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,17,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,25,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,6,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,25,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"265d80be4100fd79","q":"A function  $ f$  defined on the positive integers (and taking positive integers values) is given by:\r\n\r $ \\begin{matrix} f(1) \\equal{} 1, f(3) \\equal{} 3 \r\nf(2 \\cdot n) \\equal{} f(n) \r\nf(4 \\cdot n \\plus{} 1) \\equal{} 2 \\cdot f(2 \\cdot n \\plus{} 1) \\minus{} f(n) \r\nf(4 \\cdot n \\plus{} 3) \\equal{} 3 \\cdot f(2 \\cdot n \\plus{} 1) \\minus{} 2 \\cdot f(n), \\end{matrix}$ \r\n\r\nfor all positive integers  $ n.$  Determine with proof the number of positive integers  $ \\leq 1988$  for which  $ f(n) \\equal{} n.$","t":[{"b":0,"e":1.0,"k":"flat","v":0.80357,"x":0.85268,"p":[[0,7,0.0,0.80357,0.21651,0.71429,0.85714,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,7,0,12],[4,7,0.5714,0.84375,0.18336,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,7,0,0,6,0,15],[7,7,1.0,0.85268,0.16935,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,2,0,17]]},{"b":5,"e":0.57143,"k":"flat","v":0.69195,"x":0.96428,"p":[[0,48,0.0,0.80357,0.25191,0.67857,0.92857,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,0,4,0,16],[4,48,0.0833,0.87054,0.22406,0.82143,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,5,0,0,4,0,20],[8,48,0.1667,0.91518,0.15093,0.92857,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,0,0,24],[12,48,0.25,0.93304,0.14719,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,1,0,26],[16,48,0.3333,0.95089,0.10479,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,25],[20,48,0.4167,0.96428,0.10102,1.0,1.0,1.0,0.5714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[24,48,0.5,0.87946,0.16793,0.82143,1.0,1.0,0.57143,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,0,5,0,19],[28,48,0.5833,0.89284,0.16368,0.82132,1.0,1.0,0.5714,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,3,0,0,3,0,21],[32,48,0.6667,0.88393,0.17655,0.82143,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,4,0,20],[36,48,0.75,0.87946,0.14334,0.71429,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,5,0,17],[40,48,0.8333,0.87501,0.19147,0.71429,1.0,1.0,0.286,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,0,1,0,21],[44,48,0.9167,0.71428,0.18557,0.57143,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,10,0,0,6,0,5],[48,48,1.0,0.69195,0.16794,0.57143,0.71429,0.75,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,9,0,0,11,0,0,4,0,4]]}]},{"i":"c95a5533dcfd196d","q":"30. (BUL 3) Two students $A$ and $B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $A$ : \"Can you tell the integer written by the other student?\" If $A$ answers \"no,\" the referee puts the same question to student $B$. If $B$ answers \"no,\" the referee puts the question back to $A$, and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer \"yes.\"","t":[{"b":2,"e":1.0,"k":"volatile","v":0.43746,"x":0.96874,"p":[[0,11,0.0,0.43746,0.40237,0.0,0.42857,0.89286,0.0,1.0,11,8,2,11,0,2,0,0,2,0,0,3,0,0,4,0,0,1,0,0,1,0,8],[4,11,0.3636,0.95535,0.13093,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[8,11,0.7273,0.96874,0.08559,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27],[11,11,1.0,0.92409,0.11289,0.85714,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,20]]},{"b":6,"e":0.85714,"k":"volatile","v":0.37267,"x":0.55352,"p":[[0,3,0.0,0.37267,0.385,0.0,0.21431,0.73214,0.0,1.0,13,4,3,13,0,3,0,0,1,0,0,3,0,0,2,0,0,2,1,0,3,0,4],[3,3,1.0,0.55352,0.24678,0.39286,0.4998,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,7,0,0,8,0,0,6,0,0,3,0,0,3,0,4]]}]},{"i":"27f50e5b23ab210a","q":"Find all composite positive integers \\(m\\) such that, whenever the product of two positive integers \\(a\\) and \\(b\\) is \\(m\\), their sum is a power of  $2$ .\n\n*Proposed by Harun Khan*","t":[{"b":4,"e":0.14286,"k":"falling","v":0.18302,"x":0.49106,"p":[[0,7,0.0,0.49106,0.27649,0.14286,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,10,0,0,2,0,0,1,0,0,7,0,0,9,0,0,1,0,2],[4,7,0.5714,0.37945,0.21901,0.14286,0.35714,0.57111,0.14286,0.71429,0,0,0,0,0,11,0,0,5,0,0,7,0,0,2,0,0,7,0,0,0,0,0],[7,7,1.0,0.18302,0.10243,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,27,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.36383,"x":0.70087,"p":[[0,40,0.0,0.40157,0.20949,0.1429,0.42857,0.57143,0.14,0.85714,0,0,0,0,0,10,0,0,2,0,0,9,0,0,7,0,0,3,0,0,1,0,0],[4,40,0.1,0.37713,0.22041,0.14286,0.28571,0.57143,0.14,0.71429,0,0,0,0,0,11,0,1,5,0,0,4,0,0,5,0,0,6,0,0,0,0,0],[8,40,0.2,0.41728,0.22436,0.24999,0.42857,0.60607,0.14286,0.85714,0,0,0,0,0,8,0,0,7,0,0,5,0,1,3,0,0,7,0,0,1,0,0],[12,40,0.3,0.43301,0.18721,0.39285,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,7,0,0,1,0,0,13,0,0,6,0,0,5,0,0,0,0,0],[16,40,0.4,0.36383,0.14214,0.2857,0.32136,0.42857,0.14286,0.71429,0,0,0,0,0,4,0,0,12,0,1,11,0,0,2,0,0,2,0,0,0,0,0],[20,40,0.5,0.66518,0.1411,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,18,0,0,1,0,2],[24,40,0.6,0.70087,0.0969,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,23,0,0,4,0,0],[28,40,0.7,0.67409,0.12493,0.57143,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,20,0,0,2,0,1],[32,40,0.8,0.67407,0.0961,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,18,0,0,3,0,0],[36,40,0.9,0.65179,0.13803,0.57143,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,19,0,0,3,0,0],[40,40,1.0,0.66962,0.0974,0.57143,0.71429,0.71429,0.42857,0.857,0,0,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,20,0,0,2,0,0]]}]},{"i":"ae36008063fbc307","q":"Find all pairs  $(m,n)$  of positive integers such that  $m+n$  and  $mn+1$  are both powers of  $2$ .","t":[{"b":4,"e":0.42857,"k":"flat","v":0.44195,"x":0.58036,"p":[[0,21,0.0,0.51785,0.12752,0.42857,0.4286,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,17,0,0,8,0,0,5,0,0,1,0,0],[4,21,0.1905,0.58036,0.18536,0.42857,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,14,0,0,6,0,0,6,0,0,3,0,2],[8,21,0.381,0.49537,0.12339,0.42857,0.42857,0.57111,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,21,0,0,5,0,0,4,0,0,1,0,0],[12,21,0.5714,0.45987,0.069,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,26,0,0,5,0,0,1,0,0,0,0,0],[16,21,0.7619,0.44197,0.07457,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,2,0,0,26,0,0,3,0,0,1,0,0,0,0,0],[20,21,0.9524,0.45534,0.05573,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0],[21,21,1.0,0.44195,0.08266,0.42857,0.42857,0.42857,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,24,0,0,4,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.44643,"x":0.54015,"p":[[0,9,0.0,0.54015,0.14608,0.42857,0.4286,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,17,0,0,8,0,0,5,0,0,1,0,1],[4,9,0.4444,0.53572,0.14286,0.42857,0.42857,0.60714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,6,0,0,7,0,0,0,0,1],[8,9,0.8889,0.44643,0.08564,0.42857,0.42857,0.42858,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,28,0,0,2,0,0,0,0,0,1,0,0],[9,9,1.0,0.50888,0.14696,0.42857,0.42857,0.571,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,20,0,0,6,0,0,3,0,0,1,0,1]]}]},{"i":"a5fc710e01410ee4","q":"Find the functions $f: \\mathbb{N}_{\\geqslant 1} \\mapsto \\mathbb{N}_{\\geqslant 0}$ satisfying the following two conditions:\n\n1. $f(x y)=f(x)+f(y)$ for all integers $x \\geqslant 1$ and $y \\geqslant 1$;\n2. there exists an infinite number of integers $n \\geqslant 1$ such that the equality $f(k)=f(n-k)$ holds for every integer $k$ such that $1 \\leqslant k \\leqslant n-1$.\nWe denote $\\mathbb{N}_{\\geqslant 0}$ the set of integers greater than or equal to 0, and $\\mathbb{N}_{\\geqslant 1}$ the set of integers greater than or equal to 1.","t":[{"b":1,"e":0.42857,"k":"rising","v":0.31248,"x":0.53125,"p":[[0,13,0.0,0.31248,0.20339,0.14286,0.28571,0.42857,0.0,0.71429,6,0,1,6,0,4,0,0,7,0,0,9,0,0,5,0,0,1,0,0,0,0,0],[4,13,0.3077,0.37945,0.22758,0.14289,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,6,0,0,6,0,0,7,0,0,4,0,0,6,0,0,0,0,0],[8,13,0.6154,0.38838,0.20276,0.2857,0.35714,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,9,0,0,6,0,0,5,0,0,5,0,0,0,0,0],[12,13,0.9231,0.53125,0.1606,0.42857,0.57143,0.71407,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,5,0,0,12,0,0,9,0,0,0,0,0],[13,13,1.0,0.51337,0.17444,0.39286,0.57121,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,6,0,0,8,0,0,10,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"flat","v":0.29,"x":0.40622,"p":[[0,13,0.0,0.39283,0.23144,0.24999,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,4,0,0,5,0,0,8,0,0,5,0,0,6,0,0,0,0,0],[4,13,0.3077,0.3482,0.18876,0.14289,0.35714,0.4286,0.0,0.71429,2,0,0,2,0,7,0,0,7,0,0,9,0,0,5,0,0,2,0,0,0,0,0],[8,13,0.6154,0.40622,0.21459,0.2857,0.42857,0.571,0.0,0.71429,2,0,0,2,0,5,0,0,5,0,0,11,0,0,2,0,0,7,0,0,0,0,0],[12,13,0.9231,0.29,0.17323,0.14286,0.21428,0.42857,0.14,0.71429,0,0,0,0,0,16,0,0,4,0,0,9,0,0,1,0,0,2,0,0,0,0,0],[13,13,1.0,0.37935,0.16612,0.2857,0.42857,0.4286,0.14,0.71429,0,0,0,0,0,7,0,0,6,0,0,12,0,0,5,0,0,2,0,0,0,0,0]]}]},{"i":"922a67ad8bf9cb70","q":"Assume the  $n$  sets  $A_1, A_2..., A_n$  are a partition of the set  $A=\\{1,2,...,29\\}$ , and the sum of any elements in  $A_i$  ,  $(i=1,2,...,n)$  is not equal to  $30$ . Find the smallest possible value of  $n$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.10714,"x":0.18286,"p":[[0,13,0.0,0.10714,0.15972,0.0,0.0,0.17857,0.0,0.57143,20,0,4,20,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,13,0.3077,0.165,0.12431,0.105,0.14286,0.2857,0.0,0.42857,8,0,5,8,0,13,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.18286,0.13945,0.14,0.14286,0.28571,0.0,0.57143,7,0,3,7,0,13,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04465,"x":0.20081,"p":[[0,19,0.0,0.10259,0.16838,0.0,0.0,0.14286,0.0,0.71429,20,0,9,20,0,6,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,19,0.2105,0.1383,0.21572,0.0,0.0,0.2857,0.0,0.85714,20,0,5,20,0,3,0,0,3,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[8,19,0.4211,0.10704,0.18893,0.0,0.0,0.14286,0.0,0.57143,21,0,12,21,0,6,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[12,19,0.6316,0.20081,0.23109,0.0,0.14286,0.42857,0.0,0.71429,14,0,5,14,0,7,0,0,2,0,0,3,0,0,5,0,0,1,0,0,0,0,0],[16,19,0.8421,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,2,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.05348,0.07774,0.0,0.0,0.14286,0.0,0.2857,21,0,2,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"77524ec125e8b6e4","q":"A closed polygonal line is drawn on squared paper so that its links lie on the lines of the paper (the sides of the squares are equal to 1). The lengths of all links are odd numbers. Prove that the number of links is divisible by 4 .","t":[{"b":3,"e":1.0,"k":"flat","v":0.84375,"x":0.97545,"p":[[0,16,0.0,0.84375,0.25843,0.71429,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,7,0,0,0,0,21],[4,16,0.25,0.92856,0.12376,0.85714,1.0,1.0,0.571,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,23],[8,16,0.5,0.89286,0.21129,0.85714,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,3,0,22],[12,16,0.75,0.97545,0.07278,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,1,28],[16,16,1.0,0.95089,0.10479,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,26]]},{"b":7,"e":0.71429,"k":"falling","v":0.5223,"x":0.84375,"p":[[0,30,0.0,0.84375,0.31003,0.85714,1.0,1.0,0.0,1.0,3,23,1,3,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,23],[4,30,0.1333,0.8125,0.26592,0.71429,1.0,1.0,0.0,1.0,1,18,1,1,0,1,0,0,1,0,0,0,0,0,3,0,0,7,0,0,1,0,18],[8,30,0.2667,0.78125,0.34253,0.71429,1.0,1.0,0.0,1.0,4,19,0,4,0,0,0,0,1,0,0,1,0,0,0,0,0,5,0,0,2,0,19],[12,30,0.4,0.71875,0.3204,0.42859,0.71429,1.0,0.0,1.0,1,15,0,1,0,2,0,0,4,0,0,2,0,0,0,0,0,8,0,0,0,0,15],[16,30,0.5333,0.59375,0.33333,0.39286,0.71429,0.89286,0.0,1.0,4,8,2,4,0,1,0,0,3,0,0,5,0,0,1,0,0,9,0,0,1,0,8],[20,30,0.6667,0.5625,0.24727,0.42857,0.71429,0.71429,0.0,1.0,2,1,0,2,0,3,0,0,1,0,0,5,0,0,1,0,0,19,0,0,0,0,1],[24,30,0.8,0.64275,0.19585,0.67857,0.71429,0.71429,0.14,1.0,0,2,0,0,0,2,0,0,2,0,0,2,0,0,2,0,0,22,0,0,0,0,2],[28,30,0.9333,0.62499,0.18472,0.67857,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,1,0,0,3,0,0,24,0,0,0,0,0],[30,30,1.0,0.5223,0.25154,0.42857,0.57121,0.71429,0.0,1.0,4,1,0,4,0,0,0,0,2,0,0,8,0,0,3,0,0,14,0,0,0,0,1]]}]},{"i":"fc7dfd63c5084e13","q":"Determine whether there exists an infinite sequence of nonzero digits $a_{1}, a_{2}, a_{3}, \\ldots$ and a positive integer $N$ such that for every integer $k>N$, the number $\\overline{a_{k} a_{k-1} \\ldots a_{1}}$ is a perfect square.  (Iran)","t":[{"b":1,"e":0.0,"k":"flat","v":0.03572,"x":0.03572,"p":[[0,2,0.0,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.1429,24,0,16,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.01777,"x":0.05357,"p":[[0,42,0.0,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,14,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,9,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,42,0.1905,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,7,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.28571,23,0,4,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.0267,0.06606,0.0,0.0,0.0,0.0,0.28571,27,0,5,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.01777,0.04701,0.0,0.0,0.0,0.0,0.1429,28,0,8,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,5,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,4,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,7,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2c50aab72a343ff6","q":"Ann and Bob play a game on an infinite checkered plane making moves in turn; Ann makes the first move. A move consists in orienting any unit grid-segment that has not been oriented before. If at some stage some oriented segments form an oriented cycle, Bob wins. Does Bob have a strategy that guarantees him to win?\n\nMaxim Didin, Russia","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,24,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,6,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5a620a43ddedcb2e","q":"A sequence  $a_2, a_3, \\dots, a_n$  of positive integers is said to be *campechana*, if for each  $i$  such that  $2 \\leq i \\leq n$  it holds that exactly  $a_i$  terms of the sequence are relatively prime to  $i$ . We say that the *size* of such a sequence is  $n - 1$ . Let  $m = p_1p_2 \\dots p_k$ , where  $p_1, p_2, \\dots, p_k$  are pairwise distinct primes and  $k \\geq 2$ . Show that there exist at least two different campechana sequences of size  $m$ .","t":[{"b":3,"e":0.57143,"k":"rising","v":0.19643,"x":0.48213,"p":[[0,85,0.0,0.21875,0.29447,0.0,0.0,0.57143,0.0,1.0,18,1,12,18,0,3,0,0,1,0,0,1,0,0,6,0,0,2,0,0,0,0,1],[4,85,0.0471,0.20089,0.2854,0.0,0.0,0.57143,0.0,1.0,20,1,12,20,0,1,0,0,1,0,0,1,0,0,8,0,0,0,0,0,0,0,1],[8,85,0.0941,0.32142,0.31541,0.0,0.28571,0.57143,0.0,1.0,14,1,7,14,0,1,0,0,2,0,0,0,0,0,10,0,0,4,0,0,0,0,1],[12,85,0.1412,0.35268,0.37455,0.0,0.21435,0.71429,0.0,1.0,15,2,9,15,0,1,0,0,2,0,0,0,0,0,4,0,0,4,0,0,4,0,2],[16,85,0.1882,0.22768,0.35689,0.0,0.0,0.57143,0.0,1.0,22,1,16,22,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,5,0,1],[20,85,0.2353,0.25445,0.31689,0.0,0.0,0.57143,0.0,1.0,18,1,13,18,0,0,0,0,3,0,0,0,0,0,7,0,0,2,0,0,1,0,1],[24,85,0.2824,0.19643,0.3004,0.0,0.0,0.57143,0.0,1.0,21,1,18,21,0,1,0,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,1],[28,85,0.3294,0.29911,0.3542,0.0,0.0,0.57143,0.0,1.0,17,2,11,17,0,1,0,0,1,0,0,0,0,0,7,0,0,2,0,0,2,0,2],[32,85,0.3765,0.28571,0.35714,0.0,0.0,0.57143,0.0,1.0,18,3,18,18,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,1,0,3],[36,85,0.4235,0.33929,0.35848,0.0,0.28571,0.57143,0.0,1.0,16,3,16,16,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,0,0,3],[40,85,0.4706,0.32588,0.3916,0.0,0.0,0.57143,0.0,1.0,18,5,18,18,0,0,0,0,0,0,0,0,0,0,7,0,0,2,0,0,0,0,5],[44,85,0.5176,0.30354,0.33453,0.0,0.0,0.57143,0.0,1.0,17,1,17,17,0,0,0,0,0,0,0,0,0,0,10,0,0,3,0,0,1,0,1],[48,85,0.5647,0.48213,0.32487,0.0,0.57143,0.60714,0.0,1.0,9,1,9,9,0,0,0,0,0,0,0,0,0,0,15,0,0,1,0,0,6,0,1],[52,85,0.6118,0.375,0.3004,0.0,0.57143,0.57143,0.0,1.0,12,1,12,12,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,0,0,1],[56,85,0.6588,0.21429,0.27664,0.0,0.0,0.57143,0.0,0.57143,20,0,20,20,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0],[60,85,0.7059,0.32589,0.28846,0.0,0.57143,0.57143,0.0,0.71429,14,0,14,14,0,0,0,0,0,0,0,0,0,0,17,0,0,1,0,0,0,0,0],[64,85,0.7529,0.31696,0.30667,0.0,0.57141,0.57143,0.0,1.0,15,1,15,15,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1],[68,85,0.8,0.43749,0.25738,0.42825,0.57143,0.57143,0.0,0.85714,8,0,8,8,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,1,0,0],[72,85,0.8471,0.39286,0.26486,0.0,0.57143,0.57143,0.0,0.57143,10,0,10,10,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.04465,"x":0.29901,"p":[[0,12,0.0,0.19643,0.32878,0.0,0.0,0.46429,0.0,1.0,23,2,10,23,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,0,0,2],[4,12,0.3333,0.29901,0.36662,0.0,0.0,0.60714,0.0,1.0,17,3,7,17,0,2,0,0,0,0,0,2,0,0,3,0,0,4,0,0,1,0,3],[8,12,0.6667,0.25447,0.31488,0.0,0.0,0.57143,0.0,0.85714,18,0,9,18,0,1,0,0,1,0,0,1,0,0,6,0,0,3,0,0,2,0,0],[12,12,1.0,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,2,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"147e8b32e00c2b9b","q":"A finite sequence of natural numbers  $a_1, a_2, \\dots, a_n$  is given. A sub-sequence  $a_{k+1}, a_{k+2}, \\dots, a_l$  will be called a *repetition* if there exists a natural number  $p\\leq \\frac{l-k}2$  such that  $a_i=a_{i+p}$  for  $k+1\\leq i\\leq l-p$ , but  $a_i\\neq a_{i+p}$  for  $i=k$  (if  $k>0$ ) and  $i=l-p+1$  (if  $l<n$ ).\n\nShow that the sequence contains less than  $n$  repetitions.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.04455,"x":0.12946,"p":[[0,3,0.0,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.08027,"p":[[0,10,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,2,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,2,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"f8beef3634f589a5","q":"A triangle, a line and three rectangles, with one side parallel to the given line, are given in such a way that the rectangles completely cover the sides of the triangle. Prove that the rectangles must completely cover the interior of the triangle.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,26,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.04018,0.11425,0.0,0.0,0.0,0.0,0.57143,27,0,1,27,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,26,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8d0e2b59475a00f1","q":"Alice and Bob are playing a game of Token Tag, played on an $8 \\times 8$ chessboard. At the beginning of the game, Bob places a token for each player on the board. After this, in every round, Alice moves her token, then Bob moves his token. If at any point in a round the two tokens are on the same square, Alice immediately wins. If Alice has not won by the end of 2012 rounds, then Bob wins.\n(a) Suppose that a token can legally move to any horizontally or vertically adjacent square. Show that Bob has a winning strategy for this game.\n(b) Suppose instead that a token can legally move to any horizontally, vertically, or diagonally adjacent square. Show that Alice has a winning strategy for this game.","t":[{"b":5,"e":0.85714,"k":"rising","v":0.56251,"x":0.80803,"p":[[0,11,0.0,0.56251,0.15942,0.42857,0.57143,0.71429,0.14286,0.857,0,0,0,0,0,1,0,0,2,0,0,8,0,0,9,0,0,11,0,0,1,0,0],[4,11,0.3636,0.80803,0.13175,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,7,0,8],[8,11,0.7273,0.79014,0.12873,0.71429,0.78564,0.85714,0.571,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,11,0,5],[11,11,1.0,0.8079,0.12179,0.71429,0.85714,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,13,0,0,11,0,6]]},{"b":7,"e":0.57143,"k":"flat","v":0.54462,"x":0.58929,"p":[[0,5,0.0,0.57129,0.24734,0.39293,0.64071,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,5,0,0,3,0,0,5,0,0,10,0,0,5,0,1],[4,5,0.8,0.58929,0.12242,0.42859,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,9,0,0,11,0,0,11,0,0,1,0,0],[5,5,1.0,0.54462,0.11538,0.42857,0.57121,0.60714,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,14,0,0,10,0,0,8,0,0,0,0,0]]}]},{"i":"d4698cb2ee268de9","q":"AUT   For any integer $n \\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$. (1) If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit 0 . (2) If $1 \\leq r \\leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $n=17,151,345,543$, we will have $L=17,151, R=345,543$ and $h(n)=17,151,345,542,345,542$. Prove that, starting with an arbitrary integer $n \\geq 2$, iterated application of $h$ produces the integer 1 after finitely many steps.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.02679,"p":[[0,11,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,9,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"94fc8d644096a569","q":"Consider a  $2 \\times n$  grid of points and a path consisting of  $2n-1$  straight line segments connecting all these  $2n$  points, starting from the bottom left corner and ending at the upper right corner. Such a path is called  $\\textit{efficient}$  if each point is only passed through once and no two line segments intersect. How many efficient paths are there when  $n = 2016$ ?","t":[{"b":1,"e":0.14286,"k":"flat","v":0.07143,"x":0.08482,"p":[[0,4,0.0,0.07143,0.18898,0.0,0.0,0.03571,0.0,1.0,24,1,1,24,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,4,1.0,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04911,"x":0.125,"p":[[0,14,0.0,0.125,0.26666,0.0,0.0,0.14286,0.0,1.0,22,2,1,22,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[4,14,0.2857,0.06697,0.18205,0.0,0.0,0.03571,0.0,1.0,24,1,1,24,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,14,0.5714,0.09375,0.22758,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[12,14,0.8571,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[14,14,1.0,0.05804,0.11214,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0ea8974cf9f2c4b2","q":"4. (ARM) Let $M$ and $N$ be points inside triangle $A B C$ such that  $$ \\angle M A B=\\angle N A C \\quad \\text { and } \\quad \\angle M B A=\\angle N B C . $$  Prove that  $$ \\frac{A M \\cdot A N}{A B \\cdot A C}+\\frac{B M \\cdot B N}{B A \\cdot B C}+\\frac{C M \\cdot C N}{C A \\cdot C B}=1 $$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,13,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.14286,"k":"flat","v":0.0,"x":0.00446,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2b99b9de113abff6","q":"A rectangular array has $n$ rows and six columns, where $n>2$. In each cell there is written either 0 or 1 . All rows in the array are different from each other. For each pair of rows $\\left(x_{1}, x_{2}, \\ldots, x_{6}\\right)$ and $\\left(y_{1}, y_{2}, \\ldots, y_{6}\\right)$, the row $\\left(x_{1} y_{1}, x_{2} y_{2}, \\ldots, x_{6} y_{6}\\right)$ can also be found in the array. Prove that there is a column in which at least half of the entries are zeroes.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,20,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,20,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"flat","v":0.0,"x":0.03572,"p":[[0,21,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.02679,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1a614da21b5575ad","q":"30. C7 (IRE) Let $U$ be a finite set and let $f, g$ be bijective functions from $U$ onto itself. Let $S=\\{w \\in U: f(f(w))=g(g(w))\\}, \\quad T=\\{w \\in U: f(g(w))=g(f(w))\\}$, and suppose that $U=S \\cup T$. Prove that for $w \\in U, f(w) \\in S$ if and only if $g(w) \\in S$.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.12947,"x":0.20971,"p":[[0,12,0.0,0.16508,0.17894,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,10,0,0,6,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[4,12,0.3333,0.20971,0.19226,0.0,0.14286,0.28571,0.0,0.57143,10,0,3,10,0,8,0,0,7,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[8,12,0.6667,0.13837,0.16547,0.0,0.14286,0.1429,0.0,0.571,14,0,0,14,0,11,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[12,12,1.0,0.12947,0.13997,0.0,0.14286,0.1786,0.0,0.4286,14,0,0,14,0,10,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.4286,"k":"flat","v":0.16947,"x":0.34809,"p":[[0,16,0.0,0.22765,0.20468,0.0,0.14286,0.42857,0.0,0.57143,10,0,2,10,0,7,0,0,6,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[4,16,0.25,0.16947,0.1871,0.0,0.14143,0.2857,0.0,0.57143,14,0,3,14,0,7,0,0,4,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[8,16,0.5,0.20982,0.17491,0.0,0.21428,0.32143,0.0,0.57143,10,0,1,10,0,6,0,0,8,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[12,16,0.75,0.20954,0.16753,0.14,0.14288,0.28571,0.0,0.57143,7,0,2,7,0,12,0,0,6,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[16,16,1.0,0.34809,0.19217,0.14289,0.42857,0.4642,0.0,0.57143,4,0,3,4,0,5,0,0,4,0,0,11,0,0,8,0,0,0,0,0,0,0,0]]}]},{"i":"5f3b7c8887701573","q":"Find all primes  $p, q$  and natural numbers  $n$  such that:  $p(p+1)+q(q+1)=n(n+1)$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.1942,"x":0.51786,"p":[[0,41,0.0,0.51786,0.37244,0.14286,0.57144,0.85714,0.0,1.0,1,7,0,1,0,13,0,0,1,0,0,1,0,0,0,0,0,5,0,0,4,0,7],[4,41,0.0976,0.41517,0.36132,0.14286,0.28571,0.71429,0.0,1.0,6,5,0,6,0,9,0,0,2,0,0,2,0,0,3,0,0,3,0,0,2,0,5],[8,41,0.1951,0.31699,0.24152,0.14286,0.21428,0.42893,0.0,1.0,2,1,0,2,0,14,0,0,4,0,0,5,0,0,4,0,0,1,0,0,1,0,1],[12,41,0.2927,0.37054,0.27862,0.14286,0.28571,0.46429,0.0,1.0,1,3,0,1,0,13,0,0,4,0,0,6,0,0,2,0,0,3,0,0,0,0,3],[16,41,0.3902,0.30801,0.24247,0.14286,0.14286,0.57111,0.0,0.857,3,0,0,3,0,15,0,0,2,0,0,3,0,0,5,0,0,3,0,0,1,0,0],[20,41,0.4878,0.30806,0.26026,0.14286,0.1429,0.42893,0.0,1.0,4,1,0,4,0,14,0,0,1,0,0,6,0,0,3,0,0,2,0,0,1,0,1],[24,41,0.5854,0.29007,0.25629,0.14286,0.14286,0.32144,0.0,1.0,2,1,0,2,0,18,0,0,4,0,0,1,0,0,2,0,0,3,0,0,1,0,1],[28,41,0.6829,0.2857,0.23418,0.14286,0.14286,0.46418,0.0,0.71429,3,0,0,3,0,16,0,0,4,0,0,1,0,0,3,0,0,5,0,0,0,0,0],[32,41,0.7805,0.21875,0.15561,0.14286,0.14286,0.28571,0.0,0.57143,4,0,0,4,0,16,0,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[36,41,0.878,0.1942,0.17335,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,15,0,1,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[40,41,0.9756,0.24999,0.17125,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,20,0,0,6,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[41,41,1.0,0.2857,0.22867,0.14286,0.14286,0.42857,0.0,0.85714,1,0,0,1,0,19,0,0,3,0,0,3,0,0,1,0,0,4,0,0,1,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.12946,"x":0.51784,"p":[[0,55,0.0,0.50889,0.35162,0.14286,0.57121,0.857,0.0,1.0,4,4,0,4,0,8,0,0,0,0,0,2,0,0,3,0,0,6,0,0,5,0,4],[4,55,0.0727,0.49103,0.33869,0.14286,0.49979,0.85714,0.0,1.0,3,4,0,3,0,8,0,0,2,0,0,3,0,0,4,0,0,3,0,0,5,0,4],[8,55,0.1455,0.49544,0.3518,0.14286,0.42857,0.85714,0.0,1.0,3,5,0,3,0,7,0,0,5,0,0,3,0,0,0,0,0,4,0,0,5,0,5],[12,55,0.2182,0.5,0.3481,0.14286,0.42857,0.75,0.0,1.0,3,6,0,3,0,8,0,0,1,0,0,6,0,0,0,0,0,6,0,0,2,0,6],[16,55,0.2909,0.51784,0.36201,0.14286,0.49979,0.85714,0.0,1.0,1,7,0,1,0,12,0,0,1,0,0,2,0,0,1,0,0,5,0,0,3,0,7],[20,55,0.3636,0.20973,0.26961,0.14214,0.14286,0.14286,0.0,1.0,7,2,0,7,0,20,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[24,55,0.4364,0.20535,0.23672,0.0,0.14286,0.21429,0.0,1.0,9,1,0,9,0,15,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,1],[28,55,0.5091,0.15625,0.16888,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,16,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[32,55,0.5818,0.20981,0.23413,0.10714,0.14286,0.17857,0.0,1.0,8,1,0,8,0,16,0,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,1],[36,55,0.6545,0.12946,0.10326,0.0,0.14286,0.14286,0.0,0.4286,9,0,0,9,0,18,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.24553,0.26056,0.14286,0.14286,0.21432,0.0,1.0,6,1,0,6,0,18,0,0,0,0,0,2,0,0,1,0,0,4,0,0,0,0,1],[44,55,0.8,0.24542,0.24804,0.14214,0.14286,0.32143,0.0,1.0,7,1,0,7,0,14,0,0,3,0,0,2,0,0,3,0,0,2,0,0,0,0,1],[48,55,0.8727,0.16965,0.20341,0.0,0.14286,0.14286,0.0,1.0,9,1,0,9,0,18,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[52,55,0.9455,0.23213,0.26903,0.0,0.14286,0.32143,0.0,0.85714,10,0,0,10,0,13,0,0,1,0,0,2,0,0,1,0,0,3,0,0,2,0,0],[55,55,1.0,0.18741,0.22992,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,18,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0]]}]},{"i":"e289a57473fd9b15","q":"1. (BEL 1) Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).","t":[{"b":6,"e":0.14286,"k":"flat","v":0.09366,"x":0.13839,"p":[[0,23,0.0,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.28571,7,0,4,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,3,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.14286,11,0,6,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.12937,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.12036,"x":0.14286,"p":[[0,15,0.0,0.12036,0.06292,0.14214,0.14286,0.14286,0.0,0.28571,6,0,4,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2935c6aad40adce7","q":"Find all triples  $(p,n,k)$  of positive integers, where  $p$  is a Fermat's Prime, satisfying \\[p^n + n = (n+1)^k\\].\n\n*Observation: a Fermat's Prime is a prime number of the form  $2^{\\alpha} + 1$ , for  $\\alpha$  positive integer.*","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,10,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,8,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1780aa54bf2bb745","q":"Altitudes  $BB_1$  and  $CC_1$  of acute triangle  $ABC$  intersect at  $H$ , and  $\\angle A = 60^{o}$ ,  $AB < AC$ . The median  $AM$  intersects the circumcircle of  $ABC$  at point  $K$ ;   $L$  is the midpoint of the arc  $BC$  of the circumcircle that does not contain point  $A$ ;  lines  $B_1C_1$  and  $BC$  intersect at point  $E$ . Prove that  $\\angle EHL = \\angle ABK$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.09813,"p":[[0,47,0.0,0.09813,0.08323,0.0,0.14286,0.14286,0.0,0.28571,12,0,2,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,4,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,1,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.14286,19,0,6,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,4,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.02679,0.05577,0.0,0.0,0.0,0.0,0.143,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,25,0.0,0.09375,0.08459,0.0,0.14286,0.14286,0.0,0.28571,13,0,4,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,3,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"64130b70cdb9b678","q":"An $(n, k)$-tournament is a contest with $n$ players held in $k$ rounds such that:  (i) Each player plays in each round, and every two players meet at most once.  (ii) If player $A$ meets player $B$ in round $i$, player $C$ meets player $D$ in round $i$, and player $A$ meets player $C$ in round $j$, then player $B$ meets player $D$ in round $j$.  Determine all pairs $(n, k)$ for which there exists an $(n, k)$-tournament.  (Argentina)","t":[{"b":0,"e":0.0,"k":"flat","v":0.80804,"x":0.87052,"p":[[0,10,0.0,0.87052,0.24055,0.82143,1.0,1.0,0.0,1.0,1,21,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,21],[4,10,0.4,0.81248,0.24599,0.71429,0.85714,1.0,0.0,1.0,2,13,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,7,0,13],[8,10,0.8,0.80804,0.29582,0.71429,1.0,1.0,0.0,1.0,3,17,3,3,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,17]]},{"b":7,"e":1.0,"k":"volatile","v":0.76781,"x":0.95982,"p":[[0,2,0.0,0.76781,0.32881,0.67846,0.85714,1.0,0.0,1.0,4,15,3,4,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,0,7,0,15],[2,2,1.0,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26]]}]},{"i":"d3611fcd2ac31c71","q":"A polygon in the plane (with no self-intersections) is called equitable if every line passing through the origin divides the polygon into two (possibly disconnected) regions of equal area.\nDoes there exist an equitable polygon which is not centrally symmetric about the origin?\n(A polygon is centrally symmetric about the origin if a 180-degree rotation about the origin sends the polygon to itself.)","t":[{"b":2,"e":0.28571,"k":"flat","v":0.01786,"x":0.09821,"p":[[0,15,0.0,0.0982,0.15331,0.0,0.0,0.28571,0.0,0.571,22,0,1,22,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.09821,0.1357,0.0,0.0,0.28571,0.0,0.28571,21,0,0,21,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02679,"x":0.09375,"p":[[0,22,0.0,0.09375,0.17353,0.0,0.0,0.07143,0.0,0.57143,24,0,1,24,0,0,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,22,0.1818,0.07143,0.14286,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0625,0.11811,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4124b0e797000fe7","q":"14. G8 (RUS) Points $A, B, C$ divide the circumcircle $\\Omega$ of the triangle $A B C$ into three arcs. Let $X$ be a variable point on the $\\operatorname{arc} A B$, and let $O_{1}, O_{2}$ be the incenters of the triangles $C A X$ and $C B X$. Prove that the circumcircle of the triangle $X O_{1} O_{2}$ intersects $\\Omega$ in a fixed point.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,30,0.0,0.01116,0.05085,0.0,0.0,0.0,0.0,0.2857,30,0,3,30,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,2,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,30,0.2667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.03562,"p":[[0,19,0.0,0.03562,0.09439,0.0,0.0,0.0,0.0,0.42857,27,0,4,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"88daa8adb41fddf4","q":"For any  $h = 2^{r}$  ( $r$  is a non-negative integer), find all  $k \\in \\mathbb{N}$  which satisfy the following condition: There exists an odd natural number  $m > 1$  and  $n \\in \\mathbb{N}$ , such that  $k \\mid m^{h} - 1, m \\mid n^{\\frac{m^{h}-1}{k}} + 1$ .","t":[{"b":5,"e":0.2857,"k":"falling","v":0.32141,"x":0.52227,"p":[[0,35,0.0,0.50446,0.27428,0.28571,0.42859,0.71429,0.14286,1.0,0,2,0,0,0,7,0,0,3,0,0,8,0,0,3,0,0,4,0,0,5,0,2],[4,35,0.1143,0.52227,0.26391,0.39286,0.57121,0.71429,0.0,1.0,3,1,0,3,0,3,0,0,2,0,0,4,0,0,5,0,0,13,0,0,1,0,1],[8,35,0.2286,0.5,0.31135,0.25,0.571,0.85714,0.0,1.0,3,1,0,3,0,5,0,0,5,0,0,2,0,0,4,0,0,4,0,0,8,0,1],[12,35,0.3429,0.32141,0.32536,0.0,0.2857,0.46418,0.0,1.0,11,3,0,11,0,3,0,0,6,0,0,4,0,0,2,0,0,2,0,0,1,0,3],[16,35,0.4571,0.41517,0.27975,0.14286,0.35714,0.60714,0.0,0.85714,3,0,0,3,0,8,0,0,5,0,0,1,0,0,7,0,0,4,0,0,4,0,0],[20,35,0.5714,0.39281,0.22864,0.2857,0.42857,0.571,0.0,1.0,3,1,0,3,0,4,0,0,6,0,0,10,0,0,6,0,0,1,0,0,1,0,1],[24,35,0.6857,0.40162,0.26355,0.24999,0.42857,0.57143,0.0,0.85714,5,0,0,5,0,3,0,0,7,0,0,4,0,0,7,0,0,3,0,0,3,0,0],[28,35,0.8,0.42844,0.22313,0.24999,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,7,0,0,4,0,0,6,0,0,8,0,0,5,0,0,1,0,0],[32,35,0.9143,0.37054,0.25719,0.14286,0.28571,0.57143,0.0,1.0,2,1,0,2,0,9,0,0,8,0,0,4,0,0,2,0,0,5,0,0,1,0,1],[35,35,1.0,0.34817,0.26491,0.14286,0.28571,0.5711,0.0,0.86,6,0,0,6,0,6,0,0,6,0,0,3,0,0,6,0,0,3,0,0,2,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.10259,"x":0.65175,"p":[[0,31,0.0,0.60707,0.26976,0.42857,0.64286,0.85714,0.0,1.0,1,3,0,1,0,2,0,0,3,0,0,6,0,0,4,0,0,5,0,0,8,0,3],[4,31,0.129,0.65175,0.25238,0.57132,0.71429,0.857,0.0,1.0,2,4,0,2,0,0,0,0,2,0,0,3,0,0,5,0,0,11,0,0,5,0,4],[8,31,0.2581,0.53571,0.25,0.42857,0.5,0.71429,0.0,1.0,1,2,1,1,0,3,0,0,2,0,0,10,0,0,5,0,0,5,0,0,4,0,2],[12,31,0.3871,0.44196,0.27516,0.2857,0.49999,0.57143,0.0,1.0,4,1,2,4,0,3,0,0,7,0,0,2,0,0,9,0,0,3,0,0,3,0,1],[16,31,0.5161,0.46873,0.29065,0.14286,0.42859,0.71429,0.0,1.0,3,1,1,3,0,6,0,0,3,0,0,5,0,0,2,0,0,9,0,0,3,0,1],[20,31,0.6452,0.24107,0.22428,0.0,0.2143,0.42857,0.0,0.85714,10,0,1,10,0,6,0,0,7,0,0,4,0,0,4,0,0,0,0,0,1,0,0],[24,31,0.7742,0.10259,0.1249,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,16,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,31,0.9032,0.11607,0.15746,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,9,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[31,31,1.0,0.10714,0.13832,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,9,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"39b8a7d2faa873a5","q":"Gegeven is een verzameling $A$ van functies $f: \\mathbb{R} \\rightarrow \\mathbb{R}$. Voor alle $f_{1}, f_{2} \\in A$ bestaat er een $f_{3} \\in A$ zodat\n\n$$\nf_{1}\\left(f_{2}(y)-x\\right)+2 x=f_{3}(x+y)\n$$\n\nvoor alle $x, y \\in \\mathbb{R}$. Bewijs dat voor alle $f \\in A$ geldt:\n\n$$\nf(x-f(x))=0\n$$\n\nvoor alle $x \\in \\mathbb{R}$.","t":[{"b":1,"e":0.2857,"k":"flat","v":0.20536,"x":0.29018,"p":[[0,14,0.0,0.20536,0.08703,0.14286,0.21428,0.28571,0.0,0.286,2,0,0,2,0,14,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.22759,0.07883,0.14286,0.2857,0.28571,0.14,0.4286,0,0,0,0,0,14,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.24553,0.10248,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,10,0,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,14,0.8571,0.23214,0.13243,0.14286,0.14288,0.28571,0.14286,0.71429,0,0,0,0,0,18,0,0,11,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[14,14,1.0,0.29018,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.19644,"x":0.22768,"p":[[0,15,0.0,0.21875,0.10704,0.14286,0.1429,0.28571,0.0,0.5714,1,0,0,1,0,16,0,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.20982,0.10705,0.14286,0.14286,0.28571,0.0,0.57143,2,0,0,2,0,15,0,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.22768,0.09353,0.14286,0.2857,0.28571,0.14286,0.57143,0,0,0,0,0,15,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,15,0.8,0.19644,0.08563,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,22,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.20089,0.08645,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,21,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cd55fd33f4849819","q":"A circle passing through vertices  $A$  and  $B$  of triangle  $ABC$  intersects the sides  $AC$  and  $BC$  again at points  $P$  and  $Q$ , respectively. Given that the median from vertex  $C$  bisect the arc  $PQ$  of the circle. Prove that  $ABC$  is an isosceles triangle.","t":[{"b":1,"e":0.0,"k":"flat","v":0.04,"x":0.07143,"p":[[0,13,0.0,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,3,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.4286,19,0,3,19,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.04,0.06395,0.0,0.0,0.14,0.0,0.14286,23,0,2,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,4,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,4,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.01786,"x":0.05795,"p":[[0,13,0.0,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.14286,19,0,2,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.42857,23,0,3,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3a76cc0475183181","q":"A trapezoid is given in which one base is twice as large as the other. Use one ruler (no divisions) to draw the midline of this trapezoid.","t":[{"b":2,"e":0.4286,"k":"flat","v":0.19196,"x":0.42405,"p":[[0,52,0.0,0.34374,0.39586,0.0,0.14286,0.57143,0.0,1.0,14,7,11,14,0,3,0,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,7],[4,52,0.0769,0.19196,0.18766,0.0,0.14286,0.32143,0.0,0.57143,12,0,9,12,0,7,0,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[8,52,0.1538,0.23213,0.21352,0.0,0.14288,0.42857,0.0,0.57143,10,0,7,10,0,8,0,0,4,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[12,52,0.2308,0.28117,0.20672,0.14214,0.28571,0.4286,0.0,0.57143,7,0,4,7,0,7,0,0,4,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[16,52,0.3077,0.21871,0.19549,0.0,0.14286,0.42857,0.0,0.571,10,0,8,10,0,8,0,0,4,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[20,52,0.3846,0.29008,0.20045,0.14286,0.2857,0.42858,0.0,0.71429,5,0,1,5,0,9,0,0,4,0,0,9,0,0,4,0,0,1,0,0,0,0,0],[24,52,0.4615,0.2634,0.20238,0.14286,0.2143,0.42858,0.0,0.71429,6,0,0,6,0,10,0,0,5,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[28,52,0.5385,0.27679,0.17832,0.14286,0.14295,0.42858,0.0,0.57143,1,0,0,1,0,16,0,0,6,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[32,52,0.6154,0.35714,0.20201,0.14297,0.35714,0.57143,0.0,0.71429,3,0,0,3,0,6,0,0,7,0,0,5,0,0,10,0,0,1,0,0,0,0,0],[36,52,0.6923,0.29446,0.19876,0.14286,0.21435,0.46418,0.0,0.57143,3,0,0,3,0,13,0,0,3,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[40,52,0.7692,0.29908,0.23241,0.10714,0.28571,0.571,0.0,0.71429,8,0,0,8,0,6,0,0,3,0,0,6,0,0,8,0,0,1,0,0,0,0,0],[44,52,0.8462,0.36604,0.18185,0.14289,0.42857,0.571,0.0,0.71429,1,0,0,1,0,8,0,0,5,0,0,9,0,0,8,0,0,1,0,0,0,0,0],[48,52,0.9231,0.27668,0.17109,0.14286,0.2857,0.32143,0.0,0.57143,2,0,0,2,0,12,0,0,10,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[52,52,1.0,0.42405,0.1576,0.2857,0.42857,0.57111,0.0,0.71429,1,0,0,1,0,2,0,0,7,0,0,10,0,0,11,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.11159,"x":0.40175,"p":[[0,23,0.0,0.40175,0.33962,0.0,0.28571,0.71429,0.0,1.0,9,3,7,9,0,3,0,0,5,0,0,0,0,0,4,0,0,8,0,0,0,0,3],[4,23,0.1739,0.24112,0.26595,0.0,0.14288,0.42857,0.0,1.0,14,1,10,14,0,4,0,0,1,0,0,8,0,0,3,0,0,1,0,0,0,0,1],[8,23,0.3478,0.21865,0.20512,0.0,0.21435,0.42857,0.0,0.57143,12,0,9,12,0,4,0,0,7,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[12,23,0.5217,0.16058,0.18805,0.0,0.14143,0.2857,0.0,0.57143,15,0,12,15,0,7,0,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[16,23,0.6957,0.11159,0.17396,0.0,0.0,0.14287,0.0,0.57143,20,0,16,20,0,5,0,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"243b187eddfd15b6","q":"29. (USA 3) A number of signal lights are equally spaced along a one-way railroad track, labeled in order $1,2, \\ldots, N(N \\geq 2)$. As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume that the trains have zero length.) A series of $K$ freight trains must be driven from Signal 1 to Signal $N$. Each train travels at a distinct but constant speed (i.e., the speed is fixed and different from that of each of the other trains) at all times when it is not blocked by the safety rule. Show that regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $N$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.35268,"x":0.49116,"p":[[0,6,0.0,0.49116,0.32535,0.42857,0.42857,0.75,0.0,1.0,6,6,6,6,0,0,0,0,0,0,0,17,0,0,0,0,0,1,0,0,2,0,6],[4,6,0.6667,0.35268,0.28118,0.10714,0.42857,0.42857,0.0,1.0,8,3,8,8,0,1,0,0,5,0,0,14,0,0,0,0,0,1,0,0,0,0,3]]},{"b":6,"e":0.0,"k":"flat","v":0.375,"x":0.375,"p":[[0,3,0.0,0.375,0.33264,0.0,0.42857,0.42858,0.0,1.0,10,4,9,10,0,1,0,0,3,0,0,11,0,0,0,0,0,2,0,0,1,0,4]]}]},{"i":"a7e8078f768cbf5e","q":"27. S5 (FIN) For positive integers $n$, the numbers $f(n)$ are defined inductively as follows: $f(1)=1$, and for every positive integer $n, f(n+1)$ is the greatest integer $m$ such that there is an arithmetic progression of positive integers $a_{1}<a_{2}<\\cdots<a_{m}=n$ for which  $$ f\\left(a_{1}\\right)=f\\left(a_{2}\\right)=\\cdots=f\\left(a_{m}\\right) $$  Prove that there are positive integers $a$ and $b$ such that $f(a n+b)=n+2$ for every positive integer $n$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,4,0.0,0.10714,0.15568,0.0,0.0,0.1786,0.0,0.4286,20,0,0,20,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.16072,"p":[[0,64,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,64,0.0625,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,13,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,64,0.1875,0.05804,0.11214,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,64,0.3125,0.11607,0.1448,0.0,0.0,0.1786,0.0,0.4286,17,0,3,17,0,7,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.07141,0.16747,0.0,0.0,0.0,0.0,0.71429,25,0,1,25,0,3,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[28,64,0.4375,0.10705,0.16364,0.0,0.0,0.2857,0.0,0.57143,21,0,1,21,0,2,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[32,64,0.5,0.0625,0.11258,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.08481,0.15089,0.0,0.0,0.14286,0.0,0.571,22,0,1,22,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,64,0.625,0.09375,0.14555,0.0,0.0,0.14286,0.0,0.57143,20,0,3,20,0,6,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,64,0.6875,0.16072,0.16269,0.0,0.14286,0.28571,0.0,0.4286,14,0,1,14,0,5,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"91c84478f70ecca5","q":"$A$  is a non-empty subset of positive integers. Let   $$ f(A)=\\{abc-b-c+2\\vert a,b,c\\in A\\} $$  Determine all integers  $n$  greater than  $1$  so that we can divide the set of positive integers into  $A_1, A_2, \\dots, A_n$  ( $A_i\\neq \\emptyset (i=1, 2, \\dots , n)$ ,  $\\forall 1\\le i < j \\le n, A_i\\cap A_j = \\emptyset$  and  $\\bigcup_{i=1}^{n} A_i=\\mathbb{N}^*$ ) satisfy that  $\\forall 1\\le i\\le n, f(A_i) \\subseteq A_i$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.37496,"x":0.44636,"p":[[0,14,0.0,0.44636,0.31691,0.21427,0.4998,0.57143,0.0,1.0,8,3,8,8,0,0,0,0,3,0,0,5,0,0,9,0,0,2,0,0,2,0,3],[4,14,0.2857,0.44187,0.26327,0.24999,0.571,0.57143,0.0,0.85714,6,0,6,6,0,2,0,0,1,0,0,4,0,0,15,0,0,1,0,0,3,0,0],[8,14,0.5714,0.4062,0.30114,0.0,0.571,0.57143,0.0,1.0,9,1,9,9,0,1,0,0,3,0,0,2,0,0,10,0,0,5,0,0,1,0,1],[12,14,0.8571,0.37496,0.34761,0.0,0.4998,0.57143,0.0,1.0,13,3,13,13,0,0,0,0,2,0,0,1,0,0,10,0,0,2,0,0,1,0,3]]},{"b":7,"e":0.71429,"k":"rising","v":0.37941,"x":0.79459,"p":[[0,42,0.0,0.45528,0.20647,0.42857,0.571,0.57143,0.0,0.85714,4,0,3,4,0,1,0,0,0,0,0,10,0,0,15,0,0,1,0,0,1,0,0],[4,42,0.0952,0.37941,0.27801,0.0,0.42857,0.57111,0.0,1.0,9,1,9,9,0,1,0,0,1,0,0,9,0,0,7,0,0,4,0,0,0,0,1],[8,42,0.1905,0.42407,0.25871,0.28571,0.4998,0.57143,0.0,1.0,7,1,7,7,0,0,0,0,2,0,0,7,0,0,12,0,0,3,0,0,0,0,1],[12,42,0.2857,0.39276,0.24737,0.21429,0.42857,0.571,0.0,0.85714,8,0,8,8,0,0,0,0,1,0,0,9,0,0,12,0,0,1,0,0,1,0,0],[16,42,0.381,0.50885,0.13798,0.42859,0.571,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,6,0,0,6,0,0,17,0,0,2,0,0,1,0,0],[20,42,0.4762,0.67845,0.17135,0.571,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,3,0,0,14,0,0,8,0,0,2,0,5],[24,42,0.5714,0.70968,0.16565,0.571,0.71414,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,7,0,0,5,0,5],[28,42,0.6667,0.70085,0.16118,0.57143,0.71429,0.74996,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,12,0,0,4,0,4],[32,42,0.7619,0.6829,0.17037,0.571,0.57143,0.74996,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,6,0,0,3,0,5],[36,42,0.8571,0.75887,0.1766,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,7,0,0,6,0,8],[40,42,0.9524,0.79459,0.17478,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,7,0,0,11,0,8],[42,42,1.0,0.74994,0.17501,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,8,0,0,9,0,0,6,0,7]]}]},{"i":"8eaf10f3d90f2a92","q":"Bengt wants to put out crosses and rings in the squares of an  $n \\times n$ -square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?","t":[{"b":0,"e":0.0,"k":"flat","v":0.1875,"x":0.21875,"p":[[0,8,0.0,0.1875,0.21558,0.0,0.0,0.42857,0.0,0.57143,17,0,6,17,0,2,0,0,1,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[4,8,0.5,0.21875,0.22583,0.0,0.14286,0.42857,0.0,0.71429,16,0,4,16,0,0,0,0,1,0,0,14,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"rising","v":0.16071,"x":0.32139,"p":[[0,12,0.0,0.16071,0.24936,0.0,0.0,0.42857,0.0,1.0,21,1,8,21,0,1,0,0,0,0,0,8,0,0,1,0,0,0,0,0,0,0,1],[4,12,0.3333,0.2142,0.21432,0.0,0.14143,0.42857,0.0,0.57143,15,0,2,15,0,2,0,0,0,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[8,12,0.6667,0.32139,0.2575,0.14286,0.28571,0.571,0.0,0.85714,7,0,3,7,0,7,0,0,4,0,0,4,0,0,6,0,0,3,0,0,1,0,0]]}]},{"i":"5f5510bc515c9b17","q":"For an integer $m$, denote by $t(m)$ the unique number in $\\{1,2,3\\}$ such that $m+t(m)$ is a multiple of 3. A function $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfies $f(-1)=0, f(0)=1, f(1)=-1$ and  $$ f\\left(2^{n}+m\\right)=f\\left(2^{n}-t(m)\\right)-f(m) \\quad \\text { for all integers } m, n \\geq 0 \\text { with } 2^{n}>m \\text {. } $$  Prove that $f(3 p) \\geq 0$ holds for all integers $p \\geq 0$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,50,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"8f2ba2a1596d7875","q":"A circle touches sides  $DA$ ,  $AB$ ,  $BC$ ,  $CD$  of a quadrilateral  $ABCD$  at points  $K$ ,  $L$ ,  $M$ ,  $N$ , respectively. Let  $S_1$ ,  $S_2$ ,  $S_3$ ,  $S_4$  respectively be the incircles of triangles  $AKL$ ,  $BLM$ ,  $CMN$ ,  $DNK$ . The external common tangents distinct from the sides of  $ABCD$  are drawn to  $S_1$  and  $S_2$ ,  $S_2$  and  $S_3$ ,  $S_3$  and  $S_4$ ,  $S_4$  and  $S_1$ . Prove that these four tangents determine a rhombus.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.08036,"p":[[0,23,0.0,0.08036,0.10677,0.0,0.0,0.14286,0.0,0.42857,18,0,2,18,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.42857,22,0,1,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.03562,0.07974,0.0,0.0,0.0,0.0,0.28571,26,0,4,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,11,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.0825,"p":[[0,12,0.0,0.0825,0.1172,0.0,0.0,0.14286,0.0,0.42857,19,0,1,19,0,9,0,0,2,0,1,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,3,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"919671dddd58ba84","q":"Does there exist a sequence $a_{1}, a_{2}, \\ldots, a_{n}, \\ldots$ of positive real numbers satisfying both of the following conditions:\n(i) $\\sum_{i=1}^{n} a_{i} \\leq n^{2}$, for every positive integer $n$;\n(ii) $\\sum_{i=1}^{n} \\frac{1}{a_{i}} \\leq 2008$, for every positive integer $n$ ?","t":[{"b":2,"e":0.0,"k":"falling","v":0.07589,"x":0.32143,"p":[[0,16,0.0,0.29911,0.44084,0.0,0.0,0.89286,0.0,1.0,21,8,1,21,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,8],[4,16,0.25,0.21429,0.38796,0.0,0.0,0.17857,0.0,1.0,23,6,0,23,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[8,16,0.5,0.32143,0.43154,0.0,0.07143,1.0,0.0,1.0,16,9,0,16,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[12,16,0.75,0.13393,0.20806,0.0,0.0,0.14287,0.0,1.0,17,1,0,17,0,8,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[16,16,1.0,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"volatile","v":0.00893,"x":0.98214,"p":[[0,13,0.0,0.12054,0.2969,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[4,13,0.3077,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.17411,0.33261,0.0,0.0,0.07143,0.0,1.0,24,3,0,24,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,3],[12,13,0.9231,0.94196,0.12299,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,25],[13,13,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"89e44f883fadf241","q":"100 unit squares of an infinite squared plane form a  $ 10\\times 10$  square.  Unit segments forming these squares are coloured in several colours.  It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original  $ 10\\times 10$  square.) What maximum number of colours may appear in this colouring? \r\n\r\n*Author: S. Berlov*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,10,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,17,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,43,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,10,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,18,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9ec7275156c96503","q":"Find all integers  $k \\ge 5$  for which there is a positive integer  $n$  with exactly  $k$  positive divisors $1 = d_1 <d_2 < ... <d_k = n$  and  $d_2d_3 + d_3d_5 + d_5d_2 = n$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.71429,"x":1.0,"p":[[0,20,0.0,0.71429,0.30303,0.42857,0.85714,1.0,0.0,1.0,1,10,0,1,0,2,0,0,3,0,0,3,0,0,0,0,0,5,0,0,8,0,10],[4,20,0.2,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,20,0.4,0.90625,0.13651,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,7,0,19],[12,20,0.6,0.94643,0.12242,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,25],[16,20,0.8,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[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":0.85714,"k":"flat","v":0.49554,"x":0.9107,"p":[[0,53,0.0,0.8125,0.21558,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,7,0,0,5,0,14],[4,53,0.0755,0.72321,0.28107,0.57143,0.78571,1.0,0.14286,1.0,0,11,0,0,0,4,0,0,0,0,0,1,0,0,7,0,0,4,0,0,5,0,11],[8,53,0.1509,0.49554,0.25997,0.28571,0.42859,0.71429,0.14286,1.0,0,2,0,0,0,6,0,0,5,0,0,6,0,0,5,0,0,5,0,0,3,0,2],[12,53,0.2264,0.59817,0.30186,0.39286,0.57143,0.85714,0.0,1.0,1,6,0,1,0,4,0,0,3,0,0,3,0,0,7,0,0,3,0,0,5,0,6],[16,53,0.3019,0.68749,0.27535,0.42859,0.78571,0.89286,0.14286,1.0,0,8,0,0,0,2,0,0,2,0,0,7,0,0,2,0,0,3,0,0,8,0,8],[20,53,0.3774,0.65623,0.2693,0.42859,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,4,0,0,1,0,0,4,0,0,3,0,0,8,0,0,7,0,5],[24,53,0.4528,0.70531,0.26231,0.571,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,1,0,0,2,0,0,5,0,0,7,0,0,6,0,8],[28,53,0.5283,0.79017,0.20512,0.71429,0.78571,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,11,0,0,5,0,11],[32,53,0.6038,0.7589,0.24338,0.67857,0.857,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,5,0,0,10,0,9],[36,53,0.6792,0.86607,0.2141,0.82143,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,3,0,0,4,0,20],[40,53,0.7547,0.9107,0.14177,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,20],[44,53,0.8302,0.90178,0.11538,0.85714,0.85714,1.0,0.57143,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,14,0,15],[48,53,0.9057,0.86606,0.13803,0.82132,0.85714,1.0,0.5714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,11,0,13],[52,53,0.9811,0.88837,0.11705,0.85711,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,12,0,14],[53,53,1.0,0.83034,0.17292,0.71429,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,6,0,0,9,0,12]]}]},{"i":"e4e0bdcce27b34b5","q":"Determine all (not necessarily finite) sets  $S$  of points in the plane such that given any four distinct points in  $S$ , there is a circle passing through all four or a line passing through some three.\n\n*Carl Lian.*","t":[{"b":2,"e":0.42857,"k":"falling","v":0.21418,"x":0.4464,"p":[[0,24,0.0,0.39272,0.22874,0.2857,0.4286,0.57143,0.0,0.857,6,0,4,6,0,1,0,0,4,0,0,7,0,0,13,0,0,0,0,0,1,0,0],[4,24,0.1667,0.26786,0.24419,0.0,0.28571,0.57143,0.0,0.57143,12,0,6,12,0,3,0,0,4,0,0,3,0,0,10,0,0,0,0,0,0,0,0],[8,24,0.3333,0.33023,0.25865,0.0,0.42859,0.57143,0.0,0.71429,9,0,5,9,0,5,0,0,1,0,0,2,0,0,14,0,0,1,0,0,0,0,0],[12,24,0.5,0.308,0.23983,0.0,0.28571,0.571,0.0,0.71429,10,0,7,10,0,1,0,0,6,0,0,5,0,0,9,0,0,1,0,0,0,0,0],[16,24,0.6667,0.41955,0.16738,0.28571,0.4293,0.57143,0.14,0.57143,0,0,0,0,0,6,0,0,5,0,0,6,0,0,15,0,0,0,0,0,0,0,0],[20,24,0.8333,0.4464,0.17766,0.28571,0.4998,0.57143,0.0,0.71429,1,0,1,1,0,2,0,0,8,0,0,5,0,0,13,0,0,3,0,0,0,0,0],[24,24,1.0,0.21418,0.2113,0.0,0.14286,0.42858,0.0,0.57143,12,0,12,12,0,6,0,0,5,0,0,4,0,0,5,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.06241,"x":0.44187,"p":[[0,20,0.0,0.37945,0.20078,0.25001,0.42857,0.57143,0.0,0.71429,3,0,1,3,0,5,0,0,5,0,0,7,0,0,11,0,0,1,0,0,0,0,0],[4,20,0.2,0.44187,0.20012,0.28571,0.4998,0.57143,0.0,0.71429,2,0,1,2,0,3,0,0,5,0,0,6,0,0,12,0,0,4,0,0,0,0,0],[8,20,0.4,0.32142,0.22586,0.10714,0.28571,0.57143,0.0,0.57143,8,0,7,8,0,2,0,0,7,0,0,4,0,0,11,0,0,0,0,0,0,0,0],[12,20,0.6,0.39289,0.19232,0.28571,0.42857,0.57143,0.0,0.57143,4,0,2,4,0,1,0,0,7,0,0,7,0,0,13,0,0,0,0,0,0,0,0],[16,20,0.8,0.40606,0.17195,0.28571,0.42859,0.57143,0.0,0.57143,2,0,1,2,0,3,0,0,5,0,0,10,0,0,12,0,0,0,0,0,0,0,0],[20,20,1.0,0.06241,0.11805,0.0,0.0,0.14071,0.0,0.42857,23,0,1,23,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ac1c8e703885b1a2","q":"Decide whether it possible to cover all lattice points in $\\mathbb{R}^{2}$ by an (infinite) family of disks whose interiors are disjoint such that the radius of each disk is at least 5 .","t":[{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.20981,"p":[[0,12,0.0,0.20981,0.28566,0.0,0.0,0.42857,0.0,1.0,18,1,6,18,0,2,0,0,2,0,0,5,0,0,2,0,0,1,0,0,1,0,1],[4,12,0.3333,0.14732,0.24867,0.0,0.0,0.32143,0.0,0.857,23,0,5,23,0,0,0,0,1,0,0,3,0,0,4,0,0,0,0,0,1,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.03125,"x":0.39731,"p":[[0,24,0.0,0.14286,0.25254,0.0,0.0,0.32142,0.0,1.0,23,1,5,23,0,0,0,0,1,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[4,24,0.1667,0.16072,0.25191,0.0,0.0,0.42857,0.0,0.85714,22,0,8,22,0,0,0,0,0,0,0,7,0,0,1,0,0,1,0,0,1,0,0],[8,24,0.3333,0.0625,0.18189,0.0,0.0,0.0,0.0,0.85714,28,0,2,28,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[12,24,0.5,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,2,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.39731,0.20118,0.28571,0.42857,0.4286,0.0,1.0,2,1,2,2,0,0,0,0,12,0,0,14,0,0,1,0,0,0,0,0,2,0,1]]}]},{"i":"fd00b053b0ee242f","q":"Assume that a function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfies the following condition: For every $x, y \\in \\mathbb{R}$ such that $(f(x)+y)(f(y)+x)>0$, we have $f(x)+y=f(y)+x$. Prove that $f(x)+y \\leqslant f(y)+x$ whenever $x>y$. (Netherlands)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.09366,"x":0.13606,"p":[[0,33,0.0,0.12929,0.0807,0.14,0.14286,0.14286,0.0,0.42857,5,0,2,5,2,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.11384,0.05607,0.14286,0.14286,0.14286,0.0,0.1429,6,0,1,6,1,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.10491,0.05919,0.07143,0.14286,0.14286,0.0,0.1429,7,0,0,7,3,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.12492,0.05644,0.14286,0.14286,0.14286,0.0,0.2857,4,0,1,4,2,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.09366,0.0654,0.0,0.14286,0.14286,0.0,0.1429,10,0,3,10,2,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.11608,0.06376,0.125,0.14286,0.14286,0.0,0.2857,6,0,0,6,2,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.11808,0.06081,0.12286,0.14286,0.14286,0.0,0.28571,5,0,1,5,3,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.2857,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.13606,0.04504,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,1,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.12936,0.03761,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,2,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0714,"k":"flat","v":0.09811,"x":0.10924,"p":[[0,8,0.0,0.09811,0.06116,0.05325,0.14286,0.14286,0.0,0.1429,8,0,3,8,4,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.10924,0.0564,0.07143,0.14286,0.14286,0.0,0.14286,6,0,4,6,3,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d45909caa1ea2b9b","q":"$20$  points with no three collinear are given. How many obtuse triangles can be formed by these points? $ \\textbf{(A)}\\ 6 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 2{{10}\\choose{3}} \\qquad \\textbf{(D)}\\ 3{{10}\\choose{3}} \\qquad \\textbf{(E)}\\ {{20}\\choose{3}}$","t":[{"b":2,"e":1.0,"k":"rising","v":0.07589,"x":1.0,"p":[[0,47,0.0,0.07589,0.24996,0.0,0.0,0.0,0.0,1.0,29,2,8,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,47,0.0851,0.6875,0.44239,0.10714,1.0,1.0,0.0,1.0,8,21,5,8,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,21],[8,47,0.1702,0.91071,0.24419,1.0,1.0,1.0,0.0,1.0,1,28,0,1,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,28],[12,47,0.2553,0.96429,0.12372,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,29],[16,47,0.3404,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[20,47,0.4255,0.96429,0.13832,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,30],[24,47,0.5106,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[28,47,0.5957,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[32,47,0.6809,0.94643,0.16656,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,29],[36,47,0.766,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31],[40,47,0.8511,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,47,0.9362,0.97768,0.1017,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,30],[47,47,1.0,0.98214,0.09942,1.0,1.0,1.0,0.42857,1.0,0,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,31]]},{"b":3,"e":0.0,"k":"rising","v":0.03125,"x":0.3392,"p":[[0,13,0.0,0.16964,0.36498,0.0,0.0,0.0,0.0,1.0,26,5,7,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[4,13,0.3077,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,13,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,13,0.6154,0.25,0.36943,0.0,0.0,0.35714,0.0,1.0,20,3,17,20,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,0,3,0,3],[12,13,0.9231,0.3392,0.33269,0.105,0.21428,0.57143,0.0,1.0,8,4,8,8,0,8,0,0,4,0,0,2,0,0,5,0,0,0,0,0,1,0,4]]}]},{"i":"2342d780e209834f","q":"A hilly island has  $2023$  lookouts. It is known that each of them is in line of sight with at least  $42$  of the other lookouts. For any two distinct lookouts  $X$  and  $Y$  there is a positive integer  $n$  and lookouts  $A_1,A_2,\\dots,A_{n+1}$  such that  $A_1=X$  and  $A_{n+1}=Y$  and  $A_1$  is in line of sight with  $A_2$ ,  $A_2$  with  $A_3$ ,  $\\dots$  and  $A_n$  with  $A_{n+1}$ . The smallest such number  $n$  is called the *viewing distance* of  $X$  and  $Y$ .\n\nDetermine the largest possible viewing distance that can exist between two lookouts under these conditions.","t":[{"b":0,"e":0.0,"k":"rising","v":0.0625,"x":0.44642,"p":[[0,34,0.0,0.09821,0.23538,0.0,0.0,0.0,0.0,0.85714,27,0,21,27,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[4,34,0.1176,0.0625,0.13803,0.0,0.0,0.0,0.0,0.42857,26,0,19,26,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.29018,0.19719,0.2857,0.28571,0.32143,0.0,0.71429,7,0,7,7,0,0,0,0,17,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[12,34,0.3529,0.38391,0.14912,0.28571,0.35714,0.42857,0.0,0.71429,1,0,1,1,0,0,0,0,15,0,0,11,0,0,2,0,0,3,0,0,0,0,0],[16,34,0.4706,0.4153,0.1999,0.28571,0.28571,0.46418,0.2857,1.0,0,1,0,0,0,0,0,0,20,0,0,4,0,0,2,0,0,4,0,0,1,0,1],[20,34,0.5882,0.34826,0.10681,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,22,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[24,34,0.7059,0.41964,0.15542,0.28571,0.42857,0.46431,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,9,0,0,3,0,0,5,0,0,0,0,0],[28,34,0.8235,0.44642,0.17404,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,5,0,0,5,0,0,7,0,0,0,0,0],[32,34,0.9412,0.33481,0.1163,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,26,0,0,3,0,0,1,0,0,2,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,13,0.0,0.06696,0.18893,0.0,0.0,0.0,0.0,1.0,26,1,16,26,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,13,0.3077,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,23,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"78296dbb1d125eb4","q":"Let cevians  $AA', BB'$  and  $CC'$  of triangle  $ABC$  concur at point  $P.$  The circumcircle of triangle  $PA'B'$  meets  $AC$  and  $BC$  at points  $M$  and  $N$  respectively, and the circumcircles of triangles  $PC'B'$  and  $PA'C'$  meet  $AC$  and  $BC$  for the second time respectively at points  $K$  and  $L$ . The line  $c$  passes through the midpoints of segments  $MN$  and  $KL$ . The lines  $a$  and  $b$  are defined similarly. Prove that  $a$ ,  $b$  and  $c$  concur.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.04018,"x":0.12053,"p":[[0,9,0.0,0.04018,0.10249,0.0,0.0,0.0,0.0,0.4286,27,0,3,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.12053,0.22335,0.0,0.0,0.14286,0.0,0.71429,23,0,5,23,0,2,0,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.11607,"p":[[0,6,0.0,0.08927,0.19146,0.0,0.0,0.0,0.0,0.71429,25,0,1,25,0,1,0,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[4,6,0.6667,0.11607,0.21558,0.0,0.0,0.07143,0.0,0.71429,24,0,1,24,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"abefd09fc4ff20eb","q":"A triangle  $ABC$  is inscribed in the circle  $\\mathcal{C}(O,R)$ . Let  $\\alpha <1$  be the ratio of the radii of the circles tangent to  $\\mathcal{C}$ , and both of the rays  $(AB$  and  $(AC$ . The numbers  $\\beta <1$  and  $\\gamma <1$  are defined analogously. Prove that  $\\alpha + \\beta + \\gamma =1$ .","t":[{"b":3,"e":0.4286,"k":"flat","v":0.24105,"x":0.44642,"p":[[0,17,0.0,0.24105,0.27762,0.0,0.21429,0.42857,0.0,1.0,15,1,12,15,0,1,0,0,7,0,0,3,0,0,2,0,0,3,0,0,0,0,1],[4,17,0.2353,0.31697,0.13709,0.2857,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,5,0,0,10,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.44642,0.20123,0.28571,0.42857,0.57111,0.1429,1.0,0,2,0,0,0,1,0,0,12,0,0,10,0,0,4,0,0,3,0,0,0,0,2],[12,17,0.7059,0.41516,0.25842,0.2857,0.35714,0.42857,0.0,1.0,2,4,0,2,0,2,0,0,12,0,0,9,0,0,3,0,0,0,0,0,0,0,4],[16,17,0.9412,0.3415,0.16629,0.28571,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,1,0,0,15,0,1,6,0,0,5,0,0,1,0,0,0,0,0],[17,17,1.0,0.3125,0.1448,0.28571,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,1,0,0,18,0,0,8,0,0,1,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.23661,"x":0.35714,"p":[[0,7,0.0,0.23661,0.21902,0.0,0.2857,0.42857,0.0,0.71429,12,0,8,12,0,2,0,0,9,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[4,7,0.5714,0.25892,0.2511,0.0,0.2857,0.42857,0.0,0.85714,12,0,11,12,0,3,0,0,5,0,0,7,0,0,2,0,0,2,0,0,1,0,0],[7,7,1.0,0.35714,0.15568,0.28571,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,0,0,0,15,0,0,13,0,0,0,0,0,1,0,0,1,0,0]]}]},{"i":"92aacfd238237e57","q":"3. (GER) For each finite set $U$ of nonzero vectors in the plane we define $l(U)$ to be the length of the vector that is the sum of all vectors in $U$. Given a finite set $V$ of nonzero vectors in the plane, a subset $B$ of $V$ is said to be maximal if $l(B)$ is greater than or equal to $l(A)$ for each nonempty subset $A$ of $V$. (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that for any set $V$ consisting of $n \\geq 1$ vectors, the number of maximal subsets is less than or equal to $2 n$.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.45527,"x":0.85713,"p":[[0,9,0.0,0.45527,0.40639,0.0,0.57143,0.85714,0.0,1.0,12,2,11,12,0,2,0,0,1,0,0,1,0,0,0,0,0,3,0,0,11,0,2],[4,9,0.4444,0.85713,3e-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],[8,9,0.8889,0.85713,4e-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],[9,9,1.0,0.85713,3e-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]]},{"b":1,"e":0.571,"k":"flat","v":0.21875,"x":0.62499,"p":[[0,61,0.0,0.29911,0.35956,0.0,0.14286,0.71429,0.0,1.0,14,1,14,14,0,6,0,0,1,0,0,2,0,0,0,0,0,2,0,0,6,0,1],[4,61,0.0656,0.62499,0.32683,0.57143,0.71429,0.85714,0.0,1.0,5,3,4,5,0,2,0,0,0,0,0,0,0,0,2,0,0,11,0,0,9,0,3],[8,61,0.1311,0.46423,0.32731,0.10714,0.571,0.71429,0.0,1.0,8,1,8,8,0,1,0,0,2,0,0,4,0,0,6,0,0,4,0,0,6,0,1],[12,61,0.1967,0.30357,0.34947,0.0,0.14286,0.60714,0.0,1.0,14,2,14,14,0,4,0,0,2,0,0,3,0,0,1,0,0,3,0,0,3,0,2],[16,61,0.2623,0.23214,0.34947,0.0,0.0,0.35713,0.0,0.85714,19,0,16,19,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,0,0],[20,61,0.3279,0.38838,0.35755,0.0,0.28571,0.71429,0.0,0.85714,11,0,9,11,0,4,0,0,2,0,0,0,0,0,3,0,0,5,0,0,7,0,0],[24,61,0.3934,0.21875,0.26241,0.0,0.14286,0.42857,0.0,0.85714,14,0,8,14,0,7,0,0,1,0,0,5,0,0,1,0,0,3,0,0,1,0,0],[28,61,0.459,0.32131,0.27897,0.0,0.2857,0.60682,0.0,0.71429,9,0,7,9,0,5,0,0,5,0,0,3,0,0,2,0,0,8,0,0,0,0,0],[32,61,0.5246,0.3883,0.29072,0.14286,0.35714,0.71407,0.0,1.0,6,1,3,6,0,4,0,0,6,0,0,6,0,0,1,0,0,6,0,0,2,0,1],[36,61,0.5902,0.35267,0.28343,0.14286,0.28571,0.57143,0.0,0.85714,6,0,2,6,0,8,0,0,3,0,0,5,0,0,3,0,0,4,0,0,3,0,0],[40,61,0.6557,0.28572,0.26964,0.10714,0.2143,0.42858,0.0,0.85714,8,0,0,8,0,8,0,0,5,0,0,6,0,0,1,0,0,0,0,0,4,0,0],[44,61,0.7213,0.38395,0.27533,0.10714,0.42859,0.57143,0.0,0.85714,8,0,0,8,0,3,0,0,0,0,0,9,0,0,5,0,0,6,0,0,1,0,0],[48,61,0.7869,0.25447,0.21646,0.0,0.28571,0.42857,0.0,0.71429,11,0,0,11,0,3,0,0,4,0,0,11,0,0,2,0,0,1,0,0,0,0,0],[52,61,0.8525,0.33929,0.24936,0.10714,0.42857,0.46431,0.0,0.71429,8,0,0,8,0,3,0,0,3,0,0,10,0,0,3,0,0,5,0,0,0,0,0],[56,61,0.918,0.36161,0.22864,0.14289,0.42857,0.46431,0.0,0.71429,6,0,0,6,0,3,0,0,3,0,0,12,0,0,4,0,0,4,0,0,0,0,0],[60,61,0.9836,0.3437,0.25714,0.0,0.42857,0.57111,0.0,0.71429,10,0,0,10,0,0,0,0,4,0,0,6,0,0,9,0,0,3,0,0,0,0,0],[61,61,1.0,0.38397,0.22143,0.2857,0.42857,0.46536,0.0,0.71429,5,0,0,5,0,2,0,0,4,0,0,13,0,0,3,0,0,5,0,0,0,0,0]]}]},{"i":"ae5d27296447643e","q":"Do there exist four polynomials  $P_1(x), P_2(x), P_3(x), P_4(x)$  with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?","t":[{"b":3,"e":0.14286,"k":"falling","v":0.11607,"x":0.26786,"p":[[0,7,0.0,0.26786,0.1915,0.14286,0.21428,0.42857,0.0,0.57143,6,0,3,6,0,10,0,0,1,0,0,12,0,0,3,0,0,0,0,0,0,0,0],[4,7,0.5714,0.16518,0.17169,0.0,0.14286,0.1429,0.0,0.71429,10,0,3,10,0,15,0,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,0],[7,7,1.0,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.16517,"x":0.41516,"p":[[0,39,0.0,0.20536,0.16728,0.14286,0.14286,0.28571,0.0,0.57143,5,0,1,5,0,18,0,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[4,39,0.1026,0.20982,0.1636,0.14286,0.14286,0.1786,0.0,0.57143,3,0,0,3,0,21,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[8,39,0.2051,0.16946,0.14483,0.14214,0.14286,0.14286,0.0,0.57143,6,0,1,6,0,20,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[12,39,0.3077,0.16517,0.14769,0.10714,0.14286,0.17857,0.0,0.57143,8,0,4,8,0,16,0,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,39,0.4103,0.16518,0.14335,0.14286,0.14286,0.14286,0.0,0.57143,7,0,1,7,0,19,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[20,39,0.5128,0.20527,0.17838,0.14286,0.14286,0.14287,0.0,0.85714,3,0,0,3,0,22,0,0,2,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[24,39,0.6154,0.41516,0.23785,0.14286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,9,0,0,1,0,0,10,0,0,7,0,0,0,0,0,4,0,0],[28,39,0.7179,0.37051,0.17804,0.14286,0.42857,0.4642,0.0,0.71429,1,0,0,1,0,8,0,0,3,0,0,12,0,0,7,0,0,1,0,0,0,0,0],[32,39,0.8205,0.37054,0.23381,0.14286,0.42857,0.46429,0.0,0.85714,3,0,1,3,0,8,0,0,2,0,0,11,0,0,4,0,0,2,0,0,2,0,0],[36,39,0.9231,0.20089,0.15916,0.14286,0.14286,0.32143,0.0,0.57143,6,0,1,6,0,16,0,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[39,39,1.0,0.31697,0.1665,0.14286,0.42857,0.42857,0.14286,0.85714,0,0,0,0,0,13,0,0,2,0,0,16,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"f06a850bd3bfca64","q":"26. (USA 2) Let $P$ be a cubic polynomial with rational coefficients, and let $q_{1}, q_{2}, q_{3}, \\ldots$ be a sequence of rational numbers such that $q_{n}=P\\left(q_{n+1}\\right)$ for all $n \\geq 1$. Prove that there exists $k \\geq 1$ such that for all $n \\geq 1, q_{n+k}=q_{n}$.","t":[{"b":6,"e":0.42857,"k":"flat","v":0.12053,"x":0.2857,"p":[[0,19,0.0,0.19195,0.19431,0.0,0.14286,0.28571,0.0,0.57143,13,0,10,13,0,5,0,0,7,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[4,19,0.2105,0.12053,0.15198,0.0,0.0,0.2857,0.0,0.4286,18,0,12,18,0,4,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.22768,0.24184,0.0,0.14286,0.42857,0.0,0.857,13,0,7,13,0,5,0,0,4,0,0,4,0,0,5,0,0,0,0,0,1,0,0],[12,19,0.6316,0.17411,0.18118,0.0,0.14286,0.2857,0.0,0.57143,12,0,7,12,0,10,0,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[16,19,0.8421,0.2857,0.24997,0.14286,0.2143,0.42858,0.0,0.85714,6,0,0,6,0,10,0,0,7,0,0,2,0,0,3,0,0,2,0,0,2,0,0],[19,19,1.0,0.19179,0.19439,0.105,0.14286,0.17857,0.0,0.71429,8,0,0,8,0,16,0,0,2,0,0,3,0,0,1,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.571,"k":"rising","v":0.20972,"x":0.39726,"p":[[0,12,0.0,0.20972,0.23415,0.0,0.14286,0.42857,0.0,0.857,14,0,12,14,0,4,0,0,5,0,0,6,0,0,1,0,0,1,0,0,1,0,0],[4,12,0.3333,0.28571,0.26245,0.0,0.28571,0.46429,0.0,1.0,10,1,6,10,0,5,0,0,4,0,0,5,0,0,6,0,0,1,0,0,0,0,1],[8,12,0.6667,0.29461,0.25734,0.0,0.28571,0.4286,0.0,0.85714,10,0,7,10,0,3,0,0,6,0,0,6,0,0,3,0,0,3,0,0,1,0,0],[12,12,1.0,0.39726,0.23343,0.14286,0.42857,0.57143,0.0,1.0,3,1,0,3,0,6,0,0,4,0,0,5,0,0,12,0,0,1,0,0,0,0,1]]}]},{"i":"966a7f1f54c03d3d","q":"A sequence  $x_1, x_2, ..., x_n, ...$  consists of an initial block of  $p$  positive distinct integers that then repeat periodically. This means that  $\\{x_1, x_2, \\dots, x_p\\}$  are  $p$  distinct positive integers and  $x_{n+p}=x_n$  for every positive integer  $n$ . The terms of the sequence are not known and the goal is to find the period  $p$ . To do this, at each move it possible to reveal the value of a term of the sequence at your choice.\n\n(a) Knowing that  $1 \\le p \\le 10$ , find the least  $n$  such that there is a strategy which allows to find  $p$  revealing at most  $n$  terms of the sequence.\n(b) Knowing that  $p$  is one of the first  $k$  prime numbers, find for which values of  $k$  there exist a strategy that allows to find  $p$  revealing at most  $5$  terms of the sequence.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,18,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,27,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,28,0,18,28,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,18,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,21,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,8,0.0,0.03571,0.11294,0.0,0.0,0.0,0.0,0.42857,29,0,23,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e1e0bec6cb19239d","q":"8. (NET 1) Let $S$ be a set of $n$ points in the plane. No three points of $S$ are collinear. Prove that there exists a set $P$ containing $2 n-5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $S$ lies a point that is an element of $P$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,4,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,9,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,4,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8ce586bd2716c09d","q":"Find, with proof, all nonconstant polynomials $P(x)$ with real coefficients such that, for all nonzero real numbers $z$ with $P(z) \\neq 0$ and $P\\left(\\frac{1}{z}\\right) \\neq 0$, we have\n\n$$\n\\frac{1}{P(z)}+\\frac{1}{P\\left(\\frac{1}{z}\\right)}=z+\\frac{1}{z}\n$$","t":[{"b":5,"e":0.42857,"k":"rising","v":0.1875,"x":0.45982,"p":[[0,12,0.0,0.23214,0.2519,0.0,0.14286,0.42857,0.0,1.0,13,1,0,13,0,4,0,0,5,0,0,6,0,0,2,0,0,1,0,0,0,0,1],[4,12,0.3333,0.1875,0.15746,0.0,0.14288,0.28571,0.0,0.57143,10,0,0,10,0,7,0,0,11,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,12,0.6667,0.37936,0.15,0.28571,0.42857,0.42858,0.0,0.71429,1,0,0,1,0,3,0,0,9,0,0,13,0,0,5,0,0,1,0,0,0,0,0],[12,12,1.0,0.45982,0.09932,0.42857,0.42857,0.46431,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,21,0,0,6,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.20536,"x":0.29911,"p":[[0,9,0.0,0.29445,0.24216,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,4,0,0,5,0,0,7,0,0,5,0,0,1,0,0,1,0,0],[4,9,0.4444,0.29911,0.21237,0.14286,0.2857,0.42857,0.0,0.71429,5,0,0,5,0,8,0,0,7,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[8,9,0.8889,0.25447,0.18466,0.14286,0.2857,0.42857,0.0,0.57143,7,0,0,7,0,7,0,0,7,0,0,8,0,0,3,0,0,0,0,0,0,0,0],[9,9,1.0,0.20536,0.15126,0.10714,0.2143,0.28571,0.0,0.4286,8,0,0,8,0,8,0,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d37ea7375cf63340","q":"A jeweller makes a chain consisting of  $N>3$  numbered links.  A querulous customer then asks him to change the order of the links, in such a way that the number of links the jeweller must open is maximized.  What is the maximum number?","t":[{"b":5,"e":0.0,"k":"flat","v":0.04911,"x":0.04911,"p":[[0,3,0.0,0.04911,0.18073,0.0,0.0,0.0,0.0,1.0,28,1,24,28,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]},{"b":6,"e":0.0,"k":"flat","v":0.14732,"x":0.14732,"p":[[0,6,0.0,0.14732,0.2977,0.0,0.0,0.0,0.0,1.0,25,2,23,25,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,2]]}]},{"i":"263d924bf0d47b1e","q":"Find all positive integers  $ n $  with the following property: It is possible to fill a  $ n \\times n $  chessboard with one of arrows  $ \\uparrow, \\downarrow, \\leftarrow, \\rightarrow $  such that \n\n1. Start from any grid, if we follows the arrows, then we will eventually go back to the start point.\n\n2. For every row, except the first and the last, the number of  $ \\uparrow $  and the number of  $ \\downarrow $  are the same. \n\n3. For every column, except the first and the last, the number of  $ \\leftarrow $  and the number of  $ \\rightarrow $  are the same.","t":[{"b":2,"e":0.42857,"k":"falling","v":0.42857,"x":0.66071,"p":[[0,24,0.0,0.64286,0.26486,0.42857,0.42857,1.0,0.2857,1.0,0,10,0,0,0,0,0,0,1,0,0,17,0,0,0,0,0,3,0,0,1,0,10],[4,24,0.1667,0.64731,0.29448,0.42857,0.57121,1.0,0.0,1.0,1,10,1,1,0,0,0,0,3,0,0,11,0,0,3,0,0,0,0,0,4,0,10],[8,24,0.3333,0.66071,0.25692,0.42857,0.42857,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,2,0,0,4,0,9],[12,24,0.5,0.5625,0.23941,0.42857,0.42857,0.85714,0.0,1.0,1,3,1,1,0,0,0,0,1,0,0,18,0,0,2,0,0,1,0,0,6,0,3],[16,24,0.6667,0.58482,0.25595,0.42857,0.42857,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,0,0,0,19,0,0,1,0,0,2,0,0,3,0,6],[20,24,0.8333,0.54911,0.22619,0.42857,0.42857,0.50002,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,23,0,0,0,0,0,1,0,0,2,0,5],[24,24,1.0,0.42857,0.03571,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.65179,"x":0.91071,"p":[[0,21,0.0,0.79464,0.24984,0.42857,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,1,0,0,7,0,15],[4,21,0.1905,0.70534,0.26472,0.42857,0.85714,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,11,0,0,1,0,0,1,0,0,7,0,10],[8,21,0.381,0.69643,0.25442,0.42857,0.78571,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,12,0,0,2,0,0,1,0,0,7,0,9],[12,21,0.5714,0.65179,0.25738,0.42857,0.42857,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,2,0,0,5,0,8],[16,21,0.7619,0.87053,0.19352,0.85714,0.92857,1.0,0.2857,1.0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,12,0,16],[20,21,0.9524,0.69196,0.26271,0.42857,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,13,0,0,1,0,0,3,0,0,3,0,11],[21,21,1.0,0.91071,0.10565,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,13,0,16]]}]},{"i":"f77359a9bd19c582","q":"Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron *antipodal* if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let  $A$  be the number of antipodal pairs of vertices, and let  $B$  be the number of antipodal pairs of midpoint edges. Determine the difference  $A-B$  in terms of the numbers of vertices, edges, and faces.\n\n*Proposed by Kei Irei, Japan*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,16,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,17,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,17,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,34,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,18,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,19,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,24,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,16,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e481470cbdc08d14","q":"A set of balls contains  $ n$  balls which are labeled with numbers  $ 1,2,3,\\ldots,n.$  We are given  $ k > 1$  such sets. We want to colour the balls with two colours, black and white in such a way, that\r\n\r\n(a) the balls labeled with the same number are of the same colour,\r\n\r\n(b) any subset of  $ k\\plus{}1$  balls with (not necessarily different) labels  $ a_{1},a_{2},\\ldots,a_{k\\plus{}1}$  satisfying the condition  $ a_{1}\\plus{}a_{2}\\plus{}\\ldots\\plus{}a_{k}\\equal{} a_{k\\plus{}1}$ , contains at least one ball of each colour.\r\n\r\nFind, depending on  $ k$  the greatest possible number  $ n$  which admits such a colouring.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.08034,"p":[[0,19,0.0,0.08034,0.19209,0.0,0.0,0.0,0.0,0.71429,27,0,13,27,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[4,19,0.2105,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,15,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,14,0.0,0.0625,0.17105,0.0,0.0,0.0,0.0,0.71429,28,0,8,28,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d918edad7d42c989","q":"Euclid has a tool called cyclos which allows him to do the following:\n\n- Given three non-collinear marked points, draw the circle passing through them.\n- Given two marked points, draw the circle with them as endpoints of a diameter.\n- Mark any intersection points of two drawn circles or mark a new point on a drawn circle.\n\nShow that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos.","t":[{"b":1,"e":0.0,"k":"flat","v":0.01339,"x":0.08482,"p":[[0,55,0.0,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,10,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,11,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.14286,20,0,6,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,3,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.4286,22,0,7,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.07581,0.107,0.0,0.0,0.14286,0.0,0.4286,19,0,6,19,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.08482,0.09354,0.0,0.14286,0.14286,0.0,0.42857,15,0,3,15,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.0267,0.05558,0.0,0.0,0.0,0.0,0.1429,26,0,5,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.1429,16,0,2,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.0759,0.09439,0.0,0.0,0.14286,0.0,0.4286,17,0,1,17,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,2,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.03117,0.0589,0.0,0.0,0.0,0.0,0.143,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02679,"x":0.05804,"p":[[0,7,0.0,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,9,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.2857,20,0,5,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b416ceec49e874e3","q":"18. A4 (BLR) Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that for any two integers $x, y$ taken from two different subsets, the number $x^{2}-x y+y^{2}$ belongs to the third subset.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.02679,"p":[[0,32,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,20,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,18,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,23,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,19,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,24,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,22,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,23,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.01339,"x":0.03572,"p":[[0,14,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,20,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,22,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.03572,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,18,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,20,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f2ab4c14c0439524","q":"For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=$ $x^{n}+a_{n-1} x^{n-1}+\\cdots+a_{1} x+a_{0}$ be a polynomial, where $n \\geqslant 2$ and $a_{i}$ is a positive integer for all $0 \\leqslant i \\leqslant n-1$. Could it be the case that, for all positive integers $k, s(k)$ and $s(P(k))$ have the same parity? (Belarus)","t":[{"b":4,"e":0.14286,"k":"flat","v":0.09821,"x":0.19188,"p":[[0,50,0.0,0.16072,0.1171,0.14286,0.14286,0.2857,0.0,0.4286,7,0,0,7,0,16,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.19188,0.13654,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,50,0.16,0.14268,0.12372,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,16,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,50,0.24,0.16518,0.11904,0.14286,0.14286,0.14287,0.0,0.57143,5,0,0,5,0,20,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,50,0.32,0.17415,0.12243,0.14286,0.14286,0.2857,0.0,0.43,6,0,0,6,0,16,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.14286,0.12372,0.0,0.14286,0.1786,0.0,0.42857,10,0,0,10,0,14,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.16516,0.12423,0.14286,0.14286,0.1786,0.0,0.571,6,0,0,6,0,18,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,50,0.56,0.11607,0.10374,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.14268,0.09449,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.19179,0.13657,0.14286,0.14286,0.28571,0.0,0.57143,5,0,0,5,0,16,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[40,50,0.8,0.17848,0.10105,0.14286,0.14286,0.2857,0.0,0.42857,4,0,0,4,0,17,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.11607,0.10374,0.0,0.14286,0.14287,0.0,0.28571,12,0,0,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.09821,0.10972,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.1383,0.14053,0.0,0.14286,0.2857,0.0,0.42857,13,0,0,13,0,10,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.0625,"x":0.20536,"p":[[0,15,0.0,0.16518,0.14334,0.0,0.14286,0.2857,0.0,0.4286,11,0,0,11,0,8,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.20536,0.15542,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,15,0,0,8,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[8,15,0.5333,0.08929,0.1171,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0625,0.08703,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.10714,0.11294,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,13,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dcb804b98c87930d","q":"A pirate wants to divide a treasure consisting of 1000 gold coins, each weighing at least 1 g and together exactly 2 kg, into two parts, each of which may deviate from 1 kg in mass by at most 1 g. Prove that this is possible.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.01786,"x":0.04455,"p":[[0,20,0.0,0.04455,0.13564,0.0,0.0,0.0,0.0,0.71429,27,0,3,27,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,20,0.2,0.03125,0.08553,0.0,0.0,0.0,0.0,0.4286,27,0,2,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,4,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.02678,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01339,"x":0.08027,"p":[[0,13,0.0,0.08027,0.16339,0.0,0.0,0.14286,0.0,0.71429,22,0,1,22,0,7,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[4,13,0.3077,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,8,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,12,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e4aef9e2025d0301","q":"8. (ROM 2) $)^{\\mathrm{IMO} 3}$ In a plane two different points $O$ and $A$ are given. For each point $X \\neq O$ of the plane denote by $\\alpha(X)$ the angle $A O X$ measured in radians $(0 \\leq \\alpha(X)<2 \\pi)$ and by $C(X)$ the circle with center $O$ and radius $O X+\\frac{\\alpha(X)}{O X}$. Suppose each point of the plane is colored by one of a finite number of colors. Show that there exists a point $X$ with $\\alpha(X)>0$ such that its color appears somewhere on the circle $C(X)$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.01786,"x":0.13392,"p":[[0,32,0.0,0.08929,0.18472,0.0,0.0,0.14286,0.0,0.85714,23,0,8,23,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[4,32,0.125,0.13392,0.26947,0.0,0.0,0.03571,0.0,1.0,24,1,11,24,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,1],[8,32,0.25,0.11159,0.22791,0.0,0.0,0.14286,0.0,1.0,23,1,10,23,0,3,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[12,32,0.375,0.09813,0.20957,0.0,0.0,0.14071,0.0,1.0,23,1,5,23,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[16,32,0.5,0.08481,0.15912,0.0,0.0,0.14286,0.0,0.571,23,0,11,23,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,32,0.625,0.02679,0.10374,0.0,0.0,0.0,0.0,0.4286,30,0,6,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,2,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.71429,"k":"rising","v":0.13393,"x":0.30357,"p":[[0,10,0.0,0.13393,0.23128,0.0,0.0,0.2857,0.0,0.71429,23,0,7,23,0,0,0,0,3,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[4,10,0.4,0.17409,0.27135,0.0,0.0,0.2857,0.0,1.0,18,1,7,18,0,5,0,0,4,0,0,0,0,0,2,0,0,1,0,0,1,0,1],[8,10,0.8,0.21427,0.24998,0.0,0.07143,0.42857,0.0,0.71429,16,0,6,16,0,3,0,0,2,0,0,5,0,0,4,0,0,2,0,0,0,0,0],[10,10,1.0,0.30357,0.28065,0.0,0.28571,0.46431,0.0,1.0,10,1,5,10,0,4,0,0,5,0,0,5,0,0,4,0,0,2,0,0,1,0,1]]}]},{"i":"379ea01e4d68890d","q":"$ S$  be a set of  $ n$  points in the plane. No three points of  $ S$  are collinear. Prove that there exists a set  $ P$  containing  $ 2n \\minus{} 5$  points satisfying the following condition: In the interior of every triangle whose three vertices are elements of  $ S$  lies a point that is an element of  $ P.$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.07589,"x":0.11598,"p":[[0,7,0.0,0.07589,0.11837,0.0,0.0,0.14286,0.0,0.4286,21,0,2,21,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.11598,0.14912,0.0,0.0,0.2857,0.0,0.4286,18,0,1,18,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03571,"x":0.08482,"p":[[0,27,0.0,0.06695,0.13351,0.0,0.0,0.03571,0.0,0.571,24,0,1,24,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,27,0.1481,0.05795,0.11763,0.0,0.0,0.035,0.0,0.42857,24,0,2,24,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.05804,0.11769,0.0,0.0,0.0,0.0,0.4286,25,0,3,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,1,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.07134,0.10095,0.0,0.0,0.14286,0.0,0.28571,20,0,2,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.08482,0.12807,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0bc0199182f5fa11","q":"$N$  oligarchs built a country with  $N$  cities with each one of them owning one city. In addition, each oligarch built some roads such that the maximal amount of roads an oligarch can build between two cities is  $1$  (note that there can be more than  $1$  road going through two cities, but they would belong to different oligarchs).\nA total of  $d$  roads were built. Some oligarchs wanted to create a corporation by combining their cities and roads so that from any city of the corporation you can go to any city of the corporation using only corporation roads (roads can go to other cities outside corporation) but it turned out that no group of less than  $N$  oligarchs can create a corporation. What is the maximal amount that  $d$  can have?","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.20089,"p":[[0,60,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.20089,0.23381,0.0,0.07143,0.32143,0.0,0.85714,16,0,10,16,0,1,0,0,7,0,0,4,0,0,3,0,0,0,0,0,1,0,0],[8,60,0.1333,0.10268,0.1996,0.0,0.0,0.14286,0.0,0.85714,21,0,16,21,0,6,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[12,60,0.2,0.09821,0.15746,0.0,0.0,0.28571,0.0,0.57143,22,0,18,22,0,1,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,60,0.2667,0.13393,0.21998,0.0,0.0,0.2857,0.0,0.85714,21,0,16,21,0,1,0,0,6,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[20,60,0.3333,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,19,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.11161,0.1504,0.0,0.0,0.28571,0.0,0.42857,20,0,13,20,0,1,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,13,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.08928,0.14174,0.0,0.0,0.17857,0.0,0.42857,22,0,4,22,0,2,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.14286,0.13832,0.0,0.14286,0.28571,0.0,0.28571,15,0,4,15,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.11161,0.16262,0.0,0.0,0.28571,0.0,0.57143,21,0,9,21,0,0,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[44,60,0.7333,0.11598,0.14478,0.0,0.0,0.28571,0.0,0.42857,19,0,6,19,0,1,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.13839,0.15355,0.0,0.0,0.28571,0.0,0.42857,17,0,8,17,0,1,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.12946,0.15714,0.0,0.0,0.28571,0.0,0.42857,18,0,11,18,0,2,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.125,0.15465,0.0,0.0,0.28571,0.0,0.42857,19,0,10,19,0,0,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"rising","v":0.02679,"x":0.28125,"p":[[0,15,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,21,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.14732,0.23003,0.0,0.0,0.28571,0.0,0.71429,21,0,18,21,0,1,0,0,4,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[8,15,0.5333,0.08928,0.14617,0.0,0.0,0.2857,0.0,0.42857,23,0,21,23,0,0,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.20537,0.14699,0.0,0.28571,0.28571,0.0,0.4286,10,0,5,10,0,1,0,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.28125,0.16164,0.2857,0.28571,0.28571,0.0,0.85714,5,0,2,5,0,0,0,0,21,0,0,5,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"006f9274823f0997","q":"Am\u00e9lia and Beatriz play battleship on a  $2n\\times2n$  board, using very peculiar rules. Am\u00e9lia begins by choosing  $n$  lines and  $n$  columns of the board, placing her  $n^2$  submarines on the cells that lie on their intersections. Next, Beatriz chooses a set of cells that will explode. Which is the least number of cells that Beatriz has to choose in order to assure that at least a submarine will explode?","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.08483,"p":[[0,35,0.0,0.07589,0.17852,0.0,0.0,0.14286,0.0,0.85714,23,0,10,23,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[4,35,0.1143,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.42857,22,0,5,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,7,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.08483,0.14665,0.0,0.0,0.14286,0.0,0.71429,19,0,6,19,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,35,0.4571,0.04909,0.11065,0.0,0.0,0.03571,0.0,0.571,24,0,11,24,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,35,0.5714,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,10,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"volatile","v":0.0,"x":0.37054,"p":[[0,26,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,10,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.06247,0.14248,0.0,0.0,0.03571,0.0,0.571,24,0,10,24,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,17,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.32143,0.32341,0.0,0.28571,0.60714,0.0,0.85714,12,0,8,12,0,3,0,0,4,0,0,4,0,0,1,0,0,3,0,0,5,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.37054,0.34967,0.0,0.42857,0.71429,0.0,1.0,14,1,12,14,0,0,0,0,1,0,0,3,0,0,1,0,0,11,0,0,1,0,1]]}]},{"i":"1295caa4538e9459","q":"Given  $19$  red boxes and  $200$  blue boxes filled with balls. None of which is empty.\nSuppose that every red boxes have a maximum of  $200$  balls and every blue boxes have a maximum of  $19$  balls.\nSuppose that the sum of all balls in the red boxes is less than the sum of all the balls in the blue boxes.\nProve that there exists a subset of the red boxes and a subset of the blue boxes such that their sum is the same.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,17,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"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":"131704068b24ec05","q":"$2021$  points are given on a circle. Each point is colored by one of the  $1,2, \\cdots ,k$  colors. For all points and colors  $1\\leq r \\leq k$ , there exist an arc such that at least half of the points on it are colored with  $r$ . Find the maximum possible value of  $k$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.08482,"p":[[0,32,0.0,0.08482,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,6,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,6,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,7,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,11,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,6,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,7,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05795,"p":[[0,17,0.0,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.14286,19,0,10,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a2d5c280a9fee9c5","q":"22. G7 (RUS) For a given triangle $A B C$, let $X$ be a variable point on the line $B C$ such that $C$ lies between $B$ and $X$ and the incircles of the triangles $A B X$ and $A C X$ intersect at two distinct points $P$ and $Q$. Prove that the line $P Q$ passes through a point independent of $X$.","t":[{"b":0,"e":0.1429,"k":"flat","v":0.0,"x":0.01339,"p":[[0,32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,12,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,10,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,7,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"21479f5253f38744","q":"A infinite sequence  $\\{ a_n \\}_{n \\ge 0}$  of real numbers satisfy  $a_n \\ge n^2$ . Suppose that for each  $i, j \\ge 0$  there exist  $k, l$  with  $(i,j) \\neq (k,l)$ ,  $l - k = j - i$ , and  $a_l - a_k = a_j - a_i$ . Prove that  $a_n \\ge (n + 2016)^2$  for some  $n$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.11161,"x":0.11161,"p":[[0,8,0.0,0.11161,0.13236,0.0,0.14286,0.14286,0.0,0.57143,14,0,4,14,0,14,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.10714,"x":0.15625,"p":[[0,8,0.0,0.14504,0.1977,0.0,0.14143,0.14286,0.0,0.85714,14,0,2,14,1,11,0,0,1,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[4,8,0.5,0.10714,0.15972,0.0,0.0,0.14286,0.0,0.71429,18,0,5,18,0,8,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,8,1.0,0.15625,0.16888,0.0,0.14286,0.1429,0.0,0.71429,11,0,0,11,0,14,0,0,2,0,0,4,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"10cf925df04d49c9","q":"A set  $P$  of  $2002$  persons is given. The family of subsets of  $P$  containing exactly  $1001$  persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set  $P$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.08929,"x":0.27232,"p":[[0,10,0.0,0.16071,0.20748,0.0,0.14286,0.2857,0.0,0.85714,15,0,3,15,0,7,0,0,6,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[4,10,0.4,0.27232,0.30379,0.0,0.14286,0.46429,0.0,1.0,12,1,2,12,0,7,0,0,2,0,0,3,0,0,3,0,0,2,0,0,2,0,1],[8,10,0.8,0.09366,0.11067,0.0,0.07,0.14286,0.0,0.42857,16,0,10,16,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.08929,0.12242,0.0,0.0,0.14286,0.0,0.42857,18,0,11,18,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"volatile","v":0.19643,"x":0.69643,"p":[[0,16,0.0,0.27679,0.2878,0.0,0.14286,0.57143,0.0,0.85714,10,0,4,10,0,9,0,0,3,0,0,1,0,0,3,0,0,4,0,0,2,0,0],[4,16,0.25,0.32143,0.34626,0.0,0.21429,0.60714,0.0,1.0,13,2,5,13,0,3,0,0,4,0,0,2,0,0,2,0,0,3,0,0,3,0,2],[8,16,0.5,0.19643,0.24419,0.0,0.14286,0.32143,0.0,1.0,15,1,4,15,0,5,0,0,4,0,0,4,0,0,3,0,0,0,0,0,0,0,1],[12,16,0.75,0.33481,0.36701,0.0,0.14286,0.75,0.0,1.0,13,1,6,13,0,4,0,0,4,0,0,0,0,0,1,0,0,2,0,0,7,0,1],[16,16,1.0,0.69643,0.30671,0.57143,0.78571,0.89286,0.0,1.0,3,8,2,3,0,1,0,0,1,0,0,1,0,0,4,0,0,6,0,0,8,0,8]]}]},{"i":"977b45f7e38654ec","q":"An  $8 \\times 8$  chessboard consists of  $64$  square units. In some of the unit squares of the board, diagonals are drawn so that any two diagonals have no common points. What is the maximum number of diagonals that can be drawn?","t":[{"b":3,"e":0.0,"k":"flat","v":0.64732,"x":0.64732,"p":[[0,3,0.0,0.64732,0.45455,0.0,1.0,1.0,0.0,1.0,10,19,10,10,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,19]]},{"b":6,"e":1.0,"k":"rising","v":0.64732,"x":1.0,"p":[[0,20,0.0,0.64732,0.46289,0.0,1.0,1.0,0.0,1.0,10,20,10,10,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,20],[4,20,0.2,0.76784,0.37586,0.53539,1.0,1.0,0.0,1.0,5,22,5,5,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,22],[8,20,0.4,0.75446,0.42594,0.78571,1.0,1.0,0.0,1.0,7,24,7,7,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[12,20,0.6,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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":"f528c0fac834b909","q":"A set  $C$  of positive integers is called good if for every integer  $k$  there exist distinct  $a, b \\in C$  such that the numbers  $a+k$  and  $b+k$  are not relatively prime. Prove that if the sum of the elements of a good set  $C$  equals  $2003$ , then there exists  $c \\in C$  such that the set  $C-\\{c\\}$  is good.","t":[{"b":1,"e":0.0,"k":"flat","v":0.02679,"x":0.0625,"p":[[0,20,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,4,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.0625,0.11812,0.0,0.0,0.14286,0.0,0.4286,23,0,1,23,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.04463,0.11532,0.0,0.0,0.0,0.0,0.571,26,0,2,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,20,0.6,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.05357,"p":[[0,17,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,2,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,2,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,2,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,17,0.9412,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b560e15978636f33","q":"Circle  $\\omega$  is inscribed in quadrilateral  $ABCD$  such that  $AB$  and  $CD$  are not parallel and\nintersect at point  $O.$  Circle  $\\omega_1$  touches the side  $BC$  at  $K$  and touches line  $AB$  and  $CD$  at\npoints which are located outside quadrilateral  $ABCD;$  circle  $\\omega_2$  touches side  $AD$  at  $L$  and\ntouches line  $AB$  and  $CD$  at points which are located outside quadrilateral  $ABCD.$  If  $O,K,$ and  $L$  are collinear $,$  then show that the midpoint of side  $BC,AD,$  and the center of circle $\\omega$  are also collinear.","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.01786,"p":[[0,16,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,16,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"afb953700b30313a","q":"Consider an infinite sequence $a_{1}, a_{2}, \\ldots$ of positive integers with $a_{i} \\leqslant 2015$ for all $i \\geqslant 1$. Suppose that for any two distinct indices $i$ and $j$ we have $i+a_{i} \\neq j+a_{j}$.  Prove that there exist two positive integers $b$ and $N$ such that  $$ \\left|\\sum_{i=m+1}^{n}\\left(a_{i}-b\\right)\\right| \\leqslant 1007^{2} $$  whenever $n>m \\geqslant N$. (Australia)","t":[{"b":5,"e":0.14286,"k":"flat","v":0.05348,"x":0.17634,"p":[[0,23,0.0,0.1384,0.1803,0.0,0.0,0.1786,0.0,0.57143,17,0,7,17,0,7,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,23,0.1739,0.17634,0.22797,0.0,0.07143,0.28571,0.0,1.0,16,1,6,16,0,3,0,1,6,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[8,23,0.3478,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.08482,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0758,0.09432,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.07134,0.10709,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.07589,0.10092,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.08482,"x":0.19197,"p":[[0,7,0.0,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.42857,21,0,9,21,0,6,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.19197,0.20395,0.0,0.14286,0.42857,0.0,0.71429,12,0,7,12,0,10,0,0,0,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[7,7,1.0,0.15179,0.13803,0.0,0.14286,0.14287,0.0,0.57143,9,0,0,9,0,16,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"9a4568875d7fdebe","q":"Does there exist a pair $(g, h)$ of functions $g, h: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that the only function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x \\in \\mathbb{R}$ is the identity function $f(x) \\equiv x$ ?\n(United Kingdom) Alexander Betts","t":[{"b":1,"e":1.0,"k":"volatile","v":0.0625,"x":1.0,"p":[[0,18,0.0,0.08482,0.24964,0.0,0.0,0.0,0.0,1.0,27,2,3,27,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,18,0.2222,0.0625,0.17835,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,18,0.4444,0.07589,0.17852,0.0,0.0,0.14286,0.0,1.0,21,1,1,21,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,18,0.6667,0.07589,0.18893,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[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]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,23,0.0,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,3,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c2cfebb57b416235","q":"26. C6 (GBR) Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \\neq|y|$. Show that there are two integers of the same color whose difference has one of the following values: $x, y, x+y, x-y$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.00446,"x":0.03125,"p":[[0,31,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,15,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,2,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,3,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,3,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,6,0.0,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,10,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3985eb839176b4aa","q":"Given an acute triangle  $ABC$ , let  $AD$  be an altitude and  $H$  the orthocenter. Let  $E$  denote the reflection of  $H$  with respect to  $A$ . Point  $X$  is chosen on the circumcircle of triangle  $BDE$  such that  $AC\\| DX$  and point  $Y$  is chosen on the circumcircle of triangle  $CDE$  such that  $DY\\| AB$ . Prove that the circumcircle of triangle  $AXY$  is tangent to that of  $ABC$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.00438,"x":0.20099,"p":[[0,17,0.0,0.12053,0.18935,0.0,0.0,0.17857,0.0,0.57143,21,0,1,21,0,3,0,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[4,17,0.2353,0.20099,0.22557,0.0,0.14286,0.42857,0.0,0.71429,12,0,0,12,0,10,0,0,1,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[8,17,0.4706,0.15625,0.18681,0.0,0.07143,0.32143,0.0,0.57143,16,0,0,16,0,6,0,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[12,17,0.7059,0.10268,0.15663,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,8,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.12491,"x":0.20089,"p":[[0,18,0.0,0.12491,0.18813,0.0,0.0,0.14286,0.0,0.71429,18,0,1,18,0,8,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[4,18,0.2222,0.18304,0.18977,0.0,0.14286,0.32143,0.0,0.71429,12,0,0,12,0,9,0,0,3,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[8,18,0.4444,0.19197,0.2008,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,7,0,0,6,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[12,18,0.6667,0.20089,0.14664,0.10714,0.2143,0.28571,0.0,0.4286,8,0,1,8,0,8,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.17857,0.15972,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,8,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.17402,0.16264,0.0,0.14286,0.28571,0.0,0.4286,12,0,0,12,0,7,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b3ba5bbd18945a09","q":"Call a simple graph  $G$  *quasicolorable* if we can color each edge blue, red, green, or white such that\n\n\n- for each vertex v of degree 3 in G, the three edges incident to v are either (1) red,\ngreen, and blue, or (2) all white,\n- not all edges are white.\n\nA simple connected graph  $G$  has  $a$  vertices of degree  $4$ ,  $b$  vertices of degree  $3$ , and no other vertices, where  $a$  and  $b$  are positive integers. Find the smallest real number  $c$  so that the following statement is true: \u201cIf  $a/b > c$ , then  $G$  must be quasicolorable.\u201d","t":[{"b":3,"e":0.14286,"k":"flat","v":0.03125,"x":0.05795,"p":[[0,13,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,11,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,8,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.03125,"x":0.12946,"p":[[0,16,0.0,0.06241,0.15535,0.0,0.0,0.14071,0.0,0.857,23,0,11,23,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,16,0.25,0.12946,0.20316,0.0,0.14286,0.14286,0.0,0.85714,15,0,7,15,0,14,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[8,16,0.5,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,7,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,10,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d2f93002b84df595","q":"5. I 5 (GBR 3) Let $A_{r}, B_{r}, C_{r}$ be points on the circumference of a given circle $S$. From the triangle $A_{r} B_{r} C_{r}$, called $\\triangle_{r}$, the triangle $\\triangle_{r+1}$ is obtained by constructing the points $A_{r+1}, B_{r+1}, C_{r+1}$ on $S$ such that $A_{r+1} A_{r}$ is parallel to $B_{r} C_{r}, B_{r+1} B_{r}$ is parallel to $C_{r} A_{r}$, and $C_{r+1} C_{r}$ is parallel to $A_{r} B_{r}$. Each angle of $\\triangle_{1}$ is an integer number of degrees and those integers are not multiples of 45 . Prove that at least two of the triangles $\\triangle_{1}, \\triangle_{2}, \\ldots, \\triangle_{15}$ are congruent.","t":[{"b":4,"e":1.0,"k":"flat","v":0.63835,"x":0.78125,"p":[[0,12,0.0,0.63835,0.34067,0.42857,0.71429,1.0,0.0,1.0,3,11,3,3,0,2,0,0,2,0,0,4,0,0,4,0,0,4,0,0,2,0,11],[4,12,0.3333,0.71873,0.33785,0.53539,0.85714,1.0,0.0,1.0,4,12,4,4,0,0,0,0,1,0,0,3,0,0,2,0,0,2,0,0,8,0,12],[8,12,0.6667,0.78125,0.335,0.71429,1.0,1.0,0.0,1.0,4,17,4,4,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,6,0,17]]},{"b":5,"e":1.0,"k":"flat","v":0.6875,"x":0.73661,"p":[[0,19,0.0,0.6875,0.37701,0.53571,0.85714,1.0,0.0,1.0,6,14,5,6,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,3,0,14],[4,19,0.2105,0.73661,0.26989,0.57143,0.85714,1.0,0.0,1.0,2,9,1,2,0,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,8,0,9]]}]},{"i":"30012b62a34be728","q":"Consider a  $100\\times 100$  square unit lattice  $\\textbf{L}$  (hence  $\\textbf{L}$  has  $10000$  points). Suppose  $\\mathcal{F}$  is a set of polygons such that all vertices of polygons in  $\\mathcal{F}$  lie in  $\\textbf{L}$  and every point in  $\\textbf{L}$  is the vertex of exactly one polygon in  $\\mathcal{F}.$  Find the maximum possible sum of the areas of the polygons in  $\\mathcal{F}.$ *Michael Ren and Ankan Bhattacharya, USA*","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.00893,"p":[[0,4,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,12,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.01786,"p":[[0,3,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,9,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3fb72b7c87fbf37e","q":"4. (BUL 3) ${ }^{\\mathrm{IMO} 2} \\mathrm{~A}$ pentagonal prism $A_{1} A_{2} \\ldots A_{5} B_{1} B_{2} \\ldots B_{5}$ is given. The edges, the diagonals of the lateral walls and the internal diagonals of the prism are each colored either red or green in such a way that no triangle whose vertices are vertices of the prism has its three edges of the same color. Prove that all edges of the bases are of the same color.","t":[{"b":4,"e":0.0,"k":"flat","v":0.06695,"x":0.24999,"p":[[0,19,0.0,0.24999,0.22867,0.0,0.2143,0.4286,0.0,0.71429,11,0,3,11,0,5,0,0,4,0,0,6,0,0,5,0,0,1,0,0,0,0,0],[4,19,0.2105,0.11603,0.20016,0.0,0.0,0.14286,0.0,0.57143,22,0,5,22,0,3,0,0,2,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[8,19,0.4211,0.16518,0.21162,0.0,0.07143,0.32142,0.0,0.71429,16,0,3,16,0,7,0,0,1,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[12,19,0.6316,0.06695,0.17487,0.0,0.0,0.0,0.0,0.71429,27,0,4,27,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,19,0.8421,0.07143,0.17857,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[19,19,1.0,0.11159,0.21347,0.0,0.0,0.03571,0.0,0.57143,24,0,1,24,0,2,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.21872,"x":0.35701,"p":[[0,18,0.0,0.35701,0.28124,0.105,0.42857,0.57143,0.0,0.857,8,0,2,8,0,6,0,0,0,0,0,5,0,0,7,0,0,5,0,0,1,0,0],[4,18,0.2222,0.26784,0.21941,0.0,0.2857,0.42858,0.0,0.57143,10,0,2,10,0,3,0,0,7,0,0,5,0,0,7,0,0,0,0,0,0,0,0],[8,18,0.4444,0.23214,0.21651,0.0,0.2857,0.42857,0.0,0.85714,11,0,2,11,0,4,0,0,8,0,0,6,0,0,2,0,0,0,0,0,1,0,0],[12,18,0.6667,0.2455,0.21787,0.0,0.2857,0.28571,0.0,0.71429,9,0,0,9,0,6,0,0,10,0,0,1,0,0,4,0,0,2,0,0,0,0,0],[16,18,0.8889,0.21872,0.21419,0.0,0.14288,0.28571,0.0,0.71429,11,0,0,11,0,6,0,0,8,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[18,18,1.0,0.28124,0.1939,0.14286,0.2857,0.42857,0.0,0.71429,5,0,0,5,0,7,0,0,11,0,0,3,0,0,5,0,0,1,0,0,0,0,0]]}]},{"i":"68143e6f263600c7","q":"A *domino* is a  $2 \\times 1$  or  $1 \\times  2$  tile. Determine in how many ways exactly  $n^2$  dominoes can be placed without overlapping on a  $2n \\times  2n$  chessboard so that every  $2 \\times  2$  square contains at least two uncovered unit squares which lie in the same row or column.","t":[{"b":4,"e":0.0,"k":"flat","v":0.02223,"x":0.04018,"p":[[0,20,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,15,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,8,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,27,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,26,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,29,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00893,"x":0.03562,"p":[[0,17,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,9,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.03562,0.0713,0.0,0.0,0.0,0.0,0.28571,25,0,7,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,15,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.02455,0.06341,0.0,0.0,0.0,0.0,0.28571,27,0,12,27,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6cc620fcade35391","q":"Convex quadrilateral  $ABCD$  is inscribed in circle  $w.$ Rays  $AB$  and  $DC$  intersect at  $K.\\ L$  is chosen on the diagonal  $BD$  so that  $\\angle BAC= \\angle DAL.\\ M$  is chosen on the segment  $KL$  so that  $CM \\mid\\mid BD.$  Prove that line  $BM$  touches  $w.$ *(Kungozhin M.)*","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,29,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,29,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4822aabd4f27760f","q":"Alice has a map of Wonderland, a country consisting of  $n \\geq 2$  towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be \u201cone way\u201d only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns.\n\nAlice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most  $4n$  questions.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,20,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,5,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,22,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d5d5fecaf551fc71","q":"16. (USS 5) ${ }^{\\mathrm{IMO} 2}$ Given a convex polyhedron $P_{1}$ with 9 vertices $A_{1}, \\ldots, A_{9}$, let us denote by $P_{2}, P_{3}, \\ldots, P_{9}$ the images of $P_{1}$ under the translations mapping the vertex $A_{1}$ to $A_{2}, A_{3}, \\ldots, A_{9}$ respectively. Prove that among the polyhedra $P_{1}, \\ldots, P_{9}$ at least two have a common interior point.","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.15625,"p":[[0,17,0.0,0.15625,0.36309,0.0,0.0,0.0,0.0,1.0,27,5,1,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[4,17,0.2353,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,4,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.0,"k":"falling","v":0.0,"x":0.25,"p":[[0,11,0.0,0.25,0.43301,0.0,0.0,0.25,0.0,1.0,24,8,3,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[4,11,0.3636,0.21875,0.4134,0.0,0.0,0.0,0.0,1.0,25,7,2,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3f466b887161449a","q":"A jury of 3366 film critics are judging the Oscars. Each critic makes a single vote for his favourite actor, and a single vote for his favourite actress. It turns out that for every integer $n \\in\\{1,2, \\ldots, 100\\}$ there is an actor or actress who has been voted for exactly $n$ times. Show that there are two critics who voted for the same actor and for the same actress.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,24,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,10,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,25,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,28,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,28,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"ef44e4405eb39445","q":"Given points  $A$ ,  $B$ ,  $C$ , and  $D$  on circle  $\\omega$  such that lines  $AB$  and  $CD$  intersect on point  $T$  where  $A$  is between  $B$  and  $T$ , moreover  $D$  is between  $C$  and  $T$ . It is known that the line passing through  $D$  which is parallel to  $AB$  intersects  $\\omega$  again on point  $E$  and line  $ET$  intersects  $\\omega$  again on point  $F$ . Let  $CF$  and  $AB$  intersect on point  $G$ ,  $X$  be the  midpoint of segment  $AB$ , and  $Y$  be the reflection of point  $T$  to  $G$ . \nProve that  $X$ ,  $Y$ ,  $C$ , and  $D$  are concyclic.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.15161,"x":0.24982,"p":[[0,11,0.0,0.21875,0.15146,0.14286,0.14286,0.2857,0.0,0.71429,2,0,0,2,0,20,0,0,3,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[4,11,0.3636,0.24982,0.22025,0.14214,0.14286,0.42857,0.0,1.0,6,1,0,6,0,12,0,0,4,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[8,11,0.7273,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.1517,0.04973,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.17862,"x":0.25893,"p":[[0,14,0.0,0.25893,0.19704,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,10,0,0,8,0,0,7,0,0,0,0,0,1,0,0,1,0,0],[4,14,0.2857,0.17862,0.13372,0.14286,0.14286,0.2857,0.0,0.43,6,0,1,6,0,17,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.23661,0.18073,0.14286,0.1429,0.32143,0.0,0.85714,5,0,0,5,0,12,0,0,7,0,0,7,0,0,0,0,0,0,0,0,1,0,0],[12,14,0.8571,0.24545,0.11434,0.14286,0.2143,0.28571,0.14,0.4286,0,0,0,0,0,16,0,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.23205,0.11716,0.14286,0.1429,0.28571,0.0,0.4286,1,0,0,1,0,16,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bc89dd39d0fcc66e","q":"20. (IRE) Let $D$ be an internal point on the side $B C$ of a triangle $A B C$. The line $A D$ meets the circumcircle of $A B C$ again at $X$. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to $A B$ and $A C$, respectively, and let $\\gamma$ be the circle with diameter $X D$. Prove that the line $P Q$ is tangent to $\\gamma$ if and only if $A B=A C$.","t":[{"b":0,"e":0.0,"k":"volatile","v":0.29466,"x":0.4732,"p":[[0,15,0.0,0.4732,0.32622,0.14289,0.42857,0.85714,0.0,1.0,3,4,0,3,0,6,0,0,4,0,0,7,0,0,2,0,0,1,0,0,5,0,4],[4,15,0.2667,0.29466,0.31326,0.0,0.14286,0.46525,0.0,1.0,9,3,4,9,0,9,0,0,4,0,0,2,0,0,4,0,0,0,0,0,1,0,3],[8,15,0.5333,0.47319,0.33203,0.24999,0.42857,0.74979,0.0,1.0,5,5,3,5,0,3,0,0,4,0,0,7,0,0,4,0,0,1,0,0,3,0,5],[12,15,0.8,0.34375,0.30485,0.14286,0.28571,0.42857,0.0,1.0,6,4,2,6,0,8,0,0,3,0,0,10,0,0,0,0,0,1,0,0,0,0,4]]},{"b":2,"e":1.0,"k":"rising","v":0.33481,"x":0.86158,"p":[[0,32,0.0,0.39732,0.35307,0.14286,0.35714,0.60714,0.0,1.0,7,5,2,7,0,7,0,0,2,0,0,6,0,0,2,0,0,1,0,0,2,0,5],[4,32,0.125,0.36159,0.303,0.10714,0.28571,0.42858,0.0,1.0,8,3,2,8,0,1,0,0,9,0,0,7,0,0,1,0,0,2,0,0,1,0,3],[8,32,0.25,0.35705,0.26494,0.14286,0.28571,0.42857,0.0,1.0,4,2,2,4,0,7,0,0,6,0,0,8,0,0,3,0,0,1,0,0,1,0,2],[12,32,0.375,0.33481,0.30848,0.14286,0.2857,0.42857,0.0,1.0,6,4,1,6,0,8,0,0,7,0,0,4,0,0,2,0,0,1,0,0,0,0,4],[16,32,0.5,0.68747,0.32032,0.571,0.71429,1.0,0.0,1.0,4,10,4,4,0,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,5,0,10],[20,32,0.625,0.6607,0.30252,0.57143,0.71429,0.85714,0.0,1.0,4,6,4,4,0,0,0,0,0,0,0,2,0,0,8,0,0,4,0,0,8,0,6],[24,32,0.75,0.80354,0.17769,0.71429,0.85707,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,6,0,0,9,0,10],[28,32,0.875,0.86158,0.17312,0.82143,0.92857,1.0,0.4286,1.0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,8,0,16],[32,32,1.0,0.81918,0.18558,0.71429,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,2,0,1,3,0,0,7,0,0,6,0,13]]}]},{"i":"3c6c7c01fb74f8ee","q":"4. (CZS 1) An $n \\times n$ chessboard $(n \\geq 2)$ is numbered by the numbers $1,2, \\ldots, n^{2}$ (every number occurs once). Prove that there exist two neighboring (which share a common edge) squares such that their numbers differ by at least $n$.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.03571,"x":0.12491,"p":[[0,13,0.0,0.06464,0.06989,0.0,0.0,0.14286,0.0,0.14286,17,0,9,17,1,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.09367,0.06779,0.0,0.14286,0.14286,0.0,0.143,11,0,4,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,12,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.07143,"p":[[0,16,0.0,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,10,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0558,0.06855,0.0,0.0,0.14286,0.0,0.14286,19,0,8,19,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.04677,0.07482,0.0,0.0,0.14071,0.0,0.28571,22,0,12,22,1,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,20,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ffb0502d5dd26e06","q":"21. G6 (GBR) Let $\\mathcal{P}$ be a convex polygon. Prove that there is a convex hexagon that is contained in $\\mathcal{P}$ and that occupies at least 75 percent of the area of $\\mathcal{P}$.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,50,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,5,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,13,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,3,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"c1621d9bdc8897d2","q":"4031 lines are drawn on a plane, no two parallel or perpendicular, and no three lines meet at a point. Determine the maximum number of acute-angled triangles that may be formed.","t":[{"b":2,"e":0.0,"k":"flat","v":0.03571,"x":0.09821,"p":[[0,15,0.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,19,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,20,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.09821,0.12595,0.0,0.07143,0.14286,0.0,0.42857,16,0,14,16,0,13,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03562,"x":0.08482,"p":[[0,12,0.0,0.03562,0.0713,0.0,0.0,0.0,0.0,0.28571,25,0,17,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.42857,22,0,21,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.08482,0.1551,0.0,0.0,0.14286,0.0,0.57143,22,0,16,22,0,6,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"e7a5e0fbe91250ff","q":"1. (BRA 1) Show that there exists a finite set $A \\subset \\mathbb{R}^{2}$ such that for every $X \\in A$ there are points $Y_{1}, Y_{2}, \\ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_{i}$ is equal to 1 , for every $i$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.03562,"x":0.08929,"p":[[0,11,0.0,0.07115,0.14268,0.0,0.0,0.14,0.0,0.57143,22,0,11,22,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,11,0.3636,0.08929,0.24679,0.0,0.0,0.0,0.0,1.0,26,2,3,26,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,11,0.7273,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,8,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.4286,"k":"rising","v":0.03562,"x":0.4107,"p":[[0,30,0.0,0.05795,0.12293,0.0,0.0,0.0,0.0,0.42857,25,0,2,25,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.08482,0.17076,0.0,0.0,0.14286,0.0,0.71429,22,0,2,22,0,7,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,30,0.2667,0.03562,0.0713,0.0,0.0,0.0,0.0,0.28571,25,0,2,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.36148,0.15973,0.25001,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,7,0,0,4,0,0,14,0,0,6,0,0,0,0,0,0,0,0],[16,30,0.5333,0.39732,0.15458,0.28571,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,4,0,0,4,0,0,16,0,0,6,0,0,1,0,0,0,0,0],[20,30,0.6667,0.4107,0.17766,0.39286,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,4,0,0,2,0,0,13,0,0,10,0,0,1,0,0,0,0,0],[24,30,0.8,0.40624,0.12929,0.42857,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,3,0,0,1,0,0,22,0,0,5,0,0,0,0,0,0,0,0],[28,30,0.9333,0.39731,0.15457,0.42857,0.42857,0.42858,0.0,0.57143,2,0,0,2,0,3,0,0,2,0,0,18,0,0,7,0,0,0,0,0,0,0,0],[30,30,1.0,0.37499,0.13715,0.39286,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,5,0,0,2,0,0,21,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"f53d93d9b4ce6f89","q":"18. N5 (BUL) Prove that there exist infinitely many positive integers $n$ such that $p=n r$, where $p$ and $r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.","t":[{"b":0,"e":0.28571,"k":"rising","v":0.0625,"x":0.30803,"p":[[0,22,0.0,0.15624,0.25342,0.0,0.0,0.28571,0.0,1.0,21,1,21,21,0,0,0,0,6,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[4,22,0.1818,0.0625,0.15947,0.0,0.0,0.0,0.0,0.71429,27,0,27,27,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,22,0.3636,0.09375,0.19757,0.0,0.0,0.0,0.0,0.857,25,0,25,25,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[12,22,0.5455,0.08481,0.19836,0.0,0.0,0.0,0.0,0.85714,26,0,26,26,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[16,22,0.7273,0.1875,0.27994,0.0,0.0,0.28571,0.0,1.0,19,2,19,19,0,0,0,0,7,0,0,3,0,0,0,0,0,1,0,0,0,0,2],[20,22,0.9091,0.30803,0.27918,0.0,0.28571,0.42858,0.0,1.0,10,1,10,10,0,0,0,0,12,0,0,4,0,0,1,0,0,2,0,0,2,0,1]]},{"b":6,"e":0.0,"k":"flat","v":0.06687,"x":0.14732,"p":[[0,20,0.0,0.08482,0.1551,0.0,0.0,0.07143,0.0,0.57143,24,0,24,24,0,0,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,20,0.2,0.14732,0.23279,0.0,0.0,0.28571,0.0,1.0,20,1,19,20,0,0,0,0,9,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,20,0.4,0.06687,0.15961,0.0,0.0,0.0,0.0,0.71429,26,0,26,26,0,1,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,20,0.6,0.08036,0.17835,0.0,0.0,0.0,0.0,0.71429,26,0,25,26,0,0,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[16,20,0.8,0.06696,0.14719,0.0,0.0,0.0,0.0,0.57143,26,0,26,26,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"6064b97f8634b1dd","q":"An infinite sequence  $a_1, a_2,\\ldots$  of positive integers is such that  $a_n \\geq 2$  and  $a_{n+2}$  divides  $a_{n+1} + a_n$  for all  $n \\geq 1$ . Prove that there exists a prime which divides infinitely many terms of the sequence.","t":[{"b":3,"e":1.0,"k":"rising","v":0.08482,"x":1.0,"p":[[0,37,0.0,0.11607,0.3102,0.0,0.0,0.0,0.0,1.0,28,3,8,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[4,37,0.1081,0.08482,0.25719,0.0,0.0,0.0,0.0,1.0,28,2,5,28,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[8,37,0.2162,0.15179,0.33681,0.0,0.0,0.0,0.0,1.0,25,4,2,25,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[12,37,0.3243,0.17411,0.3169,0.0,0.0,0.17857,0.0,1.0,22,2,4,22,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,2],[16,37,0.4324,0.42847,0.45181,0.0,0.28571,1.0,0.0,1.0,14,11,1,14,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,11],[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":5,"e":0.0,"k":"volatile","v":0.0,"x":0.3214,"p":[[0,15,0.0,0.20536,0.37105,0.0,0.0,0.17857,0.0,1.0,23,5,6,23,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[4,15,0.2667,0.12946,0.28203,0.0,0.0,0.0,0.0,1.0,25,2,7,25,0,0,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[8,15,0.5333,0.05356,0.14612,0.0,0.0,0.0,0.0,0.57143,27,0,6,27,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,15,0.8,0.3214,0.2743,0.0,0.571,0.57143,0.0,0.57143,13,0,2,13,0,0,0,0,2,0,0,0,0,0,17,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"29cd9f2c2fc08b18","q":"For a fixed positive integer $m$, let $A$ be a subset of $\\left\\{0,1,2, \\ldots, 5^{m}\\right\\}$, consisting of $4 m+$ elements.\nProve that there are always three numbers $a, b, c$ in $A$ such that $a<b<c$ and $c+2 a>3 b$ holds.","t":[{"b":0,"e":1.0,"k":"rising","v":0.43302,"x":1.0,"p":[[0,28,0.0,0.43302,0.39525,0.0,0.50001,0.857,0.0,1.0,13,3,0,13,0,1,0,0,0,0,0,2,0,0,2,0,0,5,0,0,6,0,3],[4,28,0.1429,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[8,28,0.2857,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,28,0.8571,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":0.0,"k":"falling","v":0.0,"x":0.37054,"p":[[0,9,0.0,0.37054,0.40699,0.0,0.0,0.71429,0.0,1.0,17,4,5,17,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,2,0,4],[4,9,0.4444,0.03572,0.10715,0.0,0.0,0.0,0.0,0.57143,27,0,6,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f873f358198fdf3c","q":"In a mathematical competition some competitors are friends; friendship is always mutual, that is to say that when $A$ is a friend of $B$, then also $B$ is a friend of $A$. We say that $n \\geq 3$ different competitors $A_{1}, A_{2}, \\ldots, A_{n}$ form a weakly-friendly cycle if $A_{i}$ is not a friend of $A_{i+1}$ for $1 \\leq i \\leq n\\left(A_{n+1}=A_{1}\\right)$, and there are no other pairs of non-friends among the components of this cycle.\nThe following property is satisfied:\nfor every competitor $C$, and every weakly-friendly cycle $\\mathscr{S}$ of competitors not including $C$, the set of competitors $D$ in $\\mathscr{S}$ which are not friends of $C$ has at most one element.\n\nProve that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors that are in the same room are friends.\n(Serbia)\nTime allowed: 270 minutes.\nEach problem is worth 10 points.\n\n## SOLUTIONS","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,30,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"828a6ce81cc99392","q":"Given a convex quadrilateral  $ABCD$ . The points  $P$  and  $Q$  are the midpoints of the diagonals  $AC$  and  $BD$  respectively. The line  $PQ$  intersects the lines  $AB$  and  $CD$  at  $N$  and  $M$  respectively. Prove that the circumcircles of triangles  $NAP$ ,  $NBQ$ ,  $MQD$ , and  $MPC$  have a common point.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,50,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,50,0.24,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,50,0.4,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"449dd78cef2e0cc4","q":"Given a triangle  $ABC$ , let the bisector of  $\\angle BAC$  meets the side  $BC$  and circumcircle of triangle  $ABC$  at  $D$  and  $E$ , respectively. Let  $M$  and  $N$  be the midpoints of  $BD$  and  $CE$ , respectively. Circumcircle of triangle  $ABD$  meets  $AN$  at  $Q$ . Circle passing through  $A$  that is tangent to  $BC$  at  $D$  meets line  $AM$  and side  $AC$  respectively at  $P$  and  $R$ . Show that the four points  $B,P,Q,R$  lie on the same line.\n\n*Proposer: Fajar Yuliawan*","t":[{"b":2,"e":0.0,"k":"flat","v":0.02223,"x":0.12951,"p":[[0,29,0.0,0.08902,0.10551,0.0,0.07,0.14286,0.0,0.4286,16,0,0,16,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.04,0.08149,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.10259,0.12489,0.0,0.07,0.14287,0.0,0.42857,16,0,0,16,0,11,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.02223,0.06281,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.0758,0.0873,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.12951,0.14889,0.0,0.14286,0.1786,0.0,0.43,15,0,0,15,0,9,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.07589,0.10705,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.04911,"x":0.11608,"p":[[0,31,0.0,0.05795,0.11208,0.0,0.0,0.14071,0.0,0.4286,23,0,0,23,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.08902,0.10551,0.0,0.07,0.14286,0.0,0.4286,16,0,0,16,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.10706,0.10099,0.0,0.14286,0.14286,0.0,0.4286,12,0,0,12,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.08027,0.0708,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.11608,0.10375,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,20,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.08027,0.0708,0.0,0.14286,0.14286,0.0,0.143,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e3b312c2bc102a04","q":"For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that  $$ \\frac{\\varphi(d(n))}{d(\\varphi(n))} \\leqslant C $$  for all $n \\geqslant 1$ ? (Cyprus)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,29,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[29,29,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.07589,0.10705,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c37292a7ccbc4a98","q":"Fix an integer  $n \\geq 3$ . Let  $\\mathcal{S}$  be a set of  $n$  points in the plane, no three of which are collinear. Given different points  $A,B,C$  in  $\\mathcal{S}$ , the triangle  $ABC$  is *nice* for  $AB$  if  $[ABC] \\leq [ABX]$  for all  $X$  in  $\\mathcal{S}$  different from  $A$  and  $B$ . (Note that for a segment  $AB$  there could be several nice triangles). A triangle is *beautiful* if its vertices are all in  $\\mathcal{S}$  and is nice for at least two of its sides.\n\nProve that there are at least  $\\frac{1}{2}(n-1)$  beautiful triangles.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,24,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.02232,0.0724,0.0,0.0,0.0,0.0,0.2857,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ab2467da6770216c","q":"Given that  $ABC$  is a triangle where  $AB < AC$ . On the half-lines  $BA$  and  $CA$  we take points  $F$  and  $E$  respectively such that  $BF = CE = BC$ . Let  $M,N$  and  $H$  be the mid-points of the segments  $BF,CE$  and  $BC$  respectively and  $K$  and  $O$  be the circumcenters of the triangles  $ABC$  and  $MNH$  respectively. We assume that  $OK$  cuts  $BE$  and  $HN$  at the points  $A_1$  and  $B_1$  respectively and that  $C_1$  is the point of intersection of  $HN$  and  $FE$ . If the parallel line from  $A_1$  to  $OC_1$  cuts the line  $FE$  at  $D$  and the perpendicular from  $A_1$  to the line  $DB_1$  cuts  $FE$  at the point  $M_1$ , prove that  $E$  is the orthocenter of the triangle  $A_1OM_1$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.08482,"x":0.12946,"p":[[0,10,0.0,0.10714,0.11845,0.0,0.14286,0.14286,0.0,0.42857,15,0,1,15,0,11,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.12946,0.13053,0.0,0.14286,0.14287,0.0,0.57143,12,0,0,12,0,13,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,10,0.8,0.09822,0.10374,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.13831,"p":[[0,26,0.0,0.13831,0.12619,0.0,0.14286,0.14286,0.0,0.4286,10,0,0,10,0,16,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.08482,0.08645,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.06697,0.12364,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"9ead93f595f31149","q":"Given a scalene triangle  $ \\triangle ABC $ .  $ B', C' $  are points lie on the rays  $ \\overrightarrow{AB}, \\overrightarrow{AC}  $  such that  $ \\overline{AB'} = \\overline{AC}, \\overline{AC'} = \\overline{AB} $ . Now, for an arbitrary point  $ P $  in the plane. Let  $ Q $  be the reflection point of  $ P $  w.r.t  $ \\overline{BC} $ . The intersections of  $ \\odot{\\left(BB'P\\right)} $  and  $ \\odot{\\left(CC'P\\right)} $  is  $ P' $  and the intersections of  $ \\odot{\\left(BB'Q\\right)} $  and  $ \\odot{\\left(CC'Q\\right)} $  is  $ Q' $ . Suppose that  $ O, O' $  are circumcenters of  $ \\triangle{ABC}, \\triangle{AB'C'} $  Show that \n\n1.  $ O', P', Q' $  are colinear\n\n2.  $  \\overline{O'P'} \\cdot  \\overline{O'Q'} = \\overline{OA}^{2} $","t":[{"b":0,"e":0.57143,"k":"flat","v":0.4241,"x":0.56691,"p":[[0,24,0.0,0.49103,0.21108,0.571,0.57143,0.57143,0.0,1.0,3,1,0,3,0,2,0,0,1,0,0,1,0,0,24,0,0,0,0,0,0,0,1],[4,24,0.1667,0.56691,0.18723,0.571,0.57143,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,3,0,0,24,0,0,0,0,0,0,0,3],[8,24,0.3333,0.53113,0.18656,0.5713,0.57143,0.57143,0.0,1.0,2,1,0,2,0,1,0,0,0,0,0,3,0,0,24,0,0,0,0,0,1,0,1],[12,24,0.5,0.49552,0.22864,0.57132,0.57143,0.57143,0.0,1.0,4,1,0,4,0,1,0,0,1,0,0,1,0,0,23,0,0,0,0,0,1,0,1],[16,24,0.6667,0.4241,0.26361,0.14289,0.57143,0.57143,0.0,1.0,6,1,0,6,0,3,0,0,1,0,0,3,0,0,17,0,0,0,0,0,1,0,1],[20,24,0.8333,0.52672,0.11536,0.571,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,1,0,0,4,0,0,26,0,0,0,0,0,0,0,0],[24,24,1.0,0.55799,0.04162,0.57142,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.57143,"k":"flat","v":0.54907,"x":0.5714,"p":[[0,11,0.0,0.55802,0.21237,0.57142,0.57143,0.57143,0.14286,1.0,0,3,0,0,0,3,0,0,3,0,0,0,0,0,21,0,0,1,0,0,1,0,3],[4,11,0.3636,0.55354,0.05922,0.57143,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,2,0,0,29,0,0,0,0,0,0,0,0],[8,11,0.7273,0.54907,0.08072,0.57143,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,0,0,0,2,0,0,29,0,0,0,0,0,0,0,0],[11,11,1.0,0.5714,0.0001,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0]]}]},{"i":"667a8f5adb9bba0b","q":"Let  $ ABCD$  be a convex quadrilateral such that  $ AB$  and  $ CD$  are not parallel and  $ AB\\equal{}CD$ . The midpoints of the diagonals  $ AC$  and  $ BD$  are  $ E$  and  $ F$ , respectively. The line  $ EF$  meets segments  $ AB$  and  $ CD$  at  $ G$  and  $ H$ , respectively. Show that  $ \\angle AGH \\equal{} \\angle DHG$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,23,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"595b9502406789b7","q":"Find all triplets  $ (x,y,z) $  of positive integers such that\n\\[ x^y + y^x = z^y \\]\\[ x^y + 2012 = y^{z+1} \\]","t":[{"b":0,"e":0.357,"k":"flat","v":0.41964,"x":0.49103,"p":[[0,7,0.0,0.45085,0.08822,0.42857,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,19,0,0,9,0,0,0,0,0,0,0,0],[4,7,0.5714,0.49103,0.08699,0.42857,0.4998,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,14,0,0,16,0,0,0,0,0,0,0,0],[7,7,1.0,0.41964,0.07936,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,22,0,0,4,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"flat","v":0.44197,"x":0.52457,"p":[[0,20,0.0,0.50889,0.10059,0.42857,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,14,0,0,16,0,0,0,0,0,1,0,0],[4,20,0.2,0.44197,0.12037,0.39286,0.4286,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,7,0,0,12,0,0,12,0,0,0,0,0,0,0,0],[8,20,0.4,0.46427,0.09448,0.42857,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,16,0,0,12,0,0,0,0,0,0,0,0],[12,20,0.6,0.52457,0.07486,0.42964,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,8,0,1,22,0,0,0,0,0,0,0,0],[16,20,0.8,0.44643,0.08749,0.42857,0.42857,0.51786,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,1,18,0,1,8,0,0,0,0,0,0,0,0],[20,20,1.0,0.4531,0.05233,0.42857,0.42857,0.4286,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,26,0,1,5,0,0,0,0,0,0,0,0]]}]},{"i":"a72c22c9e4de4e5b","q":"Consider a ping-pong match between two teams, each consisting of 1000 players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at least one of those ten players.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.02232,"p":[[0,24,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,12,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,10,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,13,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,14,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,21,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,7,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,12,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,13,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,22,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"096d68cde136263e","q":"Given real numbers $a, b$ and $c$ such that $0 \\leq a \\leq b \\leq c$ and $a+b+c=1$. Prove that\n\n$$\na b \\sqrt{b-a}+b c \\sqrt{c-b}+a c \\sqrt{c-a}<\\frac{1}{4}\n$$","t":[{"b":6,"e":0.14286,"k":"flat","v":0.09813,"x":0.24552,"p":[[0,12,0.0,0.24552,0.25311,0.0,0.14286,0.42857,0.0,0.85714,10,0,0,10,0,9,0,0,3,0,0,5,0,0,2,0,0,1,0,0,2,0,0],[4,12,0.3333,0.22768,0.28984,0.0,0.14286,0.32143,0.0,1.0,12,1,0,12,0,11,0,0,1,0,0,3,0,0,0,0,0,2,0,0,2,0,1],[8,12,0.6667,0.09813,0.07518,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.10714,0.07143,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.05804,"x":0.375,"p":[[0,14,0.0,0.27679,0.24728,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,11,0,0,5,0,0,5,0,0,1,0,0,3,0,0,0,0,1],[4,14,0.2857,0.19634,0.24938,0.0,0.14286,0.2857,0.0,1.0,12,1,0,12,0,11,0,0,3,0,0,2,0,0,2,0,0,0,0,0,1,0,1],[8,14,0.5714,0.375,0.3004,0.14286,0.2857,0.46429,0.0,1.0,3,3,0,3,0,9,0,0,8,0,0,4,0,0,1,0,0,2,0,0,2,0,3],[12,14,0.8571,0.21429,0.20203,0.14286,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,13,0,0,7,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[14,14,1.0,0.05804,0.13767,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"c45f9ff4786ebe08","q":"Given a positive integer  $n$ , let  $D$  is the set of positive divisors of  $n$ , and let  $f: D \\to \\mathbb{Z}$  be a function. Prove that the following are equivalent:\n\n(a) For any positive divisor  $m$  of  $n$ ,\n\\[ n ~\\Big|~ \\sum_{d|m} f(d) \\binom{n/d}{m/d}. \\]\n(b) For any positive divisor  $k$  of  $n$ ,\n\\[ k ~\\Big|~ \\sum_{d|k} f(d). \\]","t":[{"b":1,"e":0.57143,"k":"flat","v":0.30801,"x":0.51784,"p":[[0,8,0.0,0.38393,0.30813,0.0,0.42857,0.71429,0.0,0.85714,9,0,0,9,0,2,0,0,4,0,0,6,0,0,0,0,0,8,0,0,3,0,0],[4,8,0.5,0.30801,0.28816,0.0,0.2143,0.571,0.0,0.85714,9,0,0,9,0,7,0,0,4,0,0,3,0,0,2,0,0,5,0,0,2,0,0],[8,8,1.0,0.51784,0.11709,0.4286,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,4,0,0,21,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"rising","v":0.33929,"x":0.59372,"p":[[0,16,0.0,0.33929,0.34209,0.0,0.28571,0.71429,0.0,0.85714,14,0,0,14,0,1,0,0,3,0,0,1,0,0,3,0,0,6,0,0,4,0,0],[4,16,0.25,0.5357,0.30722,0.28571,0.57143,0.85714,0.0,1.0,5,1,0,5,0,0,0,0,5,0,0,3,0,0,4,0,0,6,0,0,8,0,1],[8,16,0.5,0.44196,0.28652,0.14286,0.42857,0.71429,0.0,0.85714,5,0,0,5,0,4,0,0,3,0,0,6,0,0,2,0,0,9,0,0,3,0,0],[12,16,0.75,0.52677,0.16917,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,8,0,0,7,0,0,11,0,0,0,0,0],[16,16,1.0,0.59372,0.13883,0.5354,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,7,0,0,9,0,0,15,0,0,0,0,0]]}]},{"i":"51cee3f1b7ef6370","q":"Does there exist an infinite sequence of positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ such that $a_{m}$ and $a_{n}$ are coprime if and only if $|m-n|=1$ ?\n(Peru) Jorge Tipe","t":[{"b":0,"e":0.0,"k":"flat","v":0.11143,"x":0.21875,"p":[[0,24,0.0,0.1429,0.19239,0.0,0.14286,0.14286,0.0,1.0,12,1,12,12,0,15,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[4,24,0.1667,0.11143,0.13232,0.0,0.14143,0.14286,0.0,0.57143,14,0,14,14,0,14,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,24,0.3333,0.18302,0.18974,0.0,0.14286,0.2857,0.0,0.71429,9,0,9,9,0,14,0,0,5,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[12,24,0.5,0.21875,0.26959,0.0,0.14286,0.2857,0.0,1.0,11,1,10,11,0,10,0,0,6,0,0,1,0,0,0,0,0,1,0,0,2,0,1],[16,24,0.6667,0.2009,0.20782,0.0,0.14286,0.28571,0.0,1.0,9,1,8,9,0,12,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[20,24,0.8333,0.15161,0.12847,0.0,0.14286,0.28571,0.0,0.42857,10,0,9,10,0,12,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.17411,"p":[[0,41,0.0,0.11608,0.1729,0.0,0.14286,0.14286,0.0,1.0,12,1,12,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,41,0.0976,0.11152,0.10552,0.0,0.14286,0.14286,0.0,0.42857,12,0,12,12,0,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.17411,0.27137,0.0,0.14286,0.14287,0.0,1.0,14,2,14,14,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[12,41,0.2927,0.125,0.18814,0.0,0.14286,0.14286,0.0,1.0,14,1,14,14,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,41,0.3902,0.10268,0.18638,0.0,0.0,0.14286,0.0,1.0,18,1,18,18,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,41,0.4878,0.09821,0.24338,0.0,0.0,0.14286,0.0,1.0,23,2,23,23,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[24,41,0.5854,0.10714,0.19562,0.0,0.0,0.14286,0.0,1.0,19,1,19,19,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[28,41,0.6829,0.06232,0.09392,0.0,0.0,0.14286,0.0,0.28571,21,0,21,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.11161,0.24932,0.0,0.0,0.14286,0.0,1.0,22,2,22,22,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[36,41,0.878,0.05357,0.09943,0.0,0.0,0.14286,0.0,0.42857,23,0,22,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0b6ec9f6e7517a55","q":"20. N3 (FIN) ${ }^{\\text {IMO6 }}$ Find a set $A$ of positive integers such that for any infinite set $P$ of prime numbers, there exist positive integers $m \\in A$ and $n \\notin A$, both the product of the same number of distinct elements of $P$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,21,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,16,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,10,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,25,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,28,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,26,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3799f908cd0b0fdb","q":"Find all positive integers  $n$  for wich  $\\phi(\\phi (n))$  divides  $n$ .","t":[{"b":5,"e":0.42857,"k":"flat","v":0.31696,"x":0.48658,"p":[[0,56,0.0,0.4107,0.25691,0.14286,0.42857,0.57111,0.0,1.0,4,1,0,4,0,5,0,0,0,0,0,14,0,0,5,0,0,0,0,0,3,0,1],[4,56,0.0714,0.42407,0.17669,0.39286,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,3,0,0,3,0,0,11,0,0,12,0,0,1,0,0,0,0,0],[8,56,0.1429,0.39286,0.15152,0.28571,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,4,0,0,4,0,0,17,0,0,5,0,0,1,0,0,0,0,0],[12,56,0.2143,0.48658,0.17805,0.42857,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,3,0,0,10,0,0,12,0,0,3,0,0,0,0,1],[16,56,0.2857,0.4397,0.18392,0.41068,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,4,0,1,13,0,0,9,0,0,0,0,0,2,0,0],[20,56,0.3571,0.35719,0.19235,0.14286,0.42857,0.46536,0.0,0.71429,2,0,0,2,0,7,0,0,6,0,0,9,0,0,6,0,0,2,0,0,0,0,0],[24,56,0.4286,0.40621,0.16405,0.28571,0.42857,0.571,0.0,0.71429,1,0,0,1,0,4,0,0,5,0,0,12,0,0,9,0,0,1,0,0,0,0,0],[28,56,0.5,0.42405,0.14495,0.28571,0.42857,0.57025,0.14286,0.71429,0,0,0,0,0,3,0,0,6,0,0,14,0,0,7,0,0,2,0,0,0,0,0],[32,56,0.5714,0.37951,0.16603,0.2857,0.42857,0.57141,0.0,0.57143,1,0,0,1,0,6,0,0,5,0,0,11,0,0,9,0,0,0,0,0,0,0,0],[36,56,0.6429,0.31696,0.19475,0.14286,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,8,0,0,7,0,0,9,0,0,4,0,0,0,0,0,1,0,0],[40,56,0.7143,0.33482,0.16602,0.14286,0.42857,0.42858,0.0,0.57143,2,0,0,2,0,7,0,0,6,0,0,12,0,0,5,0,0,0,0,0,0,0,0],[44,56,0.7857,0.37491,0.13259,0.28571,0.42857,0.42857,0.14,0.71429,0,0,0,0,0,4,0,0,9,0,0,15,0,0,3,0,0,1,0,0,0,0,0],[48,56,0.8571,0.35713,0.1383,0.2857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,4,0,0,9,0,0,14,0,0,4,0,0,0,0,0,0,0,0],[52,56,0.9286,0.40624,0.22617,0.14289,0.42857,0.57111,0.14286,1.0,0,2,0,0,0,9,0,0,3,0,0,11,0,0,6,0,0,1,0,0,0,0,2],[56,56,1.0,0.33481,0.15813,0.24999,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,6,0,0,7,0,0,13,0,0,4,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.07813,"x":0.54015,"p":[[0,65,0.0,0.46872,0.27717,0.28571,0.4286,0.57143,0.0,1.0,3,4,0,3,0,4,0,0,2,0,0,8,0,0,10,0,0,1,0,0,0,0,4],[4,65,0.0615,0.54015,0.18465,0.42857,0.57143,0.57143,0.0,1.0,1,1,0,1,0,1,0,0,1,0,0,8,0,0,14,0,0,5,0,0,1,0,1],[8,65,0.1231,0.46423,0.24233,0.35714,0.42929,0.57143,0.0,1.0,1,2,0,1,0,7,0,0,0,0,0,9,0,0,9,0,0,4,0,0,0,0,2],[12,65,0.1846,0.38826,0.19616,0.14289,0.42857,0.4286,0.0,0.85714,1,0,0,1,0,8,0,0,1,0,0,15,0,0,4,0,0,2,0,0,1,0,0],[16,65,0.2462,0.18295,0.22085,0.0,0.14143,0.42857,0.0,0.85714,15,0,0,15,0,6,0,0,2,0,0,7,0,0,1,0,0,0,0,0,1,0,0],[20,65,0.3077,0.23643,0.21019,0.0,0.14286,0.42857,0.0,0.57143,10,0,0,10,0,7,0,0,4,0,0,6,0,0,5,0,0,0,0,0,0,0,0],[24,65,0.3692,0.15179,0.17105,0.0,0.14286,0.2857,0.0,0.57143,15,0,0,15,0,6,0,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[28,65,0.4308,0.24999,0.22302,0.0,0.14286,0.42857,0.0,0.71429,10,0,1,10,0,7,0,0,2,0,0,8,0,0,4,0,0,1,0,0,0,0,0],[32,65,0.4923,0.16069,0.21048,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,4,0,0,1,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[36,65,0.5538,0.13835,0.16562,0.0,0.07,0.2857,0.0,0.57143,16,0,0,16,0,6,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[40,65,0.6154,0.192,0.20082,0.0,0.14286,0.32143,0.0,0.57143,14,0,0,14,0,4,0,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[44,65,0.6769,0.13839,0.16554,0.0,0.07143,0.2857,0.0,0.57143,16,0,0,16,0,6,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[48,65,0.7385,0.12946,0.13997,0.0,0.14286,0.1786,0.0,0.42857,14,0,0,14,0,10,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.10714,0.14725,0.0,0.0,0.17857,0.0,0.42857,19,0,0,19,0,5,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.07813,0.10619,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,1,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.08481,0.13291,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,7,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,65,0.9846,0.13393,0.16342,0.0,0.0,0.2857,0.0,0.4286,17,0,0,17,0,5,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.11607,0.13092,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,12,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"1a18e9d3cad8c874","q":"A table tennis tournament has  $101$  contestants, where each pair of contestants will play each other exactly once. In each match, the player who gets  $11$  points first is the winner, and the other the loser. At the end of the tournament, it turns out that there exist matches with scores  $11$  to  $0$  and  $11$  to  $10$ . Show that there exists 3 contestants  $A,B,C$  such that the score of the losers in the matches between  $A,B$  and  $A,C$  are equal, but different from the score of the loser in the match between  $B,C$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,14,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,9,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,14,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,14,0.5714,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,18,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.04018,"p":[[0,12,0.0,0.04018,0.13474,0.0,0.0,0.0,0.0,0.71429,28,0,11,28,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,12,0.3333,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,10,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fc1d98e55a3ecfdc","q":"Fix an integer  $n \\geq 2$  and let  $a_1, \\ldots, a_n$  be integers, where  $a_1 = 1$ . Let  $$  f(x) = \\sum_{m=1}^n a_mm^x.  $$    Suppose that  $f(x) = 0$  for some  $K$  consecutive positive integer values of  $x$ . In terms of  $n$ , determine the maximum possible value of  $K$ .","t":[{"b":3,"e":0.28571,"k":"falling","v":0.05804,"x":0.38838,"p":[[0,29,0.0,0.23214,0.28291,0.0,0.07143,0.57143,0.0,1.0,16,1,0,16,0,4,0,0,1,0,0,1,0,0,9,0,0,0,0,0,0,0,1],[4,29,0.1379,0.29908,0.26086,0.0,0.28571,0.57143,0.0,0.57143,12,0,0,12,0,2,0,0,3,0,0,1,0,0,14,0,0,0,0,0,0,0,0],[8,29,0.2759,0.1964,0.25935,0.0,0.0,0.57111,0.0,0.57143,19,0,0,19,0,2,0,0,1,0,0,0,0,0,10,0,0,0,0,0,0,0,0],[12,29,0.4138,0.38838,0.24019,0.14286,0.57143,0.57143,0.0,0.57143,7,0,0,7,0,2,0,0,3,0,0,1,0,0,19,0,0,0,0,0,0,0,0],[16,29,0.5517,0.2857,0.31541,0.0,0.07143,0.57143,0.0,1.0,16,1,0,16,0,1,0,0,1,0,0,1,0,0,10,0,0,1,0,0,1,0,1],[20,29,0.6897,0.1875,0.24074,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,3,0,0,1,0,0,3,0,0,7,0,0,0,0,0,0,0,0],[24,29,0.8276,0.17856,0.2422,0.0,0.0,0.42857,0.0,0.71429,19,0,0,19,0,2,0,0,2,0,0,3,0,0,5,0,0,1,0,0,0,0,0],[28,29,0.9655,0.08482,0.15093,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[29,29,1.0,0.05804,0.14664,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.16964,"x":0.29464,"p":[[0,27,0.0,0.24106,0.2586,0.0,0.07145,0.57143,0.0,0.57143,16,0,0,16,0,1,0,0,2,0,0,3,0,0,10,0,0,0,0,0,0,0,0],[4,27,0.1481,0.29016,0.2684,0.0,0.35714,0.57143,0.0,0.71429,14,0,0,14,0,0,0,0,2,0,0,4,0,0,11,0,0,1,0,0,0,0,0],[8,27,0.2963,0.22765,0.26207,0.0,0.0,0.57111,0.0,0.57143,17,0,0,17,0,2,0,0,0,0,0,3,0,0,10,0,0,0,0,0,0,0,0],[12,27,0.4444,0.20536,0.25985,0.0,0.0,0.57143,0.0,0.57143,19,0,0,19,0,1,0,0,0,0,0,3,0,0,9,0,0,0,0,0,0,0,0],[16,27,0.5926,0.29464,0.26711,0.0,0.35714,0.57143,0.0,0.57143,13,0,0,13,0,2,0,0,1,0,0,2,0,0,14,0,0,0,0,0,0,0,0],[20,27,0.7407,0.20981,0.24216,0.0,0.07143,0.46418,0.0,0.57143,16,0,0,16,0,3,0,0,3,0,0,2,0,0,8,0,0,0,0,0,0,0,0],[24,27,0.8889,0.16964,0.24338,0.0,0.0,0.42858,0.0,0.57143,21,0,0,21,0,0,0,0,2,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[27,27,1.0,0.17847,0.22587,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,1,0,0,6,0,0,1,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"d9c43f6e64288d8e","q":"Given a cyclic quadrilateral  $ABCD$  with the circumcenter  $O$ , with  $BC$  and  $AD$  not parallel. Let  $P$  be the intersection of  $AC$  and  $BD$ . Let  $E$  be the intersection of the rays  $AB$  and  $DC$ . Let  $I$  be the incenter of  $EBC$  and the incircle of  $EBC$  touches  $BC$  at  $T_1$ . Let  $J$  be the excenter of  $EAD$  that touches  $AD$  and the excircle of  $EAD$  that touches  $AD$  touches  $AD$  at  $T_2$ . Let  $Q$  be the intersection between  $IT_1$  and  $JT_2$ . Prove that  $O,P,Q$  are collinear.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,58,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,58,0.069,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,58,0.1379,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,58,0.9655,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[58,58,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,34,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"655be25d5f72e110","q":"For a given chord  $MN$  of a circle discussed the triangle  $ABC$ , whose base is the diameter  $AB$  of this circle,which do not intersect the  $MN$ , and the sides  $AC$  and  $BC$  pass through the ends of  $M$  and  $N$  of the chord  $MN$ . Prove that the heights of all such triangles  $ABC$  drawn from the vertex  $C$  to the side  $AB$ , intersect at one point.","t":[{"b":3,"e":0.0,"k":"falling","v":0.00893,"x":0.36607,"p":[[0,12,0.0,0.36607,0.35344,0.0,0.28571,0.42858,0.0,1.0,10,5,1,10,0,2,0,0,6,0,0,7,0,0,0,0,0,0,0,0,2,0,5],[4,12,0.3333,0.29017,0.24085,0.0,0.28571,0.42857,0.0,0.85714,9,0,0,9,0,1,0,0,13,0,0,3,0,0,2,0,0,3,0,0,1,0,0],[8,12,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.23214,"x":0.26339,"p":[[0,3,0.0,0.26339,0.2699,0.0,0.2857,0.28571,0.0,1.0,10,2,0,10,0,3,0,0,13,0,0,2,0,0,1,0,0,0,0,0,1,0,2],[3,3,1.0,0.23214,0.08564,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,0,8,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d09db4b15b09f3c1","q":"2. N2 (ARM) Prove that every positive rational number can be represented in the form $\\frac{a^{3}+b^{3}}{c^{3}+d^{3}}$, where $a, b, c, d$ are positive integers.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,7,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f82b125148780321","q":"Find, with proof, all positive integer palindromes whose square is also a palindrome.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.1383,"x":0.27679,"p":[[0,20,0.0,0.16965,0.15335,0.0,0.14288,0.28571,0.0,0.4286,12,0,2,12,0,6,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.16509,0.12932,0.0,0.14286,0.28571,0.0,0.42857,10,0,0,10,0,8,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.1383,0.13592,0.0,0.14286,0.28571,0.0,0.42857,14,0,2,14,0,6,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.25892,0.12078,0.2857,0.2857,0.28571,0.0,0.4286,4,0,1,4,0,3,0,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.27679,0.09407,0.2857,0.28571,0.28571,0.0,0.4286,2,0,0,2,0,2,0,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.25446,0.07771,0.2857,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,3,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.03125,"x":0.18303,"p":[[0,4,0.0,0.18303,0.13474,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,4,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2847a0b7df06b5ec","q":"A test consists of  $30$  true or false questions. After the test (answering all  $30$  questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions ) several times. Can Victor work out a strategy that insure him to get a perfect score after**(a)**  $30$ th attempt?**(b)**   $25$ th attempt?\n\n(Initially, Victor does not know any answer)","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,28,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,11,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,13,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,19,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,15,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,15,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4b58d9faafc58812","q":"In a group of students, 50 students speak German, 50 students speak French, and 50 students speak Spanish. Some students speak more than one language.\nProve that the students can be divided into 5 groups such that in each group exactly 10 students speak German, 10 speak French, and 10 speak Spanish.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.16518,"x":0.17857,"p":[[0,11,0.0,0.17411,0.12745,0.14286,0.14286,0.17857,0.0,0.57143,5,0,0,5,0,19,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,11,0.3636,0.17857,0.09449,0.14286,0.14286,0.17857,0.0,0.42857,2,0,0,2,0,22,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.16518,0.13415,0.14286,0.14286,0.17857,0.0,0.71429,6,0,0,6,0,18,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[11,11,1.0,0.16947,0.0753,0.14286,0.14286,0.1786,0.0,0.28571,2,0,0,2,0,22,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.125,"x":0.14286,"p":[[0,4,0.0,0.14286,0.11845,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,16,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.125,0.07784,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fd0ac7b0a3a8e561","q":"For a prime $p$ and a positive integer $n$, denote by $\\nu_{p}(n)$ the exponent of $p$ in the prime factorization of $n$ !. Given a positive integer $d$ and a finite set $\\left\\{p_{1}, \\ldots, p_{k}\\right\\}$ of primes. Show that there are infinitely many positive integers $n$ such that $d \\mid \\nu_{p_{i}}(n)$ for all $1 \\leq i \\leq k$.  (India)","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,23,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,32,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"41be5145f0eded62","q":"Given a set  $X=\\{x_1,\\ldots,x_n\\}$  of natural numbers in which for all  $1< i \\leq n$  we have  $1\\leq x_i-x_{i-1}\\leq 2$ , call a real number  $a$  **good** if there exists  $1\\leq j \\leq n$  such that  $2|x_j-a|\\leq 1$ . Also a subset of  $X$  is called **compact** if the average of its elements is a good number. \nProve that at least  $2^{n-3}$  subsets of  $X$  are compact.\n\n*Proposed by Mahyar Sefidgaran*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a9cfe016894dd52b","q":"In a given tedrahedron  $ ABCD$  let  $ K$  and  $ L$  be the centres of edges  $ AB$  and  $ CD$  respectively. Prove that every plane that contains the line  $ KL$  divides the tedrahedron into two parts of equal volume.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.09821,"p":[[0,38,0.0,0.08929,0.1948,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,7,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[4,38,0.1053,0.09821,0.24336,0.0,0.0,0.0,0.0,1.0,25,1,2,25,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[8,38,0.2105,0.06696,0.17852,0.0,0.0,0.14286,0.0,1.0,23,1,5,23,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,38,0.3158,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,38,0.6316,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.04911,"x":0.09821,"p":[[0,21,0.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,2,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.05804,0.18161,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,21,0.381,0.09375,0.2412,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,21,0.5714,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,1,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2fc245dce3d01c95","q":"For a positive integer $m$, let $s(m)$ denote the sum of the decimal digits of $m$. A set $S$ positive integers is $k$-stable if $s\\left(\\sum_{x \\in X} x\\right)=k$ for any nonempty subset $X \\subseteq S$. For each integer $n \\geq 2$ let $f(n)$ be the minimal $k$ for which there exists a $k$-stable set with $n$ integers. Prove that there are constants $0<C_{1}<C_{2}$ with  $$ C_{1} \\log _{10} n \\leq f(n) \\leq C_{2} \\log _{10} n . $$","t":[{"b":1,"e":0.14286,"k":"flat","v":0.0758,"x":0.12053,"p":[[0,20,0.0,0.10268,0.10853,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.11152,0.09929,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.12045,0.10168,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,15,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.09813,0.08324,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.12053,0.11355,0.0,0.14286,0.17857,0.0,0.28571,13,0,0,13,0,11,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.0758,0.0873,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.09812,"x":0.11599,"p":[[0,6,0.0,0.09812,0.11535,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,6,0.6667,0.11599,0.0752,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.1159,0.07517,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"279550c534392a62","q":"For  $\\triangle ABC$  and its circumcircle  $\\omega$ , draw the tangents at  $B,C$  to  $\\omega$  meeting at  $D$ . Let the line  $AD$  meet the circle with center  $D$  and radius  $DB$  at  $E$  inside  $\\triangle ABC$ . Let  $F$  be the point on the extension of  $EB$  and  $G$  be the point on the segment  $EC$  such that  $\\angle AFB=\\angle AGE=\\angle A$ . Prove that the tangent at  $A$  to the circumcircle of  $\\triangle AFG$  is parallel to  $BC$ .\n\n*Proposed by 61plus*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,20,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,17,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"9b07e4010fe409ce","q":"For which natural numbers  $n{}$  it is possible to mark several cells of an  $n\\times n$  board in such a way that there are an even number of marked cells in all rows and columns, and an odd number on all diagonals whose length is more than one cell?\n\n*Proposed by S. Berlov*","t":[{"b":4,"e":0.42857,"k":"flat","v":0.30357,"x":0.43741,"p":[[0,53,0.0,0.33034,0.21557,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,12,0,0,6,0,0,6,0,0,5,0,0,1,0,0,0,0,1],[4,53,0.0755,0.33482,0.22191,0.14286,0.28571,0.42857,0.0,0.85714,1,0,1,1,0,13,0,0,4,0,0,7,0,0,4,0,0,1,0,0,2,0,0],[8,53,0.1509,0.43741,0.24216,0.25,0.42857,0.57143,0.0,1.0,1,1,0,1,0,7,0,0,4,0,0,6,0,0,9,0,0,2,0,0,2,0,1],[12,53,0.2264,0.40625,0.24772,0.14289,0.42857,0.46431,0.14286,1.0,0,1,0,0,0,9,0,0,6,0,0,9,0,0,3,0,0,0,0,0,4,0,1],[16,53,0.3019,0.42409,0.20972,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,6,0,0,7,0,0,7,0,0,9,0,0,1,0,0,1,0,1],[20,53,0.3774,0.37944,0.219,0.14286,0.42857,0.57111,0.0,0.85714,1,0,0,1,0,9,0,0,5,0,0,7,0,0,7,0,0,1,0,0,2,0,0],[24,53,0.4528,0.40177,0.23537,0.14289,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,8,0,0,6,0,0,6,0,0,6,0,0,2,0,0,3,0,0],[28,53,0.5283,0.375,0.20124,0.14286,0.42857,0.46431,0.14286,0.85714,0,0,0,0,0,11,0,0,2,0,0,11,0,0,5,0,0,2,0,0,1,0,0],[32,53,0.6038,0.42857,0.24743,0.25,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,8,0,0,6,0,0,8,0,0,4,0,0,0,0,0,6,0,0],[36,53,0.6792,0.36604,0.18874,0.14286,0.42857,0.4642,0.0,0.85714,1,0,1,1,0,8,0,0,5,0,0,10,0,0,7,0,0,0,0,0,1,0,0],[40,53,0.7547,0.39275,0.19895,0.2857,0.35714,0.42857,0.14,0.85714,0,0,0,0,0,5,0,0,11,0,0,10,0,0,2,0,0,1,0,0,3,0,0],[44,53,0.8302,0.37943,0.1842,0.2857,0.35714,0.571,0.14286,0.85714,0,0,0,0,0,7,0,0,9,0,0,7,0,0,7,0,0,1,0,0,1,0,0],[48,53,0.9057,0.41069,0.15869,0.28571,0.42857,0.571,0.14286,0.85714,0,0,0,0,0,4,0,0,7,0,0,12,0,0,8,0,0,0,0,0,1,0,0],[52,53,0.9811,0.35267,0.16743,0.14286,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,10,0,0,4,0,0,12,0,0,5,0,0,1,0,0,0,0,0],[53,53,1.0,0.30357,0.14174,0.14286,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,12,0,0,6,0,0,12,0,0,2,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.23214,"x":0.42409,"p":[[0,24,0.0,0.32142,0.23957,0.14286,0.21429,0.46429,0.0,0.85714,2,0,0,2,0,14,0,0,4,0,0,4,0,0,4,0,0,2,0,0,2,0,0],[4,24,0.1667,0.33918,0.1848,0.14286,0.28571,0.42857,0.14,0.85714,0,0,0,0,0,11,0,0,6,0,0,10,0,0,3,0,0,1,0,0,1,0,0],[8,24,0.3333,0.42409,0.19392,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,7,0,0,4,0,0,8,0,0,10,0,0,2,0,0,1,0,0],[12,24,0.5,0.375,0.1915,0.25,0.42857,0.46431,0.0,0.85714,1,0,0,1,0,7,0,0,6,0,0,10,0,0,6,0,0,1,0,0,1,0,0],[16,24,0.6667,0.39284,0.19232,0.2857,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,6,0,0,7,0,0,7,0,0,8,0,0,3,0,0,0,0,0],[20,24,0.8333,0.41515,0.23243,0.24999,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,8,0,0,7,0,0,6,0,0,6,0,0,1,0,0,4,0,0],[24,24,1.0,0.23214,0.10564,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,17,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6de4269e63fcf8a8","q":"Find the largest positive integer  $n$  with the following property: there are rectangles  $A_1, ... , A_n$  and  $B_1,... , B_n,$  on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles  $A_i$  and  $B_i$  are disjoint for all  $i \\in \\{1,..., n\\}$ , but the rectangles  $A_i$  and  $B_j$  have a common point for all  $i, j \\in \\{1,..., n\\}$ ,  $i \\ne j$ .\n\n*Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices*","t":[{"b":2,"e":1.0,"k":"rising","v":0.31248,"x":1.0,"p":[[0,31,0.0,0.31248,0.37016,0.0,0.14286,0.6425,0.0,1.0,15,2,11,15,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,0,6,0,2],[4,31,0.129,0.45089,0.3914,0.0,0.42857,0.85714,0.0,1.0,10,6,2,10,0,2,0,0,2,0,0,5,0,0,1,0,0,2,0,0,4,0,6],[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],[12,31,0.3871,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,31,0.5161,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[24,31,0.7742,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,31,0.9032,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[31,31,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":0.0,"k":"falling","v":0.05795,"x":0.44196,"p":[[0,20,0.0,0.44196,0.42761,0.0,0.42856,0.85714,0.0,1.0,14,7,4,14,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,4,0,7],[4,20,0.2,0.40178,0.42173,0.0,0.2857,0.89286,0.0,1.0,13,8,6,13,0,2,0,0,4,0,0,1,0,0,1,0,0,1,0,0,2,0,8],[8,20,0.4,0.2454,0.30333,0.0,0.14286,0.46429,0.0,1.0,14,1,2,14,0,7,0,0,2,0,0,1,0,0,1,0,0,6,0,0,0,0,1],[12,20,0.6,0.23658,0.26868,0.0,0.14286,0.42858,0.0,0.85714,14,0,4,14,0,4,0,0,5,0,0,2,0,0,3,0,0,3,0,0,1,0,0],[16,20,0.8,0.05795,0.07864,0.0,0.0,0.14286,0.0,0.28571,20,0,8,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c1195efc5d615eb2","q":"20. (ARG) Prove that for each positive integer $n$, there exists a positive integer with the following properties: (i) It has exactly $n$ digits. (ii) None of the digits is 0 . (iii) It is divisible by the sum of its digits.","t":[{"b":3,"e":0.42857,"k":"rising","v":0.19643,"x":0.42858,"p":[[0,44,0.0,0.2679,0.21357,0.0,0.28571,0.42857,0.0,0.85714,10,0,2,10,0,2,0,0,5,0,0,14,0,0,0,0,0,0,0,0,1,0,0],[4,44,0.0909,0.25446,0.21049,0.0,0.28571,0.42857,0.0,0.71429,11,0,4,11,0,2,0,0,5,0,0,12,0,0,1,0,0,1,0,0,0,0,0],[8,44,0.1818,0.27231,0.20933,0.0,0.42857,0.42857,0.0,0.57143,11,0,8,11,0,1,0,0,2,0,0,16,0,0,2,0,0,0,0,0,0,0,0],[12,44,0.2727,0.27231,0.22687,0.0,0.42857,0.42857,0.0,0.71429,12,0,10,12,0,1,0,0,1,0,0,15,0,0,2,0,0,1,0,0,0,0,0],[16,44,0.3636,0.29464,0.2141,0.0,0.42857,0.42857,0.0,0.71429,9,0,6,9,0,2,0,0,4,0,0,13,0,0,3,0,0,1,0,0,0,0,0],[20,44,0.4545,0.22768,0.21975,0.0,0.2857,0.42858,0.0,0.57143,14,0,12,14,0,1,0,0,5,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[24,44,0.5455,0.3393,0.19804,0.28571,0.42857,0.42857,0.0,0.71429,7,0,7,7,0,0,0,0,4,0,0,17,0,0,3,0,0,1,0,0,0,0,0],[28,44,0.6364,0.2054,0.21115,0.0,0.14285,0.42857,0.0,0.57143,16,0,15,16,0,0,0,0,3,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[32,44,0.7273,0.19643,0.21354,0.0,0.07143,0.42857,0.0,0.71429,16,0,14,16,0,1,0,0,4,0,0,10,0,0,0,0,0,1,0,0,0,0,0],[36,44,0.8182,0.36156,0.17484,0.28571,0.42857,0.42858,0.0,0.57143,5,0,4,5,0,0,0,0,5,0,0,17,0,0,5,0,0,0,0,0,0,0,0],[40,44,0.9091,0.3616,0.16359,0.42857,0.42857,0.42857,0.0,0.57143,5,0,0,5,0,0,0,0,2,0,0,23,0,0,2,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.42857,"k":"rising","v":0.1875,"x":0.42862,"p":[[0,30,0.0,0.25436,0.21352,0.0,0.42857,0.42857,0.0,0.57143,12,0,3,12,0,2,0,0,1,0,0,15,0,0,2,0,0,0,0,0,0,0,0],[4,30,0.1333,0.33481,0.17352,0.28571,0.42857,0.42857,0.0,0.57143,6,0,0,6,0,0,0,0,5,0,0,19,0,0,2,0,0,0,0,0,0,0,0],[8,30,0.2667,0.27679,0.21706,0.0,0.42857,0.42857,0.0,0.71429,11,0,1,11,0,1,0,0,2,0,0,16,0,0,1,0,0,1,0,0,0,0,0],[12,30,0.4,0.26338,0.19267,0.0,0.35714,0.42857,0.0,0.571,9,0,0,9,0,4,0,0,3,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[16,30,0.5333,0.25884,0.20657,0.0,0.42857,0.42857,0.0,0.4286,12,0,0,12,0,1,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.1875,0.19704,0.0,0.07145,0.42857,0.0,0.42857,16,0,0,16,0,1,0,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.20982,0.2082,0.0,0.21428,0.42857,0.0,0.57143,15,0,0,15,0,1,0,0,3,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[28,30,0.9333,0.21875,0.20512,0.0,0.2857,0.42857,0.0,0.57143,14,0,0,14,0,1,0,0,4,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[30,30,1.0,0.42862,0.00025,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1122d89587b72ba6","q":"A terrible group of bandits is preparing to share a loot. Each bandit aims at two other bandits with their pistols. The bandits are called in a certain order. When a bandit is called, if they are still alive, they shoot at the bandits they were aiming at. After all the bandits have been called, there have been 28 victims. Show that no matter the order in which the bandits were called, there would have been at least 10 victims.","t":[{"b":0,"e":0.0,"k":"flat","v":0.07134,"x":0.08482,"p":[[0,5,0.0,0.08482,0.25218,0.0,0.0,0.0,0.0,1.0,28,1,21,28,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[4,5,0.8,0.07134,0.1184,0.0,0.0,0.14286,0.0,0.42857,22,0,7,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.08035,"x":0.12946,"p":[[0,2,0.0,0.12946,0.25593,0.0,0.0,0.17857,0.0,0.85714,23,0,13,23,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,3,0,0],[2,2,1.0,0.08035,0.17472,0.0,0.0,0.14286,0.0,0.857,23,0,22,23,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"139a5bf97a523335","q":"From a set of 11 square integers, show that one can choose 6 numbers $a^{2}, b^{2}, c^{2}, d^{2}, e^{2}, f^{2}$ such that\n\n$$\na^{2}+b^{2}+c^{2} \\equiv d^{2}+e^{2}+f^{2} \\quad(\\bmod 12)\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,11,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,16,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"065a89b1e44b4254","q":"For any nonempty set $S$ of real numbers, let $\\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2 .","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.05804,"p":[[0,11,0.0,0.04455,0.09052,0.0,0.0,0.035,0.0,0.42857,24,0,1,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.05804,0.11214,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,11,0.7273,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.02232,"x":0.0892,"p":[[0,20,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.06241,0.094,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.0892,0.19147,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,20,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"030497198e95fce7","q":"In each square of a chessboard with  $a$  rows and  $b$  columns, a  $0$  or  $1$  is written satisfying the following conditions.\n\n- If a row and a column intersect in a square with a  $0$ , then that row and column have the same number of  $0$ s.\n- If a row and a column intersect in a square with a  $1$ , then that row and column have the same number of  $1$ s.\n\nFind all pairs  $(a,b)$  for which this is possible.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.19195,"x":0.38838,"p":[[0,34,0.0,0.29009,0.23826,0.14286,0.14286,0.28571,0.0,1.0,2,1,0,2,0,15,0,0,8,0,0,1,0,0,1,0,0,4,0,0,0,0,1],[4,34,0.1176,0.33024,0.18369,0.14286,0.35714,0.4286,0.14,0.71429,0,0,0,0,0,14,0,0,2,0,0,9,0,0,6,0,0,1,0,0,0,0,0],[8,34,0.2353,0.35268,0.20198,0.14286,0.42857,0.46431,0.0,0.71429,1,0,0,1,0,11,0,0,3,0,0,9,0,0,5,0,0,3,0,0,0,0,0],[12,34,0.3529,0.31921,0.17586,0.14286,0.35729,0.42857,0.0,0.71429,1,0,0,1,0,11,0,1,3,0,0,13,0,0,1,0,0,2,0,0,0,0,0],[16,34,0.4706,0.38838,0.23209,0.14286,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,8,0,0,5,0,0,6,0,0,4,0,0,7,0,0,0,0,0],[20,34,0.5882,0.3616,0.18204,0.14286,0.42857,0.42858,0.14286,0.71429,0,0,0,0,0,10,0,0,4,0,0,12,0,0,3,0,0,3,0,0,0,0,0],[24,34,0.7059,0.33259,0.17822,0.14286,0.28571,0.42858,0.14286,0.71429,0,0,0,0,0,10,0,1,9,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[28,34,0.8235,0.27223,0.16895,0.14286,0.21428,0.28571,0.14,0.71429,0,0,0,0,0,16,0,0,9,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[32,34,0.9412,0.23214,0.17766,0.14286,0.14286,0.28571,0.0,0.857,1,0,0,1,0,21,0,0,4,0,0,4,0,0,0,0,0,1,0,0,1,0,0],[34,34,1.0,0.19195,0.13171,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,27,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.25884,"x":0.97768,"p":[[0,37,0.0,0.32589,0.21199,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,10,0,0,4,0,0,8,0,0,4,0,0,3,0,0,0,0,0],[4,37,0.1081,0.33482,0.21902,0.14286,0.35714,0.42857,0.0,0.71429,2,0,0,2,0,12,0,0,2,0,0,10,0,0,1,0,0,5,0,0,0,0,0],[8,37,0.2162,0.29463,0.17833,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,14,0,0,4,0,0,10,0,0,1,0,0,2,0,0,0,0,0],[12,37,0.3243,0.33259,0.22841,0.14286,0.2857,0.46428,0.0,0.85714,1,0,0,1,0,13,0,1,5,0,0,4,0,0,3,0,0,4,0,0,1,0,0],[16,37,0.4324,0.31241,0.26113,0.14286,0.14286,0.42857,0.0,1.0,2,2,0,2,0,16,0,0,2,0,0,7,0,0,0,0,0,3,0,0,0,0,2],[20,37,0.5405,0.25884,0.21265,0.14286,0.14286,0.28571,0.0,1.0,2,1,0,2,0,17,0,0,6,0,0,4,0,0,0,0,0,2,0,0,0,0,1],[24,37,0.6486,0.30131,0.22424,0.14286,0.14286,0.42857,0.14286,1.0,0,1,0,0,0,17,0,0,5,0,1,4,0,0,2,0,0,1,0,0,1,0,1],[28,37,0.7568,0.84821,0.20183,0.71429,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,7,0,0,4,0,17],[32,37,0.8649,0.95089,0.15407,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,26],[36,37,0.973,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[37,37,1.0,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29]]}]},{"i":"f62d4c7d08761622","q":"Every non-negative integer is coloured white or red, so that:\n\u2022 there are at least a white number and a red number;\n\u2022 the sum of a white number and a red number is white;\n\u2022 the product of a white number and a red number is red.\nProve that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.","t":[{"b":2,"e":0.2857,"k":"falling","v":0.11161,"x":0.28572,"p":[[0,33,0.0,0.28572,0.28347,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,13,0,0,4,0,0,3,0,0,1,0,0,1,0,0,3,0,1],[4,33,0.1212,0.27678,0.27879,0.14286,0.14286,0.32143,0.0,1.0,5,2,1,5,0,15,0,0,4,0,0,3,0,0,0,0,0,2,0,0,1,0,2],[8,33,0.2424,0.20536,0.13803,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,19,0,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[12,33,0.3636,0.19634,0.24681,0.105,0.14286,0.14286,0.0,1.0,8,2,0,8,0,18,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[16,33,0.4848,0.17847,0.23957,0.0,0.14286,0.14286,0.0,1.0,9,2,0,9,0,18,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[20,33,0.6061,0.25445,0.22792,0.14286,0.14286,0.42857,0.0,1.0,6,1,0,6,0,12,0,0,5,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[24,33,0.7273,0.21429,0.2369,0.0,0.14286,0.32143,0.0,1.0,11,1,0,11,0,9,0,0,4,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[28,33,0.8485,0.14277,0.08748,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.14732,0.09094,0.14286,0.14286,0.14287,0.0,0.28571,6,0,0,6,0,19,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"falling","v":0.2366,"x":0.38839,"p":[[0,21,0.0,0.38839,0.36983,0.14286,0.14288,0.71429,0.0,1.0,6,6,1,6,0,11,0,0,3,0,0,0,0,0,3,0,0,2,0,0,1,0,6],[4,21,0.1905,0.33027,0.2777,0.14286,0.2143,0.4286,0.0,1.0,4,2,0,4,0,12,0,0,2,0,0,8,0,0,1,0,0,2,0,0,1,0,2],[8,21,0.381,0.31241,0.29765,0.105,0.14286,0.57143,0.0,1.0,8,1,0,8,0,10,0,0,1,0,0,3,0,0,5,0,0,2,0,0,2,0,1],[12,21,0.5714,0.32143,0.27894,0.14286,0.21428,0.42857,0.0,1.0,4,1,0,4,0,12,0,0,6,0,0,3,0,0,0,0,0,4,0,0,2,0,1],[16,21,0.7619,0.30357,0.32291,0.14286,0.14286,0.35714,0.0,1.0,7,3,0,7,0,12,0,0,5,0,0,0,0,0,2,0,0,1,0,0,2,0,3],[20,21,0.9524,0.25,0.11294,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,9,0,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.2366,0.07668,0.14286,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,12,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"953acd00ee4e8314","q":"Find all positive integer bases  $b \\ge 9$  so that the number \n\\[ \\frac{{\\overbrace{11 \\cdots 1}^{n-1 \\ 1's}0\\overbrace{77 \\cdots 7}^{n-1\\ 7's}8\\overbrace{11 \\cdots 1}^{n \\ 1's}}_b}{3} \\]\nis a perfect cube in base 10 for all sufficiently large positive integers  $n$ .\n\n*Proposed by Yang Liu*","t":[{"b":4,"e":0.71429,"k":"flat","v":0.72322,"x":0.79464,"p":[[0,7,0.0,0.72322,0.23673,0.42859,0.71429,1.0,0.28571,1.0,0,10,0,0,0,0,0,0,2,0,0,7,0,0,0,0,0,11,0,0,2,0,10],[4,7,0.5714,0.79464,0.20183,0.67857,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,1,0,0,2,0,0,5,0,0,5,0,0,8,0,11],[7,7,1.0,0.73661,0.22047,0.71429,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,3,0,0,3,0,0,1,0,0,12,0,0,5,0,8]]},{"b":6,"e":0.42857,"k":"flat","v":0.59821,"x":0.90179,"p":[[0,58,0.0,0.76339,0.2412,0.67857,0.71429,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,10,0,0,0,0,14],[4,58,0.069,0.7991,0.24964,0.71429,0.85714,1.0,0.14286,1.0,0,14,0,0,0,1,0,0,1,0,0,5,0,0,0,0,0,3,0,0,8,0,14],[8,58,0.1379,0.78125,0.24996,0.64286,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,6,0,0,0,0,0,6,0,0,3,0,15],[12,58,0.2069,0.76338,0.21903,0.67857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,1,0,0,0,0,0,3,0,0,4,0,0,10,0,0,3,0,11],[16,58,0.2759,0.65625,0.30485,0.42857,0.71429,1.0,0.0,1.0,1,11,0,1,0,2,0,0,2,0,0,8,0,0,0,0,0,8,0,0,0,0,11],[20,58,0.3448,0.76784,0.22233,0.67857,0.71429,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,9,0,0,3,0,12],[24,58,0.4138,0.78125,0.23954,0.67857,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,7,0,0,3,0,14],[28,58,0.4828,0.69643,0.24936,0.42859,0.71429,1.0,0.2857,1.0,0,9,0,0,0,0,0,0,4,0,0,5,0,0,3,0,0,8,0,0,3,0,9],[32,58,0.5517,0.64286,0.22868,0.42857,0.64286,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,11,0,0,3,0,0,6,0,0,5,0,5],[36,58,0.6207,0.90179,0.14032,0.85714,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,6,0,19],[40,58,0.6897,0.84821,0.15126,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,0,0,4,0,14],[44,58,0.7586,0.87054,0.15714,0.71429,1.0,1.0,0.42857,1.0,0,17,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,4,0,17],[48,58,0.8276,0.70088,0.21236,0.57143,0.71429,0.85704,0.28571,1.0,0,5,0,0,0,0,0,0,3,0,0,4,0,0,2,0,0,12,0,0,6,0,5],[52,58,0.8966,0.64732,0.2314,0.42857,0.71429,0.85714,0.1429,1.0,0,5,0,0,0,1,0,0,0,0,0,12,0,0,2,0,0,7,0,0,5,0,5],[56,58,0.9655,0.59821,0.19704,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,14,0,0,4,0,0,6,0,0,5,0,2],[58,58,1.0,0.625,0.1915,0.42857,0.64286,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,13,0,0,3,0,0,10,0,0,3,0,3]]}]},{"i":"1995a451d59dbe73","q":"Fine all positive integers  $m,n\\geq 2$ , such that \n(1)  $m+1$  is a prime number of type  $4k-1$ ;\n(2) there is a (positive) prime number  $p$  and nonnegative integer  $a$ , such that\n\\[\\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\\]","t":[{"b":4,"e":0.71429,"k":"flat","v":0.66514,"x":0.76781,"p":[[0,36,0.0,0.70079,0.14449,0.57132,0.71429,0.857,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,11,0,0,9,0,1],[4,36,0.1111,0.76781,0.11716,0.71429,0.78564,0.85714,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,12,0,0,15,0,1],[8,36,0.2222,0.74991,0.11303,0.71429,0.71429,0.85714,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,10,0,0,15,0,0],[12,36,0.3333,0.73447,0.13278,0.71429,0.71429,0.85714,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,12,1,0,13,0,0],[16,36,0.4444,0.70975,0.13597,0.57143,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,12,0,0,11,0,0],[20,36,0.5556,0.73656,0.11909,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,15,0,0,10,0,1],[24,36,0.6667,0.66514,0.17718,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,1,0,0,8,0,0,14,0,0,7,0,0],[28,36,0.7778,0.6696,0.16148,0.571,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,11,0,0,9,0,0],[32,36,0.8889,0.68063,0.11428,0.57143,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,1,10,0,0,14,0,0,6,0,0],[36,36,1.0,0.68748,0.14475,0.57142,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,7,0,0,11,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.33928,"x":0.665,"p":[[0,13,0.0,0.665,0.18765,0.57143,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,1,0,0,0,0,0,1,0,0,7,0,0,15,0,0,7,0,0],[4,13,0.3077,0.616,0.19376,0.42857,0.71414,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,5,0,0,5,0,0,12,0,0,6,0,0],[8,13,0.6154,0.65589,0.15503,0.571,0.71,0.74996,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,10,0,0,9,0,0,8,0,0],[12,13,0.9231,0.33928,0.15465,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,6,0,0,14,0,0,8,0,0,3,0,0,0,0,0,1,0,0],[13,13,1.0,0.39719,0.18475,0.2857,0.35714,0.571,0.14,0.71429,0,0,0,0,0,5,0,0,11,0,0,7,0,0,4,0,0,5,0,0,0,0,0]]}]},{"i":"5964088e6431d82b","q":"Given the equation  $$ a^b\\cdot b^c=c^a $$  in positive integers  $a$ ,  $b$ , and  $c$ .\n*(i)* Prove that any prime divisor of  $a$  divides  $b$  as well.\n*(ii)* Solve the equation under the assumption  $b\\ge a$ .\n*(iii)* Prove that the equation has infinitely many solutions.\n\n*(I. Voronovich)*","t":[{"b":0,"e":1.0,"k":"rising","v":0.76337,"x":0.98214,"p":[[0,7,0.0,0.76337,0.24383,0.57143,0.85714,1.0,0.0,1.0,1,10,1,1,0,0,0,0,1,0,0,2,0,0,6,0,0,3,0,0,9,0,10],[4,7,0.5714,0.82587,0.1371,0.71429,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,7,0,0,17,0,6],[7,7,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]},{"b":6,"e":0.857,"k":"flat","v":0.72767,"x":0.88839,"p":[[0,10,0.0,0.72767,0.26811,0.57143,0.71429,1.0,0.0,1.0,2,10,2,2,0,0,0,0,0,0,0,3,0,0,6,0,0,6,0,0,5,0,10],[4,10,0.4,0.73211,0.24937,0.57142,0.857,1.0,0.0,1.0,1,9,1,1,0,0,0,0,1,0,0,3,0,0,8,0,0,2,0,0,8,0,9],[8,10,0.8,0.88839,0.10555,0.85714,0.85714,1.0,0.71429,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,13,0,13],[10,10,1.0,0.875,0.12242,0.82143,0.85714,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,11,0,13]]}]},{"i":"5cfceeba3b1d08c6","q":"In triangle  $ABC$  let  $O$  and  $H$  be the circumcenter and the orthocenter. The point  $P$  is the reflection of  $A$  with respect to  $OH$ . Assume that  $P$  is not on the same side of  $BC$  as  $A$ . Points  $E,F$  lie on  $AB,AC$  respectively such that  $BE=PC \\ ,  CF=PB$ . Let  $K$  be the intersection point of  $AP,OH$ . Prove that  $\\angle EKF = 90 ^{\\circ}$ *Proposed by Iman Maghsoudi*","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fb9992a5b262d982","q":"Find all positive real numbers  $t$  with the following property: there exists an infinite set  $X$  of real numbers such that the inequality \\[ \\max\\{|x-(a-d)|,|y-a|,|z-(a+d)|\\}>td\\] holds for all (not necessarily distinct)  $x,y,z\\in X$ , all real numbers  $a$  and all positive real numbers  $d$ .","t":[{"b":1,"e":0.85714,"k":"volatile","v":0.43293,"x":0.68745,"p":[[0,6,0.0,0.6205,0.22477,0.5354,0.64286,0.74996,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,3,0,0,8,0,0,8,0,0,6,0,2],[4,6,0.6667,0.43293,0.34167,0.105,0.42857,0.71429,0.0,1.0,8,2,6,8,0,3,0,0,3,0,0,4,0,0,3,0,0,4,0,0,5,0,2],[6,6,1.0,0.68745,0.18709,0.57143,0.71429,0.74996,0.0,1.0,1,2,1,1,0,0,0,0,0,0,0,3,0,0,5,0,0,15,0,0,6,0,2]]},{"b":2,"e":0.0,"k":"flat","v":0.56247,"x":0.64284,"p":[[0,5,0.0,0.56247,0.35703,0.28571,0.64286,0.85714,0.0,1.0,5,6,2,5,0,2,0,0,5,0,0,0,0,0,4,0,0,4,0,0,6,0,6],[4,5,0.8,0.64284,0.28571,0.5354,0.71429,0.85714,0.0,1.0,3,4,3,3,0,1,0,0,0,0,0,4,0,0,5,0,0,7,0,0,8,0,4]]}]},{"i":"28d6d9005685c4ce","q":"For any integer $d>0$, let $f(d)$ be the smallest positive integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16$, and $f(6)=12$ ). Prove that for every integer $k \\geq 0$ the number $f\\left(2^{k}\\right)$ divides $f\\left(2^{k+1}\\right)$.","t":[{"b":2,"e":0.28571,"k":"falling","v":0.26785,"x":0.59374,"p":[[0,13,0.0,0.59374,0.33713,0.28571,0.5,1.0,0.0,1.0,1,11,0,1,0,0,0,0,14,0,0,1,0,0,1,0,0,3,0,0,1,0,11],[4,13,0.3077,0.41964,0.28333,0.28571,0.28571,0.46429,0.0,1.0,1,5,1,1,0,2,0,0,20,0,0,1,0,0,1,0,0,2,0,0,0,0,5],[8,13,0.6154,0.35714,0.19885,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,4,0,0,21,0,0,1,0,0,1,0,0,4,0,0,0,0,1],[12,13,0.9231,0.26785,0.10564,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,9,0,0,20,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[13,13,1.0,0.38839,0.20277,0.28571,0.28571,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,20,0,0,0,0,0,2,0,0,6,0,0,1,0,0]]},{"b":3,"e":0.28571,"k":"falling","v":0.31696,"x":0.54911,"p":[[0,8,0.0,0.54018,0.3402,0.28571,0.28571,1.0,0.14286,1.0,0,10,0,0,0,3,0,0,15,0,0,1,0,0,1,0,0,1,0,0,1,0,10],[4,8,0.5,0.54911,0.32362,0.28571,0.28571,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,16,0,0,2,0,0,1,0,0,1,0,0,2,0,9],[8,8,1.0,0.31696,0.18466,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,7,0,0,19,0,0,3,0,0,1,0,0,0,0,0,1,0,1]]}]},{"i":"794af645b0b50fe2","q":"For any odd positive integer $n$, let $r(n)$ be the odd positive integer such that the binary representation of $r(n)$ is the binary representation of $n$ written backwards. For example, $r(2023)=$\n$r\\left(11111100111_{2}\\right)=11100111111_{2}=1855$. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression $a_{1}, \\ldots, a_{8}$ of odd positive integers such that $r\\left(a_{1}\\right), \\ldots$, $r\\left(a_{8}\\right)$ is an arithmetic progression in that order.","t":[{"b":1,"e":0.0,"k":"falling","v":0.00893,"x":0.23213,"p":[[0,35,0.0,0.16964,0.20959,0.0,0.0,0.32143,0.0,0.71429,17,0,0,17,0,3,0,0,4,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[4,35,0.1143,0.13839,0.23003,0.0,0.0,0.32142,0.0,0.71429,23,0,0,23,0,0,0,0,1,0,0,4,0,0,3,0,0,1,0,0,0,0,0],[8,35,0.2286,0.17401,0.19143,0.0,0.14143,0.28571,0.0,0.57143,15,0,0,15,0,4,0,0,6,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[12,35,0.3429,0.23213,0.26182,0.0,0.14286,0.42857,0.0,1.0,15,1,0,15,0,2,0,0,3,0,0,8,0,0,2,0,0,1,0,0,0,0,1],[16,35,0.4571,0.16964,0.21558,0.0,0.0,0.32143,0.0,0.71429,17,0,0,17,0,4,0,0,3,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[20,35,0.5714,0.08929,0.17405,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,3,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[24,35,0.6857,0.15625,0.22968,0.0,0.0,0.2857,0.0,0.85714,19,0,0,19,0,3,0,0,4,0,0,2,0,0,3,0,0,0,0,0,1,0,0],[28,35,0.8,0.03572,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,35,0.9143,0.04464,0.12078,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[35,35,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.04009,"x":0.1607,"p":[[0,15,0.0,0.1607,0.22229,0.0,0.0,0.28571,0.0,0.85714,18,0,0,18,0,3,0,0,5,0,0,3,0,0,2,0,0,0,0,0,1,0,0],[4,15,0.2667,0.12947,0.19679,0.0,0.0,0.28571,0.0,0.57143,21,0,0,21,0,2,0,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[8,15,0.5333,0.13821,0.18723,0.0,0.0,0.17857,0.0,0.57143,17,0,0,17,0,7,0,0,3,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[12,15,0.8,0.11159,0.15036,0.0,0.0,0.1786,0.0,0.571,18,0,0,18,0,6,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[15,15,1.0,0.04009,0.08907,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"42d9bba9443efcff","q":"In an exhibition where  $2015$  paintings are shown, every participant picks a pair of paintings and writes it on the board. Then, Fake Artist (F.A.) chooses some of the pairs on the board, and marks one of the paintings in all of these pairs as \"better\". And then, Artist's Assistant (A.A.) comes and in his every move, he can mark  $A$  better then  $C$  in the pair  $(A,C)$  on the board if for a painting  $B$ ,  $A$  is marked as better than  $B$  and  $B$  is marked as better than  $C$  on the board. Find the minimum possible value of  $k$  such that, for any pairs of paintings on the board, F.A can compare  $k$  pairs of paintings making it possible for A.A to compare all of the remaining pairs of paintings.**P.S:** A.A can decide  $A_1>A_n$  if there is a sequence  $ A_1 > A_2 > A_3 > \\dots > A_{n-1} > A_n$  where  $X>Y$  means painting  $X$  is better than painting  $Y$ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.36606,"x":0.48212,"p":[[0,40,0.0,0.38839,0.10853,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,8,0,0,20,0,0,1,0,0,1,0,0,0,0,0],[4,40,0.1,0.48212,0.16655,0.42857,0.42857,0.4286,0.28571,1.0,0,2,0,0,0,0,0,0,3,0,0,22,0,0,4,0,0,0,0,0,1,0,2],[8,40,0.2,0.45089,0.16016,0.42857,0.42857,0.42857,0.2857,1.0,0,2,0,0,0,0,0,0,6,0,0,21,0,0,3,0,0,0,0,0,0,0,2],[12,40,0.3,0.43304,0.13115,0.42857,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,22,0,0,1,0,0,2,0,0,1,0,0],[16,40,0.4,0.43303,0.17307,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,4,0,0,20,0,0,2,0,0,1,0,0,1,0,1],[20,40,0.5,0.42853,0.09443,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,4,0,0,21,0,0,6,0,0,0,0,0,0,0,0],[24,40,0.6,0.43302,0.11563,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,23,0,0,1,0,0,3,0,0,0,0,0],[28,40,0.7,0.42857,0.12877,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,1,0,0,27,0,0,1,0,0,0,0,0,0,0,1],[32,40,0.8,0.36606,0.09404,0.28571,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,2,0,0,11,0,0,18,0,0,1,0,0,0,0,0,0,0,0],[36,40,0.9,0.42857,0.13363,0.42857,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,22,0,0,3,0,0,0,0,0,0,0,1],[40,40,1.0,0.41518,0.05486,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,27,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.27233,"x":0.41518,"p":[[0,10,0.0,0.41518,0.16115,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,8,0,0,18,0,0,1,0,0,2,0,0,0,0,1],[4,10,0.4,0.29464,0.07087,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,3,0,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.33928,0.15047,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,4,0,0,16,0,0,11,0,0,0,0,0,0,0,0,0,0,1],[10,10,1.0,0.27233,0.07457,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,6,0,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e717401524304b90","q":"For positive integers  $n, k, l$ , we define the number of  $l$ -tuples of positive integers  $(a_1,a_2,\\cdots a_l)$  satisfying the following as  $Q(n,k,l)$ .\n\n(i):  $n=a_1+a_2+\\cdots +a_l$ (ii):  $a_1>a_2>\\cdots > a_l > 0$ .\n\n(iii):  $a_l$  is an odd number.\n\n(iv): There are  $k$  odd numbers out of  $a_i$ .\n\nFor example, from  $9=8+1=6+3=6+2+1$ , we have  $Q(9,1,1)=1$ ,  $Q(9,1,2)=2$ ,  $Q(9,1,3)=1$ .\n\nProve that if  $n>k^2$ ,  $\\sum_{l=1}^n Q(n,k,l)$  is  $0$  or an even number.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,33,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,33,0.8485,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,54,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,54,0.7407,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,54,0.8148,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"b10069262de0f65b","q":"Each face of a  $7 \\times 7 \\times 7$  cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?\n\n*A. Chukhnov*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.06695,"p":[[0,16,0.0,0.06695,0.17848,0.0,0.0,0.0,0.0,0.57143,28,0,16,28,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[4,16,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,26,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,23,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.02679,"x":0.04911,"p":[[0,5,0.0,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,23,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,5,0.8,0.04911,0.14555,0.0,0.0,0.0,0.0,0.57143,28,0,18,28,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"edde1a53aeabcc9d","q":"For every $n \\geq 3$, determine all the configurations of $n$ distinct points $X_{1}, X_{2}, \\ldots, X_{n}$ in the plane, with the property that for any pair of distinct points $X_{i}, X_{j}$ there exists a permutation $\\sigma$ of the integers $\\{1, \\ldots, n\\}$, such that $\\mathrm{d}\\left(X_{i}, X_{k}\\right)=\\mathrm{d}\\left(X_{j}, X_{\\sigma(k)}\\right)$ for all $1 \\leq k \\leq n$.\n(We write $\\mathrm{d}(X, Y)$ to denote the distance between points $X$ and $Y$.)\n(United Kingdom) LuKe BetTs","t":[{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.2545,"p":[[0,22,0.0,0.2545,0.21941,0.0,0.2857,0.42858,0.0,0.71429,11,0,3,11,0,4,0,0,2,0,0,12,0,0,2,0,0,1,0,0,0,0,0],[4,22,0.1818,0.1696,0.20032,0.0,0.0,0.32143,0.0,0.57143,17,0,1,17,0,2,0,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[8,22,0.3636,0.22766,0.23379,0.0,0.2857,0.42857,0.0,0.71429,15,0,0,15,0,0,0,0,6,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[12,22,0.5455,0.05357,0.12752,0.0,0.0,0.0,0.0,0.42857,27,0,3,27,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.20978,"p":[[0,17,0.0,0.20978,0.18214,0.0,0.2857,0.42857,0.0,0.43,12,0,2,12,0,3,0,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.12945,0.19016,0.0,0.0,0.2857,0.0,0.571,21,0,3,21,0,1,0,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.12947,0.18681,0.0,0.0,0.28571,0.0,0.57143,20,0,1,20,0,3,0,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[12,17,0.7059,0.15618,0.21522,0.0,0.0,0.28571,0.0,0.57143,20,0,5,20,0,0,0,0,5,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"b407239e230d89b6","q":"Given a cyclic quadrilateral  $ABCD$ . There is a point  $P$  on side  $BC$  such that  $\\angle PAB=\\angle PDC=90^\\circ$ . The medians of vertexes  $A$  and  $D$  in triangles  $PAB$  and  $PDC$  meet at  $K$  and the bisectors of  $\\angle PAB$  and  $\\angle PDC$  meet at  $L$ . Prove that  $KL\\perp BC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.02232,"p":[[0,21,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,2,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":2,"e":0.14286,"k":"flat","v":0.0,"x":0.02223,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,4,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"efe57d5bafd5f687","q":"Given is a graph  $G$  of  $n+1$  vertices, which is constructed as follows: initially there is only one vertex  $v$ , and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that  $v$  has weight  $0$  and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by  $1$ . Prove that no weight can exceed  $n^2$ .","t":[{"b":2,"e":0.28571,"k":"flat","v":0.13839,"x":0.27232,"p":[[0,11,0.0,0.21428,0.20203,0.0,0.21428,0.28571,0.0,1.0,9,1,0,9,0,7,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[4,11,0.3636,0.13839,0.14053,0.0,0.14286,0.2857,0.0,0.42857,15,0,0,15,0,4,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.27232,0.17261,0.2857,0.28571,0.28571,0.0,1.0,4,1,0,4,0,3,0,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[11,11,1.0,0.26339,0.18249,0.14286,0.2857,0.28571,0.0,0.8571,6,0,0,6,0,3,0,0,18,0,0,2,0,0,2,0,0,0,0,0,1,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.17411,"x":0.24545,"p":[[0,13,0.0,0.18295,0.20279,0.0,0.14286,0.2857,0.0,1.0,10,1,0,10,0,12,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[4,13,0.3077,0.17856,0.22013,0.0,0.14286,0.28571,0.0,1.0,14,1,0,14,0,6,0,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[8,13,0.6154,0.17411,0.19799,0.0,0.14286,0.28571,0.0,0.85714,14,0,0,14,0,5,0,0,8,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[12,13,0.9231,0.17857,0.15152,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,6,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.24545,0.17587,0.14214,0.28571,0.42857,0.0,0.71429,7,0,0,7,0,6,0,0,10,0,0,8,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"434b20ccd6fd0b24","q":"Find the greatest natural number  $N$  such that, for any arrangement of the numbers  $1, 2, \\ldots, 400$  in a chessboard  $20 \\times 20$ , there exist two numbers in the same row or column, which differ by at least  $N.$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,13,0.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":2,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,9,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a1223bc9fd23b7d0","q":"Let  $ X$ ,  $ Y$ ,  $ Z$  be distinct positive integers having exactly two digits in such a way that:\r\n\r $ X\\equal{}10a \\plus{}b$ \r $ Y\\equal{}10b \\plus{}c$ \r $ Z\\equal{}10c \\plus{}a$ \r\n\r\n( $ a,b,c$  are digits)\r\n\r\nFind all posible values of  $ gcd(X,Y,Z)$","t":[{"b":3,"e":0.57143,"k":"flat","v":0.58479,"x":0.74553,"p":[[0,81,0.0,0.74553,0.18466,0.71429,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,14,0,0,6,0,6],[4,81,0.0494,0.69641,0.18815,0.57143,0.71429,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,14,0,0,6,0,3],[8,81,0.0988,0.69197,0.14334,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,19,0,0,5,0,1],[12,81,0.1481,0.70982,0.10999,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,19,0,0,7,0,0],[16,81,0.1975,0.67857,0.16366,0.57143,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,16,0,0,6,0,1],[20,81,0.2469,0.70089,0.14445,0.71429,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,0,16,0,0,9,0,0],[24,81,0.2963,0.64732,0.16746,0.57143,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,3,0,0,8,0,0,13,0,0,6,0,0],[28,81,0.3457,0.62052,0.15407,0.57143,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,16,0,0,2,0,0],[32,81,0.3951,0.64731,0.11838,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,15,0,0,3,0,0],[36,81,0.4444,0.61605,0.18012,0.42857,0.64286,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,8,0,0,6,0,0,12,0,0,2,0,2],[40,81,0.4938,0.65177,0.10063,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,22,0,0,0,0,0],[44,81,0.5432,0.625,0.14174,0.42859,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,4,0,0,18,0,0,0,0,1],[48,81,0.5926,0.67857,0.11294,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,25,0,0,2,0,0],[52,81,0.642,0.6875,0.1448,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,16,0,0,4,0,2],[56,81,0.6914,0.66964,0.14914,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,17,0,0,4,0,1],[60,81,0.7407,0.67411,0.1439,0.67857,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,3,0,0,20,0,0,3,0,1],[64,81,0.7901,0.63393,0.15126,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,17,0,0,3,0,0],[68,81,0.8395,0.58479,0.13534,0.53539,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,6,0,0,12,0,0,11,0,0,1,0,0],[72,81,0.8889,0.63388,0.13805,0.571,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,6,0,0,5,0,0,18,0,0,2,0,0],[76,81,0.9383,0.70088,0.11497,0.67857,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,20,0,0,2,0,2],[80,81,0.9877,0.66964,0.15335,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,18,0,0,4,0,1],[81,81,1.0,0.63393,0.13333,0.57143,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,4,0,0,4,0,0,22,0,0,0,0,0]]},{"b":7,"e":0.85714,"k":"flat","v":0.72321,"x":0.83482,"p":[[0,12,0.0,0.72321,0.17835,0.71429,0.71429,0.75,0.2857,1.0,0,5,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,19,0,0,3,0,5],[4,12,0.3333,0.72768,0.20628,0.57143,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,2,0,0,6,0,0,10,0,0,8,0,5],[8,12,0.6667,0.73213,0.23351,0.57143,0.71429,0.85714,0.0,1.0,1,7,0,1,0,0,0,0,1,0,0,3,0,0,4,0,0,8,0,0,8,0,7],[12,12,1.0,0.83482,0.13882,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,9,0,0,12,0,9]]}]},{"i":"98bea03dc8d97b4a","q":"For positive integers  $a,b,c$  let  $ \\alpha, \\beta, \\gamma$  be pairwise distinct positive integers such that\n    \\[ \\begin{cases}{c} \\displaystyle a &= \\alpha + \\beta + \\gamma,  \n    b &= \\alpha \\cdot \\beta + \\beta \\cdot \\gamma + \\gamma \\cdot \\alpha, \n    c^2 &= \\alpha\\beta\\gamma. \\end{cases} \\]\n    Also, let  $ \\lambda$  be a real number that satisfies the condition\n    \\[\\lambda^4 -2a\\lambda^2 + 8c\\lambda + a^2 - 4b = 0.\\]\n    Prove that  $\\lambda$  is an integer if and only if  $\\alpha, \\beta, \\gamma$  are all perfect squares.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.57139,"x":0.62054,"p":[[0,9,0.0,0.62054,0.3722,0.28571,0.64286,1.0,0.0,1.0,4,13,0,4,0,2,0,0,3,0,0,4,0,0,3,0,0,2,0,0,1,0,13],[4,9,0.4444,0.5982,0.37019,0.28571,0.71429,1.0,0.0,1.0,6,9,0,6,0,1,0,0,2,0,0,3,0,0,1,0,0,7,0,0,3,0,9],[8,9,0.8889,0.57139,0.19233,0.42857,0.57143,0.71429,0.0,1.0,1,1,0,1,0,0,0,0,2,0,0,8,0,0,9,0,0,9,0,0,2,0,1],[9,9,1.0,0.61161,0.12492,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,13,0,0,12,0,0,0,0,1]]},{"b":6,"e":0.71429,"k":"rising","v":0.47765,"x":0.81694,"p":[[0,27,0.0,0.47765,0.33428,0.14286,0.4286,0.71429,0.0,1.0,5,5,0,5,0,5,0,0,0,0,0,8,0,0,4,0,0,3,0,0,2,0,5],[4,27,0.1481,0.62054,0.34369,0.42857,0.71429,1.0,0.0,1.0,4,10,0,4,0,2,0,0,0,0,0,6,0,0,2,0,0,7,0,0,1,0,10],[8,27,0.2963,0.79017,0.25251,0.57143,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,0,0,0,2,0,0,5,0,0,4,0,0,4,0,15],[12,27,0.4444,0.81694,0.19639,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,6,0,0,5,0,14],[16,27,0.5926,0.74554,0.25187,0.57143,0.71429,1.0,0.14286,1.0,0,13,0,0,0,1,0,0,1,0,0,4,0,0,6,0,0,5,0,0,2,0,13],[20,27,0.7407,0.66075,0.19145,0.57143,0.71429,0.75,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,4,0,0,7,0,0,11,0,0,6,0,2],[24,27,0.8889,0.75892,0.13572,0.71429,0.71429,0.85714,0.571,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,18,0,0,3,0,6],[27,27,1.0,0.68749,0.24337,0.57143,0.71429,0.85704,0.0,1.0,1,5,0,1,0,2,0,0,0,0,0,2,0,0,4,0,0,13,0,0,5,0,5]]}]},{"i":"a23b114d81463e6e","q":"Find all triples  $(x,y,z)$  of positive integers such that  $x \\leq y \\leq z$  and \n\\[x^3(y^3+z^3)=2012(xyz+2).\\]","t":[{"b":1,"e":0.57143,"k":"flat","v":0.52231,"x":0.81694,"p":[[0,95,0.0,0.53571,0.29233,0.42857,0.4286,0.71429,0.0,1.0,3,5,1,3,0,1,0,0,3,0,0,10,0,0,4,0,0,4,0,0,2,0,5],[4,95,0.0421,0.58482,0.25843,0.42857,0.57143,0.71429,0.0,1.0,1,5,1,1,0,0,0,0,6,0,0,6,0,0,6,0,0,6,0,0,2,0,5],[8,95,0.0842,0.8125,0.22711,0.71429,0.85714,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0,4,0,0,6,0,15],[12,95,0.1263,0.81694,0.20278,0.57143,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,2,0,0,7,0,14],[16,95,0.1684,0.75443,0.1864,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,8,0,0,4,0,9],[20,95,0.2105,0.74997,0.19564,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,2,0,0,12,0,0,3,0,0,6,0,9],[24,95,0.2526,0.73659,0.19598,0.57143,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,10,0,0,7,0,0,3,0,9],[28,95,0.2947,0.81249,0.20342,0.57143,0.85714,1.0,0.42857,1.0,0,13,0,0,0,0,0,0,0,0,0,3,0,0,7,0,0,0,0,0,9,0,13],[32,95,0.3368,0.62944,0.19187,0.42859,0.57143,0.75,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,8,0,0,11,0,0,4,0,0,5,0,3],[36,95,0.3789,0.73213,0.18815,0.57143,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,3,0,0,9,0,6],[40,95,0.4211,0.6562,0.20159,0.571,0.57143,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,6,0,0,13,0,0,1,0,0,7,0,4],[44,95,0.4632,0.62942,0.1851,0.5354,0.57143,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,8,0,0,14,0,0,3,0,0,3,0,4],[48,95,0.5053,0.63838,0.18206,0.57143,0.57143,0.71429,0.2857,1.0,0,4,0,0,0,0,0,0,2,0,0,2,0,0,17,0,0,5,0,0,2,0,4],[52,95,0.5474,0.58487,0.1571,0.53605,0.57143,0.57143,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,7,0,0,18,0,0,2,0,0,2,0,2],[56,95,0.5895,0.63393,0.19212,0.5357,0.57143,0.71429,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,7,0,0,12,0,0,5,0,0,3,0,4],[60,95,0.6316,0.57142,0.16366,0.42857,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,9,0,0,13,0,0,4,0,0,3,0,1],[64,95,0.6737,0.5938,0.19265,0.4286,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,9,0,0,12,0,0,3,0,0,3,0,3],[68,95,0.7158,0.63836,0.18893,0.4286,0.57143,0.85704,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,11,0,0,3,0,0,6,0,3],[72,95,0.7579,0.60712,0.17497,0.4286,0.57143,0.60714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,15,0,0,2,0,0,3,0,3],[76,95,0.8,0.61153,0.15664,0.571,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,7,0,0,15,0,0,7,0,0,0,0,3],[80,95,0.8421,0.58924,0.15871,0.4286,0.57143,0.60714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,14,0,0,4,0,0,2,0,2],[84,95,0.8842,0.60265,0.15041,0.5713,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,17,0,0,6,0,0,1,0,2],[88,95,0.9263,0.56691,0.13592,0.571,0.57143,0.60714,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,5,0,0,17,0,0,7,0,0,1,0,0],[92,95,0.9684,0.52231,0.1107,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,11,0,0,15,0,0,4,0,0,0,0,0],[95,95,1.0,0.55357,0.14169,0.42857,0.57141,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,10,0,0,17,0,0,1,0,0,2,0,1]]},{"b":6,"e":0.42857,"k":"flat","v":0.55801,"x":0.70087,"p":[[0,14,0.0,0.5624,0.28347,0.28571,0.64286,0.71429,0.0,1.0,2,3,0,2,0,2,0,0,5,0,0,4,0,0,3,0,0,9,0,0,4,0,3],[4,14,0.2857,0.64714,0.27427,0.42857,0.71214,0.85714,0.0,1.0,1,7,0,1,0,1,0,0,3,0,0,5,0,0,5,0,0,6,0,0,4,0,7],[8,14,0.5714,0.58035,0.3213,0.28593,0.57143,0.85714,0.0,1.0,2,6,1,2,0,3,0,0,4,0,0,5,0,0,5,0,0,0,0,0,7,0,6],[12,14,0.8571,0.70087,0.13999,0.57143,0.71429,0.75,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,7,0,0,15,0,0,7,0,1],[14,14,1.0,0.55801,0.16114,0.4286,0.57143,0.60714,0.0,0.85714,1,0,0,1,0,0,0,0,1,0,0,7,0,0,15,0,0,6,0,0,2,0,0]]}]},{"i":"726c28091365f071","q":"Given trapezoid  $ABCD$  with parallel sides  $AB$  and  $CD$ , let  $E$  be a point on line  $BC$  outside segment  $BC$ , such that segment  $AE$  intersects segment  $CD$ . Assume that there exists a point  $F$  inside segment  $AD$  such that  $\\angle EAD=\\angle CBF$ . Denote by  $I$  the point of intersection of  $CD$  and  $EF$ , and by  $J$  the point of intersection of  $AB$  and  $EF$ . Let  $K$  be the midpoint of segment  $EF$ , and assume that  $K$  is different from  $I$  and  $J$ .\n\nProve that  $K$  belongs to the circumcircle of  $\\triangle ABI$  if and only if  $K$  belongs to the circumcircle of  $\\triangle CDJ$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.00893,"x":0.25893,"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],[4,35,0.1143,0.0357,0.10708,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,35,0.2286,0.13839,0.27776,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[12,35,0.3429,0.08027,0.21107,0.0,0.0,0.035,0.0,1.0,24,1,0,24,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[16,35,0.4571,0.04018,0.08918,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.25893,0.27067,0.14286,0.14286,0.32143,0.0,1.0,5,3,0,5,0,16,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,3],[28,35,0.8,0.12483,0.0778,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.12482,0.06912,0.14214,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.12054,"p":[[0,15,0.0,0.06696,0.18205,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,15,0.2667,0.12054,0.28818,0.0,0.0,0.14286,0.0,1.0,23,3,0,23,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,15,0.5333,0.09375,0.21902,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,15,0.8,0.0625,0.12846,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"06120a5b1279ba71","q":"In the acute-angled triangle  $ABC$ , with  $AB \\ne AC$ ,  $D$  is the foot of the angle bisector of angle  $A$ , and  $E, F$  are the feet of the altitudes from  $B$  and  $C$ , respectively. The circumcircles of triangles  $DBF$  and  $DCE$  intersect for the second time at  $M$ . Prove that  $ME = MF$ .\n\nLeonard Giugiuc","t":[{"b":6,"e":1.0,"k":"rising","v":0.62945,"x":0.80804,"p":[[0,23,0.0,0.64729,0.223,0.57143,0.71429,0.71429,0.0,1.0,2,2,1,2,0,1,0,0,0,0,0,0,0,0,8,0,0,16,0,0,3,0,2],[4,23,0.1739,0.62945,0.19516,0.57143,0.71429,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,0,0,0,0,0,0,9,0,0,18,0,0,1,0,1],[8,23,0.3478,0.64283,0.15154,0.57143,0.71429,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,0,0,0,10,0,0,18,0,0,2,0,0],[12,23,0.5217,0.66071,0.22799,0.57143,0.71429,0.71429,0.0,1.0,1,4,0,1,0,2,0,0,0,0,0,1,0,0,7,0,0,15,0,0,2,0,4],[16,23,0.6957,0.66069,0.37072,0.42857,0.85714,1.0,0.0,1.0,6,9,6,6,0,0,0,0,1,0,0,3,0,0,2,0,0,0,0,0,11,0,9],[20,23,0.8696,0.79915,0.34411,0.75036,1.0,1.0,0.14286,1.0,0,23,0,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,23],[23,23,1.0,0.80804,0.32461,0.57143,1.0,1.0,0.14286,1.0,0,23,0,0,0,5,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,23]]},{"b":7,"e":0.71429,"k":"flat","v":0.54009,"x":0.67856,"p":[[0,20,0.0,0.60266,0.23347,0.57143,0.57143,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,0,0,0,2,0,0,12,0,0,8,0,0,4,0,2],[4,20,0.2,0.61606,0.24856,0.57143,0.64286,0.71429,0.14286,1.0,0,4,0,0,0,5,0,0,0,0,0,1,0,0,10,0,0,10,0,0,2,0,4],[8,20,0.4,0.63839,0.24218,0.57143,0.71429,0.71429,0.0,1.0,1,3,0,1,0,3,0,0,0,0,0,1,0,0,8,0,0,12,0,0,4,0,3],[12,20,0.6,0.54009,0.27384,0.35714,0.57143,0.71429,0.14,1.0,0,3,0,0,0,8,0,0,0,0,0,4,0,0,7,0,0,8,0,0,2,0,3],[16,20,0.8,0.67408,0.27488,0.57143,0.71429,1.0,0.0,1.0,1,9,1,1,0,3,0,0,0,0,0,1,0,0,8,0,0,10,0,0,0,0,9],[20,20,1.0,0.67856,0.09451,0.71429,0.71429,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,27,0,0,0,0,0]]}]},{"i":"0d6f62dcb02713d1","q":"Given a non-isoceles triangle  $ABC$  inscribes a circle  $(O,R)$  (center  $O$ , radius  $R$ ). Consider a varying line  $l$  such that  $l\\perp OA$  and  $l$  always intersects the rays  $AB,AC$  and these intersectional points are called  $M,N$ . Suppose that the lines  $BN$  and  $CM$  intersect, and if the intersectional point is called  $K$  then the lines  $AK$  and  $BC$  intersect.\r $1$ , Assume that   $P$  is the intersectional point of  $AK$  and  $BC$ . Show that the circumcircle of the triangle  $MNP$  is always through a fixed point.\r $2$ , Assume that  $H$  is the orthocentre of the triangle  $AMN$ . Denote  $BC=a$ , and  $d$  is the distance between  $A$  and the line  $HK$ . Prove that  $d\\leq\\sqrt{4R^2-a^2}$  and the equality occurs iff the line  $l$  is through the intersectional point of two lines  $AO$  and  $BC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.05804,"x":0.10259,"p":[[0,11,0.0,0.07125,0.08734,0.0,0.0,0.14286,0.0,0.28571,18,0,3,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0892,0.09274,0.0,0.14143,0.14286,0.0,0.2857,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.10259,0.08914,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.04911,"x":0.16518,"p":[[0,44,0.0,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,3,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.10705,0.08745,0.0,0.14286,0.14286,0.0,0.28571,11,0,1,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.07589,0.09438,0.0,0.0,0.14286,0.0,0.2857,18,0,1,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.07589,0.10092,0.0,0.0,0.14286,0.0,0.28571,19,0,6,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.12938,0.07455,0.14286,0.14286,0.14286,0.0,0.28571,6,0,1,6,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.12937,0.07455,0.14286,0.14286,0.14286,0.0,0.28571,6,0,2,6,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.15616,0.0827,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,24,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.16054,0.07788,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,25,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.14723,0.04352,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.16518,0.06297,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.16296,0.05985,0.14286,0.14286,0.1429,0.0,0.28571,1,0,0,1,0,25,0,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.14706,0.04355,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8ee7f5c2559d9e79","q":"For any odd prime $p$ and any integer $n$, let $d_{p}(n) \\in\\{0,1, \\ldots, p-1\\}$ denote the remainder when $n$ is divided by $p$. We say that $\\left(a_{0}, a_{1}, a_{2}, \\ldots\\right)$ is a $p$-sequence, if $a_{0}$ is a positive integer coprime to $p$, and $a_{n+1}=a_{n}+d_{p}\\left(a_{n}\\right)$ for $n \\geqslant 0$. (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $\\left(a_{0}, a_{1}, a_{2}, \\ldots\\right)$ and $\\left(b_{0}, b_{1}, b_{2}, \\ldots\\right)$ such that $a_{n}>b_{n}$ for infinitely many $n$, and $b_{n}>a_{n}$ for infinitely many $n$ ? (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $\\left(a_{0}, a_{1}, a_{2}, \\ldots\\right)$ and $\\left(b_{0}, b_{1}, b_{2}, \\ldots\\right)$ such that $a_{0}<b_{0}$, but $a_{n}>b_{n}$ for all $n \\geqslant 1$ ? (United Kingdom)","t":[{"b":2,"e":0.14286,"k":"flat","v":0.19643,"x":0.28555,"p":[[0,45,0.0,0.20525,0.19542,0.0,0.14286,0.28571,0.0,0.71429,9,0,1,9,0,11,0,0,6,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[4,45,0.0889,0.22321,0.12846,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,0,19,0,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[8,45,0.1778,0.26786,0.10564,0.14286,0.2857,0.28571,0.14286,0.57143,0,0,0,0,0,10,0,0,17,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,45,0.2667,0.28555,0.09476,0.2857,0.2857,0.28579,0.14,0.4286,0,0,0,0,0,7,0,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.26339,0.08827,0.14296,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,9,0,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.25893,0.10374,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,11,0,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[24,45,0.5333,0.27229,0.1203,0.14286,0.2857,0.28571,0.14286,0.571,0,0,0,0,0,11,0,0,15,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[28,45,0.6222,0.25,0.16751,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,11,0,0,14,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[32,45,0.7111,0.28554,0.14743,0.14286,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,10,0,0,12,0,0,7,0,0,1,0,0,1,0,0,0,0,0],[36,45,0.8,0.27232,0.13534,0.14286,0.2857,0.42857,0.0,0.57143,1,0,0,1,0,12,0,0,9,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[40,45,0.8889,0.22306,0.13817,0.14286,0.14286,0.28571,0.0,0.57143,3,0,0,3,0,14,0,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[44,45,0.9778,0.20071,0.09365,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,19,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.19643,0.08564,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,19,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.16964,"x":0.27223,"p":[[0,23,0.0,0.27223,0.22694,0.14214,0.21428,0.42857,0.0,0.71429,7,0,0,7,0,9,0,0,5,0,0,5,0,0,3,0,0,3,0,0,0,0,0],[4,23,0.1739,0.23651,0.19105,0.14286,0.14286,0.42857,0.0,0.71429,5,0,0,5,0,15,0,0,3,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[8,23,0.3478,0.22767,0.23105,0.0,0.14286,0.32143,0.0,0.71429,9,0,0,9,0,12,0,0,3,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[12,23,0.5217,0.17858,0.21129,0.0,0.14286,0.17857,0.0,0.85714,11,0,0,11,0,13,0,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[16,23,0.6957,0.22768,0.21387,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,9,0,0,10,0,0,2,0,0,0,0,0,2,0,0,1,0,0],[20,23,0.8696,0.16964,0.14914,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,0,11,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.21874,0.17488,0.14286,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,11,0,0,7,0,0,4,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"1af1e063f6341c45","q":"For every positive integer  $n$ , let  $p(n)$  denote the number of sets  $\\{x_1, x_2, \\dots, x_k\\}$  of integers with  $x_1 > x_2 > \\dots > x_k > 0$  and  $n = x_1 + x_3 + x_5 + \\dots$  (the right hand side here means the sum of all odd-indexed elements). As an example,  $p(6) = 11$  because all satisfying sets are as follows:   $$ \\{6\\}, \\{6, 5\\}, \\{6, 4\\}, \\{6, 3\\}, \\{6, 2\\}, \\{6, 1\\}, \\{5, 4, 1\\}, \\{5, 3, 1\\}, \\{5, 2, 1\\}, \\{4, 3, 2\\}, \\{4, 3, 2, 1\\}. $$   Show that  $p(n)$  equals to the number of partitions of  $n$  for every positive integer  $n$ .","t":[{"b":0,"e":0.0,"k":"volatile","v":0.04018,"x":0.4241,"p":[[0,7,0.0,0.34375,0.37433,0.0,0.2143,0.71429,0.0,1.0,14,3,0,14,0,2,0,0,3,0,0,2,0,0,2,0,0,2,0,0,4,0,3],[4,7,0.5714,0.4241,0.38045,0.0,0.42857,0.857,0.0,1.0,11,3,2,11,0,3,0,0,0,0,0,4,0,0,2,0,0,3,0,0,6,0,3],[7,7,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.71429,"k":"volatile","v":0.38838,"x":0.72308,"p":[[0,8,0.0,0.38838,0.37155,0.0,0.28571,0.71429,0.0,1.0,12,3,0,12,0,2,0,0,3,0,0,1,0,0,4,0,0,3,0,0,4,0,3],[4,8,0.5,0.70076,0.09004,0.71429,0.71429,0.71429,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,25,0,0,1,0,1],[8,8,1.0,0.72308,0.07938,0.71429,0.71429,0.71429,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,0,0,2]]}]},{"i":"a7e7811dbc10f383","q":"Given a triangle  $ \\triangle ABC $ . Denote its incenter and orthocenter by  $ I, H $ , respectively. If there is a point  $ K $  with   $$  AH+AK = BH+BK = CH+CK  $$   Show that  $ H, I, K $  are collinear.\n\n*Proposed by Evan Chen*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,10,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,7,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5be7281d5e6bb3d6","q":"Given an acute-angled triangle  $ABC$  with orthocenter  $H$ . Reflection of nine-point circle about  $AH$  intersects circumcircle at points  $X$  and  $Y$ . Prove that  $AH$  is the external bisector of  $\\angle XHY$ . \n\n*Proposed by Mohammad Javad Shabani*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,19,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,31,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"bd529fa17a5ee5f5","q":"In a circle  $\\Gamma_{1}$ , centered at  $O$ ,  $AB$  and  $CD$  are two unequal in length chords intersecting at  $E$  inside  $\\Gamma_{1}$ . A circle  $\\Gamma_{2}$ , centered at  $I$  is tangent to  $\\Gamma_{1}$  internally at  $F$ , and also tangent to  $AB$  at  $G$  and  $CD$  at  $H$ . A line  $l$  through  $O$  intersects  $AB$  and  $CD$  at  $P$  and  $Q$  respectively such that  $EP = EQ$ . The line  $EF$  intersects  $l$  at  $M$ . Prove that the line through  $M$  parallel to  $AB$  is tangent to  $\\Gamma_{1}$","t":[{"b":0,"e":0.1429,"k":"falling","v":0.12045,"x":0.38392,"p":[[0,37,0.0,0.37044,0.267,0.14286,0.28571,0.42858,0.0,1.0,2,3,0,2,0,8,0,0,9,0,0,6,0,0,2,0,0,2,0,0,0,0,3],[4,37,0.1081,0.38392,0.2822,0.14286,0.28571,0.57111,0.0,1.0,3,3,0,3,0,8,0,0,6,0,0,6,0,0,3,0,0,3,0,0,0,0,3],[8,37,0.2162,0.3125,0.21852,0.14286,0.28571,0.28571,0.0,1.0,1,1,0,1,0,11,0,0,13,0,0,1,0,0,2,0,0,3,0,0,0,0,1],[12,37,0.3243,0.30357,0.20439,0.14286,0.28571,0.42857,0.0,1.0,1,1,0,1,0,12,0,0,10,0,0,4,0,0,3,0,0,1,0,0,0,0,1],[16,37,0.4324,0.25445,0.12741,0.14286,0.28571,0.28571,0.0,0.571,2,0,0,2,0,10,0,0,14,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[20,37,0.5405,0.12045,0.10169,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,15,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.18302,0.1566,0.0,0.14286,0.28571,0.0,0.571,10,0,0,10,0,8,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[28,37,0.7568,0.21875,0.14279,0.14286,0.2857,0.28571,0.0,0.57143,6,0,0,6,0,8,0,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[32,37,0.8649,0.19196,0.16982,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,7,0,0,11,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[36,37,0.973,0.20536,0.16342,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,10,0,0,12,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[37,37,1.0,0.21875,0.14719,0.14286,0.28571,0.28571,0.0,0.71429,6,0,0,6,0,7,0,0,17,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.19188,"x":0.35259,"p":[[0,10,0.0,0.35259,0.25257,0.14286,0.28571,0.46428,0.0,1.0,3,1,0,3,0,8,0,0,8,0,0,5,0,0,4,0,0,1,0,0,2,0,1],[4,10,0.4,0.29909,0.30168,0.14286,0.14286,0.57111,0.0,1.0,6,3,3,6,0,14,0,0,2,0,0,1,0,0,5,0,0,1,0,0,0,0,3],[8,10,0.8,0.19188,0.13654,0.14286,0.14286,0.28571,0.0,0.4286,5,0,3,5,0,17,0,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.25429,0.10574,0.14286,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,10,0,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ec2a97140d8b39c8","q":"Given is a trapezoid  $ ABCD$  where  $ AB$  and  $ CD$  are parallel, and  $ A,B,C,D$  are clockwise in this order. Let  $ \\Gamma_1$  be the circle with center  $ A$  passing through  $ B$ ,  $ \\Gamma_2$  be the circle with center  $ C$  passing through  $ D$ . The intersection of line  $ BD$  and  $ \\Gamma_1$  is  $ P$   $ ( \\ne B,D)$ . Denote by  $ \\Gamma$  the circle with diameter  $ PD$ , and let  $ \\Gamma$  and  $ \\Gamma_1$  meet at  $ X$  $ ( \\ne P)$ .  $ \\Gamma$  and  $ \\Gamma_2$  meet at  $ Y$ . If the circumcircle of triangle  $ XBY$  and  $ \\Gamma_2$  meet at  $ Q$ , prove that  $ B,D,Q$  are collinear.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,24,0.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.02223,0.06281,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.03562,0.08737,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,26,0.0,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,1,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,26,0.1538,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,1,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.04009,0.08907,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a71bc09e0caffa86","q":"For every point on the plane, one of  $ n$  colors are colored to it such that:\r\n\r $ (1)$  Every color is used infinitely many times.\r\n\r $ (2)$  There exists one line such that all points on this lines are colored exactly by one of two colors.\r\n\r\nFind the least value of  $ n$  such that there exist four concyclic points with pairwise distinct colors.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,30,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,8,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,6,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"715b8d3476d3d2fa","q":"For integral $m$, let $p(m)$ be the greatest prime divisor of $m$. By convention, we set $p( \\pm 1)=1$ and $p(0)=\\infty$. Find all polynomials $f$ with integer coefficients such that the sequence  $$ \\left\\{p\\left(f\\left(n^{2}\\right)\\right)-2 n\\right\\}_{n \\geq 0} $$  is bounded above. (In particular, this requires $f\\left(n^{2}\\right) \\neq 0$ for $n \\geq 0$.)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,34,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.01339,"x":0.0625,"p":[[0,25,0.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c66488512bce696d","q":"For a binary string  $S$  (i.e. a string of 0 's and 1's) that contains at least one 0 , we produce a binary string  $f(S)$  as follows:\n- If the substring 110 occurs in  $S$ , replace each instance of 110 with 01 to produce  $f(S)$ ;\n- Otherwise, replace the leftmost occurrence of 0 in  $S$  by 1 to produce  $f(S)$ .\nGiven binary string  $S$  of length  $n$ , we define the lifetime of  $S$  to be the number of times  $f$  can be applied to  $S$  until the resulting string contains no more 0 's. For example, $$","t":[{"b":1,"e":0.14286,"k":"falling","v":0.05786,"x":0.30801,"p":[[0,9,0.0,0.30801,0.32945,0.0,0.28571,0.4286,0.0,1.0,11,3,4,11,0,4,0,0,7,0,0,3,0,0,1,0,0,1,0,0,2,0,3],[4,9,0.4444,0.19634,0.23625,0.0,0.14286,0.28571,0.0,1.0,13,1,1,13,0,7,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[8,9,0.8889,0.05786,0.14213,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,3,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[9,9,1.0,0.08464,0.15909,0.0,0.0,0.14,0.0,0.57143,23,0,0,23,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"rising","v":0.22308,"x":0.83705,"p":[[0,60,0.0,0.38838,0.34299,0.0,0.28571,0.71429,0.0,1.0,9,3,1,9,0,2,0,0,8,0,0,1,0,0,3,0,0,3,0,0,3,0,3],[4,60,0.0667,0.28116,0.27779,0.0,0.2857,0.32144,0.0,1.0,9,1,0,9,0,6,0,0,9,0,0,1,0,0,3,0,0,1,0,0,2,0,1],[8,60,0.1333,0.27669,0.20503,0.14214,0.2857,0.28571,0.0,0.71429,7,0,0,7,0,3,0,0,15,0,0,1,0,0,4,0,0,2,0,0,0,0,0],[12,60,0.2,0.22308,0.15121,0.14286,0.2857,0.28571,0.0,0.571,5,0,1,5,0,10,0,0,14,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[16,60,0.2667,0.31247,0.24335,0.14286,0.28571,0.42857,0.0,1.0,5,1,0,5,0,5,0,0,13,0,0,4,0,0,2,0,0,0,0,0,2,0,1],[20,60,0.3333,0.33035,0.32229,0.10714,0.2857,0.4642,0.0,1.0,8,4,1,8,0,6,0,0,8,0,0,2,0,0,2,0,0,2,0,0,0,0,4],[24,60,0.4,0.299,0.25096,0.14286,0.2857,0.28571,0.0,1.0,5,1,0,5,0,7,0,0,13,0,0,2,0,0,1,0,0,1,0,0,2,0,1],[28,60,0.4667,0.46648,0.28457,0.28571,0.42857,0.71429,0.0,1.0,2,3,0,2,0,3,0,0,9,0,1,5,0,0,3,0,0,3,0,0,3,0,3],[32,60,0.5333,0.30349,0.14184,0.2857,0.28571,0.28571,0.14,0.85714,0,0,0,0,0,5,0,0,23,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[36,60,0.6,0.73654,0.16023,0.57143,0.71429,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,12,0,0,2,0,7],[40,60,0.6667,0.72756,0.13545,0.57132,0.71429,0.85704,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,12,0,0,7,0,3],[44,60,0.7333,0.70534,0.18878,0.57143,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,12,0,0,8,0,3],[48,60,0.8,0.80343,0.12251,0.71429,0.85707,0.85714,0.4286,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,15,0,4],[52,60,0.8667,0.79685,0.21512,0.71429,0.85714,1.0,0.0,1.0,1,10,1,1,0,0,0,0,0,0,0,1,0,0,4,1,0,5,0,0,10,0,10],[56,60,0.9333,0.80353,0.14178,0.71429,0.78564,0.89286,0.571,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,8,0,8],[60,60,1.0,0.83705,0.11467,0.71429,0.85714,0.89286,0.643,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,11,0,0,12,0,8]]}]},{"i":"e2ccf22a32de79ae","q":"Given is a triangle $A B C$. The bisector of $\\angle C A B$ intersects $B C$ at $L$. On the interiors of sides $A C$ and $A B$ lie points $M$ and $N$, respectively, such that $A L$, $B M$, and $C N$ are concurrent and $\\angle A M N=\\angle A L B$. Prove that $\\angle N M L=90^{\\circ}$.","t":[{"b":6,"e":0.14286,"k":"flat","v":0.10268,"x":0.13375,"p":[[0,9,0.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,1,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.10706,0.06181,0.105,0.14286,0.14286,0.0,0.1429,8,0,1,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.1159,"x":0.13384,"p":[[0,9,0.0,0.1159,0.05568,0.14214,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e978fac045cea47f","q":"In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order:\n\n\n- Choose two adjacent points with <u>different</u> colors, and add a point between them with one of the two colors only.\n- Choose two adjacent points with <u>the same</u> color, and add a point between them with any of the three colors.\n- Choose three adjacent points, at least two of them having the same color, and delete the middle point.\n\nCan Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?","t":[{"b":1,"e":1.0,"k":"rising","v":0.29464,"x":1.0,"p":[[0,79,0.0,0.33918,0.42524,0.0,0.0,0.78571,0.0,1.0,17,8,0,17,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,8],[4,79,0.0506,0.37946,0.46649,0.0,0.0,1.0,0.0,1.0,18,10,0,18,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,10],[8,79,0.1013,0.29464,0.44741,0.0,0.0,1.0,0.0,1.0,22,9,0,22,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9],[12,79,0.1519,0.47768,0.47462,0.0,0.35714,1.0,0.0,1.0,15,13,0,15,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,13],[16,79,0.2025,0.32589,0.45769,0.0,0.0,1.0,0.0,1.0,20,10,0,20,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[20,79,0.2532,0.375,0.45702,0.0,0.0,1.0,0.0,1.0,18,10,1,18,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,10],[24,79,0.3038,0.875,0.33072,1.0,1.0,1.0,0.0,1.0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[28,79,0.3544,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,79,0.4051,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,79,0.4557,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[40,79,0.5063,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[44,79,0.557,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[56,79,0.7089,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[60,79,0.7595,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[64,79,0.8101,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[68,79,0.8608,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[72,79,0.9114,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":0.71429,"k":"volatile","v":0.29018,"x":0.96875,"p":[[0,11,0.0,0.53571,0.47916,0.0,0.71429,1.0,0.0,1.0,14,15,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,15],[4,11,0.3636,0.29018,0.43811,0.0,0.0,0.89286,0.0,1.0,21,8,0,21,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,8],[8,11,0.7273,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[11,11,1.0,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]}]},{"i":"c2aae415b4a51118","q":"Given quadrilateral  $ABCD$  such that  $\\angle BAD+2 \\angle BCD=180 ^ \\circ .$  \nLet  $E$  be the intersection of  $BD$  and the internal bisector of   $\\angle BAD$ . \nThe perpendicular bisector of  $AE$  intersects  $CB,CD$  at  $X,Y,$  respectively.\nProve that  $A,C,X,Y$  are concyclic.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.1429,"k":"flat","v":0.0,"x":0.1383,"p":[[0,32,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,32,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.143,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0cf68db903b65e88","q":"Find the smallest natural number  $n$  that the following statement holds :\r\nLet  $A$  be a finite subset of  $\\mathbb R^{2}$ . For each  $n$  points in  $A$  there are two lines including these  $n$  points. All of the points lie on two lines.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,7,0.0,0.06696,0.15561,0.0,0.0,0.0,0.0,0.42857,27,0,6,27,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.07589,0.15146,0.0,0.0,0.0,0.0,0.4286,25,0,6,25,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.03125,"x":0.29022,"p":[[0,25,0.0,0.08929,0.21651,0.0,0.0,0.0,0.0,1.0,26,1,3,26,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[4,25,0.16,0.10714,0.18558,0.0,0.0,0.10714,0.0,0.42857,24,0,3,24,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.08034,0.15943,0.0,0.0,0.0,0.0,0.571,25,0,5,25,0,0,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,25,0.48,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,8,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.29022,0.19722,0.0,0.42857,0.42857,0.0,0.43,10,0,10,10,0,0,0,0,1,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.18304,0.20897,0.0,0.0,0.42857,0.0,0.4286,18,0,18,18,0,0,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"18e6102a71b8355f","q":"Given six positive numbers $a, b, c, d, e, f$ such that $a<b<c<d<e<f$. Let $a+c+e=S$ and $b+d+f=T$. Prove that  $$ 2 S T>\\sqrt{3(S+T)(S(b d+b f+d f)+T(a c+a e+c e))} $$  (South Korea)","t":[{"b":6,"e":0.28571,"k":"flat","v":0.10268,"x":0.20973,"p":[[0,25,0.0,0.11161,0.12234,0.0,0.14286,0.14286,0.0,0.4286,15,0,0,15,0,10,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.1384,0.15355,0.0,0.14286,0.1786,0.0,0.4286,14,0,0,14,0,10,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.18732,0.15341,0.105,0.14286,0.2857,0.0,0.4286,8,0,0,8,0,13,0,0,4,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.15625,0.10926,0.14286,0.14286,0.14286,0.0,0.4286,5,0,0,5,0,22,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.10268,0.10853,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.17411,0.14167,0.14286,0.14286,0.2857,0.0,0.57143,7,0,0,7,0,16,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[24,25,0.96,0.16518,0.11904,0.14286,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,15,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.20973,0.15565,0.14286,0.14286,0.28571,0.0,0.71429,5,0,0,5,0,14,0,0,8,0,0,4,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04464,"x":0.14286,"p":[[0,22,0.0,0.14286,0.16366,0.0,0.14286,0.17857,0.0,0.57143,14,0,0,14,0,10,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[4,22,0.1818,0.09375,0.14987,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,5,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.09375,0.14987,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,22,0.5455,0.08924,0.09953,0.0,0.14143,0.14286,0.0,0.43,15,0,0,15,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0759,0.09439,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c1ce7bdc84e38a37","q":"For any finite sets $X$ and $Y$ of positive integers, denote by $f_{X}(k)$ the $k^{\\text {th }}$ smallest positive integer not in $X$, and let  $$ X * Y=X \\cup\\left\\{f_{X}(y): y \\in Y\\right\\} $$  Let $A$ be a set of $a>0$ positive integers, and let $B$ be a set of $b>0$ positive integers. Prove that if $A * B=B * A$, then  $$ \\underbrace{A *(A * \\cdots *(A *(A * A)) \\ldots)}_{A \\text { appears } b \\text { times }}=\\underbrace{B *(B * \\cdots *(B *(B * B)) \\ldots)}_{B \\text { appears } a \\text { times }} . $$","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,24,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.00446,"x":0.05357,"p":[[0,15,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f0b5f0f8d2c6b27b","q":"Given is trapezoid  $ABCD$ ,  $M$  and  $N$  being the midpoints of the bases of  $AD$  and  $BC$ , respectively.\na) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment  $MN$ .\nb) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line  $MN$ ?","t":[{"b":0,"e":0.57143,"k":"flat","v":0.47762,"x":0.558,"p":[[0,14,0.0,0.49557,0.26481,0.42857,0.5005,0.71429,0.0,1.0,3,1,3,3,0,4,0,0,0,0,0,9,0,0,4,0,0,9,0,0,2,0,1],[4,14,0.2857,0.47762,0.31258,0.14286,0.571,0.71429,0.0,1.0,5,2,4,5,0,4,0,0,3,0,0,1,0,0,9,0,0,4,0,0,4,0,2],[8,14,0.5714,0.558,0.23244,0.42859,0.57143,0.71429,0.0,0.85714,1,0,1,1,0,3,0,0,2,0,0,4,0,0,10,0,0,6,0,0,6,0,0],[12,14,0.8571,0.50443,0.27194,0.28571,0.4998,0.71429,0.0,1.0,2,2,2,2,0,3,0,0,5,0,0,6,0,0,6,0,0,4,0,0,4,0,2],[14,14,1.0,0.51775,0.25455,0.39286,0.571,0.71429,0.0,1.0,1,2,0,1,0,4,0,0,3,0,0,7,0,0,7,0,0,5,0,0,3,0,2]]},{"b":4,"e":0.2857,"k":"flat","v":0.30346,"x":0.51337,"p":[[0,26,0.0,0.51337,0.27399,0.28571,0.57143,0.71429,0.0,1.0,3,2,3,3,0,1,0,0,6,0,0,4,0,0,8,0,0,4,0,0,4,0,2],[4,26,0.1538,0.49995,0.31742,0.24999,0.571,0.71429,0.0,1.0,6,1,5,6,0,2,0,0,2,0,0,4,0,0,4,0,0,7,0,0,6,0,1],[8,26,0.3077,0.49538,0.31528,0.2857,0.4998,0.74996,0.0,1.0,4,2,4,4,0,3,0,0,6,0,0,3,0,0,3,0,0,5,0,0,6,0,2],[12,26,0.4615,0.42415,0.29771,0.14289,0.35714,0.71429,0.0,1.0,4,2,4,4,0,5,0,0,7,0,0,4,0,0,2,0,0,6,0,0,2,0,2],[16,26,0.6154,0.43299,0.23549,0.2857,0.42857,0.57141,0.0,0.85714,1,0,1,1,0,6,0,0,6,0,0,7,0,0,5,0,0,4,0,0,3,0,0],[20,26,0.7692,0.30346,0.18475,0.14286,0.2857,0.42857,0.0,0.71429,1,0,0,1,0,12,0,0,9,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[24,26,0.9231,0.39955,0.2341,0.19643,0.35714,0.57143,0.0,0.85714,1,0,0,1,0,7,0,1,7,0,0,5,0,0,6,0,0,2,0,0,3,0,0],[26,26,1.0,0.3839,0.23536,0.14289,0.28571,0.571,0.0,0.85714,1,0,0,1,0,8,0,0,9,0,0,5,0,0,2,0,0,5,0,0,2,0,0]]}]},{"i":"1a87dc26c03b2865","q":"In triangle  $ABC$ ,  $\\omega$  is its circumcircle and  $O$  is the center of this circle. Points  $M$  and  $N$  lie on sides  $AB$  and  $AC$  respectively.  $\\omega$  and the circumcircle of triangle  $AMN$  intersect each other for the second time in  $Q$ . Let  $P$  be the intersection point of  $MN$  and  $BC$ . Prove that  $PQ$  is tangent to  $\\omega$  iff  $OM=ON$ .\n\n*proposed by Mr.Etesami*","t":[{"b":5,"e":0.1429,"k":"flat","v":0.11599,"x":0.15161,"p":[[0,47,0.0,0.13393,0.10062,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,22,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.11608,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.12054,0.06298,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.11599,0.05572,0.14286,0.14286,0.14286,0.0,0.1429,6,0,1,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.14286,0.07143,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.14723,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.143,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.12054,"x":0.24554,"p":[[0,15,0.0,0.12054,0.08073,0.10714,0.14286,0.14286,0.0,0.28571,8,0,1,8,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.12054,0.11355,0.0,0.14286,0.14286,0.0,0.57143,10,0,1,10,0,19,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.13831,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.12938,0.06544,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.24554,0.07349,0.14289,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,10,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"42c036c648f2fedf","q":"For which  $n\\ge 3$  does there exist positive integers  $a_1<a_2<\\cdots <a_n$ , such that:   $$ a_n=a_1+...+a_{n-1}, \\hspace{0.5cm} \\frac{1}{a_1}=\\frac{1}{a_2}+...+\\frac{1}{a_n} $$   are both true?\n\n*Proposed by Ivan Chan Kai Chin*","t":[{"b":4,"e":0.0,"k":"flat","v":0.38839,"x":0.44643,"p":[[0,6,0.0,0.44643,0.1171,0.42857,0.42857,0.42857,0.0,0.71429,1,0,1,1,0,0,0,0,0,0,0,27,0,0,1,0,0,3,0,0,0,0,0],[4,6,0.6667,0.38839,0.16065,0.42857,0.42857,0.42857,0.0,0.71429,3,0,3,3,0,2,0,0,0,0,0,25,0,0,0,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.4286,"k":"flat","v":0.3884,"x":0.46874,"p":[[0,41,0.0,0.3884,0.12492,0.42857,0.42857,0.42857,0.0,0.4286,3,0,3,3,0,0,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.42858,0.07143,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0],[8,41,0.1951,0.45982,0.08552,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0],[12,41,0.2927,0.44197,0.05486,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0],[16,41,0.3902,0.46874,0.10247,0.42857,0.42857,0.42857,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,27,0,0,2,0,0,2,0,0,1,0,0],[20,41,0.4878,0.42862,0.00025,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.42411,0.04351,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0],[28,41,0.6829,0.4375,0.04971,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0],[32,41,0.7805,0.44643,0.04724,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0],[36,41,0.878,0.42857,1e-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],[40,41,0.9756,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.42857,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6feec30b978ebe0c","q":"In a convex quadrilateral  $ABCD$ , point  $M$  is the midpoint of the diagonal  $AC$ . Prove that if  $\\angle BAD=\\angle BMC=\\angle CMD$ , then a circle can be inscribed in quadrilateral  $ABCD$ .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.13393,"x":0.28572,"p":[[0,19,0.0,0.13393,0.14258,0.0,0.0,0.28571,0.0,0.28571,17,0,6,17,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.14733,0.13593,0.0,0.14286,0.28571,0.0,0.286,14,0,2,14,0,3,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.14286,0.13833,0.0,0.14286,0.28571,0.0,0.286,15,0,3,15,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.25893,0.07523,0.2857,0.28571,0.28571,0.0,0.28571,2,0,1,2,0,2,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.28572,5e-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],[19,19,1.0,0.27232,0.05486,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.13839,"p":[[0,17,0.0,0.13839,0.13592,0.0,0.14286,0.28571,0.0,0.28571,15,0,5,15,0,3,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.07589,0.12364,0.0,0.0,0.17857,0.0,0.28571,23,0,5,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.09821,0.13092,0.0,0.0,0.2857,0.0,0.28571,20,0,6,20,0,2,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"1421fd86b5169f0e","q":"Let  $ APD $  be an acute-angled triangle and let  $ B,C $  be two points on the segments (excluding their endpoints)  $ AP,PD, $  respectively. The diagonals of  $ ABCD $  meet at  $ Q. $  Denote by  $ H_1,H_2 $  the orthocenters of  $ APD,BPC, $  respectively. The circumcircles of  $ ABQ $  and  $ CDQ $  intersect at  $ X\\neq Q, $  and the circumcircles of  $ ADQ,BCQ $  meet at  $ Y\\neq Q. $  Prove that if the line  $ H_1H_2 $  passes through  $ X, $  then it also passes through  $ Y. $","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,34,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,43,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"74a8e88e52be2176","q":"Let   $f: (0,+\\infty)\\rightarrow (0,+\\infty)$  be a function satisfying the following condition: for arbitrary positive real numbers  $x$  and  $y$ , we have  $f(xy)\\le f(x)f(y)$ . Show that for arbitrary positive real number  $x$  and natural number  $n$ , inequality  $f(x^n)\\le f(x)f(x^2)^{\\dfrac{1}{2}}\\dots f(x^n)^{\\dfrac{1}{n}}$  holds.","t":[{"b":3,"e":0.2857,"k":"flat","v":0.09822,"x":0.1517,"p":[[0,8,0.0,0.12045,0.09521,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,20,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.1517,0.10063,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,22,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.15179,"x":0.15625,"p":[[0,4,0.0,0.15625,0.10926,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,21,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,4,1.0,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e1277c25b0807f1f","q":"Given a natural number  $n$ , call a divisor  $d$  of  $n$  to be  $\\textit{nontrivial}$  if  $d>1$ . A natural number  $n$  is  $\\textit{good}$  if one or more distinct nontrivial divisors of  $n$  sum up to  $n-1$ . \n\nProve that every natural number  $n$  has a multiple that is good.","t":[{"b":3,"e":0.0,"k":"volatile","v":0.0,"x":0.40625,"p":[[0,9,0.0,0.22321,0.36585,0.0,0.0,0.28571,0.0,1.0,20,5,0,20,0,2,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[4,9,0.4444,0.25446,0.36375,0.0,0.0,0.42857,0.0,1.0,18,4,0,18,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0,2,0,4],[8,9,0.8889,0.40625,0.45612,0.0,0.14285,1.0,0.0,1.0,16,11,0,16,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,11],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"volatile","v":0.28561,"x":0.90625,"p":[[0,16,0.0,0.40625,0.43757,0.0,0.28571,1.0,0.0,1.0,14,10,0,14,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,10],[4,16,0.25,0.28561,0.37459,0.0,0.0,0.46418,0.0,1.0,17,5,0,17,0,1,0,0,4,0,0,2,0,0,2,0,0,0,0,0,1,0,5],[8,16,0.5,0.29464,0.42399,0.0,0.0,0.75,0.0,1.0,20,7,0,20,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,7],[12,16,0.75,0.89285,0.15152,0.82132,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,5,0,19],[16,16,1.0,0.90625,0.11633,0.85711,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,7,0,18]]}]},{"i":"08dba4d4c3b83e21","q":"Given  $\\vartriangle ABC$  with circumcircle  $\\Omega$ . Assume  $\\omega_a, \\omega_b, \\omega_c$  are circles which tangent internally to  $\\Omega$  at  $T_a,T_b, T_c $  and tangent to  $BC,CA,AB$  at  $P_a, P_b, P_c$ , respectively. If  $AT_a,BT_b,CT_c$  are collinear, prove that  $AP_a,BP_b,CP_c$  are collinear.","t":[{"b":2,"e":1.0,"k":"flat","v":0.47768,"x":0.59821,"p":[[0,4,0.0,0.59821,0.38038,0.24999,0.71429,1.0,0.0,1.0,6,9,0,6,0,2,0,0,1,0,0,4,0,0,0,0,0,5,0,0,5,0,9],[4,4,1.0,0.47768,0.26633,0.28571,0.42857,0.60714,0.0,1.0,2,3,0,2,0,0,0,0,13,0,0,3,0,0,6,0,0,3,0,0,2,0,3]]},{"b":4,"e":0.571,"k":"volatile","v":0.29464,"x":0.62496,"p":[[0,10,0.0,0.59825,0.377,0.24999,0.71429,1.0,0.0,1.0,5,11,0,5,0,3,0,0,2,0,0,2,0,0,1,0,0,8,0,0,0,0,11],[4,10,0.4,0.29464,0.29437,0.0,0.28571,0.42857,0.0,1.0,9,2,1,9,0,5,0,0,9,0,0,4,0,0,0,0,0,1,0,0,2,0,2],[8,10,0.8,0.45981,0.38088,0.14286,0.35714,0.85714,0.0,1.0,6,7,1,6,0,7,0,0,3,0,0,2,0,0,2,0,0,3,0,0,2,0,7],[10,10,1.0,0.62496,0.2165,0.57132,0.57143,0.74996,0.0,1.0,1,3,1,1,0,0,0,0,2,0,0,3,0,0,14,0,0,4,0,0,5,0,3]]}]},{"i":"3fbcdfaf070435ec","q":"Find maximum value of number  $a$  such that for any arrangement of numbers  $1,2,\\ldots ,10$  on a circle, we can find three consecutive numbers such their sum bigger or equal than  $a$ .","t":[{"b":4,"e":0.42857,"k":"rising","v":0.20982,"x":0.63391,"p":[[0,15,0.0,0.26329,0.2205,0.14286,0.14286,0.42857,0.0,0.71429,4,0,3,4,0,16,0,0,2,0,0,5,0,0,1,0,0,4,0,0,0,0,0],[4,15,0.2667,0.20982,0.22011,0.0,0.14286,0.42857,0.0,0.71429,10,0,9,10,0,12,0,0,1,0,0,6,0,0,0,0,0,3,0,0,0,0,0],[8,15,0.5333,0.33482,0.30222,0.14286,0.14286,0.42857,0.0,1.0,4,3,4,4,0,13,0,0,2,0,0,7,0,0,1,0,0,0,0,0,2,0,3],[12,15,0.8,0.63391,0.17474,0.42857,0.64286,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,6,0,0,10,0,0,4,0,2],[15,15,1.0,0.61161,0.17941,0.42857,0.57143,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,8,0,0,4,0,0,7,0,1]]},{"b":5,"e":0.14286,"k":"flat","v":0.14723,"x":0.29018,"p":[[0,11,0.0,0.26339,0.21162,0.14286,0.14286,0.42857,0.0,0.71429,2,0,2,2,0,19,0,0,2,0,0,4,0,0,1,0,0,4,0,0,0,0,0],[4,11,0.3636,0.29018,0.23279,0.14286,0.14286,0.42857,0.0,0.71429,3,0,3,3,0,16,0,0,2,0,0,4,0,0,2,0,0,5,0,0,0,0,0],[8,11,0.7273,0.17411,0.11143,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[11,11,1.0,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ed98be486ed651b8","q":"Fifteen stones are placed on a  $4 \\times 4$  board, one in each cell, the remaining cell being empty. Whenever two stones are on neighbouring cells (having a common side), one may jump over the other to the opposite neighbouring cell, provided this cell is empty. The stone jumped over is removed from the board.\n\nFor which initial positions of the empty cell is it possible to end up with exactly one stone on the board?","t":[{"b":1,"e":0.42857,"k":"flat","v":0.04232,"x":0.1875,"p":[[0,101,0.0,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.42857,21,0,4,21,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,101,0.0396,0.15179,0.18189,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,5,0,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[8,101,0.0792,0.04232,0.0639,0.0,0.0,0.14071,0.0,0.1429,22,0,0,22,1,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,101,0.1188,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,22,0,2,22,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,101,0.1584,0.10268,0.14827,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,101,0.198,0.06696,0.12869,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,101,0.2376,0.0892,0.11706,0.0,0.0,0.14287,0.0,0.42857,18,0,0,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,101,0.2772,0.125,0.15872,0.0,0.0,0.2857,0.0,0.42857,18,0,1,18,0,4,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,101,0.3168,0.10696,0.13358,0.0,0.0,0.2857,0.0,0.42857,18,0,1,18,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,101,0.3564,0.10706,0.14284,0.0,0.0,0.1429,0.0,0.4286,18,0,0,18,0,7,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,101,0.396,0.10268,0.13939,0.0,0.0,0.1786,0.0,0.42857,19,0,0,19,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,101,0.4356,0.17857,0.17128,0.0,0.14288,0.28571,0.0,0.4286,13,0,0,13,0,5,0,0,7,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[48,101,0.4752,0.08036,0.11812,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,10,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,101,0.5149,0.14714,0.15355,0.0,0.14143,0.28571,0.0,0.42857,14,0,1,14,0,7,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,101,0.5545,0.16072,0.16656,0.0,0.14286,0.28571,0.0,0.4286,15,0,1,15,0,3,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[60,101,0.5941,0.13839,0.16554,0.0,0.0,0.28571,0.0,0.4286,17,0,2,17,0,4,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[64,101,0.6337,0.10268,0.1439,0.0,0.0,0.2857,0.0,0.4286,20,0,0,20,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[68,101,0.6733,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.42857,21,0,1,21,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[72,101,0.7129,0.12491,0.14616,0.0,0.07,0.2857,0.0,0.42857,16,0,0,16,0,7,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,101,0.7525,0.12491,0.16268,0.0,0.0,0.2857,0.0,0.4286,18,0,0,18,0,5,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[80,101,0.7921,0.1875,0.17655,0.0,0.14286,0.32143,0.0,0.42857,13,0,0,13,0,4,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[84,101,0.8317,0.16072,0.16656,0.0,0.14286,0.28571,0.0,0.4286,14,0,0,14,0,6,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[88,101,0.8713,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,101,0.9109,0.11374,0.13582,0.0,0.1055,0.14286,0.0,0.42857,15,0,0,15,1,10,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[96,101,0.9505,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,1,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[100,101,0.9901,0.09813,0.12593,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[101,101,1.0,0.12045,0.14772,0.0,0.07,0.14292,0.0,0.4286,16,0,0,16,0,9,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.05804,"p":[[0,41,0.0,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.05804,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,3,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.04911,0.09852,0.0,0.0,0.03571,0.0,0.4286,24,0,1,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1e087b6e20ecb8e3","q":"In a country, there are 100 cities. Each of these cities is connected to exactly three other cities by direct two-way roads. Prove that there exists a city $A$ from which one can travel from city to city and return to $A$, without ever using the same road twice, and using a total number of roads that is not divisible by 3 (it is not required that all cities in the country be visited during this journey).","t":[{"b":0,"e":0.0,"k":"flat","v":0.04911,"x":0.1384,"p":[[0,15,0.0,0.04911,0.11633,0.0,0.0,0.0,0.0,0.42857,27,0,1,27,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0758,0.13351,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.1384,0.13593,0.0,0.14286,0.28571,0.0,0.42857,14,0,0,14,0,6,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.09822,0.13092,0.0,0.0,0.1786,0.0,0.42857,19,0,0,19,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.11606,0.15331,0.0,0.0,0.2857,0.0,0.571,18,0,0,18,0,5,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.01339,"x":0.04911,"p":[[0,25,0.0,0.0357,0.11839,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,25,0.16,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.02233,0.05188,0.0,0.0,0.0,0.0,0.143,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"223b45c399a71edc","q":"Given a set  $S$  of  $n$  variables, a binary operation  $\\times$  on  $S$  is called *simple* if it satisfies  $(x \\times y) \\times z = x \\times (y \\times z)$  for all  $x,y,z \\in S$  and  $x \\times y \\in \\{x,y\\}$  for all  $x,y \\in S$ .  Given a simple operation  $\\times$  on  $S$ , any string of elements in  $S$  can be reduced to a single element, such as  $xyz \\to x \\times (y \\times z)$ .  A string of variables in  $S$  is called*full*if it contains each variable in  $S$  at least once, and two strings are *equivalent* if they evaluate to the same variable regardless of which simple  $\\times$  is chosen.  For example  $xxx$ ,  $xx$ , and  $x$  are equivalent, but these are only full if  $n=1$ .  Suppose  $T$  is a set of strings such that any full string is equivalent to exactly one element of  $T$ .  Determine the number of elements of  $T$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,7,0.0,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,6,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01777,"p":[[0,5,0.0,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6315d95c2e6ca287","q":"For any integer $k \\geqslant 0$, we denote $a_{k}$ as the first digit of the number $2^{k}$, written in base 10. For example, $a_{5}=3$ is the first digit of $2^{5}=32$.\nLet $n \\geqslant 1$ be an integer. Prove that, among the digits from 1 to 9, there is one that is equal to at most $n / 17$ of the $n$ digits $a_{0}, a_{1}, a_{2}, \\ldots, a_{n-1}$.","t":[{"b":1,"e":1.0,"k":"volatile","v":0.0,"x":0.97321,"p":[[0,18,0.0,0.08929,0.27837,0.0,0.0,0.0,0.0,1.0,29,2,2,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[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],[8,18,0.4444,0.08929,0.23077,0.0,0.0,0.0,0.0,0.85714,27,0,1,27,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0],[12,18,0.6667,0.85713,0.27666,0.85714,1.0,1.0,0.0,1.0,2,22,1,2,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,3,0,22],[16,18,0.8889,0.95089,0.09852,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,24],[18,18,1.0,0.97321,0.09062,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,29]]},{"b":7,"e":0.0,"k":"falling","v":0.00446,"x":0.19196,"p":[[0,5,0.0,0.19196,0.34921,0.0,0.0,0.07143,0.0,1.0,24,1,1,24,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,1],[4,5,0.8,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,2,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[5,5,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6ec9b7eeb68d0fc9","q":"Given are  $n$  pairwise intersecting convex  $k$ -gons on the plane. Any of them can be transferred to any other by a homothety with a positive coefficient. Prove that there is a point in a plane belonging to at least  $1 +\\frac{n-1}{2k}$  of these  $k$ -gons.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"efcebd9402da51d7","q":"Given  $n\\geq2$ , let  $\\mathcal{A}$  be a family of subsets of the set  $\\{1,2,\\dots,n\\}$  such that, for any  $A_1,A_2,A_3,A_4 \\in \\mathcal{A}$ , it holds that  $|A_1 \\cup A_2 \\cup A_3 \\cup A_4| \\leq n -2$ .\nProve that  $|\\mathcal{A}| \\leq 2^{n-2}.$","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.0,"p":[[0,8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f97b965a29a7741f","q":"Let  $1 = x_{1} < x_{2} < \\dots < x_{k} = n$  denote the sequence of all divisors  $x_{1}, x_{2} \\dots x_{k}$  of  $n$  in increasing order. Find the smallest possible value of  $n$  such that  $$ n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}. $$  *Proposed by Justin Lee*","t":[{"b":5,"e":1.0,"k":"flat","v":0.89286,"x":0.95982,"p":[[0,19,0.0,0.94643,0.09942,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,24],[4,19,0.2105,0.92857,0.11294,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,4,0,22],[8,19,0.4211,0.89286,0.15972,0.82143,1.0,1.0,0.28571,1.0,0,19,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,5,0,19],[12,19,0.6316,0.93304,0.10705,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,22],[16,19,0.8421,0.95982,0.08918,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[19,19,1.0,0.95981,0.06425,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23]]},{"b":6,"e":1.0,"k":"flat","v":0.93304,"x":0.98661,"p":[[0,60,0.0,0.93304,0.11285,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,3,0,23],[4,60,0.0667,0.97768,0.0724,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,29],[8,60,0.1333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[12,60,0.2,0.96875,0.07771,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,27],[16,60,0.2667,0.97767,0.05188,1.0,1.0,1.0,0.857,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[20,60,0.3333,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[24,60,0.4,0.97321,0.05577,1.0,1.0,1.0,0.857,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[28,60,0.4667,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[32,60,0.5333,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[36,60,0.6,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[40,60,0.6667,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[44,60,0.7333,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[48,60,0.8,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[52,60,0.8667,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[56,60,0.9333,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[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":"07076f25c171827b","q":"In a triangle  $ABC$ , with  $AB\\neq AC$  and  $A\\neq 60^{0},120^{0}$ ,  $D$  is a point on line  $AC$  different from  $C$ . Suppose that the circumcentres and orthocentres of triangles  $ABC$  and  $ABD$  lie on a circle. Prove that  $\\angle ABD=\\angle ACB$ .","t":[{"b":2,"e":0.42857,"k":"flat","v":0.38829,"x":0.60266,"p":[[0,38,0.0,0.45081,0.29048,0.14286,0.64286,0.71429,0.0,0.71429,4,0,0,4,0,8,0,0,0,0,0,3,0,0,1,0,0,16,0,0,0,0,0],[4,38,0.1053,0.51784,0.2519,0.28571,0.71429,0.71429,0.0,0.85714,2,0,0,2,0,4,0,0,3,0,0,4,0,0,2,0,0,16,0,0,1,0,0],[8,38,0.2105,0.60266,0.18809,0.5713,0.71429,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,3,0,0,4,0,0,21,0,0,0,0,0],[12,38,0.3158,0.55804,0.19351,0.42857,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,3,0,0,2,0,0,7,0,0,3,0,0,17,0,0,0,0,0],[16,38,0.4211,0.54909,0.22335,0.42857,0.71429,0.71429,0.0,0.71429,2,0,0,2,0,1,0,0,4,0,0,4,0,0,3,0,0,18,0,0,0,0,0],[20,38,0.5263,0.38829,0.22663,0.24999,0.42857,0.57143,0.0,0.71429,5,0,1,5,0,3,0,0,3,0,0,10,0,0,7,0,0,4,0,0,0,0,0],[24,38,0.6316,0.46871,0.15661,0.42857,0.42859,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,5,0,0,12,0,0,8,0,0,5,0,0,0,0,0],[28,38,0.7368,0.42845,0.18909,0.28571,0.42857,0.57111,0.0,0.71429,1,0,0,1,0,4,0,0,5,0,0,11,0,0,6,0,0,5,0,0,0,0,0],[32,38,0.8421,0.41071,0.21943,0.2857,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,4,0,0,4,0,0,10,0,0,5,0,0,6,0,0,0,0,0],[36,38,0.9474,0.41518,0.23787,0.2857,0.42857,0.60714,0.0,0.71429,5,0,0,5,0,1,0,0,5,0,0,10,0,0,3,0,0,8,0,0,0,0,0],[38,38,1.0,0.4598,0.15456,0.42857,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,3,0,0,16,0,0,7,0,0,4,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.45534,"x":0.71429,"p":[[0,43,0.0,0.55357,0.22232,0.42857,0.71429,0.71429,0.0,0.71429,2,0,0,2,0,2,0,0,1,0,0,6,0,0,3,0,0,18,0,0,0,0,0],[4,43,0.093,0.45534,0.27534,0.25,0.57121,0.71429,0.0,0.71429,5,0,1,5,0,3,0,0,4,0,0,3,0,0,3,0,0,14,0,0,0,0,0],[8,43,0.186,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[12,43,0.2791,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[16,43,0.3721,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[20,43,0.4651,0.70089,0.05486,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0],[24,43,0.5581,0.71415,0.00075,0.71429,0.71429,0.71429,0.71,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[28,43,0.6512,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[32,43,0.7442,0.71429,0.0,0.71429,0.71429,0.71429,0.71429,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0],[36,43,0.8372,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[40,43,0.9302,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0],[43,43,1.0,0.70982,0.02486,0.71429,0.71429,0.71429,0.57143,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0]]}]},{"i":"cde3feb1be9b65e7","q":"Given is an acute angled triangle  $ABC$  with orthocenter  $H$  and circumcircle  $k$ . Let  $\\omega$  be the circle with diameter  $AH$  and  $P$  be the point of intersection of  $\\omega$  and  $k$  other than  $A$ . Assume that  $BP$  and  $CP$  intersect  $\\omega$  for the second time at points  $Q$  and  $R$ , respectively. If  $D$  is the foot of the altitude from  $A$  to  $BC$  and  $S$  is the point of the intersection of  $\\omega$  and  $QD$ , prove that  $HR = HS$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,10,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,30,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c8703c2d228e5a41","q":"In triangle $A B C$, let $\\omega$ be the excircle opposite $A$. Let $D, E$, and $F$ be the points where $\\omega$ is tangent to lines $B C, C A$, and $A B$, respectively. The circle $A E F$ intersects line $B C$ at $P$ and $Q$. Let $M$ be the midpoint of $A D$. Prove that the circle $M P Q$ is tangent to $\\omega$. (Denmark)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,25,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,25,0.32,0.04018,0.12492,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,12,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"04d009e08dc1f7bb","q":"In the acute triangle  $ABC$  point  $I$  is the incenter,  $O$  is the circumcenter, while  $I_a$  is the excenter opposite the vertex  $A$ . Point  $A'$  is the reflection of  $A$  across the line  $BC$ . Prove that angles  $\\angle IOI_a$  and  $\\angle IA'I_a$  are equal.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.12054,"p":[[0,33,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,1,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,33,0.2424,0.04464,0.15746,0.0,0.0,0.0,0.0,0.85714,28,0,4,28,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,33,0.3636,0.12054,0.23987,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,3,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,1],[16,33,0.4848,0.07143,0.22016,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[20,33,0.6061,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.03115,0.10539,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,33,0.8485,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,42,0.0,0.03563,0.10706,0.0,0.0,0.0,0.0,0.4286,28,0,1,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,42,0.1905,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce652994ae2d8a94","q":"In a  $2^n \\times 2^n$  square with  $n$  positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions  (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,24,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.01107,0.03588,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,18,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"bce863c66270930c","q":"Given a scalene triangle  $ABC$  inscribed in the circle  $(O)$ . Let  $(I)$  be its incircle and  $BI,CI$  cut  $AC,AB$  at  $E,F$  respectively. A circle passes through  $E$  and touches  $OB$  at  $B$  cuts  $(O)$  again at  $M$ . Similarly, a circle passes through  $F$  and touches  $OC$  at  $C$  cuts  $(O)$  again at  $N$ .  $ME,NF$  cut  $(O)$  again at  $P,Q$ . Let  $K$  be the intersection of  $EF$  and  $BC$  and let  $PQ$  cuts  $BC$  and  $EF$  at  $G,H$ , respectively. Show that the median correspond to  $G$  of the triangle  $GHK$  is perpendicular to  $IO$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,60,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,60,0.1333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,60,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,60,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,20,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"ba162b5439731b4c","q":"In triangle $A B C$, $I$ is the center of the inscribed circle. A circle is tangent to $A I$ at $I$ and also passes through $B$. This circle intersects $A B$ again at $P$ and $B C$ again at $Q$. The line $Q I$ intersects $A C$ at $R$. Prove that $|A R| \\cdot|B Q|=|P I|^{2}$.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.12946,"x":0.16518,"p":[[0,14,0.0,0.16072,0.14617,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,0,13,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.16518,0.11904,0.14286,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,15,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.16071,0.08564,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,23,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.14732,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,25,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.12946,0.07457,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"rising","v":0.1563,"x":0.41079,"p":[[0,20,0.0,0.1563,0.14454,0.0,0.14286,0.1786,0.0,0.43,10,0,1,10,0,14,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.17857,0.17128,0.0,0.14286,0.28571,0.0,0.57143,11,0,4,11,0,9,0,0,7,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[8,20,0.4,0.2633,0.14342,0.14286,0.2857,0.28571,0.0,0.85714,1,0,0,1,0,10,0,0,17,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[12,20,0.6,0.27204,0.14467,0.14286,0.2857,0.42857,0.0,0.57143,2,0,0,2,0,10,0,0,11,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[16,20,0.8,0.41079,0.14616,0.28571,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,3,0,0,6,0,0,18,0,0,3,0,0,1,0,0,1,0,0],[20,20,1.0,0.34369,0.14214,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,4,0,0,13,0,0,9,0,0,5,0,0,0,0,0,0,0,0]]}]},{"i":"b7dda98f63ebbc6f","q":"In a word formed with the letters  $a,b$  we can change some blocks:  $aba$  in  $b$  and back,  $bba$  in  $a$  and backwards. If the initial word is  $aaa\\ldots ab$  where  $a$  appears 2003 times can we reach the word  $baaa\\ldots a$ , where  $a$  appears 2003 times.","t":[{"b":0,"e":0.42857,"k":"rising","v":0.38839,"x":0.81696,"p":[[0,26,0.0,0.40179,0.43512,0.0,0.07143,0.78571,0.0,1.0,16,8,0,16,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,8],[4,26,0.1538,0.40624,0.4361,0.0,0.21429,1.0,0.0,1.0,15,9,0,15,0,1,0,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,9],[8,26,0.3077,0.38839,0.41378,0.0,0.21429,0.74996,0.0,1.0,15,6,0,15,0,1,0,0,1,0,0,2,0,0,1,0,0,4,0,0,2,0,6],[12,26,0.4615,0.48214,0.42521,0.0,0.64286,0.89286,0.0,1.0,11,8,0,11,0,3,0,0,1,0,0,0,0,0,1,0,0,5,0,0,3,0,8],[16,26,0.6154,0.45536,0.44383,0.0,0.35714,1.0,0.0,1.0,13,9,0,13,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,9],[20,26,0.7692,0.42856,0.37964,0.0,0.35714,0.71429,0.0,1.0,11,5,1,11,0,1,0,0,4,0,0,1,0,0,2,0,0,7,0,0,1,0,5],[24,26,0.9231,0.81696,0.14827,0.71429,0.78571,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,0,6,0,10],[26,26,1.0,0.78124,0.14278,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,17,0,0,7,0,6]]},{"b":3,"e":0.71429,"k":"flat","v":0.43301,"x":0.51784,"p":[[0,9,0.0,0.46876,0.46323,0.0,0.42871,1.0,0.0,1.0,15,11,0,15,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,2,0,11],[4,9,0.4444,0.51784,0.43264,0.0,0.57143,1.0,0.0,1.0,10,12,0,10,0,3,0,0,0,0,0,1,0,0,4,0,0,2,0,0,0,0,12],[8,9,0.8889,0.45087,0.16014,0.42857,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,3,0,0,12,0,0,10,0,0,3,0,0,0,0,0],[9,9,1.0,0.43301,0.18377,0.28571,0.4286,0.57143,0.0,0.857,1,0,0,1,0,4,0,0,5,0,0,7,0,0,14,0,0,0,0,0,1,0,0]]}]},{"i":"0f9705a538f5e596","q":"In each unit square of an infinite square grid a natural number is written. The polygons of area  $n$  with sides going along the gridlines are called *admissible*, where  $n > 2$  is a given natural number. The *value* of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon  $P$  is congruent to  $P$ .)","t":[{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.10265,"p":[[0,17,0.0,0.0625,0.12846,0.0,0.0,0.0,0.0,0.4286,25,0,1,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.10265,0.18631,0.0,0.0,0.07143,0.0,0.571,24,0,1,24,0,0,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,17,0.7059,0.04464,0.14033,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,17,0.9412,0.03571,0.11294,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.571,"k":"rising","v":0.03125,"x":0.42857,"p":[[0,13,0.0,0.06695,0.14714,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,13,0.3077,0.04465,0.12596,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.25445,0.23617,0.0,0.28571,0.28571,0.0,1.0,10,1,0,10,0,1,0,0,16,0,0,1,0,0,1,0,0,2,0,0,0,0,1],[13,13,1.0,0.42857,0.26245,0.2857,0.42857,0.57143,0.0,1.0,4,2,0,4,0,1,0,0,7,0,0,11,0,0,3,0,0,2,0,0,2,0,2]]}]},{"i":"5ffe034b42273190","q":"Initially, three non-collinear points, $A, B$, and $C$, are marked on the plane. You have a pencil and a double-edged ruler of width 1. Using them, you may perform the following operations:\n\n- Mark an arbitrary point in the plane.\n- Mark an arbitrary point on an already drawn line.\n- If two points $P_{1}$ and $P_{2}$ are marked, draw the line connecting $P_{1}$ and $P_{2}$.\n- If two non-parallel lines $\\ell_{1}$ and $\\ell_{2}$ are drawn, mark the intersection of $\\ell_{1}$ and $\\ell_{2}$.\n- If a line $\\ell$ is drawn, draw a line parallel to $\\ell$ that is at distance 1 away from $\\ell$ (note that two such lines may be drawn).\n\nProve that it is possible to mark the orthocenter of $A B C$ using these operations.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.03125,"x":0.08928,"p":[[0,36,0.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0625,0.11259,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0625,0.1234,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,36,0.3333,0.06696,0.11837,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.03563,0.08738,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.04884,0.09155,0.0,0.0,0.14,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.08928,0.14174,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,11,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,36,1.0,0.03125,0.08552,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.03116,"x":0.1026,"p":[[0,7,0.0,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.04909,0.12677,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[7,7,1.0,0.1026,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0d14fb5a360ce210","q":"Given is a triangle  $ABC$  with its circumscribed circle and  $| AC | <| AB |$ . On the short arc  $AC$ , there is a variable point  $D\\ne A$ . Let  $E$  be the reflection of  $A$  wrt the inner bisector of  $\\angle BDC$ . Prove that the line  $DE$  passes through a fixed point, regardless of point  $D$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.04018,"x":0.15616,"p":[[0,26,0.0,0.05803,0.09353,0.0,0.0,0.14286,0.0,0.28571,22,0,13,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,8,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,22,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.14277,0.08748,0.14286,0.14286,0.14287,0.0,0.28571,6,0,0,6,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.13393,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.15616,0.07458,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,23,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.13367,0.03452,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"rising","v":0.01786,"x":0.35481,"p":[[0,36,0.0,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,4,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,12,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.15629,0.14454,0.0,0.14286,0.28571,0.0,0.43,12,0,9,12,0,8,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.12947,0.12557,0.0,0.14286,0.2857,0.0,0.42857,13,0,5,13,0,10,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.19196,0.20394,0.0,0.14286,0.2857,0.0,0.85714,10,0,8,10,0,10,0,0,9,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[20,36,0.5556,0.10268,0.11425,0.0,0.07143,0.14286,0.0,0.28571,16,0,12,16,0,9,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.125,0.14174,0.0,0.0,0.28571,0.0,0.42857,17,0,12,17,0,3,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.16518,0.13415,0.0,0.14286,0.2857,0.0,0.57143,9,0,8,9,0,11,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,36,0.8889,0.35481,0.24976,0.2857,0.28571,0.32143,0.0,1.0,1,2,1,1,1,5,0,0,17,0,0,3,0,0,0,0,0,1,0,0,2,0,2],[36,36,1.0,0.31697,0.21349,0.24999,0.28571,0.32164,0.0,1.0,2,2,0,2,0,6,0,0,16,0,0,5,0,0,1,0,0,0,0,0,0,0,2]]}]},{"i":"466ec5af6b867c39","q":"Find the largest integer  $n$  for which there exist  $n$  different integers such that none of them are divisible by either of  $7,11$  or  $13$ , but the sum of any two of them is divisible by at least one of  $7,11$  and  $13$ .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.21428,"x":0.38838,"p":[[0,12,0.0,0.37499,0.26183,0.2857,0.28571,0.4642,0.0,1.0,5,2,5,5,0,1,0,0,13,0,0,5,0,0,2,0,0,4,0,0,0,0,2],[4,12,0.3333,0.21428,0.20825,0.0,0.2857,0.28571,0.0,0.71429,13,0,12,13,0,1,0,0,11,0,0,5,0,0,0,0,0,2,0,0,0,0,0],[8,12,0.6667,0.33927,0.2389,0.28571,0.28571,0.46418,0.0,1.0,6,1,3,6,0,0,0,0,16,0,0,2,0,0,4,0,0,3,0,0,0,0,1],[12,12,1.0,0.38838,0.15249,0.28571,0.28571,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,20,0,0,5,0,0,3,0,0,4,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.33482,"x":0.41964,"p":[[0,15,0.0,0.35268,0.27889,0.24999,0.28571,0.4286,0.0,1.0,7,1,6,7,0,1,0,0,12,0,0,5,0,0,1,0,0,2,0,0,3,0,1],[4,15,0.2667,0.41964,0.3213,0.2857,0.28571,0.60714,0.0,1.0,6,4,5,6,0,1,0,0,11,0,0,3,0,0,3,0,0,2,0,0,2,0,4],[8,15,0.5333,0.36159,0.26721,0.2857,0.28571,0.57111,0.0,1.0,6,1,6,6,0,1,0,0,14,0,0,2,0,0,2,0,0,5,0,0,1,0,1],[12,15,0.8,0.33482,0.21011,0.2857,0.28571,0.42857,0.0,1.0,4,1,4,4,0,1,0,0,17,0,0,5,0,0,2,0,0,2,0,0,0,0,1]]}]},{"i":"8c7602a01bec505e","q":"Initally a pair  $(x, y)$  is written on the board, such that exactly one of it's coordinates is odd. On such a pair we perform an operation to get pair  $(\\frac x 2, y+\\frac x 2)$  if  $2|x$  and  $(x+\\frac y 2, \\frac y 2)$  if  $2|y$ . Prove that for every odd  $n>1$  there is a even positive integer  $b<n$  such that starting from the pair  $(n, b)$  we will get the pair  $(b, n)$  after finitely many operations.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.42847,"x":0.82143,"p":[[0,22,0.0,0.76776,0.31711,0.42857,1.0,1.0,0.14,1.0,0,19,0,0,0,3,0,0,2,0,0,4,0,0,2,0,0,0,0,0,2,0,19],[4,22,0.1818,0.76786,0.31894,0.67857,1.0,1.0,0.0,1.0,1,18,1,1,0,3,0,0,1,0,0,2,0,0,1,0,0,5,0,0,1,0,18],[8,22,0.3636,0.82143,0.26964,0.71429,1.0,1.0,0.14286,1.0,0,18,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,5,0,0,4,0,18],[12,22,0.5455,0.73214,0.31491,0.42857,0.92857,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,2,0,0,6,0,0,0,0,0,3,0,0,2,0,16],[16,22,0.7273,0.49553,0.26483,0.28571,0.42857,0.71429,0.0,1.0,1,3,1,1,0,3,0,0,7,0,0,8,0,0,3,0,0,5,0,0,2,0,3],[20,22,0.9091,0.45987,0.19474,0.39286,0.42857,0.60714,0.0,0.71429,1,0,1,1,0,3,0,0,4,0,0,12,0,0,4,0,0,8,0,0,0,0,0],[22,22,1.0,0.42847,0.20215,0.28571,0.42857,0.60714,0.14,0.71429,0,0,0,0,0,6,0,0,6,0,0,10,0,0,2,0,0,8,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"volatile","v":0.34375,"x":0.88392,"p":[[0,22,0.0,0.62932,0.33282,0.39286,0.64071,1.0,0.0,1.0,1,11,1,1,0,4,0,0,3,0,0,5,0,0,3,0,0,3,0,0,2,0,11],[4,22,0.1818,0.77679,0.32328,0.53571,1.0,1.0,0.0,1.0,1,20,1,1,0,3,0,0,0,0,0,4,0,0,1,0,0,3,0,0,0,0,20],[8,22,0.3636,0.88392,0.20958,0.82132,1.0,1.0,0.14286,1.0,0,22,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,5,0,0,2,0,22],[12,22,0.5455,0.65625,0.37263,0.28571,0.71429,1.0,0.0,1.0,2,15,2,2,0,5,0,0,3,0,0,1,0,0,2,0,0,4,0,0,0,0,15],[16,22,0.7273,0.81696,0.27718,0.71429,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,4,0,0,3,0,19],[20,22,0.9091,0.74107,0.26591,0.64286,0.71429,1.0,0.14286,1.0,0,12,0,0,0,2,0,0,1,0,0,5,0,0,0,0,0,9,0,0,3,0,12],[22,22,1.0,0.34375,0.1063,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,5,0,0,9,0,0,18,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"215315cd2531a7d6","q":"In convex hexagon  $ABCDEF$ ,  $AB \\parallel DE$ ,  $BC \\parallel EF$ ,  $CD \\parallel FA$ , and \\[ AB+DE = BC+EF = CD+FA. \\] The midpoints of sides  $AB$ ,  $BC$ ,  $DE$ ,  $EF$  are  $A_1$ ,  $B_1$ ,  $D_1$ ,  $E_1$ , and segments  $A_1D_1$  and  $B_1E_1$  meet at  $O$ . Prove that  $\\angle D_1OE_1 = \\frac12 \\angle DEF$ .","t":[{"b":1,"e":0.14286,"k":"rising","v":0.08027,"x":0.29924,"p":[[0,12,0.0,0.08027,0.10057,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.22765,0.21679,0.14286,0.14286,0.28571,0.0,0.85714,6,0,1,6,0,16,0,0,3,0,0,2,0,0,3,0,0,1,0,0,1,0,0],[8,12,0.6667,0.29924,0.19687,0.14286,0.28571,0.42857,0.0,1.0,2,1,0,2,0,10,0,0,9,0,0,9,0,0,0,0,0,1,0,0,0,0,1],[12,12,1.0,0.2767,0.1555,0.14286,0.2857,0.42857,0.0,0.57143,1,0,0,1,0,14,0,0,6,0,0,8,0,0,3,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.08927,"x":0.20981,"p":[[0,21,0.0,0.18741,0.18015,0.105,0.14286,0.1786,0.0,0.71429,8,0,4,8,0,16,0,0,1,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[4,21,0.1905,0.20981,0.15143,0.14286,0.14286,0.2857,0.0,0.57143,3,0,1,3,0,20,0,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[8,21,0.381,0.19643,0.17405,0.14286,0.14286,0.2857,0.0,0.71429,6,0,1,6,0,17,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[12,21,0.5714,0.08927,0.11146,0.0,0.14286,0.14286,0.0,0.571,15,0,4,15,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,21,0.7619,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,1,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.12491,0.07782,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,25,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"40c84cc668b06d5a","q":"Given is a triangle  $ABC$  and a point  $X$  inside its circumcircle. If  $I_B, I_C$  denote the  $B, C$  excenters, then prove that  $XB \\cdot XC <XI_B \\cdot XI_C$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.23214,"p":[[0,16,0.0,0.23214,0.10564,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,2,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.17848,0.13365,0.0,0.2857,0.28571,0.0,0.28571,11,0,3,11,0,2,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.16955,0.12597,0.0,0.21428,0.28571,0.0,0.28571,10,0,2,10,0,6,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.19196,0.12682,0.0,0.2857,0.28571,0.0,0.28571,9,0,0,9,0,3,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.2857,"k":"flat","v":0.22768,"x":0.26776,"p":[[0,15,0.0,0.22768,0.1063,0.24999,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,3,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.24553,0.08917,0.2857,0.28571,0.28571,0.0,0.28571,3,0,2,3,0,3,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.25,0.09449,0.2857,0.28571,0.28571,0.0,0.28571,4,0,4,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.26339,0.0724,0.28571,0.28571,0.28571,0.0,0.28571,2,0,2,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.26776,0.05941,0.2857,0.28571,0.28571,0.0,0.28571,1,0,1,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a0f5104717bf2ec6","q":"Given a positive integer $n=\\prod_{i=1}^{s} p_{i}^{\\alpha_{i}}$, we write $\\Omega(n)$ for the total number $\\sum_{i=1}^{s} \\alpha_{i}$ of prime factors of $n$, counted with multiplicity. Let $\\lambda(n)=(-1)^{\\Omega(n)}$ (so, for example, $\\left.\\lambda(12)=\\lambda\\left(2^{2} \\cdot 3^{1}\\right)=(-1)^{2+1}=-1\\right)$.\n\nProve the following two claims:\ni) There are infinitely many positive integers $n$ such that $\\lambda(n)=\\lambda(n+1)=+1 ;$\nii) There are infinitely many positive integers $n$ such that $\\lambda(n)=\\lambda(n+1)=-1$.\n(ROMANIA) DAN SchwARZ","t":[{"b":1,"e":0.0,"k":"flat","v":0.13393,"x":0.32141,"p":[[0,98,0.0,0.32141,0.33881,0.0,0.2857,0.42858,0.0,1.0,11,5,1,11,0,2,0,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,5],[4,98,0.0408,0.23653,0.31058,0.0,0.14286,0.28571,0.0,1.0,13,3,0,13,0,7,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[8,98,0.0816,0.25447,0.24153,0.0,0.28571,0.42857,0.0,1.0,11,1,0,11,0,2,0,0,10,0,0,5,0,0,2,0,0,1,0,0,0,0,1],[12,98,0.1224,0.24106,0.22709,0.0,0.2857,0.28571,0.0,1.0,10,1,0,10,0,4,0,0,11,0,0,3,0,0,3,0,0,0,0,0,0,0,1],[16,98,0.1633,0.26351,0.29474,0.0,0.2857,0.42857,0.0,1.0,13,3,0,13,0,1,0,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[20,98,0.2041,0.18302,0.18636,0.0,0.21428,0.28571,0.0,0.571,15,0,0,15,0,1,0,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[24,98,0.2449,0.27677,0.28332,0.0,0.2857,0.28571,0.0,1.0,10,2,0,10,0,4,0,0,11,0,0,1,0,0,1,0,0,3,0,0,0,0,2],[28,98,0.2857,0.21865,0.28569,0.0,0.14286,0.28571,0.0,1.0,12,3,0,12,0,8,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[32,98,0.3265,0.25446,0.26901,0.0,0.2857,0.32143,0.0,1.0,12,2,0,12,0,2,0,0,10,0,0,3,0,0,3,0,0,0,0,0,0,0,2],[36,98,0.3673,0.19643,0.20437,0.0,0.2143,0.32164,0.0,0.57143,15,0,0,15,0,1,0,0,8,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[40,98,0.4082,0.20527,0.18192,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,4,0,0,12,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[44,98,0.449,0.16071,0.26184,0.0,0.0,0.2857,0.0,1.0,19,2,1,19,0,3,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[48,98,0.4898,0.17409,0.22225,0.0,0.07143,0.2857,0.0,1.0,16,1,0,16,0,2,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[52,98,0.5306,0.20515,0.19213,0.0,0.2143,0.28571,0.0,0.71429,11,0,0,11,0,5,0,0,11,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[56,98,0.5714,0.20084,0.20478,0.0,0.1429,0.28571,0.0,0.71429,13,0,0,13,0,4,0,0,8,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[60,98,0.6122,0.15624,0.16502,0.0,0.14286,0.28571,0.0,0.571,15,0,0,15,0,3,0,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[64,98,0.6531,0.22767,0.2763,0.0,0.1429,0.28571,0.0,1.0,14,2,0,14,0,3,0,0,8,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[68,98,0.6939,0.19196,0.16213,0.0,0.2857,0.28571,0.0,0.57143,11,0,0,11,0,4,0,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[72,98,0.7347,0.23661,0.25155,0.0,0.21428,0.32143,0.0,0.71429,13,0,0,13,0,3,0,0,8,0,0,3,0,0,0,0,0,5,0,0,0,0,0],[76,98,0.7755,0.13393,0.17835,0.0,0.0,0.2857,0.0,0.71429,17,0,0,17,0,6,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[80,98,0.8163,0.22321,0.18189,0.0,0.2857,0.28571,0.0,0.71429,10,0,0,10,0,3,0,0,12,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[84,98,0.8571,0.18744,0.16923,0.0,0.21428,0.28571,0.0,0.571,12,0,0,12,0,4,0,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[88,98,0.898,0.14286,0.15972,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,1,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[92,98,0.9388,0.19197,0.20705,0.0,0.2857,0.28571,0.0,0.71429,15,0,0,15,0,0,0,0,12,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[96,98,0.9796,0.19186,0.21902,0.0,0.14143,0.28571,0.0,0.71429,15,0,0,15,0,2,0,0,10,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[98,98,1.0,0.20081,0.1781,0.0,0.2857,0.28571,0.0,0.71429,11,0,0,11,0,4,0,0,12,0,0,4,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.10268,"x":0.27231,"p":[[0,26,0.0,0.17856,0.24482,0.0,0.0,0.2857,0.0,1.0,17,1,1,17,0,3,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,1],[4,26,0.1538,0.27231,0.32014,0.0,0.2857,0.32143,0.0,1.0,13,4,0,13,0,2,0,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,4],[8,26,0.3077,0.24554,0.32778,0.0,0.07143,0.42857,0.0,1.0,16,4,0,16,0,2,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,4],[12,26,0.4615,0.21875,0.1636,0.0,0.28571,0.28571,0.0,0.57143,10,0,1,10,0,1,0,0,16,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[16,26,0.6154,0.10268,0.12992,0.0,0.0,0.2857,0.0,0.28571,19,0,1,19,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.18303,0.14826,0.0,0.2857,0.28571,0.0,0.42857,11,0,0,11,0,4,0,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.15168,0.16725,0.0,0.07,0.2857,0.0,0.571,16,0,0,16,0,2,0,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[26,26,1.0,0.15178,0.16728,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,0,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e2e9ddcbf1f050f1","q":"In acute  $\\triangle ABC$  with circumcircle  $\\Gamma$  and incentre  $I$ , the incircle touches side  $AB$  at  $F$ . The external angle bisector of  $\\angle ACB$  meets ray  $AB$  at  $L$ . Point  $K$  lies on the arc  $CB$  of  $\\Gamma$  not containing  $A$ , such that  $\\angle CKI=\\angle IKL$ .  Ray  $KI$  meets  $\\Gamma$  again at  $D\\ne K$ . Prove that  $\\angle ACF =\\angle DCB$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.05348,"p":[[0,6,0.0,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.05795,"p":[[0,22,0.0,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0533,0.06882,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.01768,0.04678,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"ce3a8efb2786221c","q":"Find all positive integers  $n\\geq 3$  such that it is possible to triangulate a convex  $n$ -gon such that all vertices of the  $n$ -gon have even degree.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.06249,"p":[[0,32,0.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,11,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.06249,0.15537,0.0,0.0,0.0,0.0,0.57143,26,0,9,26,0,3,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,32,0.25,0.04465,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,11,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,10,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,9,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"rising","v":0.01339,"x":0.32588,"p":[[0,26,0.0,0.05803,0.13291,0.0,0.0,0.0,0.0,0.571,25,0,11,25,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,26,0.1538,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,18,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,15,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.03125,0.12745,0.0,0.0,0.0,0.0,0.71429,29,0,16,29,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,26,0.6154,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,13,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.32588,0.16455,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,2,0,0,16,0,0,6,0,0,4,0,0,1,0,0,0,0,0],[24,26,0.9231,0.29901,0.13541,0.25,0.28571,0.42857,0.0,0.571,2,0,1,2,0,6,0,0,12,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[26,26,1.0,0.30802,0.12425,0.2857,0.28571,0.42857,0.0,0.571,3,0,1,3,0,0,0,0,19,0,0,9,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"6b1d527366253a40","q":"In an acute triangle  $ABC$ , the circle with diameter  $AC$  intersects  $AB$  and  $AC$  at  $K$  and  $L$  different from  $A$  and  $C$  respectively. The circumcircle of  $ABC$  intersects the line  $CK$  at the point  $F$  different from  $C$  and the line  $AL$  at the point  $D$  different from  $A$ . A point  $E$  is choosen on the smaller arc of  $AC$  of the circumcircle of  $ABC$  . Let  $N$  be the intersection of the lines  $BE$  and  $AC$  . If  $AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}$   prove that  $\\angle KNB= \\angle BNL$  .","t":[{"b":2,"e":0.14286,"k":"falling","v":0.21875,"x":0.64729,"p":[[0,16,0.0,0.64729,0.23954,0.5354,0.71429,0.74996,0.0,1.0,1,5,1,1,0,0,0,0,3,0,0,4,0,0,6,0,0,10,0,0,3,0,5],[4,16,0.25,0.63837,0.22584,0.57132,0.71429,0.71429,0.0,1.0,1,4,0,1,0,0,0,0,4,0,0,1,0,0,7,0,0,14,0,0,1,0,4],[8,16,0.5,0.26767,0.25946,0.14,0.14286,0.4642,0.0,0.85714,7,0,0,7,0,13,0,0,3,0,0,1,0,0,3,0,0,4,0,0,1,0,0],[12,16,0.75,0.24084,0.23783,0.14286,0.14286,0.2857,0.0,1.0,5,1,0,5,0,16,0,0,6,0,0,0,0,0,1,0,0,3,0,0,0,0,1],[16,16,1.0,0.21875,0.2082,0.14286,0.14286,0.2857,0.0,1.0,3,1,0,3,0,20,0,0,6,0,0,0,0,0,0,0,0,2,0,0,0,0,1]]},{"b":5,"e":0.57143,"k":"volatile","v":0.58925,"x":0.9241,"p":[[0,4,0.0,0.58925,0.32684,0.42857,0.57143,0.85714,0.0,1.0,4,7,0,4,0,2,0,0,1,0,0,3,0,0,8,0,0,4,0,0,3,0,7],[4,4,1.0,0.9241,0.16361,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,4,0,24]]}]},{"i":"6ba4b97fb425af72","q":"Let  $ ABC $  be an acute triangle having  $ AB<AC, I $  be its incenter,  $ D,E,F $  be intersection of the incircle with  $ BC, CA, $  respectively,  $ AB, X $  be the middle of the arc  $ BAC, $  which is an arc of the circumcicle of it,  $ P $  be the projection of  $ D $  on  $ EF $  and  $ Q $  be the projection of  $ A $  on  $ ID. $ **a)** Show that  $ IX $  and  $ PQ $  are parallel.**b)** If the circle of diameter  $ AI $  intersects the circumcircle of  $ ABC $  at  $ Y\\neq A, $  prove that  $ XQ $  intersects  $ PI $  at  $ Y. $","t":[{"b":2,"e":0.0,"k":"flat","v":0.02214,"x":0.05358,"p":[[0,26,0.0,0.04455,0.09052,0.0,0.0,0.0,0.0,0.28571,25,0,4,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,2,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.02679,0.05577,0.0,0.0,0.0,0.0,0.1429,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.02214,0.05146,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.05358,0.06917,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.06241,"p":[[0,23,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.2857,24,0,1,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.06241,0.11805,0.0,0.0,0.035,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.05348,0.09934,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b160c286b878fad8","q":"Given a triangle  $ ABC$ . A circle passes through vertices  $ B$  and  $ C$  and intersects sides  $ AB$  and  $ AC$  at points  $ D$  and  $ E$ , respectively. Segments  $ CD$  and  $ BE$  intersect at point  $ O$ . Denote the incenters of triangles  $ ADE$  and  $ ODE$  by  $ M$  and  $ N$ , respectiely. Prove that the midpoint of the smaller arc  $ DE$  lies on line  $ MN$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.00446,"x":0.05804,"p":[[0,19,0.0,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.04884,0.06749,0.0,0.0,0.14071,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.02679,0.05577,0.0,0.0,0.0,0.0,0.143,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.08911,"p":[[0,13,0.0,0.05795,0.07863,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.08911,0.07772,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"35edadd0dae024e5","q":"Let  $ a_{1},a_{2},\\cdots,a_{n}$  be positive real numbers satisfying  $ a_{1} \\plus{} a_{2} \\plus{} \\cdots \\plus{} a_{n} \\equal{} 1$ . Prove that\n\\[\\left(a_{1}a_{2} \\plus{} a_{2}a_{3} \\plus{} \\cdots \\plus{} a_{n}a_{1}\\right)\\left(\\frac {a_{1}}{a_{2}^2 \\plus{} a_{2}} \\plus{} \\frac {a_{2}}{a_{3}^2 \\plus{} a_{3}} \\plus{} \\cdots \\plus{} \\frac {a_{n}}{a_{1}^2 \\plus{} a_{1}}\\right)\\ge\\frac {n}{n \\plus{} 1}\\]","t":[{"b":1,"e":0.0,"k":"flat","v":0.03571,"x":0.1875,"p":[[0,51,0.0,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,51,0.0784,0.05804,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,51,0.1569,0.14286,0.22303,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,3,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[12,51,0.2353,0.08036,0.14698,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,1,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,51,0.3137,0.1875,0.16536,0.0,0.2857,0.28571,0.0,0.71429,11,0,0,11,0,4,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,51,0.3922,0.11607,0.14032,0.0,0.0,0.28571,0.0,0.42857,18,0,0,18,0,3,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,51,0.4706,0.13393,0.16727,0.0,0.0,0.2857,0.0,0.71429,17,0,0,17,0,3,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,51,0.549,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,51,0.6275,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,51,0.7059,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[44,51,0.8627,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,51,0.9412,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[51,51,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"rising","v":0.08036,"x":0.29464,"p":[[0,60,0.0,0.08036,0.12846,0.0,0.0,0.28571,0.0,0.28571,23,0,0,23,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.09821,0.14914,0.0,0.0,0.2857,0.0,0.57143,21,0,0,21,0,2,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,60,0.1333,0.13393,0.13803,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,5,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.11161,0.20119,0.0,0.0,0.2857,0.0,1.0,21,1,0,21,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,60,0.2667,0.13839,0.14934,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,3,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.13839,0.14934,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,3,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,60,0.4,0.10268,0.12492,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.125,0.15872,0.0,0.0,0.2857,0.0,0.57143,18,0,0,18,0,3,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,60,0.5333,0.27678,0.07087,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,2,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.27678,0.07087,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,2,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.27678,0.07936,0.28571,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9a79f8ee433f2346","q":"Let  $ m$  and  $ n$  be two positive integers. Let  $ a_1$ ,  $ a_2$ ,  $ \\ldots$ ,  $ a_m$  be  $ m$  different numbers from the set  $ \\{1, 2,\\ldots, n\\}$  such that for any two indices  $ i$  and  $ j$  with  $ 1\\leq i \\leq j \\leq m$  and  $ a_i \\plus{} a_j \\leq n$ , there exists an index  $ k$  such that  $ a_i \\plus{} a_j \\equal{} a_k$ . Show that\r\n\\[ \\frac {a_1 \\plus{} a_2 \\plus{} ... \\plus{} a_m}{m} \\geq \\frac {n \\plus{} 1}{2}.\r\n\\]","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.12054,"p":[[0,20,0.0,0.12054,0.30117,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[4,20,0.2,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[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],[16,20,0.8,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.11607,"p":[[0,28,0.0,0.11607,0.2683,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[4,28,0.1429,0.04911,0.13175,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.05357,0.14174,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.04022,0.12506,0.0,0.0,0.0,0.0,0.43,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"80f859c595944440","q":"Given is a triangle  $ABC$  with circumcircle  $\\gamma$ . Points  $E, F$  lie on  $AB, AC$  such that  $BE=CF$ . Let  $(AEF)$  meet  $\\gamma$  at  $D$ . The perpendicular from  $D$  to  $EF$  meets  $\\gamma$  at  $G$  and  $AD$  meets  $EF$  at  $P$ . If  $PG$  meets  $\\gamma$  at  $J$ , prove that  $\\frac {JE} {JF}=\\frac{AE} {AF}$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.10268,"x":0.13384,"p":[[0,15,0.0,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.12473,0.11148,0.0,0.14286,0.14286,0.0,0.42857,11,0,1,11,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.13384,0.0612,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.04018,"x":0.13393,"p":[[0,31,0.0,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.13393,0.07087,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.12045,0.06295,0.14286,0.14286,0.14286,0.0,0.28571,6,0,1,6,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.11161,0.11143,0.0,0.14286,0.14286,0.0,0.4286,12,0,0,12,0,17,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.09357,0.06773,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.2857,19,0,1,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ca1e6988ba4b1790","q":"For two positive integers  $n$  and  $d$ , let  $S_n(d)$  be the set of all ordered  $d$ -tuples  $(x_1,x_2,\\dots ,x_d)$  that satisfy all of the following conditions:\ni.  $x_i\\in \\{1,2,\\dots ,n\\}$  for every  $i\\in\\{1,2,\\dots ,d\\}$ ;\nii.  $x_i\\ne x_{i+1}$  for every  $i\\in\\{1,2,\\dots ,d-1\\}$ ;\niii. There does not exist  $i,j,k,l\\in\\{1,2,\\dots ,d\\}$  such that  $i<j<k<l$  and  $x_i=x_k,\\, x_j=x_l$ ;\na. Compute  $|S_3(5)|$ b. Prove that  $|S_n(d)|>0$  if and only if  $d\\leq 2n-1$ .","t":[{"b":4,"e":0.42857,"k":"flat","v":0.44193,"x":0.56249,"p":[[0,6,0.0,0.44193,0.24833,0.39286,0.42857,0.57141,0.0,1.0,4,1,4,4,0,2,0,0,2,0,0,13,0,0,5,0,0,3,0,0,2,0,1],[4,6,0.6667,0.56249,0.17105,0.42857,0.57143,0.57143,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,9,0,0,15,0,0,4,0,0,2,0,1],[6,6,1.0,0.51782,0.13242,0.42857,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,11,0,0,3,0,0,0,0,1]]},{"b":5,"e":0.71429,"k":"flat","v":0.51339,"x":0.6964,"p":[[0,24,0.0,0.51339,0.23107,0.42857,0.57143,0.71429,0.0,0.85714,3,0,3,3,0,2,0,0,0,0,0,8,0,0,7,0,0,11,0,0,1,0,0],[4,24,0.1667,0.56691,0.21572,0.4286,0.57143,0.71429,0.0,1.0,1,3,1,1,0,2,0,0,0,0,0,6,0,0,14,0,0,6,0,0,0,0,3],[8,24,0.3333,0.61603,0.19704,0.571,0.57143,0.71429,0.0,1.0,1,3,1,1,0,0,0,0,0,0,0,6,0,0,13,0,0,7,0,0,2,0,3],[12,24,0.5,0.57583,0.10999,0.5354,0.57143,0.60714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,8,0,0,16,0,0,7,0,0,1,0,0],[16,24,0.6667,0.6964,0.09282,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,24,0,0,1,0,1],[20,24,0.8333,0.66959,0.09067,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,22,0,0,1,0,0],[24,24,1.0,0.65624,0.09355,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,19,0,0,1,0,0]]}]},{"i":"5f2590f6ef27c5d8","q":"Four positive integers $x, y, z$, and $t$ satisfy the relations  $$ x y-z t=x+y=z+t . $$  Is it possible that both $x y$ and $z t$ are perfect squares?  (Russia)","t":[{"b":2,"e":0.28571,"k":"flat","v":0.04018,"x":0.09375,"p":[[0,39,0.0,0.04902,0.08643,0.0,0.0,0.07143,0.0,0.28571,21,0,1,21,6,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.09146,0.10897,0.0,0.07141,0.08929,0.0,0.28571,13,0,2,13,11,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.08705,0.11809,0.0,0.0,0.16071,0.0,0.28571,19,0,0,19,2,3,0,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.08482,0.11769,0.0,0.0,0.16071,0.0,0.28571,19,0,0,19,3,2,0,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.09375,0.11355,0.0,0.07143,0.16071,0.0,0.28571,15,0,2,15,7,2,0,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.08482,0.12428,0.0,0.0,0.08929,0.0,0.42857,18,0,1,18,6,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.08034,0.10527,0.0,0.0355,0.08929,0.0,0.28571,16,0,2,16,8,1,0,2,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.09152,0.106,0.0,0.07143,0.14286,0.0,0.28571,13,0,5,13,10,2,0,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.04018,0.07771,0.0,0.0,0.07141,0.0,0.28571,23,0,11,23,4,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.04679,0.08667,0.0,0.0,0.07143,0.0,0.28571,22,0,8,22,5,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.09152,0.11042,0.0,0.07143,0.14286,0.0,0.28571,14,0,3,14,9,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00223,"x":0.06696,"p":[[0,8,0.0,0.06696,0.11425,0.0,0.0,0.08929,0.0,0.28571,23,0,1,23,1,1,0,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.02674,0.07084,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,4,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1cbc25ac6829ced4","q":"In an acute triangle  $ABC$  the points  $D,E$  and  $F$  are the feet of the altitudes through  $A,B$  and  $C$  respectively. The incenters of the triangles  $AEF$  and  $BDF$  are  $I_1$  and  $I_2$  respectively; the circumcenters of the triangles  $ACI_1$  and  $BCI_2$  are  $O_1$  and  $O_2$  respectively. Prove that  $I_1I_2$  and  $O_1O_2$  are parallel.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02223,"p":[[0,22,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,9,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,3,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,63,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,63,0.0635,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,63,0.127,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,63,0.1905,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,63,0.254,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,63,0.3175,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,63,0.381,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,63,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,63,0.5079,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,63,0.5714,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,63,0.6349,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,63,0.6984,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,63,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,63,0.8254,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,63,0.8889,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,63,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[63,63,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0e754fd304c5ee1d","q":"Given a positive integer number $n \\geq 3$, colour each cell of an $n \\times n$ square array one of $\\left[(n+2)^{2} / 3\\right]$ colours, each colour being used at least once. Prove that the cells of some $1 \\times 3$ or $3 \\times 1$ rectangular subarray have pairwise distinct colours.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.00893,"p":[[0,12,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,10,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,9,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,12,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,11,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,4,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"158de208e5156d15","q":"In triangle $A B C$, let $M$ be the midpoint of $B C$ and $D$ be a point on segment $A M$. Distinct points $Y$ and $Z$ are chosen on rays $\\overrightarrow{C A}$ and $\\overrightarrow{B A}$, respectively, such that $\\angle D Y C=\\angle D C B$ and $\\angle D B C=\\angle D Z B$. Prove that the circumcircle of $\\triangle D Y Z$ is tangent to the circumcircle of $\\triangle D B C$.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,22,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.00893,"x":0.03571,"p":[[0,14,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f403ff64ee1987df","q":"Given any nine integers show that it is possible to choose, from among them, four integers $a, b, c, d$ such that $a+b-c-d$ is divisible by 20 . Further show that such a selection is not possible if we start with eight integers instead of nine.","t":[{"b":0,"e":0.4286,"k":"flat","v":0.47767,"x":0.60265,"p":[[0,6,0.0,0.47767,0.384,0.0,0.4998,0.85714,0.0,1.0,10,4,8,10,0,1,0,0,2,0,0,3,0,0,1,0,0,5,0,0,6,0,4],[4,6,0.6667,0.51337,0.25469,0.39286,0.57143,0.60714,0.0,1.0,3,2,3,3,0,1,0,0,4,0,0,4,0,0,12,0,0,4,0,0,2,0,2],[6,6,1.0,0.60265,0.15459,0.57132,0.57143,0.71429,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,5,0,0,11,0,0,14,0,0,1,0,0]]},{"b":5,"e":0.2857,"k":"falling","v":0.32142,"x":0.60714,"p":[[0,6,0.0,0.60714,0.29667,0.42857,0.71429,0.85714,0.0,0.85714,5,0,5,5,0,0,0,0,0,0,0,4,0,0,3,0,0,8,0,0,12,0,0],[4,6,0.6667,0.39728,0.22805,0.24999,0.42857,0.4286,0.0,0.85714,1,0,1,1,0,7,0,0,5,0,0,13,0,0,1,0,0,1,0,0,4,0,0],[6,6,1.0,0.32142,0.17495,0.14286,0.28571,0.4286,0.0,0.57143,2,0,1,2,0,9,0,0,6,0,0,9,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"670d9f16bb2a8b25","q":"Let  $ G(V,E)$  be a connected graph, and let  $ d_G(x,y)$  denote the length of the shortest path joining  $ x$  and  $ y$  in  $ G$ . Let  $ r_G(x)\\equal{} \\max \\{ d_G(x,y) : \\; y \\in V \\ \\}$  for  $ x \\in V$ , and let  $ r(G)\\equal{} \\min \\{ r_G(x) : \\;x \\in V\\ \\}$ . Show that if  $ r(G) \\geq 2$ , then  $ G$  contains a path of length  $ 2r(G)\\minus{}2$  as an induced subgraph. \r\n\r\n*V. T. Sos*","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.09375,"p":[[0,23,0.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.09375,0.11633,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.06679,0.07957,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.04902,"x":0.10688,"p":[[0,15,0.0,0.04902,0.09844,0.0,0.0,0.035,0.0,0.4286,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.06688,0.08729,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.1025,0.11419,0.0,0.14,0.14287,0.0,0.42857,15,0,0,15,0,12,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.07134,0.07978,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.10688,0.08738,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"463f5c003a43e82b","q":"Let  $ f$  be a finite real function of one variable. Let  $ \\overline{D}f$  and  $ \\underline{D}f$  be its upper and lower derivatives, respectively, that is, \\[ \\overline{D}f\\equal{}\\limsup_{{h,k\\rightarrow 0}_{{h,k \\geq 0}_{h\\plus{}k>0}}} \\frac{f(x\\plus{}h)\\minus{}f(x\\minus{}k)}{h\\plus{}k}\\] ,\r\n\\[ \\underline{D}f\\equal{}\\liminf_{{h,k\\rightarrow 0}_{{h,k \\geq 0}_{h\\plus{}k>0}}} \\frac{f(x\\plus{}h)\\minus{}f(x\\minus{}k)}{h\\plus{}k}.\\] Show that  $ \\overline{D}f$  and  $ \\underline{D}f$  are Borel-measurable functions. [A. Csaszar]","t":[{"b":1,"e":0.14286,"k":"flat","v":0.05804,"x":0.16295,"p":[[0,14,0.0,0.16295,0.16486,0.0,0.14286,0.2857,0.0,0.71429,10,0,0,10,1,12,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[4,14,0.2857,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.11607,0.13092,0.0,0.0,0.2857,0.0,0.28571,17,0,0,17,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04911,"x":0.11607,"p":[[0,5,0.0,0.09812,0.14911,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,9,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,5,0.8,0.11607,0.14032,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,9,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ea145cb93f6876cc","q":"Let  $(a_n)_{n\\ge 1}$  be a sequence of positive numbers. If there is a constant  $M > 0$  such that  $a_2^2 + a_2^2 +\\ldots  + a_n^2 < Ma_{n+1}^2$  for all  $n$ , then prove that there is a constant  $M ' > 0$  such that  $a_1 + a_2 +\\ldots + a_n < M ' a_{n+1}$  .","t":[{"b":3,"e":0.57143,"k":"volatile","v":0.37043,"x":0.83929,"p":[[0,13,0.0,0.66293,0.32745,0.28571,0.71429,1.0,0.0,1.0,1,13,0,1,0,1,0,0,7,0,0,3,0,0,3,0,0,2,1,0,1,0,13],[4,13,0.3077,0.5446,0.34522,0.25001,0.57121,0.85714,0.0,1.0,5,6,0,5,0,3,0,0,1,0,0,4,0,0,6,0,0,3,0,0,4,0,6],[8,13,0.6154,0.37043,0.3442,0.14214,0.2857,0.60714,0.0,1.0,7,4,0,7,0,8,0,0,4,0,0,3,0,0,2,0,0,2,0,0,2,0,4],[12,13,0.9231,0.83929,0.23623,0.71429,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,2,0,0,4,0,19],[13,13,1.0,0.80578,0.22677,0.69643,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,2,0,1,1,0,0,3,1,0,3,1,0,6,0,14]]},{"b":4,"e":1.0,"k":"rising","v":0.53123,"x":0.99107,"p":[[0,39,0.0,0.53123,0.38338,0.14286,0.4998,1.0,0.0,1.0,2,10,0,2,0,10,0,0,3,0,0,1,0,0,1,0,0,4,0,0,1,0,10],[4,39,0.1026,0.8125,0.2126,0.71429,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,7,0,0,7,0,13],[8,39,0.2051,0.70979,0.31841,0.53539,0.857,1.0,0.0,1.0,1,13,0,1,0,3,0,0,2,0,0,2,0,0,4,0,0,3,0,0,4,0,13],[12,39,0.3077,0.9241,0.18206,1.0,1.0,1.0,0.2857,1.0,0,25,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,3,0,25],[16,39,0.4103,0.94643,0.1915,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,29],[20,39,0.5128,0.92857,0.18211,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,26],[24,39,0.6154,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,39,0.7179,0.94195,0.14668,1.0,1.0,1.0,0.42857,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,27],[32,39,0.8205,0.95536,0.12078,1.0,1.0,1.0,0.4286,1.0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,27],[36,39,0.9231,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[39,39,1.0,0.96429,0.13363,1.0,1.0,1.0,0.28571,1.0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,29]]}]},{"i":"e5975614777fed66","q":"For a given integer  $n\\ge3$ , let  $S_1, S_2,\\ldots,S_m$  be distinct three-element subsets of the set  $\\{1,2,\\ldots,n\\}$  such that for each  $1\\le i,j\\le m; i\\neq j$  the sets  $S_i\\cap S_j$  contain exactly one element. Determine the maximal possible value of  $m$  for each  $n$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,26,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,30,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,27,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,26,0.3077,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,25,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.08481,"p":[[0,14,0.0,0.08481,0.18158,0.0,0.0,0.0,0.0,0.71429,25,0,24,25,0,1,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,14,0.2857,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,26,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,28,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cf18dcc53b590480","q":"For  $n\\geq k\\geq 3$ , let  $X=\\{1,2,...,n\\}$  and let  $F_{k}$  a the family of  $k$ -element subsets of  $X$ , any two of which have at most  $k-2$  elements in common. Show that there exists a subset  $M_{k}$  of  $X$  with at least  $[\\log_{2}{n}]+1$  elements containing no subset in  $F_{k}$ .","t":[{"b":2,"e":0.0,"k":"volatile","v":0.01339,"x":0.77232,"p":[[0,7,0.0,0.63392,0.42699,0.10714,0.85714,1.0,0.0,1.0,8,15,2,8,0,2,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,15],[4,7,0.5714,0.77232,0.37262,0.75001,1.0,1.0,0.0,1.0,3,21,0,3,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,21],[7,7,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"volatile","v":0.51338,"x":0.98214,"p":[[0,15,0.0,0.72768,0.40777,0.39285,1.0,1.0,0.0,1.0,6,20,1,6,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,20],[4,15,0.2667,0.54463,0.44811,0.0,0.78564,1.0,0.0,1.0,11,12,1,11,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,12],[8,15,0.5333,0.56696,0.41724,0.0,0.78571,1.0,0.0,1.0,9,10,0,9,0,0,0,0,3,0,0,2,0,0,1,0,0,1,0,0,6,0,10],[12,15,0.8,0.51338,0.4328,0.0,0.5712,1.0,0.0,1.0,11,11,0,11,0,1,0,0,1,0,0,2,0,0,3,0,0,1,0,0,2,0,11],[15,15,1.0,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29]]}]},{"i":"de58a0b75e79e755","q":"Let\n\\[f = X^{n}+a_{n-1}X^{n-1}+\\ldots+a_{1}X+a_{0}\\]\nbe an integer polynomial of degree  $n \\geq 3$  such that  $a_{k}+a_{n-k}$  is even for all  $k \\in \\overline{1,n-1}$  and  $a_{0}$  is even.\nSuppose that  $f = gh$ , where  $g,h$  are integer polynomials and  $\\deg g \\leq \\deg h$  and all the coefficients of  $h$  are odd.\nProve that  $f$  has an integer root.","t":[{"b":3,"e":0.2857,"k":"flat","v":0.32143,"x":0.53575,"p":[[0,36,0.0,0.44197,0.32015,0.14286,0.42857,0.71429,0.0,1.0,3,5,2,3,0,8,0,0,2,0,0,9,0,0,0,0,0,5,0,0,0,0,5],[4,36,0.1111,0.39286,0.2369,0.14286,0.42857,0.42857,0.0,1.0,1,2,1,1,0,8,0,0,4,0,0,13,0,0,1,0,0,3,0,0,0,0,2],[8,36,0.2222,0.49554,0.2923,0.28571,0.42857,0.71429,0.0,1.0,1,5,1,1,0,6,0,0,3,0,0,10,0,0,1,0,0,6,0,0,0,0,5],[12,36,0.3333,0.53575,0.25503,0.28571,0.4293,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,6,0,0,8,0,0,2,0,0,9,0,0,0,0,4],[16,36,0.4444,0.33482,0.2412,0.14286,0.28571,0.42857,0.0,1.0,3,2,3,3,0,7,0,0,10,0,0,8,0,0,0,0,0,2,0,0,0,0,2],[20,36,0.5556,0.37054,0.27632,0.14286,0.28571,0.42858,0.0,1.0,4,3,3,4,0,6,0,0,7,0,0,8,0,0,2,0,0,2,0,0,0,0,3],[24,36,0.6667,0.32143,0.23419,0.14286,0.2857,0.42857,0.0,1.0,2,2,1,2,0,10,0,0,9,0,0,7,0,0,1,0,0,1,0,0,0,0,2],[28,36,0.7778,0.51783,0.26426,0.28571,0.42859,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,9,0,0,5,0,0,4,0,0,6,0,0,1,0,4],[32,36,0.8889,0.37052,0.2078,0.2857,0.28571,0.42858,0.14286,1.0,0,1,0,0,0,7,0,0,12,0,0,7,0,0,1,0,0,4,0,0,0,0,1],[36,36,1.0,0.43293,0.24878,0.28571,0.28571,0.57143,0.14,1.0,0,3,0,0,0,4,0,0,14,0,0,4,0,0,3,0,0,4,0,0,0,0,3]]},{"b":5,"e":0.14286,"k":"falling","v":0.22321,"x":0.44643,"p":[[0,19,0.0,0.44643,0.34947,0.14286,0.28571,0.78571,0.0,1.0,2,8,2,2,0,9,0,0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,8],[4,19,0.2105,0.44643,0.25191,0.2857,0.42857,0.71429,0.0,1.0,1,1,1,1,0,6,0,0,5,0,0,10,0,0,0,0,0,7,0,0,2,0,1],[8,19,0.4211,0.32588,0.21198,0.14286,0.28571,0.42857,0.0,1.0,1,1,1,1,0,9,0,0,12,0,0,6,0,0,1,0,0,1,0,0,1,0,1],[12,19,0.6316,0.27232,0.13054,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,12,0,0,13,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[16,19,0.8421,0.23214,0.12242,0.14286,0.14288,0.28571,0.0,0.57143,1,0,0,1,0,16,0,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[19,19,1.0,0.22321,0.08702,0.14286,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,13,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"87d074536086ca77","q":"In the triangle  $ ABC $  the point  $ I $  is the center of the inscribed circle. On rays  $ AI $  and  $ BI $ , points  $ A_1 $  and  $ B_1 $  respectively are taken for  $ I $  and such that  $ \\angle ACA_1 = \\angle BCB_1 = 90 ^\\circ. $  Let  $ M $  be the midpoint of the segment  $ A_1B_1 $ . Prove that the lines  $ IM $  and  $ AB $  are perpendicular.","t":[{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.16964,"p":[[0,8,0.0,0.08929,0.19804,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,8,0.5,0.16964,0.2683,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,1,0,0,1,0,0,1,0,0,5,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.01339,"x":0.17857,"p":[[0,14,0.0,0.06701,0.15156,0.0,0.0,0.0,0.0,0.57143,26,0,1,26,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,14,0.2857,0.17857,0.29451,0.0,0.0,0.28571,0.0,1.0,20,2,2,20,0,3,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,2],[8,14,0.5714,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dff7805086bdc3e4","q":"Let  $ ABCD$  be a cyclic quadrilateral and  $ O$  be the intersection of diagonal  $ AC$  and  $ BD$ . The circumcircles of triangle  $ ABO$  and the triangle  $ CDO$  intersect at  $ K$ . Let  $ L$  be a point such that the triangle  $ BLC$  is similar to  $ AKD$  (in that order). Prove that if  $ BLCK$  is a convex quadrilateral, then it has an incircle.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,17,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,24,0.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"367913ece47f7a2c","q":"In triangle $A B C$, the incircle touches sides $B C, C A, A B$ at $D, E, F$ respectively. Assume there exists a point $X$ on the line $E F$ such that\n\n$$\n\\angle X B C=\\angle X C B=45^{\\circ} .\n$$\n\nLet $M$ be the midpoint of the arc $B C$ on the circumcircle of $A B C$ not containing $A$. Prove that the line $M D$ passes through $E$ or $F$.","t":[{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.17409,"p":[[0,16,0.0,0.15179,0.2111,0.0,0.0,0.42857,0.0,0.57143,20,0,0,20,0,2,0,0,0,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[4,16,0.25,0.17409,0.25437,0.0,0.0,0.42858,0.0,0.71429,21,0,0,21,0,1,0,0,0,0,0,3,0,0,6,0,0,1,0,0,0,0,0],[8,16,0.5,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.42857,"k":"rising","v":0.06696,"x":0.49103,"p":[[0,31,0.0,0.06696,0.15561,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,31,0.129,0.09375,0.18766,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,1,0,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[8,31,0.2581,0.24105,0.27531,0.0,0.0,0.4642,0.0,0.71429,17,0,1,17,0,1,0,0,0,0,0,6,0,0,5,0,0,3,0,0,0,0,0],[12,31,0.3871,0.44643,0.14174,0.42857,0.42857,0.46431,0.0,0.71429,2,0,0,2,0,0,0,0,0,0,0,22,0,0,6,0,0,2,0,0,0,0,0],[16,31,0.5161,0.40625,0.12428,0.42857,0.42857,0.42857,0.0,0.57143,2,0,0,2,0,1,0,0,0,0,0,26,0,0,3,0,0,0,0,0,0,0,0],[20,31,0.6452,0.49103,0.11256,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,18,0,0,10,0,0,3,0,0,0,0,0],[24,31,0.7742,0.46874,0.08916,0.42857,0.42857,0.4286,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,26,0,0,3,0,0,3,0,0,0,0,0],[28,31,0.9032,0.47768,0.09182,0.42857,0.42857,0.46431,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,24,0,0,5,0,0,3,0,0,0,0,0],[31,31,1.0,0.47767,0.07666,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0]]}]},{"i":"58ae6632a974f4f8","q":"Fix a circle  $\\Gamma$ , a line  $\\ell$  to tangent  $\\Gamma$ , and another circle  $\\Omega$  disjoint from  $\\ell$  such that  $\\Gamma$  and  $\\Omega$  lie on opposite sides of  $\\ell$ . The tangents to  $\\Gamma$  from a variable point  $X$  on  $\\Omega$  meet  $\\ell$  at  $Y$  and  $Z$ . Prove that, as  $X$  varies over  $\\Omega$ , the circumcircle of  $XYZ$  is tangent to two fixed circles.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0758,"x":0.11147,"p":[[0,21,0.0,0.09348,0.06766,0.0,0.14286,0.14286,0.0,0.14286,11,0,3,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.11147,0.05628,0.12286,0.14286,0.14286,0.0,0.14286,6,0,5,6,2,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,2,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.10241,0.06407,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,1,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01783,"x":0.12722,"p":[[0,28,0.0,0.1183,0.0524,0.14286,0.14286,0.14286,0.0,0.1429,5,0,2,5,1,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.11603,0.05286,0.14286,0.14286,0.14286,0.0,0.143,5,0,3,5,2,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.12722,0.04278,0.14286,0.14286,0.14286,0.0,0.14286,3,0,1,3,1,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.09357,0.06773,0.0,0.14286,0.14286,0.0,0.14286,11,0,3,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.08464,0.07002,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.07812,0.06998,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,1,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.08036,0.06858,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,2,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.01783,0.04371,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"94dc234d465051b5","q":"For a positive integer  $n$ , and a non empty subset  $A$  of  $\\{1,2,...,2n\\}$ , call  $A$  good if the set  $\\{u\\pm v|u,v\\in A\\}$  does not contain the set  $\\{1,2,...,n\\}$ . Find the smallest real number  $c$ , such that for any positive integer  $n$ , and any good subset  $A$  of   $\\{1,2,...,2n\\}$ ,  $|A|\\leq cn$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01562,"p":[[0,10,0.0,0.01562,0.04276,0.0,0.0,0.0,0.0,0.14286,28,0,9,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,26,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,18,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,11,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.01562,0.05571,0.0,0.0,0.0,0.0,0.28571,29,0,8,29,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,25,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"58670239679b3704","q":"Find a right triangle that can be cut into  $365$  equal triangles.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.15625,"p":[[0,17,0.0,0.03572,0.15152,0.0,0.0,0.0,0.0,0.85714,29,0,9,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,17,0.2353,0.15625,0.26088,0.0,0.07143,0.14286,0.0,1.0,16,2,10,16,0,10,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[8,17,0.4706,0.11161,0.24675,0.0,0.0,0.14286,0.0,1.0,22,2,19,22,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,32,0.0,0.05804,0.18161,0.0,0.0,0.0,0.0,1.0,26,1,9,26,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,32,0.125,0.05357,0.17768,0.0,0.0,0.0,0.0,1.0,26,1,5,26,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,32,0.25,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,10,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,16,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,25,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,27,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,32,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"19a45fa371a064c1","q":"Let  $ G$  be a simple graph with  $ 2 \\cdot n$  vertices and  $ n^{2}+1$  edges. Show that this graph  $ G$  contains a  $ K_{4}-\\text{one edge}$ , that is, two triangles with a common edge.","t":[{"b":3,"e":0.0,"k":"flat","v":0.00438,"x":0.01339,"p":[[0,16,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,16,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,19,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"89368ad227ed19cf","q":"For all integers $n \\geqslant 1$, we define $u_{n}=1!+2!+\\ldots+n!$. Show that there are infinitely many prime numbers dividing at least one of the terms of the sequence ( $\\mathfrak{u}_{n}$ ).","t":[{"b":3,"e":0.28571,"k":"flat","v":0.09812,"x":0.14286,"p":[[0,9,0.0,0.09812,0.12075,0.0,0.0,0.14286,0.0,0.42857,17,0,1,17,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.10714,0.12999,0.0,0.0,0.16071,0.0,0.42857,17,0,3,17,0,7,0,1,5,0,1,1,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.12491,0.1115,0.0,0.14286,0.1429,0.0,0.42857,11,0,0,11,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.14286,0.11845,0.0,0.14286,0.14287,0.0,0.42857,9,0,0,9,0,16,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.0267,"x":0.12946,"p":[[0,32,0.0,0.10715,0.11845,0.0,0.14286,0.14286,0.0,0.4286,14,0,4,14,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.2857,14,0,1,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,10,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.28571,21,0,8,21,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.0267,0.08316,0.0,0.0,0.0,0.0,0.42857,28,0,4,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.12946,0.14445,0.0,0.07143,0.2857,0.0,0.42857,16,0,9,16,0,5,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,14,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.02901,0.07545,0.0,0.0,0.0,0.0,0.357,27,0,23,27,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ff7cb87f99343e47","q":"Given a circle and a point  $P$  inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point  $P$ . Find the locus of the points of intersection of the common external tangents to these circles.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.13375,"x":0.47321,"p":[[0,12,0.0,0.47321,0.41563,0.14286,0.28571,1.0,0.0,1.0,5,11,1,5,0,10,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,11],[4,12,0.3333,0.15183,0.07952,0.14286,0.14286,0.14286,0.0,0.43,2,0,1,2,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.13813,0.02482,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"rising","v":0.45089,"x":0.96429,"p":[[0,36,0.0,0.45089,0.39947,0.14286,0.2857,1.0,0.0,1.0,4,9,1,4,0,11,0,0,5,0,0,0,0,0,0,0,0,1,0,0,2,0,9],[4,36,0.1111,0.59357,0.39967,0.14286,0.71429,1.0,0.0,1.0,3,13,1,3,0,7,0,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,13],[8,36,0.2222,0.82134,0.31359,0.82143,1.0,1.0,0.14,1.0,0,23,0,0,0,3,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,23],[12,36,0.3333,0.96429,0.11294,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,29],[16,36,0.4444,0.92857,0.21724,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,28],[20,36,0.5556,0.95536,0.16536,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,29],[24,36,0.6667,0.88839,0.24932,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,25],[28,36,0.7778,0.9375,0.19541,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[32,36,0.8889,0.89286,0.24223,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,25],[36,36,1.0,0.90625,0.19759,0.96429,1.0,1.0,0.14286,1.0,0,24,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,2,0,24]]}]},{"i":"fa4cebeb8ccadb6c","q":"In a scalene triangle  $ABC$ , let the angle bisector of  $A$  meets side  $BC$  at  $D$ . Let  $E, F$  be the circumcenter of the triangles  $ABD$  and  $ADC$ , respectively. Suppose that the circumcircles of the triangles  $BDE$  and  $DCF$  intersect at  $P(\\neq D)$ , and denote by  $O, X, Y$  the circumcenters of the triangles  $ABC, BDE, DCF$ , respectively. Prove that   $OP$  and  $XY$  are parallel.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.14285,"p":[[0,33,0.0,0.14285,0.25753,0.0,0.0,0.17857,0.0,0.85714,22,0,0,22,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,2,0,0],[4,33,0.1212,0.08929,0.15047,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,4,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.12946,0.19352,0.0,0.0,0.42857,0.0,0.42857,22,0,0,22,0,0,0,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.08482,0.18851,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[16,33,0.4848,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.16071,"p":[[0,27,0.0,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.16071,0.28065,0.0,0.0,0.2857,0.0,1.0,22,1,0,22,0,0,0,0,4,0,0,3,0,0,0,0,0,0,0,0,2,0,1],[8,27,0.2963,0.09366,0.19432,0.0,0.0,0.035,0.0,0.85714,24,0,1,24,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[12,27,0.4444,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.03125,0.10555,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"11e2d7d4a1889d0d","q":"Let  $ A_{1}A_{2}A_{3}A_{4}A_{5}$  be a convex pentagon, such that\r\n\\[ [A_{1}A_{2}A_{3}] \\equal{} [A_{2}A_{3}A_{4}] \\equal{} [A_{3}A_{4}A_{5}] \\equal{} [A_{4}A_{5}A_{1}] \\equal{} [A_{5}A_{1}A_{2}].\\]\r\nProve that there exists a point   $ M$  in the plane of the pentagon such that\r\n\\[ [A_{1}MA_{2}] \\equal{} [A_{2}MA_{3}] \\equal{} [A_{3}MA_{4}] \\equal{} [A_{4}MA_{5}] \\equal{} [A_{5}MA_{1}].\\]\r\nHere  $ [XYZ]$  stands for the area of the triangle  $ \\Delta XYZ$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.32143,"x":0.8125,"p":[[0,21,0.0,0.8125,0.35614,0.85714,1.0,1.0,0.0,1.0,4,23,0,4,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,23],[4,21,0.1905,0.57813,0.41587,0.10714,0.71429,1.0,0.0,1.0,8,12,1,8,0,2,0,0,1,0,0,2,0,0,1,0,0,5,0,0,0,1,12],[8,21,0.381,0.54018,0.41301,0.10714,0.64286,1.0,0.0,1.0,8,11,0,8,0,4,0,0,0,0,0,1,0,0,3,0,0,5,0,0,0,0,11],[12,21,0.5714,0.38385,0.2512,0.14286,0.42857,0.57143,0.0,0.71429,5,0,0,5,0,5,0,0,4,0,0,6,0,0,5,0,0,7,0,0,0,0,0],[16,21,0.7619,0.32143,0.24743,0.14286,0.28571,0.42858,0.0,0.71429,7,0,0,7,0,5,0,0,6,0,0,7,0,0,1,0,0,6,0,0,0,0,0],[20,21,0.9524,0.47766,0.24382,0.28571,0.57121,0.71429,0.0,0.71429,3,0,0,3,0,2,0,0,6,0,0,4,0,0,4,0,0,13,0,0,0,0,0],[21,21,1.0,0.5089,0.26471,0.28571,0.64286,0.71429,0.0,0.71429,4,0,0,4,0,3,0,0,2,0,0,1,0,0,6,0,0,16,0,0,0,0,0]]},{"b":4,"e":0.1429,"k":"volatile","v":0.03126,"x":0.77232,"p":[[0,8,0.0,0.77232,0.38937,0.71429,1.0,1.0,0.0,1.0,6,22,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,22],[4,8,0.5,0.71429,0.3977,0.39286,1.0,1.0,0.0,1.0,6,18,0,6,0,0,0,0,2,0,0,1,0,0,1,0,0,1,0,0,3,0,18],[8,8,1.0,0.03126,0.08555,0.0,0.0,0.0,0.0,0.286,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63be07ec1e561079","q":"In an acute triangle  $\\triangle ABC$  the points  $C_m, A_m, B_m$  are the midpoints of  $AB, BC, CA$  respectively. Inside the triangle  $\\triangle ABC$  a point  $P$  is chosen so that  $\\angle PCB = \\angle B_mBC$  and  $\\angle PAB = \\angle ABB_m.$  A line passing through  $P$  and perpendicular to  $AC$  meets the median  $BB_m$  at  $E.$  Prove that  $E$  lies on the circumcircle of the triangle  $\\triangle A_mB_mC_m.$ *(K. Ivanov )*","t":[{"b":3,"e":0.0,"k":"falling","v":0.00446,"x":0.2857,"p":[[0,18,0.0,0.16964,0.27302,0.0,0.0,0.28571,0.0,1.0,20,1,1,20,0,2,0,0,4,0,0,2,0,0,1,0,0,1,0,0,1,0,1],[4,18,0.2222,0.2857,0.31134,0.0,0.2857,0.42857,0.0,1.0,13,3,0,13,0,0,0,0,10,0,0,3,0,0,1,0,0,2,0,0,0,0,3],[8,18,0.4444,0.0625,0.13333,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.06241,0.16723,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[16,18,0.8889,0.12945,0.18678,0.0,0.0,0.2857,0.0,0.71429,19,0,0,19,0,3,0,0,7,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[18,18,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.1741,"x":0.35714,"p":[[0,31,0.0,0.24553,0.30143,0.0,0.14285,0.42857,0.0,1.0,16,2,0,16,0,0,0,0,7,0,0,4,0,0,0,0,0,3,0,0,0,0,2],[4,31,0.129,0.29464,0.35524,0.0,0.2857,0.4286,0.0,1.0,15,5,0,15,0,0,0,0,7,0,0,4,0,0,0,0,0,1,0,0,0,0,5],[8,31,0.2581,0.33482,0.36875,0.0,0.2857,0.71429,0.0,1.0,14,3,0,14,0,1,0,0,6,0,0,1,0,0,0,0,0,4,0,0,3,0,3],[12,31,0.3871,0.1741,0.1504,0.0,0.2857,0.28571,0.0,0.42857,13,0,0,13,0,1,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.28794,0.22195,0.25,0.2857,0.28571,0.0,0.85714,7,0,0,7,0,1,0,0,18,0,0,1,0,1,1,0,0,1,0,0,2,0,0],[20,31,0.6452,0.35714,0.1675,0.2857,0.28571,0.42857,0.14286,0.857,0,0,0,0,0,3,0,0,19,0,0,6,0,0,0,0,0,3,0,0,1,0,0],[24,31,0.7742,0.26786,0.16269,0.2857,0.28571,0.28571,0.0,0.71429,6,0,0,6,0,1,0,0,19,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[28,31,0.9032,0.29016,0.10399,0.2857,0.28571,0.28571,0.0,0.571,2,0,0,2,0,1,0,0,24,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[31,31,1.0,0.28572,0.12877,0.2857,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,1,0,0,18,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dee97d353c0e6145","q":"Let  $ C$  and  $ D$  be two intersection points of circle  $ O_1$  and circle  $ O_2$ . A line, passing through  $ D$ , intersects the circle  $ O_1$  and the circle  $ O_2$  at the points  $ A$  and  $ B$  respectively. The points  $ P$  and  $ Q$  are on circles  $ O_1$  and  $ O_2$  respectively. The lines  $ PD$  and  $ AC$  intersect at  $ H$ , and the lines  $ QD$  and  $ BC$  intersect at  $ M$ . Suppose that  $ O$  is the circumcenter of the triangle  $ ABC$ . Prove that  $ OD\\perp MH$  if and only if  $ P,Q,M$  and  $ H$  are concyclic.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,33,0.0,0.08036,0.12339,0.0,0.0,0.1786,0.0,0.28571,22,0,0,22,0,2,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.06696,0.12363,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.04911,0.08458,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,16,0.0,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.08482,0.13767,0.0,0.0,0.14287,0.0,0.42857,22,0,0,22,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.09375,0.14555,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,4,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"65fbe1269369db4d","q":"Given a triangle  $ABC$ . Consider all the tetrahedrons  $PABC$  with  $PH$  -- the smallest of all tetrahedron's heights. Describe the set of all possible points  $H$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.81696,"x":0.94196,"p":[[0,6,0.0,0.875,0.15047,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,10,0,15],[4,6,0.6667,0.81696,0.1931,0.71429,0.85714,0.89286,0.0,1.0,1,8,1,1,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,15,0,8],[6,6,1.0,0.94196,0.09354,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,7,0,22]]},{"b":4,"e":0.0,"k":"flat","v":0.71429,"x":0.71429,"p":[[0,4,0.0,0.71429,0.39448,0.67857,0.92857,1.0,0.0,1.0,7,16,7,7,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,4,0,16]]}]},{"i":"ce338d5a03a57024","q":"Given is a positive integer $n$ and a game board consisting of $n+1$ square fields arranged side by side, numbered from 0 to $n$ from left to right. At the beginning of the game, $n$ game pieces are located on field number 0, and the other fields are empty.\nA patient player now chooses for each move a field with $k \\neq 0$ pieces and moves one of them at most $k$ fields to the right. The piece must remain on the board. His goal is to move all $n$ pieces to field number $n$ with a sequence of such moves.\nProve that the player cannot achieve this goal in fewer than $\\left\\lceil\\frac{n}{1}\\right\\rceil+\\left\\lceil\\frac{n}{2}\\right\\rceil+\\left\\lceil\\frac{n}{3}\\right\\rceil+\\ldots+\\left\\lceil\\frac{n}{n}\\right\\rceil$ moves. (Here, $\\lceil x\\rceil$ denotes the smallest integer not less than $x$.)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.08929,"p":[[0,19,0.0,0.05357,0.15872,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,19,0.2105,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,1,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.08929,0.10564,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.42857,17,0,2,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.03571,"x":0.09375,"p":[[0,33,0.0,0.06241,0.09399,0.0,0.0,0.14286,0.0,0.28571,21,0,3,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.03572,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,1,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.09375,0.12682,0.0,0.0,0.14286,0.0,0.4286,18,0,1,18,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4cbade62a0450e1f","q":"Let  $ ABC$  be a triangle and  $ O$  its circumcenter. Lines  $ AB$  and  $ AC$  meet the circumcircle of  $ OBC$  again in  $ B_1\\neq B$  and  $ C_1 \\neq C$ , respectively, lines  $ BA$  and  $ BC$  meet the circumcircle of  $ OAC$  again in  $ A_2\\neq A$  and  $ C_2\\neq C$ , respectively, and lines  $ CA$  and  $ CB$  meet the circumcircle of  $ OAB$  in  $ A_3\\neq A$  and  $ B_3\\neq B$ , respectively. Prove that lines  $ A_2A_3$ ,  $ B_1B_3$  and  $ C_1C_2$  have a common point.","t":[{"b":1,"e":1.0,"k":"flat","v":0.55803,"x":0.60711,"p":[[0,15,0.0,0.55803,0.43647,0.0,0.71429,1.0,0.0,1.0,9,14,2,9,0,1,0,0,4,0,0,1,0,0,0,0,0,3,0,0,0,0,14],[4,15,0.2667,0.60711,0.3993,0.28571,0.64264,1.0,0.0,1.0,6,14,4,6,0,1,0,0,3,0,0,4,0,0,2,0,0,1,0,0,1,0,14],[8,15,0.5333,0.56693,0.36331,0.28571,0.4998,1.0,0.0,1.0,5,11,3,5,0,0,0,0,5,0,0,6,0,0,3,0,0,2,0,0,0,0,11]]},{"b":5,"e":0.1429,"k":"volatile","v":0.08482,"x":0.7366,"p":[[0,9,0.0,0.54018,0.41147,0.24999,0.42857,1.0,0.0,1.0,7,13,2,7,0,1,0,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,13],[4,9,0.4444,0.7366,0.38317,0.4286,1.0,1.0,0.0,1.0,5,20,4,5,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,0,1,0,20],[8,9,0.8889,0.5491,0.43757,0.0,0.4998,1.0,0.0,1.0,9,14,3,9,0,1,0,0,4,0,0,2,0,0,1,0,0,0,0,0,1,0,14],[9,9,1.0,0.08482,0.14223,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"49d1ec139dfaa55e","q":"Given a triangle $A B C$, let its incircle touch the sides $B C, C A, A B$ at $D, E, F$, respectively. Let $G$ be the midpoint of the segment $D E$. Prove that $\\angle E F C=$ $\\angle G F D$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.11607,"p":[[0,33,0.0,0.11607,0.11538,0.0,0.14286,0.17857,0.0,0.28571,14,0,1,14,0,10,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,6,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.08464,0.11205,0.0,0.0,0.14286,0.0,0.28571,19,0,1,19,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,2,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.28571,21,0,2,21,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.05804,"x":0.15169,"p":[[0,28,0.0,0.14733,0.2004,0.0,0.07143,0.28571,0.0,1.0,16,1,0,16,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,28,0.1429,0.11607,0.19704,0.0,0.0,0.17857,0.0,1.0,19,1,2,19,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,28,0.2857,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,4,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.10268,0.1931,0.0,0.0,0.14286,0.0,1.0,20,1,3,20,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,28,0.5714,0.15169,0.15126,0.0,0.14286,0.2857,0.0,0.71429,11,0,2,11,0,11,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,28,0.7143,0.11598,0.09737,0.0,0.14286,0.14286,0.0,0.28571,11,0,2,11,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.13384,0.09406,0.105,0.14286,0.14287,0.0,0.28571,8,0,0,8,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9f1b739224df7eba","q":"Let  $ n$  be a positive integer, and let  $ x$  and  $ y$  be a positive real number such that  $ x^n \\plus{} y^n \\equal{} 1.$  Prove that\n\\[ \\left(\\sum^n_{k \\equal{} 1} \\frac {1 \\plus{} x^{2k}}{1 \\plus{} x^{4k}} \\right) \\cdot \\left( \\sum^n_{k \\equal{} 1} \\frac {1 \\plus{} y^{2k}}{1 \\plus{} y^{4k}} \\right) < \\frac {1}{(1 \\minus{} x) \\cdot (1 \\minus{} y)}.\n\\]\n\n*Author: Juhan Aru, Estonia*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,73,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,73,0.0548,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,73,0.1096,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,73,0.1644,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,73,0.2192,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,73,0.3288,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,73,0.3836,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,73,0.4384,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[40,73,0.5479,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,73,0.6027,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,73,0.6575,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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],[64,73,0.8767,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[72,73,0.9863,0.02679,0.09061,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[73,73,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6f47fc1f58c0f163","q":"For every ordered pair of integers  $(i,j)$ , not necessarily positive, we wish to select a point  $P_{i,j}$  in the Cartesian plane whose coordinates lie inside the unit square defined by\n\\[ i < x < i+1, \\qquad j < y < j+1. \\]\nFind all real numbers  $c > 0$  for which it's possible to choose these points such that for all integers  $i$  and  $j$ , the (possibly concave or degenerate) quadrilateral  $P_{i,j} P_{i+1,j} P_{i+1,j+1} P_{i,j+1}$  has perimeter strictly less than  $c$ .\n\n*Karthik Vedula*","t":[{"b":0,"e":0.0,"k":"rising","v":0.03571,"x":0.3259,"p":[[0,28,0.0,0.16062,0.27141,0.0,0.0,0.17857,0.0,0.85714,21,0,17,21,0,3,0,0,2,0,0,1,0,0,0,0,0,4,0,0,1,0,0],[4,28,0.1429,0.09375,0.17353,0.0,0.0,0.14286,0.0,0.71429,22,0,17,22,0,4,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,28,0.2857,0.03571,0.12877,0.0,0.0,0.0,0.0,0.71429,28,0,16,28,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,28,0.4286,0.0625,0.17474,0.0,0.0,0.0,0.0,0.71429,26,0,14,26,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[16,28,0.5714,0.31695,0.34019,0.0,0.2857,0.57143,0.0,1.0,15,3,15,15,0,0,0,0,3,0,0,2,0,0,7,0,0,2,0,0,0,0,3],[20,28,0.7143,0.3259,0.26543,0.0,0.42857,0.57143,0.0,0.85714,11,0,11,11,0,0,0,0,4,0,0,7,0,0,7,0,0,2,0,0,1,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.06696,"x":0.21866,"p":[[0,26,0.0,0.20536,0.30917,0.0,0.0,0.46429,0.0,0.85714,21,0,21,21,0,1,0,0,1,0,0,1,0,0,1,0,0,6,0,0,1,0,0],[4,26,0.1538,0.21866,0.31134,0.0,0.0,0.42858,0.0,0.85714,19,0,18,19,0,3,0,0,0,0,0,3,0,0,0,0,0,5,0,0,2,0,0],[8,26,0.3077,0.11607,0.26107,0.0,0.0,0.0,0.0,0.85714,25,0,23,25,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,0],[12,26,0.4615,0.06696,0.18205,0.0,0.0,0.0,0.0,0.71429,27,0,22,27,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0]]}]},{"i":"aefbc79555291e51","q":"Let  $ I$  be the incenter of triangle  $ ABC.$  Let  $ M,N$  be the midpoints of  $ AB,AC,$  respectively. Points  $ D,E$  lie on  $ AB,AC$  respectively such that  $ BD\\equal{}CE\\equal{}BC.$  The line perpendicular to  $ IM$  through  $ D$  intersects the line perpendicular to  $ IN$  through  $ E$  at  $ P.$  Prove that  $ AP\\perp BC.$","t":[{"b":4,"e":0.0,"k":"flat","v":0.09812,"x":0.2857,"p":[[0,15,0.0,0.23206,0.20443,0.0,0.21428,0.42857,0.0,0.57143,11,0,0,11,0,5,0,0,4,0,0,9,0,0,3,0,0,0,0,0,0,0,0],[4,15,0.2667,0.2232,0.20804,0.0,0.14286,0.42857,0.0,0.57143,12,0,0,12,0,5,0,0,3,0,0,9,0,0,3,0,0,0,0,0,0,0,0],[8,15,0.5333,0.22768,0.21387,0.0,0.2857,0.42857,0.0,0.57143,13,0,0,13,0,2,0,0,6,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[12,15,0.8,0.2857,0.19883,0.10714,0.28571,0.42857,0.0,0.57143,8,0,0,8,0,3,0,0,6,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[15,15,1.0,0.09812,0.16912,0.0,0.0,0.14286,0.0,0.714,21,0,1,21,0,5,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03125,"x":0.16516,"p":[[0,37,0.0,0.16516,0.20548,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,2,0,0,3,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[4,37,0.1081,0.11607,0.1729,0.0,0.0,0.1786,0.0,0.57143,20,0,0,20,0,4,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[8,37,0.2162,0.08929,0.14174,0.0,0.0,0.14287,0.0,0.4286,21,0,0,21,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.06688,0.12358,0.0,0.0,0.14071,0.0,0.42857,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.05803,0.09353,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"59475e3a3f103741","q":"Gleb picked positive integers  $N$  and  $a$  ( $a < N$ ). He wrote the number  $a$  on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number  $N$  by this last number with remainder and wrote the remainder onto the board. When he wrote the number  $0$  onto the board, he stopped. Could he pick  $N$  and  $a$  such that the sum of the numbers on the blackboard would become greater than  $100N$  ?\n\nIvan Mitrofanov","t":[{"b":1,"e":0.14286,"k":"flat","v":0.12946,"x":0.33469,"p":[[0,15,0.0,0.26767,0.21363,0.14286,0.14286,0.42858,0.0,0.71429,4,0,1,4,0,15,0,0,3,0,0,3,0,0,5,0,0,2,0,0,0,0,0],[4,15,0.2667,0.33469,0.18078,0.14286,0.28571,0.4642,0.0,0.57143,2,0,0,2,0,8,0,0,7,0,0,7,0,0,8,0,0,0,0,0,0,0,0],[8,15,0.5333,0.1875,0.1448,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,23,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,15,0.8,0.1517,0.08704,0.14286,0.14286,0.14286,0.0,0.57143,2,0,1,2,0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[15,15,1.0,0.12946,0.07457,0.14286,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,23,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"rising","v":0.26784,"x":0.54899,"p":[[0,16,0.0,0.26786,0.19805,0.14286,0.14286,0.32143,0.0,0.71429,2,0,0,2,0,16,0,0,6,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[4,16,0.25,0.39732,0.26901,0.14286,0.35714,0.71429,0.0,0.85714,1,0,0,1,0,14,0,0,1,0,0,2,0,0,4,0,0,9,0,0,1,0,0],[8,16,0.5,0.35255,0.22588,0.14286,0.28571,0.57111,0.0,0.71429,2,0,0,2,0,11,0,0,4,0,0,4,0,0,7,0,0,4,0,0,0,0,0],[12,16,0.75,0.26784,0.20436,0.14286,0.14286,0.28571,0.0,0.71429,2,0,1,2,0,16,0,0,7,0,0,2,0,0,1,0,0,4,0,0,0,0,0],[16,16,1.0,0.54899,0.18269,0.57132,0.57143,0.71429,0.0,0.71429,2,0,1,2,0,1,0,0,0,0,0,3,0,0,17,0,0,9,0,0,0,0,0]]}]},{"i":"d1d1f2bc0658155a","q":"Fix a nonnegative integer  $a_0$  to define a sequence of integers  $a_0,a_1,\\ldots$  by letting  $a_k,k\\geq 1$  be the smallest integer (strictly) greater than  $a_{k-1}$  making  $a_{k-1}+a_k{}$  into a perfect square. Let  $S{}$  be the set of positive integers not expressible as the difference of two terms of the sequence  $(a_k)_{k\\geq 0}.$  Prove that  $S$  is finite and determine its size in terms of  $a_0.$","t":[{"b":4,"e":0.0,"k":"flat","v":0.14732,"x":0.17411,"p":[[0,12,0.0,0.14732,0.23551,0.0,0.0,0.17857,0.0,0.71429,20,0,4,20,0,4,0,0,2,0,0,2,0,0,1,0,0,3,0,0,0,0,0],[4,12,0.3333,0.17411,0.19799,0.0,0.14286,0.28571,0.0,0.71429,15,0,10,15,0,4,0,0,7,0,0,4,0,0,1,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"rising","v":0.21874,"x":0.5491,"p":[[0,11,0.0,0.21874,0.29009,0.0,0.14286,0.32143,0.0,1.0,15,2,0,15,0,6,0,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,2],[4,11,0.3636,0.24768,0.31341,0.0,0.14143,0.42857,0.0,1.0,15,1,2,15,0,4,0,1,3,0,0,2,0,0,2,0,0,1,0,0,3,0,1],[8,11,0.7273,0.48217,0.1847,0.42857,0.57141,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,3,0,0,8,0,0,13,0,0,5,0,0,0,0,0],[11,11,1.0,0.5491,0.13415,0.42857,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,3,0,0,8,0,0,12,0,0,9,0,0,0,0,0]]}]},{"i":"4c7422aef7dfb438","q":"Let  $0<c<1$  and  $n$  a positive integer. Alice and Bob are playing a game. Bob writes  $n$  integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most  $cn$  moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins.\nFor which values of  $c$  does Alice win for all large enough  $n$ ?","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.11152,"p":[[0,19,0.0,0.11152,0.16648,0.0,0.0,0.14287,0.0,0.57143,19,0,0,19,0,6,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,19,0.2105,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04455,"p":[[0,14,0.0,0.04455,0.09731,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"422b2449b748f51e","q":"Given a prime number  $p > 2$ . Find the least  $n\\in Z_+$ , for which every set of  $n$  perfect squares not divisible by  $p$  contains nonempty subset with product of all it's elements equal to  $1\\ (\\text{mod}\\ p)$","t":[{"b":0,"e":1.0,"k":"rising","v":0.15179,"x":1.0,"p":[[0,36,0.0,0.30804,0.42425,0.0,0.0,0.64286,0.0,1.0,20,7,5,20,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,7],[4,36,0.1111,0.15179,0.3008,0.0,0.0,0.0,0.0,1.0,25,2,6,25,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,2],[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],[12,36,0.3333,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,36,0.4444,0.97768,0.08073,1.0,1.0,1.0,0.57143,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[20,36,0.5556,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,36,0.6667,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[28,36,0.7778,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[32,36,0.8889,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[36,36,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.29911,"p":[[0,21,0.0,0.29911,0.44658,0.0,0.0,0.89286,0.0,1.0,22,8,7,22,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,8],[4,21,0.1905,0.08036,0.25738,0.0,0.0,0.0,0.0,1.0,29,2,10,29,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[8,21,0.381,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,10,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,21,0.5714,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,1,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0625,0.20806,0.0,0.0,0.0,0.0,1.0,29,1,1,29,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ae7bb37d8f5ffde4","q":"Given are circles $\\Gamma_{1}$ with center $A$ and $\\Gamma_{2}$ with center $B$, where $A$ lies on $\\Gamma_{2}$. On $\\Gamma_{2}$, there is a variable point $P$, not on $A B$. A line through $P$ that is tangent to $\\Gamma_{1}$ at $S$, intersects $\\Gamma_{2}$ again at $Q$, where $P$ and $Q$ lie on the same side of $A B$. Another line through $Q$ is tangent to $\\Gamma_{1}$ at $T$. Let $M$ be the foot of the perpendicular from $P$ to $A B$. Let $N$ be the intersection of $A Q$ and $M T$. Prove that $N$ lies on a line that is independent of the position of $P$ on $\\Gamma_{2}$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.09375,"x":0.125,"p":[[0,15,0.0,0.10714,0.09449,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.11598,0.09059,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.11598,0.09737,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.125,0.05923,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.07589,"x":0.13393,"p":[[0,33,0.0,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.07589,0.07974,0.0,0.07143,0.14286,0.0,0.28571,16,0,1,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,1,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,1,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c0100d4a1c33ca2f","q":"Given a ring  $\\left( A,+,\\cdot  \\right)$  that meets both of the following conditions:\n(1)  $A$  is not a field, and\n(2) For every non-invertible element  $x$  of  $ A$ , there is an integer  $m>1$  (depending on  $x$ ) such that  $x=x^2+x^3+\\ldots+x^{2^m}$ .\nShow that \n(a)  $x+x=0$  for every  $x \\in A$ , and\n(b)  $x^2=x$  for every non-invertible  $x\\in A$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.125,"p":[[0,7,0.0,0.11161,0.11143,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,14,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.125,0.08564,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.12927,"x":0.18723,"p":[[0,5,0.0,0.12927,0.10919,0.105,0.14286,0.14286,0.0,0.571,8,0,0,8,0,21,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,5,0.8,0.17848,0.0799,0.14286,0.14286,0.1786,0.0,0.42857,1,0,0,1,0,23,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.18723,0.0754,0.14286,0.14286,0.2857,0.14,0.4286,0,0,0,0,0,23,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3955b52050d8d9df","q":"In a row of  $2009$  weights, the weight of each weight is an integer grams and does not exceed  $1$  kg. The weights of any two adjacent weights differ by exactly  $1$  g, and the total weight of all weights in grams is an even number. Prove that weights can be separated into two piles, the sums of the weights in which are equal.","t":[{"b":1,"e":1.0,"k":"rising","v":0.30356,"x":1.0,"p":[[0,29,0.0,0.32143,0.42107,0.0,0.0,0.67857,0.0,1.0,18,8,0,18,0,1,0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,8],[4,29,0.1379,0.30356,0.38257,0.0,0.0,0.42858,0.0,1.0,17,6,0,17,0,1,0,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,6],[8,29,0.2759,0.61161,0.4199,0.32143,0.78571,1.0,0.0,1.0,8,16,0,8,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,16],[12,29,0.4138,0.63839,0.42104,0.32143,0.92857,1.0,0.0,1.0,8,16,0,8,0,0,0,0,0,0,0,5,0,0,1,0,0,0,0,0,2,0,16],[16,29,0.5517,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,29,0.6897,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,29,0.8276,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[29,29,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":6,"e":1.0,"k":"volatile","v":0.1875,"x":0.99553,"p":[[0,37,0.0,0.47768,0.4512,0.0,0.42857,1.0,0.0,1.0,13,12,0,13,0,1,0,0,0,0,0,4,0,0,1,0,0,0,0,0,1,0,12],[4,37,0.1081,0.54018,0.48673,0.0,0.92857,1.0,0.0,1.0,14,16,0,14,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16],[8,37,0.2162,0.59821,0.47573,0.0,1.0,1.0,0.0,1.0,12,18,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,18],[12,37,0.3243,0.70089,0.40777,0.39286,1.0,1.0,0.0,1.0,6,20,0,6,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,20],[16,37,0.4324,0.54911,0.44908,0.0,0.42857,1.0,0.0,1.0,11,15,0,11,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,15],[20,37,0.5405,0.22321,0.38949,0.0,0.0,0.32142,0.0,1.0,23,6,0,23,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,6],[24,37,0.6486,0.38393,0.42324,0.0,0.28571,1.0,0.0,1.0,15,9,0,15,0,1,0,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,9],[28,37,0.7568,0.29911,0.42462,0.0,0.0,0.57145,0.0,1.0,20,8,0,20,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,8],[32,37,0.8649,0.1875,0.33964,0.0,0.0,0.32142,0.0,1.0,23,4,0,23,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[36,37,0.973,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[37,37,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"88d025f809b73cd6","q":"Given are triangle  $ABC$  and line  $\\ell$  intersecting  $BC, CA$  and  $AB$  at points  $A_1, B_1$  and  $C_1$  respectively. Point  $A'$  is the midpoint of the segment between the projections of  $A_1$  to  $AB$  and  $AC$ . Points  $B'$  and  $C'$  are defined similarly.\n(a) Prove that  $A', B'$  and  $C'$  lie on some line  $\\ell'$ .\n(b) Suppose  $\\ell$   passes through the circumcenter of  $\\triangle ABC$ . Prove that in this case  $\\ell'$  passes through the center of its nine-points circle.\n\n*M. Marinov and N. Beluhov*","t":[{"b":3,"e":0.42857,"k":"flat","v":0.37945,"x":0.5267,"p":[[0,19,0.0,0.37945,0.18422,0.24999,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,6,0,0,7,0,0,10,0,0,1,0,0,0,0,0],[4,19,0.2105,0.5267,0.11535,0.53539,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,0,0,0,0,0,0,7,0,0,23,0,0,1,0,0,0,0,0],[8,19,0.4211,0.49106,0.10676,0.42857,0.57121,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,11,0,0,16,0,0,1,0,0,0,0,0],[12,19,0.6316,0.5045,0.08732,0.42857,0.5712,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,14,0,0,16,0,0,1,0,0,0,0,0],[16,19,0.8421,0.50444,0.1009,0.42857,0.49979,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,15,0,0,15,0,0,0,0,0,1,0,0],[19,19,1.0,0.50444,0.10703,0.42859,0.57143,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,2,0,0,8,0,0,21,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"falling","v":0.19634,"x":0.39281,"p":[[0,21,0.0,0.39281,0.19895,0.28571,0.42857,0.57143,0.14,1.0,0,1,0,0,0,7,0,0,8,0,0,7,0,0,8,0,0,1,0,0,0,0,1],[4,21,0.1905,0.34795,0.22594,0.14286,0.28571,0.42858,0.14,1.0,0,2,0,0,0,11,0,0,9,0,0,5,0,0,5,0,0,0,0,0,0,0,2],[8,21,0.381,0.26786,0.11152,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,11,0,0,15,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[12,21,0.5714,0.23634,0.11655,0.14286,0.14288,0.28571,0.14,0.57143,0,0,0,0,0,17,0,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[16,21,0.7619,0.23205,0.13722,0.14286,0.14286,0.28571,0.14,0.71429,0,0,0,0,0,19,0,0,9,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[20,21,0.9524,0.20536,0.09406,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,21,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.19634,0.10569,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,0,21,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"ba87e6ac4e61e99a","q":"For every positive integer  $k>1$  prove that there exist a real number  $x$  so that for every positive integer  $n<1398$ :  $$ \\left\\{x^n\\right\\}<\\left\\{x^{n-1}\\right\\} \\Longleftrightarrow k\\mid n. $$  *Proposed by Mohammad Amin Sharifi*","t":[{"b":3,"e":0.0,"k":"flat","v":0.01786,"x":0.03795,"p":[[0,9,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,18,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,15,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,9,0.8889,0.03795,0.11006,0.0,0.0,0.0,0.0,0.4286,28,0,13,28,0,1,0,1,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01786,"x":0.02679,"p":[[0,15,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,16,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,9,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,10,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,12,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4350a521ebee8b54","q":"Inside of convex quadrilateral  $ABCD$  found a point  $M$  such that  $\\angle AMB=\\angle ADM+\\angle BCM$  and  $\\angle AMD=\\angle ABM+\\angle DCM$ .Prove that   $$ AM\\cdot CM+BM\\cdot DM\\ge \\sqrt{AB\\cdot BC\\cdot CD\\cdot DA}. $$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,7,0.0,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,28,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.04018,0.13474,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,28,0.7143,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f2bb0c0316bc4117","q":"For a positive integer $m$, we denote $d(m)$ as the number of positive divisors of $m$ (including 1 and $m$). Let $k$ be a strictly positive integer. Show that there are infinitely many positive integers $n$ such that $n$ has exactly $k$ distinct prime divisors and for all positive integers $a, b$ with $n=a+b$, $d(n)$ does not divide $d\\left(a^{2}+b^{2}\\right)$","t":[{"b":0,"e":0.0,"k":"falling","v":0.02679,"x":0.21429,"p":[[0,49,0.0,0.21429,0.31944,0.0,0.0,0.28571,0.0,1.0,18,3,15,18,0,3,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,3],[4,49,0.0816,0.125,0.26426,0.0,0.0,0.0,0.0,1.0,25,1,19,25,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,1],[8,49,0.1633,0.14277,0.28794,0.0,0.0,0.14071,0.0,1.0,23,1,21,23,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,1],[12,49,0.2449,0.14286,0.2369,0.0,0.0,0.28571,0.0,0.85714,20,0,19,20,0,3,0,0,5,0,0,1,0,0,1,0,0,0,0,0,2,0,0],[16,49,0.3265,0.15165,0.26443,0.0,0.0,0.2857,0.0,0.85714,22,0,20,22,0,1,0,0,4,0,0,0,0,0,2,0,0,1,0,0,2,0,0],[20,49,0.4082,0.11161,0.24932,0.0,0.0,0.0,0.0,1.0,25,1,23,25,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[24,49,0.4898,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,26,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,26,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.07143,0.14725,0.0,0.0,0.0,0.0,0.57143,25,0,23,25,0,1,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,49,0.7347,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,27,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.03125,0.1504,0.0,0.0,0.0,0.0,0.85714,30,0,28,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.19196,"x":0.24544,"p":[[0,12,0.0,0.24544,0.3278,0.0,0.0,0.46429,0.0,1.0,17,1,13,17,0,3,0,0,3,0,0,1,0,0,2,0,0,2,0,0,3,0,1],[4,12,0.3333,0.19641,0.31891,0.0,0.0,0.28571,0.0,1.0,21,2,15,21,0,1,0,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,2],[8,12,0.6667,0.19196,0.28483,0.0,0.0,0.28571,0.0,1.0,19,1,15,19,0,2,0,0,4,0,0,1,0,0,3,0,0,1,0,0,1,0,1]]}]},{"i":"db4ef428676281db","q":"Given convex hexagon  $ABCDEF$  with  $AB \\parallel DE$ ,  $BC \\parallel EF$ , and  $CD \\parallel FA$  . The distance between the lines  $AB$  and  $DE$  is equal to the distance between the lines  $BC$  and  $EF$  and to the distance between the lines  $CD$  and  $FA$ . Prove that the sum  $AD+BE+CF$  does not exceed the perimeter of hexagon  $ABCDEF$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,13,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,5,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02223,"p":[[0,23,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"10c7a9fcd5f4dbb3","q":"Initially, a non-constant polynomial $S(x)$ with real coefficients is written down on a board. Whenever the board contains a polynomial $P(x)$, not necessarily alone, one can write down on the board any polynomial of the form $P(C+x)$ or $C+P(x)$, where $C$ is a real constant. Moreover, if the board contains two (not necessarily distinct) polynomials $P(x)$ and $Q(x)$, one can write $P(Q(x))$ and $P(x)+Q(x)$ down on the board. No polynomial is ever erased from the board.\n\nGiven two sets of real numbers, $A=\\left\\{a_{1}, a_{2}, \\ldots, a_{n}\\right\\}$ and $B=\\left\\{b_{1}, b_{2}, \\ldots, b_{n}\\right\\}$, a polynomial $f(x)$ with real coefficients is $(A, B)$-nice if $f(A)=B$, where $f(A)=\\left\\{f\\left(a_{i}\\right): i=1,2, \\ldots, n\\right\\}$.\n\nDetermine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A|=|B|$, one can produce an $(A, B)$-nice polynomial in a finite number of steps.\n\nIran, Navid SafaEi","t":[{"b":0,"e":0.14286,"k":"flat","v":0.07116,"x":0.11125,"p":[[0,9,0.0,0.11125,0.09257,0.0,0.14286,0.14286,0.0,0.4286,10,0,8,10,0,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.10259,0.09602,0.0,0.14286,0.14286,0.0,0.42857,12,0,8,12,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.07116,0.07117,0.0,0.07,0.14286,0.0,0.14286,16,0,1,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14,"k":"flat","v":0.08918,"x":0.14259,"p":[[0,32,0.0,0.08918,0.06675,0.0,0.14286,0.14286,0.0,0.1429,11,0,9,11,2,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.10268,0.09606,0.0,0.14286,0.14286,0.0,0.42857,12,0,11,12,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.1429,7,0,5,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.12947,0.04164,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.13366,0.03452,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.143,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.13348,0.03448,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a6cc29cb652b87c4","q":"Let  $A$  be a finite set of positive integers.  Prove that there exists a finite set  $B$  of positive integers such that  $A \\subseteq B$  and\n\n\\[\\prod_{x\\in B} x = \\sum_{x\\in B} x^2.\\]","t":[{"b":3,"e":0.0,"k":"flat","v":0.09822,"x":0.12054,"p":[[0,10,0.0,0.12054,0.14335,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,13,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,10,0.4,0.11152,0.13234,0.0,0.07,0.14287,0.0,0.4286,16,0,0,16,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.10697,0.12367,0.0,0.14,0.14286,0.0,0.4286,15,0,0,15,0,12,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.05795,"x":0.125,"p":[[0,48,0.0,0.09803,0.10964,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.10259,0.1299,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.09375,0.11633,0.0,0.07143,0.14286,0.0,0.4286,16,0,0,16,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.08482,0.16311,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,6,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[16,48,0.3333,0.125,0.1171,0.0,0.14286,0.1429,0.0,0.42857,12,0,2,12,0,13,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.10714,0.09449,0.0,0.14286,0.14286,0.0,0.4286,11,0,0,11,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.05795,0.08636,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.07143,0.10101,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.08018,0.0869,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.08473,0.10007,0.0,0.07,0.14286,0.0,0.42857,16,0,0,16,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.08473,0.10626,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b8eeb40324dc81b1","q":"Given points  $O$  and  $A$  in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point  $X$  in the plane, the circle  $C(X)$  has center  $O$  and radius  $OX+{\\angle AOX\\over OX}$ , where  $\\angle AOX$  is measured in radians in the range  $[0,2\\pi)$ . Prove that we can find a point  $X$ , not on  $OA$ , such that its color appears on the circumference of the circle  $C(X)$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.02008,"x":0.04464,"p":[[0,7,0.0,0.04464,0.16534,0.0,0.0,0.0,0.0,0.857,29,0,6,29,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,7,0.5714,0.02008,0.05424,0.0,0.0,0.0,0.0,0.214,28,0,10,28,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.0133,"x":0.07134,"p":[[0,7,0.0,0.0133,0.04137,0.0,0.0,0.0,0.0,0.1429,29,0,6,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02451,0.07275,0.0,0.0,0.0,0.0,0.28571,28,0,7,28,1,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.07134,0.10857,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,1,7,0,1,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"03aa3547d5404d03","q":"Let  $(a_n)$  be sequnce of positive integers such that first  $k$  members  $a_1,a_2,...,a_k$  are distinct positive integers, and for each  $n>k$ , number  $a_n$  is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers  $a_1,a_2,...,a_{n-1}$ . Prove that  $a_n=2a_{n-1}$  for all sufficently large  $n$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.16964,"x":0.20982,"p":[[0,5,0.0,0.20982,0.17122,0.14286,0.14286,0.2857,0.0,1.0,1,1,1,1,0,22,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,5,0.8,0.16964,0.08328,0.14286,0.14286,0.2857,0.0,0.28571,3,0,2,3,0,20,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.14277,"x":0.37054,"p":[[0,12,0.0,0.22759,0.20785,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,24,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[4,12,0.3333,0.37054,0.329,0.14286,0.14286,0.46431,0.14286,1.0,0,5,0,0,0,18,0,0,4,0,0,2,0,0,1,0,0,0,0,0,2,0,5],[8,12,0.6667,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e1cdc4ba7555b23a","q":"In a class of at least four people, the following applies: if four of them sit down at a round table, there is always someone who knows both of their neighbors or does not know both of their neighbors. Prove that it is possible to divide the people into two groups (one of which may be empty) such that in one group everyone knows each other and in the other group no one knows each other.\n(If person $A$ knows person $B$, then $B$ also knows $A$.)","t":[{"b":0,"e":0.28571,"k":"flat","v":0.17858,"x":0.24108,"p":[[0,23,0.0,0.21428,0.12372,0.21427,0.28571,0.28571,0.0,0.28571,8,0,1,8,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.24108,0.10374,0.2857,0.28571,0.28571,0.0,0.286,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.23214,0.11152,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.19643,0.13244,0.0,0.2857,0.28571,0.0,0.286,10,0,0,10,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.21428,0.12372,0.21427,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.17858,0.13833,0.0,0.28571,0.28571,0.0,0.286,12,0,0,12,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.23214,"x":0.25,"p":[[0,7,0.0,0.25,0.09449,0.2857,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.23214,0.11152,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.24108,0.10374,0.2857,0.28571,0.28571,0.0,0.286,5,0,0,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7912353f19c86c5a","q":"Given a positive integer  $n$ , determine the largest real number  $\\mu$  satisfying the following condition: for every set  $C$  of  $4n$  points in the interior of the unit square  $U$ , there exists a rectangle  $T$  contained in  $U$  such that $\\bullet$  the sides of  $T$  are parallel to the sides of  $U$ ; $\\bullet$  the interior of  $T$  contains exactly one point of  $C$ ; $\\bullet$  the area of  $T$  is at least  $\\mu$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,20,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,20,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0625,0.08517,0.0,0.0,0.14286,0.0,0.2857,19,0,2,19,2,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02455,"p":[[0,25,0.0,0.02455,0.07278,0.0,0.0,0.0,0.0,0.35714,28,0,19,28,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,22,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,28,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,27,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,16,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2b86d7a470419305","q":"Given a positive integer  $n$ , there are  $n$  boxes  $B_1,...,B_n$ . The following procedure can be used to add balls.  $$ \\text{(Procedure) Chosen two positive integers }n\\geq i\\geq j\\geq 1\\text{, we add one ball each to the boxes }B_k\\text{ that }i\\geq k\\geq j. $$  For positive integers  $x_1,...,x_n$  let  $f(x_1,...,x_n)$  be the minimum amount of procedures to get all boxes have its amount of balls to be a multiple of 3, starting with  $x_i$  balls for  $B_i(i=1,...,n)$ . Find the largest possible value of  $f(x_1,...,x_n)$ . (If  $x_1,...,x_n$  are all multiples of 3,  $f(x_1,...,x_n)=0$ .)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,14,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,21,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,26,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,25,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0133,"x":0.02679,"p":[[0,5,0.0,0.02679,0.05577,0.0,0.0,0.0,0.0,0.143,26,0,16,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,17,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4f5bf41d84956ad7","q":"Let  $A, B, C, D, E$  be five points on a circle such that  $|AB| = |CD|$  and  $|BC| = |DE|$ . The segments  $AD$  and  $BE$  intersect at  $F$ . Let  $M$  denote the midpoint of segment  $CD$ . Prove that the circle of center  $M$  and radius  $ME$  passes through the midpoint of segment  $AF$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,17,0.0,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,4,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"da33627613938bd5","q":"Given a triangle  $ABC$ .\r\nLet  $M$  and  $N$  be the points where the angle bisectors of the angles  $ABC$  and  $BCA$  intersect the sides  $CA$  and  $AB$ , respectively.\r\nLet  $D$  be the point where the ray  $MN$  intersects the circumcircle of triangle  $ABC$ .\r\n\r\nProve that  $\\frac{1}{BD}=\\frac{1}{AD}+\\frac{1}{CD}$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,9,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.03125,0.13236,0.0,0.0,0.0,0.0,0.71429,30,0,2,30,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[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]]}]},{"i":"9903450640f0f800","q":"Let  $ p $  and  $ q $  be given primes and the sequence  $ \\{ p_n \\}_{n = 1}^{\\infty} $  defined recursively as follows: $ p_1 = p $ ,  $ p_2 = q $ , and  $ p_{n+2} $  is the largest prime divisor of the number  $( p_n + p_{n + 1} + 2016) $  for all  $ n \\geq \u00151 $ . Prove that this sequence is bounded. That is, there exists a positive real number  $ M $  such that  $ p_n < M $  for all positive integers  $ n $ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.0,"x":0.03572,"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],[4,18,0.2222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,56,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,56,0.1429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,56,0.2143,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,56,0.6429,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce29e3a46a46c583","q":"In an acute-angled triangle $ABC$, it holds that $|AB| > |CA| > |BC|$. The points $D, E$, and $F$ are the feet of the altitudes from $A, B$, and $C$ respectively. The line through $F$ parallel to $DE$ intersects $BC$ at $M$. The bisector of $\\angle MFE$ intersects $DE$ at $N$. Prove that $F$ is the circumcenter of $\\triangle DMN$ if and only if $B$ is the circumcenter of $\\triangle FMN$.","t":[{"b":3,"e":0.57143,"k":"rising","v":0.20536,"x":0.58029,"p":[[0,17,0.0,0.29462,0.27647,0.0,0.21428,0.46418,0.0,0.85714,10,0,5,10,0,6,0,0,3,0,0,5,0,0,2,0,0,5,0,0,1,0,0],[4,17,0.2353,0.27221,0.26575,0.0,0.2857,0.42857,0.0,0.85714,11,0,10,11,0,4,0,0,6,0,0,5,0,0,2,0,0,2,0,0,2,0,0],[8,17,0.4706,0.20536,0.19541,0.10714,0.14286,0.32143,0.0,0.71429,8,0,3,8,0,14,0,0,2,0,0,6,0,0,0,0,0,2,0,0,0,0,0],[12,17,0.7059,0.58029,0.24468,0.42857,0.57121,0.74996,0.0,0.85714,2,0,1,2,0,1,0,0,2,0,0,6,0,0,6,0,0,7,0,0,8,0,0],[16,17,0.9412,0.54893,0.18236,0.42857,0.57121,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,7,0,0,6,0,0,6,0,0,11,0,0,2,0,0],[17,17,1.0,0.47757,0.16619,0.42857,0.4286,0.57143,0.14,0.85714,0,0,0,0,0,2,0,0,4,0,0,14,0,0,6,0,0,5,0,0,1,0,0]]},{"b":5,"e":0.57143,"k":"rising","v":0.23642,"x":0.62497,"p":[[0,17,0.0,0.24997,0.26241,0.0,0.14286,0.42857,0.0,0.85714,11,0,9,11,0,8,0,0,3,0,0,3,0,0,3,0,0,3,0,0,1,0,0],[4,17,0.2353,0.30802,0.27457,0.0,0.28571,0.57143,0.0,0.85714,11,0,3,11,0,3,0,0,3,0,0,5,0,0,6,0,0,3,0,0,1,0,0],[8,17,0.4706,0.23642,0.23318,0.0,0.14286,0.4286,0.0,0.71429,12,0,12,12,0,5,0,0,4,0,0,6,0,0,3,0,0,2,0,0,0,0,0],[12,17,0.7059,0.49551,0.33308,0.10714,0.57121,0.85714,0.0,0.85714,8,0,8,8,0,1,0,0,0,0,0,5,0,0,4,0,0,5,0,0,9,0,0],[16,17,0.9412,0.58478,0.26811,0.571,0.64286,0.74996,0.0,0.85714,4,0,3,4,0,0,0,0,2,0,0,1,0,0,9,0,0,8,0,0,8,0,0],[17,17,1.0,0.62497,0.20748,0.5354,0.64286,0.85704,0.0,0.85714,1,0,0,1,0,0,0,0,2,0,0,5,0,0,8,0,0,7,0,0,9,0,0]]}]},{"i":"88dfbfcb9a37e9da","q":"In the acute triangle $ABC$, $\\angle C$ is greater than $\\angle A$. Let $E$ be such that $AE$ is a diameter of the circumcircle $\\Gamma$ of $\\triangle ABC$. Let $K$ be the intersection of $AC$ and the tangent to $\\Gamma$ at $B$. Let $L$ be the foot of the perpendicular from $K$ to $AE$, and let $D$ be the intersection of $KL$ and $AB$.\nProve that $CE$ is the angle bisector of $\\angle BCD$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.08482,"p":[[0,31,0.0,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,2,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.04241,0.07755,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.08482,0.12807,0.0,0.0,0.14286,0.0,0.4286,20,0,2,20,0,7,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.07358,0.13297,0.0,0.0,0.14286,0.0,0.71429,19,0,1,19,1,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,31,0.5161,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.07357,0.07459,0.0,0.07,0.14286,0.0,0.21429,16,0,0,16,0,15,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.05802,"x":0.09822,"p":[[0,18,0.0,0.07134,0.07978,0.0,0.0,0.14286,0.0,0.28571,17,0,4,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.05802,0.11763,0.0,0.0,0.14286,0.0,0.571,23,0,4,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,18,0.4444,0.08258,0.09296,0.0,0.07143,0.14286,0.0,0.357,16,0,0,16,0,14,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.09821,0.09062,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f4da7a381ead55c8","q":"In a matrix  $2n \\times 2n$ ,  $n \\in N$ , are  $4n^2$  real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than  $1$ . Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed  $n$ .","t":[{"b":6,"e":0.14286,"k":"falling","v":0.05348,"x":0.2857,"p":[[0,19,0.0,0.2857,0.30721,0.0,0.14288,0.46418,0.0,1.0,11,2,2,11,0,6,0,0,5,0,0,2,0,0,3,0,0,2,0,0,1,0,2],[4,19,0.2105,0.20536,0.26711,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,8,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0,0,1],[8,19,0.4211,0.13839,0.23551,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,7,0,0,2,0,0,0,0,0,1,0,0,2,0,0,1,0,0],[12,19,0.6316,0.09821,0.18013,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,8,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[16,19,0.8421,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.11384,"x":0.24552,"p":[[0,10,0.0,0.24552,0.25058,0.0,0.14288,0.32143,0.0,0.85714,10,0,1,10,0,8,0,0,6,0,0,2,0,0,2,0,0,3,0,0,1,0,0],[4,10,0.4,0.20978,0.23409,0.0,0.14286,0.32143,0.0,0.85714,11,0,0,11,0,11,0,0,2,0,0,3,0,0,3,0,0,1,0,0,1,0,0],[8,10,0.8,0.19195,0.25655,0.0,0.14286,0.28571,0.0,1.0,14,2,0,14,0,6,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[10,10,1.0,0.11384,0.1427,0.0,0.14286,0.14286,0.0,0.71429,14,0,0,14,0,13,0,1,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"6880119b27f129da","q":"Is it true that for integer  $n\\ge 2$ , and given any non-negative reals  $\\ell_{ij}$ ,  $1\\le i<j\\le n$ , we can find a sequence  $0\\le a_1,a_2,\\ldots,a_n$  such that for all  $1\\le i<j\\le n$  to have  $|a_i-a_j|\\ge \\ell_{ij}$ , yet still  $\\sum_{i=1}^n a_i\\le \\sum_{1\\le i<j\\le n}\\ell_{ij}$ ?","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,6,0.0,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.11152,"p":[[0,38,0.0,0.04463,0.10965,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,38,0.1053,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.08929,0.13716,0.0,0.0,0.14287,0.0,0.42857,21,0,0,21,0,4,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.11152,0.13234,0.0,0.14143,0.14286,0.0,0.42857,15,0,0,15,0,12,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.0625,0.11259,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9624b6e10f1f91bf","q":"In triangle  $ ABC$  the bisector of  $ \\angle ACB$  intersects  $ AB$  at  $ D$ . Consider an arbitrary circle  $ O$  passing through  $ C$  and  $ D$ , so that it is not tangent to  $ BC$  or  $ CA$ . Let  $ O\\cap BC \\equal{} \\{M\\}$  and  $ O\\cap CA \\equal{} \\{N\\}$ .\r\na) Prove that there is a circle  $ S$  so that  $ DM$  and  $ DN$  are tangent to  $ S$  in  $ M$  and  $ N$ , respectively.\r\nb) Circle  $ S$  intersects lines  $ BC$  and  $ CA$  in  $ P$  and  $ Q$  respectively. Prove that the lengths of  $ MP$  and  $ NQ$  do not depend on the choice of circle  $ O$ .","t":[{"b":5,"e":0.42857,"k":"flat","v":0.37944,"x":0.49996,"p":[[0,15,0.0,0.37944,0.23311,0.14286,0.42857,0.4642,0.0,0.85714,2,0,0,2,0,8,0,0,4,0,0,10,0,0,4,0,0,1,0,0,3,0,0],[4,15,0.2667,0.49996,0.18897,0.42857,0.4286,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,11,0,0,10,0,0,2,0,0,2,0,1],[8,15,0.5333,0.4286,0.07984,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,5,0,0,22,0,0,5,0,0,0,0,0,0,0,0],[12,15,0.8,0.41963,0.07933,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,3,0,0,25,0,0,3,0,0,0,0,0,0,0,0],[15,15,1.0,0.41516,0.13994,0.39286,0.42857,0.42858,0.14286,0.857,0,0,0,0,0,3,0,0,5,0,0,18,0,0,5,0,0,0,0,0,1,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.34369,"x":0.52232,"p":[[0,22,0.0,0.41959,0.20814,0.28571,0.42857,0.57025,0.0,1.0,1,1,0,1,0,5,0,0,4,0,0,13,0,0,6,0,0,1,0,0,1,0,1],[4,22,0.1818,0.41063,0.21661,0.25,0.42857,0.57143,0.0,0.85714,1,0,1,1,0,7,0,0,3,0,0,12,0,0,4,0,0,3,0,0,2,0,0],[8,22,0.3636,0.34369,0.19525,0.24999,0.28571,0.42893,0.0,0.85714,2,0,1,2,0,6,0,0,11,0,0,6,0,0,5,0,0,1,0,0,1,0,0],[12,22,0.5455,0.52232,0.28928,0.42857,0.42857,0.74996,0.0,1.0,3,4,3,3,0,1,0,0,2,0,0,15,0,0,0,0,0,3,0,0,4,0,4],[16,22,0.7273,0.40167,0.17666,0.28571,0.42857,0.57143,0.0,0.71429,1,0,1,1,0,6,0,0,3,0,0,11,0,0,10,0,0,1,0,0,0,0,0],[20,22,0.9091,0.49991,0.10708,0.42857,0.571,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,12,0,0,17,0,0,1,0,0,0,0,0],[22,22,1.0,0.41504,0.17636,0.2857,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,6,0,0,9,0,0,10,0,0,2,0,0,0,0,0]]}]},{"i":"e4afa50669bbc912","q":"In the city of Flensburg there is a single, infinitely long, street with houses numbered 2, 3, .. The police in Flensburg is trying to catch a thief who every night moves from the house where she is currently hiding to one of its neighbouring houses.\n\nTo taunt the local law enforcement the thief reveals every morning the highest prime divisor of the number of the house she has moved to.\n\nEvery Sunday afternoon the police searches a single house, and they catch the thief if they search the house she is currently occupying. Does the police have a strategy to catch the thief in finite time?","t":[{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.04456,"p":[[0,12,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.04456,0.06609,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.02679,"p":[[0,9,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"779cf414361b8373","q":"Given is a parallelogram  $ABCD$ , with  $AB <AC <BC$ . Points  $E$  and  $F$  are selected on the circumcircle   $\\omega$  of  $ABC$  so that the tangenst to  $\\omega$  at these points pass through point  $D$  and the segments  $AD$  and  $CE$  intersect. \nIt turned out that  $\\angle ABF = \\angle DCE$ . Find the angle  $\\angle{ABC}$ .\nA. Yakubov, S. Berlov","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,46,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,91,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,91,0.044,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,91,0.0879,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,91,0.1319,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,91,0.1758,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,91,0.2198,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,91,0.2637,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,91,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,91,0.3516,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,91,0.3956,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,91,0.4396,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,91,0.4835,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,91,0.5275,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,91,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,91,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,91,0.6593,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,91,0.7033,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,91,0.7473,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,91,0.7912,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,91,0.8352,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,91,0.8791,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,91,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b57bd6c67df7fea4","q":"Let  $ABC$  be a fixed acute-angled triangle. Consider some points  $E$  and  $F$  lying on the sides  $AC$  and  $AB$ , respectively, and let  $M$  be the midpoint of  $EF$ . Let the perpendicular bisector of  $EF$  intersect the line  $BC$  at  $K$ , and let the perpendicular bisector of  $MK$  intersect the lines  $AC$  and  $AB$  at  $S$  and  $T$ , respectively. If the quadrilateral  $KSAT$  is cycle, prove that  $\\angle{KEF}=\\angle{KFE}=\\angle{A}$ .","t":[{"b":0,"e":0.2857,"k":"flat","v":0.12937,"x":0.16072,"p":[[0,25,0.0,0.12937,0.10925,0.0,0.14286,0.17857,0.0,0.28571,11,0,2,11,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.14723,0.12104,0.105,0.14286,0.14286,0.0,0.4286,8,0,1,8,0,18,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.12946,0.10926,0.0,0.14286,0.17857,0.0,0.28571,11,0,1,11,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.13392,0.11253,0.10714,0.14286,0.14286,0.0,0.571,8,0,1,8,0,20,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,25,0.64,0.16071,0.09279,0.14286,0.14286,0.14286,0.0,0.57143,2,0,2,2,0,26,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,25,0.8,0.15624,0.07449,0.14286,0.14286,0.14286,0.14286,0.571,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,25,0.96,0.14723,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.16072,0.04725,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.14732,"p":[[0,34,0.0,0.14732,0.14054,0.0,0.14286,0.1786,0.0,0.42857,11,0,1,11,0,13,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.09821,0.10374,0.0,0.14286,0.14286,0.0,0.28571,15,0,2,15,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.11607,0.0974,0.0,0.14286,0.14286,0.0,0.4286,10,0,1,10,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.10714,0.11845,0.0,0.14286,0.14286,0.0,0.42857,14,0,1,14,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,1,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.57143,20,0,4,20,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,34,0.7059,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e938db1dd54e2916","q":"In a triangle  $ABC$ , it is drawn a circumference with center in the incenter  $I$  and that meet twice each of the sides of the triangle: the segment  $BC$  on  $D$  and  $P$  (where  $D$  is nearer two  $B$ ); the segment  $CA$  on  $E$  and  $Q$  (where  $E$  is nearer to  $C$ ); and the segment  $AB$  on  $F$  and  $R$  ( where  $F$  is nearer to  $A$ ).\n\nLet  $S$  be the point of intersection of the diagonals of the quadrilateral  $EQFR$ . Let  $T$  be the point of intersection of the diagonals of the quadrilateral  $FRDP$ . Let  $U$  be the point of intersection of the diagonals of the quadrilateral  $DPEQ$ .\n\nShow that the circumcircle to the triangle  $\\triangle{FRT}$ ,  $\\triangle{DPU}$  and  $\\triangle{EQS}$  have a unique point in common.","t":[{"b":2,"e":0.57143,"k":"flat","v":0.26784,"x":0.74107,"p":[[0,31,0.0,0.52899,0.39884,0.0,0.71429,0.85714,0.0,1.0,9,4,2,9,1,1,0,0,1,0,0,1,0,0,2,0,0,2,0,0,11,0,4],[4,31,0.129,0.58035,0.38785,0.21427,0.71429,0.85714,0.0,1.0,8,7,2,8,0,0,0,0,2,0,0,2,0,0,2,0,0,3,0,0,8,0,7],[8,31,0.2581,0.26784,0.37243,0.0,0.0,0.64254,0.0,1.0,19,1,2,19,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,7,0,1],[12,31,0.3871,0.30781,0.37124,0.0,0.07,0.71429,0.0,1.0,16,1,1,16,0,3,0,0,2,0,0,0,0,0,0,0,0,5,0,0,5,0,1],[16,31,0.5161,0.74107,0.19703,0.71429,0.78571,0.85714,0.1429,1.0,0,3,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0,11,0,0,13,0,3],[20,31,0.6452,0.7232,0.17835,0.67857,0.85707,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,3,0,0,3,0,0,7,0,0,17,0,0],[24,31,0.7742,0.71424,0.1713,0.67857,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,11,0,0,12,0,1],[28,31,0.9032,0.7098,0.19393,0.67857,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,3,0,0,2,0,0,9,0,0,14,0,1],[31,31,1.0,0.67852,0.189,0.57132,0.71429,0.85714,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,4,0,0,6,0,0,10,0,0,11,0,0]]},{"b":3,"e":0.0,"k":"volatile","v":0.0,"x":0.375,"p":[[0,3,0.0,0.375,0.41304,0.0,0.07143,0.85714,0.0,1.0,16,2,5,16,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,2],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"934a698560754171","q":"Let  $D$  be the foot of the  $A$ -altitude of an acute triangle  $ABC$ . The internal bisector of the angle  $DAC$  intersects  $BC$  at  $K$ . Let  $L$  be the projection of  $K$  onto  $AC$ . Let  $M$  be the intersection point of  $BL$  and  $AD$ . Let  $P$  be the intersection point of  $MC$  and  $DL$ . Prove that  $PK \\perp AB$ .","t":[{"b":6,"e":1.0,"k":"flat","v":0.87052,"x":0.95089,"p":[[0,25,0.0,0.92856,0.15568,0.96429,1.0,1.0,0.42857,1.0,0,24,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,24],[4,25,0.16,0.87052,0.2953,0.96429,1.0,1.0,0.0,1.0,3,24,1,3,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,24],[8,25,0.32,0.90614,0.2043,0.85714,1.0,1.0,0.14,1.0,0,23,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,23],[12,25,0.48,0.88383,0.21883,0.857,1.0,1.0,0.14,1.0,0,21,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,5,0,0,4,0,21],[16,25,0.64,0.94196,0.08645,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,21],[20,25,0.8,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[24,25,0.96,0.91517,0.17076,0.85714,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,8,0,21],[25,25,1.0,0.90176,0.22144,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,5,0,23]]},{"b":7,"e":1.0,"k":"flat","v":0.83928,"x":1.0,"p":[[0,40,0.0,0.9375,0.16728,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,26],[4,40,0.1,0.83928,0.30462,0.85711,1.0,1.0,0.0,1.0,2,21,1,2,0,2,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,21],[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],[12,40,0.3,0.93308,0.17839,1.0,1.0,1.0,0.42857,1.0,0,28,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,28],[16,40,0.4,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[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],[24,40,0.6,0.96429,0.14286,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[28,40,0.7,0.95982,0.15251,1.0,1.0,1.0,0.14286,1.0,0,28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[32,40,0.8,0.9375,0.19865,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[36,40,0.9,0.97321,0.08329,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,28],[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":"d7eec3572620b9f3","q":"In a triangle  $ABC$ , let  $BD$  be its altitude and let  $H$  be its orthocenter. The perpendicular bisector of of  $HD$  meets  $(BCD)$  at  $P, Q$ . Prove that  $\\angle APB+\\angle AQB=180^{o}$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,30,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,18,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3517ce1d7c90dbfc","q":"Let  $A'\\in(BC),$   $B'\\in(CA),C'\\in(AB)$  be the points of tangency of the excribed circles of triangle  $\\triangle ABC$  with the sides of  $\\triangle ABC.$  Let  $R'$  be the circumradius of triangle  $\\triangle A'B'C'.$  Show that \\[\nR'=\\frac{1}{2r}\\sqrt{2R\\left(2R-h_{a}\\right)\\left(2R-h_{b}\\right)\\left(2R-h_{c}\\right)}\\]\nwhere as usual,  $R$  is the circumradius of  $\\triangle ABC,$  r is the inradius of  $\\triangle ABC,$  and  $h_{a},h_{b},h_{c}$  are the lengths of altitudes of  $\\triangle ABC.$","t":[{"b":2,"e":1.0,"k":"rising","v":0.41072,"x":0.96429,"p":[[0,19,0.0,0.50447,0.31132,0.28571,0.42859,0.71429,0.0,1.0,2,5,0,2,0,5,0,0,5,0,0,5,0,0,4,0,0,4,0,0,2,0,5],[4,19,0.2105,0.41072,0.28291,0.14286,0.35714,0.60714,0.0,1.0,2,3,1,2,0,8,0,0,6,0,0,6,0,0,2,0,0,5,0,0,0,0,3],[8,19,0.4211,0.81248,0.28447,0.82132,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,6,0,18],[12,19,0.6316,0.9375,0.1234,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,24],[16,19,0.8421,0.96429,0.07143,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[19,19,1.0,0.8683,0.18044,0.82143,1.0,1.0,0.4286,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,1,4,0,0,2,0,0,6,0,18]]},{"b":4,"e":0.0,"k":"falling","v":0.04464,"x":0.54911,"p":[[0,31,0.0,0.50429,0.34457,0.14286,0.42859,0.85714,0.0,1.0,2,6,1,2,0,8,0,0,3,0,0,5,0,0,2,0,0,2,0,0,4,0,6],[4,31,0.129,0.54911,0.37306,0.14286,0.49999,1.0,0.0,1.0,2,11,0,2,0,7,0,0,5,0,0,2,0,0,2,0,0,3,0,0,0,0,11],[8,31,0.2581,0.29018,0.26841,0.14286,0.14286,0.32143,0.0,1.0,3,3,1,3,0,15,0,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,3],[12,31,0.3871,0.22768,0.27166,0.0,0.14286,0.28571,0.0,1.0,10,2,0,10,0,11,0,0,5,0,0,2,0,0,0,0,0,2,0,0,0,0,2],[16,31,0.5161,0.0758,0.13351,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,31,0.6452,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6bffa753d6f99e6e","q":"For any two coprime positive integers  $p, q$ , define  $f(i)$  to be the remainder of  $p\\cdot i$  divided by  $q$  for  $i = 1, 2,\\ldots,q -1$ . The number  $i$  is called a**large**number (resp. **small** number) when  $f(i)$  is the maximum (resp. the minimum) among the numbers  $f(1), f(2),\\ldots,f(i)$ . Note that  $1$  is both large and small. Let  $a, b$  be two fixed positive integers. Given that there are exactly  $a$  large numbers and  $b$  small numbers among  $1, 2,\\ldots , q - 1$ , find the least possible number for  $q$ .\n*Proposed by usjl*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.09152,"x":0.14723,"p":[[0,23,0.0,0.09152,0.07625,0.0,0.14286,0.14286,0.0,0.2857,12,0,9,12,1,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.10929,0.07777,0.0,0.14286,0.14286,0.0,0.28571,9,0,7,9,1,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.2857,8,0,5,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,3,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,4,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.14714,0.0249,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.14723,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.07589,"x":0.09598,"p":[[0,5,0.0,0.09598,0.07493,0.0,0.14286,0.14286,0.0,0.28571,11,0,5,11,1,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.07589,0.07973,0.0,0.07143,0.14286,0.0,0.28571,16,0,8,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"80308b8e0e02cf77","q":"Let  $ABC$  be a non-isosceles triangle, let  $AA_1$  be its angle bisector and  $A_2$  be the touching point of the incircle with side  $BC$ . The points  $B_1,B_2,C_1,C_2$  are defined similarly. Let  $O$  and  $I$  be the circumcenter and the incenter of triangle  $ABC$ . Prove that the radical center of the circumcircle of the triangles  $AA_1A_2, BB_1B_2, CC_1C_2$  lies on the line  $OI$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c90c679b80559b07","q":"Let  $ABC$  be a triangle such that  $AB=AC$  and  $\\angle BAC$  is obtuse. Point  $O$  is the circumcenter of triangle  $ABC$ , and  $M$  is the reflection of  $A$  in  $BC$ . Let  $D$  be an arbitrary point on line  $BC$ , such that  $B$  is in between  $D$  and  $C$ . Line  $DM$  cuts the circumcircle of  $ABC$  in  $E,F$ . Circumcircles of triangles  $ADE$  and  $ADF$  cut  $BC$  in  $P,Q$  respectively. Prove that  $DA$  is tangent to the circumcircle of triangle  $OPQ$ .","t":[{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.18747,"p":[[0,12,0.0,0.18747,0.2799,0.0,0.0,0.32144,0.0,0.71429,21,0,0,21,0,0,0,0,3,0,0,1,0,0,2,0,0,5,0,0,0,0,0],[4,12,0.3333,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,2,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.71429,"k":"rising","v":0.09822,"x":0.54904,"p":[[0,17,0.0,0.09822,0.21558,0.0,0.0,0.0,0.0,0.71429,25,0,1,25,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[4,17,0.2353,0.11161,0.24153,0.0,0.0,0.0,0.0,0.71429,26,0,3,26,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0,0,0,0,0],[8,17,0.4706,0.15624,0.27745,0.0,0.0,0.14287,0.0,0.71429,23,0,1,23,0,2,0,0,0,0,0,0,0,0,2,0,0,5,0,0,0,0,0],[12,17,0.7059,0.47312,0.22424,0.28571,0.571,0.60714,0.0,0.71429,3,0,0,3,0,1,0,0,7,0,0,1,0,0,12,0,0,8,0,0,0,0,0],[16,17,0.9412,0.54904,0.1992,0.28571,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,0,0,0,8,0,0,0,0,0,8,0,0,15,0,0,0,0,0],[17,17,1.0,0.5312,0.19637,0.39286,0.57143,0.71429,0.0,0.71429,1,0,0,1,0,1,0,0,6,0,0,2,0,0,10,0,0,12,0,0,0,0,0]]}]},{"i":"815879c37691e151","q":"Let  $ABC$  be a given acute triangle, in which  $BC$  is the longest side. Let  $H$  be the orthocenter of the triangle, and let  $D$  and  $E$  be the feet of the altitudes from  $B$  and  $C$ , respectively. Let  $F$  and  $G$  be the midpoints of sides  $AB$  and  $AC$ , respectively.  $X$  is the point of intersection of lines  $DF$  and  $EG$ . Let  $O_1$  and  $O_2$  be the circumcenters of triangles  $EFX$  and  $DGX$ , respectively. Finally,  $M$  is the midpoint of line segment  $O_1O_2$ . Prove that points  $X, H$  and  $M$  are collinear.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,31,0.0,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,31,0.7742,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[31,31,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00665,"x":0.04009,"p":[[0,36,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.01563,0.04277,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.00665,0.02733,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.04009,0.0816,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3c9ef2eb67f16cd8","q":"Let  $AB=AC$  in  $\\triangle ABC$ , and let  $D$  be a point on segment  $AB$ . The tangent at  $D$  to the circumcircle  $\\omega$  of  $BCD$  hits  $AC$  at  $E$ . The other tangent from  $E$  to  $\\omega$  touches it at  $F$ , and  $G=BF \\cap CD$ ,  $H=AG \\cap BC$ . Prove that  $BH=2HC$ .\n\n*Proposed by David Stoner*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,39,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"4232ff6febb00eaa","q":"Let  $ \\Omega$  be the circumcircle of triangle  $ ABC$ . Let  $ D$  be the point at which the incircle of  $ ABC$  touches its side  $ BC$ . Let  $ M$  be the point on  $ \\Omega$  such that the line  $ AM$  is parallel to  $ BC$ . Also, let  $ P$  be the point at which the circle tangent to the segments  $ AB$  and  $ AC$  and to the circle  $ \\Omega$  touches  $ \\Omega$ . Prove that the points  $ P$ ,  $ D$ ,  $ M$  are collinear.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,19,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,45,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,45,0.1778,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,45,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[45,45,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"574038c722d120db","q":"Let  $ a_0,a_1,\\dots,a_{n \\plus{} 1}$  be natural numbers such that  $ a_0 \\equal{} a_{n \\plus{} 1} \\equal{} 1$ ,  $ a_i>1$  for all  $ 1\\leq i \\leq n$ , and for each  $ 1\\leq j\\leq n$ ,  $ a_i|a_{i \\minus{} 1} \\plus{} a_{i \\plus{} 1}$ . Prove that there exist one  $ 2$  in the sequence.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.26781,"x":0.3481,"p":[[0,9,0.0,0.3481,0.24408,0.1429,0.28571,0.44645,0.0,1.0,4,1,1,4,0,6,0,0,8,0,0,6,0,2,2,0,0,2,0,0,1,0,1],[4,9,0.4444,0.31471,0.17753,0.14289,0.2857,0.42857,0.0,0.71429,3,0,1,3,1,5,0,0,9,0,1,8,0,1,3,0,0,1,0,0,0,0,0],[8,9,0.8889,0.26781,0.16651,0.14286,0.24999,0.42857,0.0,0.57143,4,0,0,4,0,7,0,5,5,0,2,5,0,1,3,0,0,0,0,0,0,0,0],[9,9,1.0,0.30129,0.18091,0.1429,0.2857,0.42857,0.0,0.71429,3,0,0,3,1,6,0,1,9,0,1,6,0,0,4,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.21429,"k":"flat","v":0.22983,"x":0.29452,"p":[[0,16,0.0,0.29452,0.20344,0.14286,0.2857,0.42857,0.0,0.85714,4,0,0,4,1,6,0,2,8,0,1,4,0,0,5,0,0,0,0,0,1,0,0],[4,16,0.25,0.26113,0.17086,0.14286,0.2857,0.42857,0.0,0.57143,5,0,2,5,0,8,0,1,7,0,2,6,0,0,3,0,0,0,0,0,0,0,0],[8,16,0.5,0.29132,0.16365,0.14286,0.24999,0.42857,0.14,0.71429,0,0,0,0,0,13,1,2,5,0,0,7,0,0,3,0,0,1,0,0,0,0,0],[12,16,0.75,0.26991,0.13228,0.14286,0.24999,0.35704,0.14,0.57143,0,0,0,0,0,12,0,4,7,0,2,4,0,1,2,0,0,0,0,0,0,0,0],[16,16,1.0,0.22983,0.1195,0.14286,0.14286,0.30354,0.14,0.4286,0,0,0,0,0,20,0,0,4,0,1,7,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3e50669c02dc1040","q":"In the unit squares of a transparent  $1 \\times 100$  tape, numbers  $1,2,\\cdots,100$  are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a  $1 \\times1$  tape with  $100$  layers.A permutation of the numbers  $1,2,\\cdots,100$  can be seen on the tape, from the top to the bottom.\nProve that the number of possible permutations is between  $2^{100}$  and  $4^{100}$ .\n(*e.g.* We can produce all permutations of numbers  $1,2,3$  with a  $1\\times3$  tape)\n\n*Proposed by Morteza Saghafian*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0267,"x":0.04911,"p":[[0,4,0.0,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.02233,"p":[[0,16,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,3,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.02233,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d14f437e7114d9a6","q":"Let  $ABC$  be a triangle and  $M$  the midpoint of  $AB$ . Let circumcircles of triangles  $CMO$  and  $ABC$  intersect at  $K$  where  $O$  is the circumcenter of  $ABC$ . Let  $P$  be the intersection of lines  $OM$  and  $CK$ . Prove that  $\\angle{PAK} = \\angle{MCB}$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.09367,"p":[[0,22,0.0,0.05353,0.09952,0.0,0.0,0.14071,0.0,0.43,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.09367,0.14552,0.0,0.0,0.14287,0.0,0.71429,18,0,0,18,0,10,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,22,0.3636,0.09151,0.16959,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,1,4,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[12,22,0.5455,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"flat","v":0.08701,"x":0.23652,"p":[[0,30,0.0,0.1607,0.21648,0.0,0.07143,0.2857,0.0,0.71429,16,0,1,16,0,7,0,0,4,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[4,30,0.1333,0.08701,0.12208,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,1,11,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,30,0.2667,0.23652,0.26636,0.0,0.14286,0.28571,0.0,0.85714,9,0,0,9,0,14,0,0,2,0,0,0,0,0,1,0,0,5,0,0,1,0,0],[12,30,0.4,0.10936,0.1703,0.0,0.0,0.14286,0.0,0.71429,17,0,1,17,1,10,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,30,0.5333,0.22981,0.22359,0.14286,0.14286,0.2857,0.0,0.857,5,0,0,5,1,17,0,0,2,0,0,2,0,0,2,0,0,2,0,0,1,0,0],[20,30,0.6667,0.14507,0.19104,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,1,15,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[24,30,0.8,0.09356,0.1162,0.0,0.14,0.14286,0.0,0.571,15,0,0,15,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,30,0.9333,0.183,0.23746,0.0,0.07143,0.2857,0.0,0.857,16,0,0,16,0,5,0,0,4,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[30,30,1.0,0.14499,0.12689,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,1,12,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2023e256c1ff2fff","q":"Let  $ABC$  be an acute scalene triangle with orthocentre  $H{}$  and circumcentre  $O.{}$  Let  $P{}$  be an arbitrary point on the segment  $OH$  and  $O_a$  be the circumcentre of  $PBC.{}$  The line  $PO_a$  intersects the line  $HA$  at  $X_a.{}$  Define  $X_b$  and  $X_c$  similarly. Let  $Q{}$  be the isogonal conjugate of  $P{}$  and  $X{}$  be the circumcentre of  $X_aX_bX_c.{}$  Prove that  $PQ$  and  $HX$  are parallel.\n\n*Proposed by David Anghel*","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04465,"p":[[0,6,0.0,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.04465,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14,"k":"flat","v":0.02679,"x":0.06697,"p":[[0,8,0.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.2857,25,0,1,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.06697,0.09439,0.0,0.0,0.14287,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1c03b2aca95c04dc","q":"Let  $ G=(V,E)$  be a simple graph.\n\na) Let  $ A,B$  be a subsets of  $ E$ , and spanning subgraphs of  $ G$  with edges  $ A,B,A\\cup B$  and  $ A\\cap B$  have  $ a,b,c$  and  $ d$  connected components respectively. Prove that  $ a+b\\leq c+d$ .\n\nWe say that subsets  $ A_1,A_2,\\dots,A_m$  of  $ E$  have  $ (R)$  property if and only if for each  $ I\\subset\\{1,2,\\dots,m\\}$  the spanning subgraph of  $ G$  with edges  $ \\cup_{i\\in I}A_i$  has at most  $ n-|I|$  connected components.\nb) Prove that when  $ A_1,\\dots,A_m,B$  have  $ (R)$  property, and  $ |B|\\geq2$ , there exists an  $ x\\in B$  such that  $ A_1,A_2,\\dots,A_m,B\\backslash\\{x\\}$  also have property  $ (R)$ .\n\nSuppose that edges of  $ G$  are colored arbitrarily. A spanning subtree in  $ G$  is called colorful if and only if it does not have any two edges with the same color.\nc) Prove that  $ G$  has a colorful subtree if and only if for each partition of  $ V$  to  $ k$  non-empty subsets such as  $ V_1,\\dots,V_k$ , there are at least  $ k\\minus{}1$  edges with distinct colors that each of these edges has its two ends in two different  $ V_i$ s.\nd) Assume that edges of  $ K_n$  has been colored such that each color is repeated  $ \\left[\\frac n2\\right]$  times. Prove that there exists a colorful subtree.\ne) Prove that in part d) if  $ n\\geq5$  there is a colorful subtree that is non-isomorphic to  $ K_{1,n-1}$ .\nf) Prove that in part e) there are at least two non-intersecting colorful subtrees.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.0267,"x":0.04902,"p":[[0,7,0.0,0.04902,0.09173,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.01786,"x":0.04018,"p":[[0,19,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.03571,0.07986,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"60edae287015bc10","q":"In an acute-angled triangle  $ABC$ , point  $M$  is the midpoint of side  $BC$  and the centers of the  $M$ - excircles of triangles  $AMB$  and  $AMC$  are  $D$  and  $E$ , respectively. The circumcircle of triangle  $ABD$  intersects line  $BC$  at points  $B$  and  $F$ . The circumcircle of triangle  $ACE$  intersects line  $BC$  at points  $C$  and  $G$ . Prove that  $BF\\hspace{0.25mm} = \\hspace{0.25mm} CG$  .","t":[{"b":3,"e":0.57143,"k":"rising","v":0.11143,"x":0.58482,"p":[[0,29,0.0,0.21429,0.25254,0.0,0.14286,0.28571,0.0,1.0,10,2,4,10,0,9,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[4,29,0.1379,0.16518,0.22618,0.0,0.14286,0.2857,0.0,1.0,13,1,5,13,0,10,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[8,29,0.2759,0.16964,0.14032,0.0,0.14286,0.28571,0.0,0.57143,9,0,1,9,0,11,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,29,0.4138,0.11143,0.06894,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.26764,0.28068,0.14,0.14286,0.46418,0.0,1.0,7,1,0,7,0,15,0,0,1,0,0,1,0,0,3,0,0,3,0,0,1,0,1],[20,29,0.6897,0.25892,0.27763,0.14286,0.14286,0.17868,0.0,1.0,4,1,0,4,0,20,0,0,2,0,0,0,0,0,1,0,0,1,0,0,3,0,1],[24,29,0.8276,0.41962,0.30079,0.14286,0.28571,0.71429,0.14286,1.0,0,2,0,0,0,14,0,0,3,0,0,3,0,0,3,0,0,3,0,0,4,0,2],[28,29,0.9655,0.45535,0.3163,0.14286,0.42857,0.74996,0.0,1.0,2,4,0,2,0,8,0,0,3,0,0,9,0,0,1,0,0,1,0,0,4,0,4],[29,29,1.0,0.58482,0.25092,0.42857,0.42857,0.75,0.14286,1.0,0,6,0,0,0,1,0,0,2,0,0,16,0,0,1,0,0,4,0,0,2,0,6]]},{"b":5,"e":0.28571,"k":"flat","v":0.16518,"x":0.29017,"p":[[0,13,0.0,0.16955,0.18014,0.0,0.14286,0.17857,0.0,0.71429,10,0,3,10,0,14,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[4,13,0.3077,0.16518,0.12931,0.0,0.21428,0.28571,0.0,0.28571,11,0,6,11,0,5,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.26785,0.09279,0.2857,0.28571,0.28571,0.0,0.42857,3,0,3,3,0,0,0,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.28572,5e-05,0.2857,0.28571,0.28571,0.2857,0.286,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cbb431eeb2387944","q":"Given are $2 n-1$ two-element subsets of the set $\\{1,2, \\ldots, n\\}$. Prove that one can choose $n$ of these subsets whose union contains no more than $\\frac{2}{3} n+1$ elements.\n\n(Dushan Dukic)","t":[{"b":3,"e":0.0,"k":"flat","v":0.02679,"x":0.10268,"p":[[0,11,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,11,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.09822,0.12078,0.0,0.0,0.14287,0.0,0.42857,17,0,5,17,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.10268,0.11426,0.0,0.14286,0.14286,0.0,0.42857,14,0,3,14,0,15,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.04009,"x":0.08473,"p":[[0,19,0.0,0.05786,0.08628,0.0,0.0,0.14286,0.0,0.2857,21,0,4,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.2857,23,0,6,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.14286,23,0,2,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0625,0.18189,0.0,0.0,0.0,0.0,1.0,25,1,6,25,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,19,0.8421,0.08473,0.11765,0.0,0.0,0.14286,0.0,0.4286,19,0,2,19,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0e9b7c39a8e03ced","q":"In triangle  $ABC$ , points  $E$  and  $F$  are the feet in altitudes from  $ B$  and  $C$  respectively. Let  $D$  be a point such that  $ABCD$  is a parallelogram with  $A$  and  $D$  in different half planes with respect to the line  $BC$ , and  $T$  a point such that  $AEFT$  is a parallelogram with  $T$  and  $A$  in different half-planes with respect to the line  $EF$ . Prove that  $T, D$ , and the orthocenter of the triangle  $ABC$  are collinear.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,19,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"02cad0d588e154fe","q":"In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city  $A$  to city  $B$ , from city  $B$  to city  $C$ , and from city  $C$  to city  $A$ . Prove that the kings order was followed incorrectly.","t":[{"b":0,"e":0.71429,"k":"rising","v":0.0692,"x":0.29228,"p":[[0,15,0.0,0.10705,0.15566,0.0,0.0,0.14287,0.0,0.57143,19,0,2,19,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0692,0.1467,0.0,0.0,0.08929,0.0,0.71429,23,0,2,23,1,4,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,15,0.5333,0.08036,0.12339,0.0,0.0,0.14286,0.0,0.42857,21,0,5,21,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.10272,0.16466,0.0,0.0,0.2857,0.0,0.57143,22,0,3,22,0,1,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[15,15,1.0,0.29228,0.24247,0.0,0.2857,0.51775,0.0,0.71429,9,0,0,9,0,4,0,0,7,0,0,3,0,1,5,0,0,3,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02679,"x":0.1116,"p":[[0,5,0.0,0.1116,0.17029,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,3,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,5,0.8,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,5,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"175708ab09621825","q":"Lavaman versus the Flea. Let $A, B$, and $F$ be positive integers, and assume $A<B<2 A$. A flea is at the number 0 on the number line. The flea can move by jumping to the right by $A$ or by $B$. Before the flea starts jumping, Lavaman chooses finitely many intervals $\\{m+$ $1, m+2, \\ldots, m+A\\}$ consisting of $A$ consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that:\n\n(i) any two distinct intervals are disjoint and not adjacent;\n\n(ii) there are at least $F$ positive integers with no lava between any two intervals; and\n\n(iii) no lava is placed at any integer less than $F$.\n\nProve that the smallest $F$ for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does,\nis $F=(n-1) A+B$, where $n$ is the positive integer such that $\\frac{A}{n+1} \\leq B-A<\\frac{A}{n}$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.03563,"x":0.12947,"p":[[0,11,0.0,0.08027,0.09401,0.0,0.0,0.14286,0.0,0.28571,17,0,6,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,6,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.12947,0.13997,0.0,0.14286,0.1429,0.0,0.57143,13,0,0,13,0,12,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[11,11,1.0,0.125,0.10564,0.0,0.14286,0.14292,0.0,0.28571,11,0,0,11,0,14,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.04447,"x":0.11606,"p":[[0,23,0.0,0.10259,0.10245,0.0,0.14286,0.14287,0.0,0.28571,14,0,6,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.11606,0.13087,0.0,0.14286,0.14286,0.0,0.571,14,0,4,14,0,12,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,23,0.3478,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.04447,0.075,0.0,0.0,0.14,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.04464,0.0974,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.05799,0.09363,0.0,0.0,0.14286,0.0,0.43,21,0,0,21,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"70ded5eaa51d2434","q":"Given are positive integers  $ n>1$  and  $ a$  so that  $ a>n^2$ , and among the integers  $ a\\plus{}1, a\\plus{}2, \\ldots, a\\plus{}n$  one can find a multiple of each of the numbers  $ n^2\\plus{}1, n^2\\plus{}2, \\ldots, n^2\\plus{}n$ . Prove that  $ a>n^4\\minus{}n^3$ .","t":[{"b":6,"e":0.0,"k":"flat","v":0.04464,"x":0.11161,"p":[[0,7,0.0,0.04911,0.13175,0.0,0.0,0.0,0.0,0.42857,28,0,5,28,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.11161,0.14166,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,10,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[7,7,1.0,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.714,"k":"flat","v":0.0,"x":0.12053,"p":[[0,30,0.0,0.04464,0.1357,0.0,0.0,0.0,0.0,0.71429,27,0,4,27,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,30,0.1333,0.07589,0.18205,0.0,0.0,0.0,0.0,0.85714,25,0,3,25,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.04911,0.13175,0.0,0.0,0.0,0.0,0.57143,27,0,1,27,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,30,0.5333,0.03571,0.10714,0.0,0.0,0.0,0.0,0.42857,28,0,2,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.05357,0.14617,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,30,0.8,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.05795,0.10005,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.12053,0.11355,0.0,0.14286,0.1786,0.0,0.28571,13,0,0,13,0,11,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0b5577dc1a3b94a1","q":"In isoceles  $\\triangle ABC$ ,  $AB=AC$ ,  $I$  is its incenter,  $D$  is a point inside  $\\triangle ABC$  such that  $I,B,C,D$  are concyclic. The line through  $C$  parallel to  $BD$  meets  $AD$  at  $E$ . Prove that  $CD^2=BD\\cdot CE$ .","t":[{"b":3,"e":0.85714,"k":"volatile","v":0.18304,"x":0.80801,"p":[[0,8,0.0,0.22313,0.30083,0.0,0.14286,0.28571,0.0,1.0,14,2,13,14,0,8,0,0,3,0,0,1,0,0,2,0,0,1,0,0,1,0,2],[4,8,0.5,0.18304,0.24545,0.0,0.0,0.28571,0.0,0.71429,17,0,17,17,0,3,0,0,7,0,0,0,0,0,1,0,0,4,0,0,0,0,0],[8,8,1.0,0.80801,0.24122,0.71429,0.85714,1.0,0.0,1.0,1,13,0,1,0,0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,9,0,13]]},{"b":5,"e":0.0,"k":"falling","v":0.04009,"x":0.2543,"p":[[0,31,0.0,0.20982,0.28568,0.0,0.0,0.32143,0.0,1.0,17,1,10,17,0,4,0,0,3,0,0,1,0,0,3,0,0,3,0,0,0,0,1],[4,31,0.129,0.25,0.27664,0.0,0.14286,0.32144,0.0,1.0,11,1,2,11,0,8,0,0,5,0,0,1,0,0,2,0,0,4,0,0,0,0,1],[8,31,0.2581,0.2543,0.29586,0.0,0.14286,0.571,0.0,1.0,12,1,5,12,0,9,0,0,2,0,0,0,0,0,3,0,0,5,0,0,0,0,1],[12,31,0.3871,0.11607,0.12079,0.0,0.14286,0.14286,0.0,0.4286,13,0,13,13,0,14,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.04009,0.07337,0.0,0.0,0.035,0.0,0.28571,24,0,24,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3da771ffddd4eeb6","q":"Inside the triangle $A B C$ a point $M$ is given. The line $B M$ meets the side $A C$ at $N$. The point $K$ is symmetrical to $M$ with respect to $A C$. The line $B K$ meets $A C$ at $P$. If $\\angle A M P=\\angle C M N$, prove that $\\angle A B P=\\angle C B N$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.00893,"p":[[0,11,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.0346,0.0,0.0,0.0,0.0,0.143,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,9,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ecb5061bce4f863a","q":"John has a string of paper where $n$ real numbers $a_{i} \\in[0,1]$, for all $i \\in\\{1, \\ldots, n\\}$, are written in a row. Show that for any given $k<n$, he can cut the string of paper into $k$ non-empty pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than 1 .\n\n## Baltic Way\n\nReykjav\u00edk, November 11th - 15th\n\nSolutions","t":[{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.0625,"p":[[0,18,0.0,0.06241,0.11805,0.0,0.0,0.14286,0.0,0.57143,22,0,5,22,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,18,0.2222,0.05802,0.11763,0.0,0.0,0.14286,0.0,0.571,23,0,8,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0625,0.12846,0.0,0.0,0.14286,0.0,0.57143,23,0,3,23,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,18,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,5,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.02224,0.05168,0.0,0.0,0.0,0.0,0.143,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04464,"x":0.06696,"p":[[0,10,0.0,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.28571,22,0,8,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.4286,23,0,6,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.06696,0.15559,0.0,0.0,0.14286,0.0,0.857,22,0,5,22,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[10,10,1.0,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8c95d5288215d0dd","q":"In a simple graph  $G$ , an operation is defined as taking two neighbor vertices  $u,v$  which have a common neighbor, deleting the edge between  $u,v$  and adding a new vertex  $w$  whose neighbors are exactly the common neighbors of  $u$  and  $v$ . Starting with the complete graph  $G=K_n$  where  $n\\ge 3$  is a positive integer, find the maximum number of operations that can be applied.\n\nProposed by*Deniz Can Kara\u00e7elebi*","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,7,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,9,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,7,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.01339,"p":[[0,8,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,6,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,10,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"357c288beef4e9a7","q":"In triangle  $\\triangle ABC$  ,  $M$  is the midpoint of arc  $(BAC)$  and  $N$  is the antipode of  $A$  in  $(ABC)$ . The line through  $B$  perpendicular to  $AM$  , intersects  $AM , (ABC)$  at  $D,P$  respectively and a line through  $D$  perpendicular to  $AC$  , intersects  $BC,AC$  at  $F,E$  respectively. Prove that  $PE,MF,ND$  are concurrent.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,30,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,20,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,20,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cbb8bf0590b4dbfb","q":"Let  $ABC$  be a triangle and let  $M$  be the midpoint of the segment  $BC$ . Let  $X$  be a point on the ray  $AB$  such that  $2 \\angle CXA=\\angle CMA$ . Let  $Y$  be a point on the ray  $AC$  such that  $2 \\angle AYB=\\angle AMB$ . The line  $BC$  intersects the circumcircle of the triangle  $AXY$  at  $P$  and  $Q$ , such that the points  $P, B, C$ , and  $Q$  lie in this order on the line  $BC$ . Prove that  $PB=QC$ .\n\n*Proposed by Dominik Burek, Poland*","t":[{"b":1,"e":0.0,"k":"falling","v":0.04018,"x":0.34801,"p":[[0,22,0.0,0.34801,0.36067,0.0,0.14286,0.71429,0.0,1.0,10,3,0,10,0,9,0,0,0,0,0,1,0,0,2,0,0,5,0,0,2,0,3],[4,22,0.1818,0.21853,0.32121,0.0,0.07,0.21429,0.0,1.0,16,2,0,16,0,8,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,2],[8,22,0.3636,0.17857,0.30093,0.0,0.07143,0.14286,0.0,1.0,16,3,0,16,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[12,22,0.5455,0.21213,0.32036,0.0,0.14286,0.14286,0.0,1.0,14,2,0,14,1,11,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,2],[16,22,0.7273,0.33034,0.38702,0.0,0.14286,0.71429,0.0,1.0,13,5,0,13,0,6,0,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,5],[20,22,0.9091,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.143,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"volatile","v":0.22319,"x":0.80356,"p":[[0,8,0.0,0.22319,0.31928,0.0,0.14286,0.21429,0.0,1.0,15,3,0,15,0,9,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,3],[4,8,0.5,0.47757,0.37227,0.14286,0.42835,0.74996,0.0,1.0,4,6,1,4,0,10,0,0,2,0,0,0,0,0,1,0,0,7,0,0,2,0,6],[8,8,1.0,0.80356,0.25444,0.71429,0.85714,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,1,0,0,1,0,0,3,0,0,4,0,0,6,0,15]]}]},{"i":"35cef5b40177ff5a","q":"In an acute-angled triangle  $ABC$ , we consider the feet  $H_a$  and  $H_b$  of the altitudes from  $A$  and  $B$ , and the intersections  $W_a$  and  $W_b$  of the angle bisectors from  $A$  and  $B$  with the opposite sides  $BC$  and  $CA$  respectively. Show that the centre of the incircle  $I$  of triangle  $ABC$  lies on the segment  $H_aH_b$  if and only if the centre of the circumcircle  $O$  of triangle  $ABC$  lies on the segment  $W_aW_b$ .","t":[{"b":0,"e":0.71429,"k":"flat","v":0.72318,"x":0.82588,"p":[[0,19,0.0,0.82588,0.27604,0.71429,1.0,1.0,0.0,1.0,2,19,2,2,0,0,0,0,0,0,0,2,0,0,2,0,0,4,0,0,3,0,19],[4,19,0.2105,0.75889,0.33013,0.67836,0.92857,1.0,0.0,1.0,4,16,4,4,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,3,0,16],[8,19,0.4211,0.78555,0.28127,0.71321,0.92857,1.0,0.0,1.0,2,16,2,2,0,0,0,0,1,0,0,1,0,0,3,0,0,7,0,0,2,0,16],[12,19,0.6316,0.72318,0.35164,0.571,0.85714,1.0,0.0,1.0,5,13,5,5,0,0,0,0,0,0,0,2,0,0,2,0,0,3,0,0,7,0,13],[16,19,0.8421,0.73658,0.16016,0.71429,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,14,0,0,9,0,3],[19,19,1.0,0.75888,0.15337,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,13,0,0,10,0,4]]},{"b":6,"e":1.0,"k":"flat","v":0.85266,"x":1.0,"p":[[0,32,0.0,0.85266,0.2487,0.71429,1.0,1.0,0.0,1.0,1,21,1,1,0,0,0,0,1,0,0,1,0,0,3,0,0,3,0,0,2,0,21],[4,32,0.125,0.88393,0.29329,1.0,1.0,1.0,0.0,1.0,3,26,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,26],[8,32,0.25,0.96875,0.13236,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,30],[12,32,0.375,0.95534,0.1871,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[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],[20,32,0.625,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[24,32,0.75,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[28,32,0.875,0.97321,0.10972,1.0,1.0,1.0,0.42857,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,30],[32,32,1.0,0.96874,0.12238,1.0,1.0,1.0,0.4286,1.0,0,30,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,30]]}]},{"i":"38d584f5ce224607","q":"Let  $ABC$   be a scalene triangle. Let  $\\ell$  be a line passing through point  $B$  that lies outside of the triangle  $ABC$  and creates different angles with sides  $AB$  and  $BC$  . The point  $M$  is the midpoint of side  $AC$  and the ponts  $H_a$  and  $H_c$  are the bases of the perpendicular lines on the line  $\\ell$  drawn from points  $A$  and  $C$  respectively. The circle circumscribing triangle  $MBH_a$  intersects AB at the point  $A_1$  and the circumscribed circle of triangle  $MBH_c$  intersects  $BC$  at point  $C_1$ . The point  $A_2$  is symmetric to the point  $A$  relative to the point  $A_1$  and the point  $C_2$  is symmetric to the point  $C_1$  relative to the point  $C_1$ . Prove that the lines  $\\ell$ ,  $AC_2$  and  $CA_2$  intersect at one point. \n\n(Yana Kolodach)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,28,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.03125,0.10555,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,28,0.5714,0.02231,0.10163,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,28,0.7143,0.03571,0.15567,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,28,0.8571,0.03563,0.13356,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,28,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.01339,"x":0.07143,"p":[[0,18,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.03116,0.08541,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c60fc4c50860cb90","q":"In an acute-angled triangle  $ABC$ ,  $D$  and  $E$  are interior points of side  $BC$ , and  $D$  lies between  $B$  and  $E$ ,  $|AD| = |CD|$  and  $|AE| = |BE|$ . Suppose that the bisector of angle  $\\angle DAE$  has a single point in common with the peprendicular bisector of segment  $BC$ , which we will denote by  $F$ . Prove the equality  $|\\angle BAC| +|\\angle  DFE| = 180^o.$","t":[{"b":3,"e":0.0,"k":"falling","v":0.01786,"x":0.30807,"p":[[0,36,0.0,0.30807,0.21163,0.14286,0.35714,0.4286,0.0,0.71429,7,0,0,7,0,4,0,0,5,0,0,10,0,0,5,0,0,1,0,0,0,0,0],[4,36,0.1111,0.22304,0.22576,0.0,0.14143,0.42857,0.0,0.71429,14,0,1,14,0,3,0,0,2,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[8,36,0.2222,0.24997,0.1988,0.10714,0.2143,0.42857,0.0,0.57143,8,0,0,8,0,8,0,0,4,0,0,8,0,0,4,0,0,0,0,0,0,0,0],[12,36,0.3333,0.27223,0.23522,0.0,0.28571,0.42857,0.0,0.85714,10,0,0,10,0,4,0,0,5,0,0,7,0,0,5,0,0,0,0,0,1,0,0],[16,36,0.4444,0.21874,0.21122,0.0,0.14286,0.42857,0.0,0.71429,11,0,0,11,0,7,0,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[20,36,0.5556,0.22766,0.22546,0.0,0.21428,0.32143,0.0,0.71429,12,0,0,12,0,4,0,0,8,0,0,3,0,0,3,0,0,2,0,0,0,0,0],[24,36,0.6667,0.14286,0.21428,0.0,0.0,0.2857,0.0,0.71429,20,0,0,20,0,2,0,0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,0],[28,36,0.7778,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.05357,0.16656,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[36,36,1.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"falling","v":0.15616,"x":0.37492,"p":[[0,13,0.0,0.37492,0.21063,0.25,0.42857,0.46431,0.0,1.0,2,1,0,2,0,6,0,0,6,0,0,10,0,0,6,0,0,1,0,0,0,0,1],[4,13,0.3077,0.26786,0.1948,0.14286,0.2857,0.42857,0.0,0.71429,7,0,2,7,0,5,0,0,10,0,0,6,0,0,3,0,0,1,0,0,0,0,0],[8,13,0.6154,0.17402,0.09935,0.14286,0.14286,0.2857,0.0,0.42857,4,0,0,4,0,18,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.15616,0.07458,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.18732,0.0907,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,19,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4cce4d09a54979c5","q":"In a triangle  $A_1A_2A_3$ , the excribed circles corresponding to sides  $A_2A_3$ ,  $A_3A_1$ ,  $A_1A_2$  touch these sides at  $T_1$ ,  $T_2$ ,  $T_3$ , respectively. If  $H_1$ ,  $H_2$ ,  $H_3$  are the orthocenters of triangles  $A_1T_2T_3$ ,  $A_2T_3T_1$ ,  $A_3T_1T_2$ , respectively, prove that lines  $H_1T_1$ ,  $H_2T_2$ ,  $H_3T_3$  are concurrent.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,23,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,11,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,9,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,14,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.07125,"p":[[0,34,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,10,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.28571,22,0,1,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.07125,0.0797,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"22ba44deff445f1f","q":"Let  $ABC$  be a triangle and let  $P$  be a point on  $BC$ . Points  $M$  and  $N$  lie on  $AB$  and  $AC$ , respectively such that  $MN$  is not parallel to  $BC$  and  $AMP N$  is a parallelogram. Line  $MN$  meets the circumcircle of  $ABC$  at  $R$  and  $S$ . Prove that the circumcircle of triangle  $RP S$  is tangent to  $BC$ .","t":[{"b":2,"e":0.42857,"k":"flat","v":0.03571,"x":0.42411,"p":[[0,30,0.0,0.26338,0.26751,0.0,0.2857,0.42858,0.0,0.85714,12,0,0,12,0,3,0,0,8,0,0,2,0,0,2,0,0,4,0,0,1,0,0],[4,30,0.1333,0.22768,0.21975,0.0,0.2857,0.32143,0.0,0.71429,12,0,0,12,0,3,0,0,9,0,0,4,0,0,2,0,0,2,0,0,0,0,0],[8,30,0.2667,0.21429,0.23958,0.0,0.14286,0.42857,0.0,0.71429,15,0,0,15,0,2,0,0,6,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[12,30,0.4,0.04464,0.14914,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,30,0.5333,0.03572,0.10715,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.03571,0.14286,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[24,30,0.8,0.42411,0.20666,0.28571,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,1,0,0,12,0,0,5,0,0,5,0,0,7,0,0,0,0,0],[28,30,0.9333,0.36156,0.15161,0.28571,0.28571,0.42858,0.14,0.71429,0,0,0,0,0,4,0,0,14,0,0,10,0,0,1,0,0,3,0,0,0,0,0],[30,30,1.0,0.36606,0.15945,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,4,0,0,15,0,0,7,0,0,3,0,0,3,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.08036,"x":0.19174,"p":[[0,16,0.0,0.19174,0.26368,0.0,0.0,0.32143,0.0,0.71429,18,0,0,18,0,3,0,0,3,0,0,3,0,0,0,0,0,5,0,0,0,0,0],[4,16,0.25,0.11607,0.19704,0.0,0.0,0.17857,0.0,0.71429,22,0,0,22,0,2,0,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[8,16,0.5,0.15622,0.19013,0.0,0.0,0.28571,0.0,0.571,18,0,0,18,0,0,0,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,16,0.75,0.08036,0.15126,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,6,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[16,16,1.0,0.11598,0.18362,0.0,0.0,0.14286,0.0,0.71429,19,0,0,19,0,7,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"460adbfa26f54e12","q":"In scalene triangle $A B C$, let the feet of the perpendiculars from $A$ to $\\overline{B C}, B$ to $\\overline{C A}$, $C$ to $\\overline{A B}$ be $A_{1}, B_{1}, C_{1}$, respectively. Denote by $A_{2}$ the intersection of lines $B C$ and $B_{1} C_{1}$. Define $B_{2}$ and $C_{2}$ analogously. Let $D, E, F$ be the respective midpoints of sides $\\overline{B C}, \\overline{C A}, \\overline{A B}$. Show that the perpendiculars from $D$ to $\\overline{A A_{2}}, E$ to $\\overline{B B_{2}}$ and $F$ to $\\overline{C C_{2}}$ are concurrent.","t":[{"b":5,"e":0.85714,"k":"rising","v":0.20088,"x":0.8616,"p":[[0,21,0.0,0.32589,0.42594,0.0,0.0,0.85714,0.0,1.0,18,4,11,18,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,4],[4,21,0.1905,0.20088,0.32704,0.0,0.0,0.24992,0.0,0.85714,21,0,14,21,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,5,0,0],[8,21,0.381,0.58486,0.31822,0.24999,0.71429,0.85714,0.0,0.86,3,0,0,3,0,5,0,0,1,0,0,0,0,0,6,0,0,2,0,0,15,0,0],[12,21,0.5714,0.85714,2e-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],[16,21,0.7619,0.85714,3e-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],[20,21,0.9524,0.8616,0.02486,0.85714,0.85714,0.85714,0.857,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,1],[21,21,1.0,0.85714,2e-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]]},{"b":7,"e":0.857,"k":"rising","v":0.34366,"x":0.85714,"p":[[0,16,0.0,0.44196,0.41398,0.0,0.21428,0.85714,0.0,1.0,11,2,8,11,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,2],[4,16,0.25,0.34366,0.42241,0.0,0.0,0.85714,0.0,1.0,17,5,13,17,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,5,0,5],[8,16,0.5,0.5714,0.40406,0.10714,0.85714,0.85714,0.0,1.0,8,6,3,8,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0,12,0,6],[12,16,0.75,0.85267,0.06667,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,25,0,3],[16,16,1.0,0.85714,0.06186,0.85714,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,26,0,3]]}]},{"i":"470dadee65f8658a","q":"Let  $AA_1, BB_1, CC_1$  be the heights of triangle  $ABC$  and  $H$  be its orthocenter. Liune  $\\ell$  parallel to  $AC$ , intersects straight lines  $AA_1$  and  $CC_1$  at points  $A_2$  and  $C_2$ , respectively. Suppose that point  $B_1$  lies outside the circumscribed circle of triangle  $A_2 HC_2$ . Let  $B_1P$  and  $B_1T$  be tangent to of this circle. Prove that points  $A_1, C_1, P$ , and  $T$  are cyclic.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,47,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"95903360516683f2","q":"Given a triangle  $ABC$  inscribed in circle  $(O)$  and let  $P$  be a point on the interior angle bisector of  $BAC$ .  $PB$ ,  $PC$  cut  $CA$ ,  $AB$  at  $E,F$  respectively. Let  $EF$  meet  $(O)$  at  $M,N$ . The line that is perpendicular to  $PM$ ,  $PN$  at  $M,N$  respectively intersect  $(O)$  at  $S, T$  different from  $M,N$ . Prove that  $ST \\parallel BC$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,35,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,6,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"11b60b04c3d5d0cb","q":"In a party, there are  $2n + 1$   people. It's well known that for every group of  $n$  people, there exist a person(out of the group) who knows all them(the  $n$  people of the group). Show that there exist a person who knows all the people in the party.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,17,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.03125,0.12234,0.0,0.0,0.0,0.0,0.57143,30,0,1,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2aee73a390166071","q":"In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most  $100$  cities at distance exactly three from it. Prove that  there is no city such that more than  $2550$  other cities have distance exactly four from it.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.08928,"x":0.12723,"p":[[0,14,0.0,0.11159,0.21643,0.0,0.0,0.0,0.0,0.71429,25,0,24,25,0,0,0,0,0,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[4,14,0.2857,0.08928,0.19804,0.0,0.0,0.0,0.0,0.57143,26,0,25,26,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[8,14,0.5714,0.1183,0.21622,0.0,0.0,0.03571,0.0,0.57143,24,0,22,24,0,1,0,0,0,0,0,2,0,1,4,0,0,0,0,0,0,0,0],[12,14,0.8571,0.12723,0.22213,0.0,0.0,0.10714,0.0,0.57143,24,0,22,24,0,0,0,0,0,0,0,2,0,3,3,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.71429,"k":"rising","v":0.04241,"x":0.54464,"p":[[0,17,0.0,0.15177,0.25173,0.0,0.0,0.42857,0.0,0.78571,23,0,22,23,0,0,0,0,0,0,0,4,0,1,2,0,0,1,1,0,0,0,0],[4,17,0.2353,0.07366,0.17266,0.0,0.0,0.0,0.0,0.57143,27,0,19,27,0,0,0,0,0,0,0,3,0,1,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.04241,0.13226,0.0,0.0,0.0,0.0,0.5,29,0,24,29,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.47322,0.18536,0.42857,0.5,0.57143,0.0,1.0,3,1,2,3,0,0,0,0,0,0,0,11,0,4,13,0,0,0,0,0,0,0,1],[16,17,0.9412,0.54464,0.21484,0.42857,0.57143,0.60714,0.0,1.0,2,2,2,2,0,0,0,0,0,0,0,11,0,2,9,0,0,4,0,0,2,0,2],[17,17,1.0,0.46433,0.23555,0.42857,0.5,0.57143,0.0,1.0,5,1,5,5,0,0,0,0,0,0,0,9,0,4,10,0,0,2,0,0,1,0,1]]}]},{"i":"82a9a81f6f8330d2","q":"Let  $ ABC$  be a triangle with circumcentre  $ O$ . The points  $ P$  and  $ Q$  are interior points of the sides  $ CA$  and  $ AB$  respectively. Let  $ K,L$  and  $ M$  be the midpoints of the segments  $ BP,CQ$  and  $ PQ$ . respectively, and let  $ \\Gamma$  be the circle passing through  $ K,L$  and  $ M$ . Suppose that the line  $ PQ$  is tangent to the circle  $ \\Gamma$ . Prove that  $ OP \\equal{} OQ.$ \r\n\r\n*Proposed by Sergei Berlov, Russia*","t":[{"b":4,"e":0.85714,"k":"volatile","v":0.51784,"x":0.94642,"p":[[0,8,0.0,0.51784,0.3809,0.24999,0.42857,1.0,0.0,1.0,6,9,0,6,0,2,0,0,5,0,0,5,0,0,2,0,0,0,0,0,3,0,9],[4,8,0.5,0.94642,0.06917,0.85714,1.0,1.0,0.857,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[8,8,1.0,0.91964,0.07936,0.85714,0.85714,1.0,0.71429,1.0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,15]]},{"b":7,"e":1.0,"k":"volatile","v":0.22772,"x":0.99554,"p":[[0,16,0.0,0.54909,0.39141,0.24999,0.5712,1.0,0.0,1.0,7,11,1,7,0,1,0,0,3,0,0,4,0,0,4,0,0,1,0,0,1,0,11],[4,16,0.25,0.41061,0.3876,0.0,0.35715,0.71429,0.0,1.0,11,6,1,11,0,4,0,0,1,0,0,1,0,0,4,0,0,5,0,0,0,0,6],[8,16,0.5,0.22772,0.32708,0.0,0.07143,0.32143,0.0,1.0,16,4,3,16,0,5,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[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]]}]},{"i":"dc18bd9ad56cf18a","q":"Let  $ABCD$  be a trapezoid with  $AB\\parallel CD$  and inscribed in a circumference  $\\Gamma$ . Let  $P$  and  $Q$  be two points on segment  $AB$  ( $A$ ,  $P$ ,  $Q$ ,  $B$  appear in that order and are distinct) such that  $AP=QB$ . Let  $E$  and  $F$  be the second intersection points of lines  $CP$  and  $CQ$  with  $\\Gamma$ , respectively. Lines  $AB$  and  $EF$  intersect at  $G$ . Prove that line  $DG$  is tangent to  $\\Gamma$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,34,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.02233,0.05189,0.0,0.0,0.0,0.0,0.143,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.04018,0.0735,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.01786,"p":[[0,6,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a2083f80af4f578e","q":"Let  $ABC$  be a triangle. Let  $ABC_1, BCA_1, CAB_1$  be three equilateral triangles that do not overlap with  $ABC$ .\nLet  $P$  be the intersection of the circumcircles of triangle  $ABC_1$  and  $CAB_1$ .\nLet  $Q$  be the point on the circumcircle of triangle  $CAB_1$  so that  $PQ$  is parallel to  $BA_1$ . Let  $R$  be the point on the circumcircle of triangle  $ABC_1$  so that  $PR$  is parallel to  $CA_1$ .\n\nShow that the line connecting the centroid of triangle  $ABC$  and the centroid of triangle  $PQR$  is parallel to  $BC$ .\n\n*Proposed by usjl*","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,25,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,16,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"7e8212be4a2c273c","q":"Let  $ABC$  be a triangle with circumcircle  $\\omega$  and let  $\\omega_A$  be a circle drawn outside  $ABC$  and tangent to side  $BC$  at  $A_1$  and tangent to  $\\omega$  at  $A_2$ . Let the circles  $\\omega_B$  and  $\\omega_C$  and the points  $B_1, B_2, C_1, C_2$  are defined similarly. Prove that if the lines  $AA_1, BB_1, CC_1$  are concurrent, then the lines  $AA_2, BB_2, CC_2$  are also concurrent.","t":[{"b":2,"e":0.14286,"k":"falling","v":0.25001,"x":0.49999,"p":[[0,8,0.0,0.45981,0.36896,0.14286,0.42857,0.85704,0.0,1.0,7,4,0,7,0,5,0,0,3,0,0,2,0,0,3,0,0,2,0,0,6,0,4],[4,8,0.5,0.49999,0.32927,0.14286,0.4998,0.71429,0.14286,1.0,0,5,0,0,0,12,0,0,2,0,0,2,0,0,2,0,0,7,0,0,2,0,5],[8,8,1.0,0.25001,0.16751,0.14286,0.14286,0.42857,0.14286,0.71429,0,0,0,0,0,21,0,0,2,0,0,7,0,0,0,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.23214,"x":0.54911,"p":[[0,29,0.0,0.54911,0.36962,0.14286,0.57143,0.89286,0.0,1.0,4,8,1,4,0,6,0,0,2,0,0,1,0,0,4,0,0,4,0,0,3,0,8],[4,29,0.1379,0.53124,0.36111,0.2857,0.4998,0.85714,0.0,1.0,6,7,1,6,0,1,0,0,4,0,0,5,0,0,3,0,0,2,0,0,4,0,7],[8,29,0.2759,0.23214,0.28064,0.0,0.14286,0.42857,0.0,1.0,14,1,0,14,0,7,0,0,0,0,0,4,0,0,5,0,0,0,0,0,1,0,1],[12,29,0.4138,0.29909,0.20626,0.14286,0.2857,0.57111,0.0,0.57143,4,0,0,4,0,11,0,0,4,0,0,4,0,0,9,0,0,0,0,0,0,0,0],[16,29,0.5517,0.33479,0.20391,0.14286,0.28571,0.57141,0.0,0.57143,3,0,0,3,0,9,0,0,5,0,0,4,0,0,11,0,0,0,0,0,0,0,0],[20,29,0.6897,0.33916,0.21357,0.14286,0.28571,0.57143,0.0,0.57143,3,0,0,3,0,10,0,0,4,0,0,2,0,0,13,0,0,0,0,0,0,0,0],[24,29,0.8276,0.26338,0.22897,0.14286,0.14286,0.57143,0.0,0.57143,7,0,0,7,0,13,0,0,0,0,0,2,0,0,10,0,0,0,0,0,0,0,0],[28,29,0.9655,0.33024,0.2003,0.14286,0.2857,0.57143,0.0,0.57143,2,0,0,2,0,11,0,0,5,0,0,3,0,0,11,0,0,0,0,0,0,0,0],[29,29,1.0,0.31694,0.21936,0.14286,0.35714,0.57111,0.0,0.57143,6,0,0,6,0,7,0,0,3,0,0,6,0,0,10,0,0,0,0,0,0,0,0]]}]},{"i":"5a527af1ef2c33a8","q":"Let  $ABC$  be a scalene triangle, with circumcircle  $\\omega$  and incentre  $I.{}$  The tangent line at  $C$  to  $\\omega$  intersects the line  $AB$  at  $D.{}$  The angle bisector of  $BDC$  meets  $BI$  at  $P{}$  and  $AI{}$  at  $Q{}.$  Let  $M{}$  be the midpoint of the segment  $PQ.$  Prove that the line  $IM$  passes through the middle of the arc  $ACB$  of  $\\omega.$ *Dana Heuberger*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,26,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,15,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"90838f710a37e114","q":"In the scalene triangle  $ABC$ , $\\angle ACB=60$  and  $\\Omega$  is its cirumcirle.On the bisectors of the angles  $BAC$  and  $CBA$  points  $A^\\prime$ , $B^\\prime$  are chosen respectively such that  $AB^\\prime \\parallel  BC$  and  $BA^\\prime \\parallel  AC$ . $A^\\prime B^\\prime$  intersects with  $\\Omega$  at  $D,E$ .Prove that triangle  $CDE$  is isosceles.(A. Kuznetsov)","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,44,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.00447,"p":[[0,5,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"defb27596750970b","q":"Let  $ n $  be a positive integer. Consider words of length  $n$  composed of letters from the set  $ \\{ M, E, O \\} $ . Let  $ a $  be the number of such words containing an even number (possibly 0) of blocks  $ ME $  and an even number (possibly 0) blocks of  $ MO $  . Similarly let  $ b $  the number of such words containing an odd number of blocks  $ ME $  and an odd number of blocks  $ MO $ . Prove that  $ a>b $ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.13821,"p":[[0,25,0.0,0.13821,0.16553,0.0,0.14,0.2857,0.0,0.71429,15,0,0,15,0,7,0,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,25,0.16,0.0625,0.11258,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0625,0.18189,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,25,0.48,0.125,0.20748,0.0,0.0,0.2857,0.0,1.0,20,1,0,20,0,2,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,25,0.64,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.08482,0.22263,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1]]},{"b":5,"e":0.28571,"k":"flat","v":0.10268,"x":0.29018,"p":[[0,71,0.0,0.11161,0.12745,0.0,0.07143,0.17857,0.0,0.42857,16,0,0,16,0,8,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,71,0.0563,0.16964,0.20341,0.0,0.14286,0.2857,0.0,1.0,14,1,0,14,0,4,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,71,0.1127,0.17857,0.20203,0.0,0.14286,0.28571,0.0,1.0,13,1,0,13,0,4,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,71,0.169,0.13393,0.17835,0.0,0.14286,0.17857,0.0,0.85714,15,0,0,15,0,9,0,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,71,0.2254,0.12491,0.17404,0.0,0.07,0.17857,0.0,0.85714,16,0,0,16,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,71,0.2817,0.10268,0.13474,0.0,0.0,0.2857,0.0,0.28571,20,0,0,20,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,71,0.338,0.12947,0.13055,0.0,0.14286,0.28571,0.0,0.286,15,0,0,15,0,5,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,71,0.3944,0.14732,0.13115,0.0,0.14286,0.28571,0.0,0.28571,13,0,0,13,0,5,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,71,0.4507,0.18741,0.12598,0.0,0.2857,0.28571,0.0,0.28571,9,0,0,9,0,4,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,71,0.507,0.13393,0.14698,0.0,0.07143,0.28571,0.0,0.42857,16,0,0,16,0,4,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,71,0.5634,0.14286,0.16366,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,4,0,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[44,71,0.6197,0.21875,0.14719,0.0,0.28571,0.28571,0.0,0.42857,9,0,0,9,0,1,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[48,71,0.6761,0.17857,0.18898,0.0,0.14286,0.28571,0.0,0.85714,13,0,0,13,0,4,0,0,12,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[52,71,0.7324,0.23214,0.10564,0.2857,0.28571,0.28571,0.0,0.28571,5,0,0,5,0,2,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,71,0.7887,0.26785,0.05922,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.27232,0.07457,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,0,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.26339,0.0724,0.28571,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,4,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[68,71,0.9577,0.29018,0.07563,0.28571,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,1,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.22321,0.10677,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,4,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c6d4b4f5fb1e6c4d","q":"Let  $ABC$  be a triangle and  $\\Omega$  its circumcircle. Let the internal angle bisectors of  $\\angle BAC, \\angle ABC, \\angle BCA$  intersect  $BC,CA,AB$  on  $D,E,F$ , respectively. The perpedincular line to  $EF$  through  $D$  intersects  $EF$  on  $X$  and  $AD$  intersects  $EF$  on  $Z$ . The circle internally tangent to  $\\Omega$  and tangent to  $AB,AC$  touches  $\\Omega$  on  $Y$ . Prove that  $(XYZ)$  is tangent to  $\\Omega$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,46,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"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],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"cd2a8490f6039856","q":"Let  $ABC$  be a triangle with orthocenter  $H$ . Let  $P$  be any point of the plane of the triangle. Let  $\\Omega$  be the circle with the diameter  $AP$  . The circle  $\\Omega$   cuts  $CA$  and  $AB$  again at  $E$  and  $F$  , respectively. The line  $PH$  cuts  $\\Omega$   again at  $G$ . The tangent lines to  $\\Omega$   at  $E, F$  intersect at  $T$ . Let  $M$  be the midpoint of  $BC$  and  $L$  be the point on  $MG$  such that  $AL$  and  $MT$  are parallel. Prove that  $LA$  and  $LH$  are orthogonal.\n\nL\u00ea Ph\u00fac L\u1eef","t":[{"b":2,"e":0.14286,"k":"flat","v":0.07589,"x":0.16062,"p":[[0,24,0.0,0.125,0.11709,0.0,0.14286,0.14287,0.0,0.42857,12,0,0,12,0,13,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.14286,0.09449,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,21,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.12947,0.09689,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,20,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.11161,0.10555,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.07589,0.10092,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,12,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.16045,0.04735,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.16062,0.04728,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.10259,"x":0.125,"p":[[0,6,0.0,0.10259,0.09603,0.0,0.14286,0.14286,0.0,0.4286,12,0,0,12,0,18,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.10268,0.10248,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.125,0.08564,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,23,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"233eb7b9bfd00bf8","q":"In parallellogram  $ABCD$ , on the arc  $BC$  of the circumcircle  $(ABC)$ , not containing the point  $A$ , we take a point  $P$  and on the  $[AC$ , we take a point  $Q$  such that  $\\angle PBC= \\angle CDQ$ . Prove that  $(APQ)$  is tangent to  $AB$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.01777,"x":0.13384,"p":[[0,17,0.0,0.09366,0.11629,0.0,0.0,0.14286,0.0,0.42857,17,0,10,17,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.07143,0.13832,0.0,0.0,0.14286,0.0,0.57143,23,0,11,23,0,5,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,11,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.12937,0.04161,0.14286,0.14286,0.14286,0.0,0.14286,3,0,1,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,1,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.08929,"p":[[0,14,0.0,0.05339,0.11138,0.0,0.0,0.14,0.0,0.57143,23,0,10,23,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,14,0.2857,0.06696,0.08737,0.0,0.0,0.14286,0.0,0.2857,19,0,9,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.05795,0.08637,0.0,0.0,0.14286,0.0,0.28571,21,0,11,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.28571,14,0,5,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e572286f391e137a","q":"Let  $ I $  be the incenter of a triangle  $ ABC $  with  $ AB \\neq AC $ , and let  $ M $  be the midpoint of the arc  $ BAC $  of the circumcircle of the triangle. The perpendicular line to  $ AI $  passing through  $ I $  intersects line  $ BC $  at point  $ D $ . The line  $ MI $  intersects the circumcircle of triangle  $ BIC $  at point  $ N $ . Prove that line  $ DN $  is tangent to the circumcircle of triangle  $ BIC $ .","t":[{"b":3,"e":0.28571,"k":"flat","v":0.25893,"x":0.29454,"p":[[0,19,0.0,0.29454,0.23133,0.14214,0.28571,0.42858,0.0,0.85714,7,0,4,7,0,5,0,0,9,0,0,5,0,0,3,0,0,2,0,0,1,0,0],[4,19,0.2105,0.28099,0.21883,0.14286,0.14295,0.28571,0.14,1.0,0,2,0,0,0,17,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,2],[8,19,0.4211,0.25893,0.09061,0.14289,0.2857,0.28571,0.14286,0.57143,0,0,0,0,0,9,0,0,21,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,19,0.6316,0.27678,0.10062,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,6,0,0,24,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,19,0.8421,0.25893,0.05576,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.26339,0.05187,0.2857,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.19196,"x":0.33906,"p":[[0,20,0.0,0.30808,0.26029,0.14286,0.28571,0.42857,0.0,1.0,6,2,3,6,0,5,0,0,12,0,0,4,0,0,2,0,0,0,0,0,1,0,2],[4,20,0.2,0.20982,0.18893,0.14286,0.14286,0.28571,0.0,1.0,7,1,4,7,0,10,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[8,20,0.4,0.33906,0.27383,0.14286,0.2857,0.42857,0.0,1.0,3,3,2,3,0,8,0,0,12,0,0,3,0,0,2,0,0,0,0,0,1,0,3],[12,20,0.6,0.29017,0.27311,0.14286,0.2143,0.32143,0.0,1.0,5,3,5,5,0,11,0,0,8,0,0,3,0,0,2,0,0,0,0,0,0,0,3],[16,20,0.8,0.19196,0.11071,0.14286,0.14286,0.2857,0.0,0.42857,4,0,4,4,0,15,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.25893,0.08328,0.14289,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,9,0,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"610330e92a3e920c","q":"In the  $n \\times n$  table in every cell there is one child. Every child looks in neigbour cell. So every child sees ear or back of the head of neighbour.  What is minimal number children, that see  ear ?","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,13,0.0,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,18,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.02679,"x":0.12053,"p":[[0,50,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,18,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.06687,0.10699,0.0,0.0,0.14286,0.0,0.28571,22,0,8,22,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.09821,0.12595,0.0,0.0,0.2857,0.0,0.28571,19,0,1,19,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.12053,0.13415,0.0,0.0,0.2857,0.0,0.28571,17,0,0,17,0,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.07589,0.11836,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.28571,20,0,2,20,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.12053,0.13415,0.0,0.0,0.2857,0.0,0.28571,17,0,1,17,0,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,2,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.28571,22,0,1,22,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.05348,0.09934,0.0,0.0,0.035,0.0,0.28571,24,0,1,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[50,50,1.0,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,2,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"21e082327c69e304","q":"Let  $ a $  and  $ b $  be natural numbers with property  $ gcd(a,b)=1 $  . Find the least natural number  $ k $  such that for every natural number  $ r \\ge k $  , there exist natural numbers  $ m,n >1 $  in such a way that the number  $ m^a n^b $  has exactly  $ r+1 $  positive divisors.","t":[{"b":3,"e":1.0,"k":"flat","v":0.80802,"x":0.92857,"p":[[0,9,0.0,0.80802,0.29367,0.71429,1.0,1.0,0.0,1.0,3,18,2,3,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,1,0,18],[4,9,0.4444,0.92187,0.12798,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,7,1,20],[8,9,0.8889,0.92857,0.12877,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0,22],[9,9,1.0,0.88839,0.13709,0.71429,1.0,1.0,0.57143,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,4,0,18]]},{"b":6,"e":0.71429,"k":"flat","v":0.6607,"x":0.79015,"p":[[0,9,0.0,0.79015,0.16364,0.71429,0.71429,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,14,0,0,3,0,10],[4,9,0.4444,0.67634,0.09447,0.71429,0.71429,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,1,0,26,0,0,0,0,0],[8,9,0.8889,0.68749,0.09063,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,24,0,0,0,0,1],[9,9,1.0,0.6607,0.0928,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,23,0,0,0,0,0]]}]},{"i":"0d7b681236e81150","q":"In the acute triangle $A B C$, the bisectors of $\\angle A, \\angle B$ and $\\angle C$ intersect the circumcircle again in $A_{1}, B_{1}$ and $C_{1}$, respectively. Let $M$ be the point of intersection of $A B$ and $B_{1} C_{1}$, and let $N$ be the point of intersection of $B C$ and $A_{1} B_{1}$. Prove that $M N$ passes through the incentre of triangle $A B C$.","t":[{"b":0,"e":0.0,"k":"rising","v":0.36161,"x":0.66961,"p":[[0,29,0.0,0.40177,0.43365,0.0,0.28571,1.0,0.0,1.0,15,9,4,15,0,0,0,0,4,0,0,0,0,0,1,0,0,3,0,0,0,0,9],[4,29,0.1379,0.52229,0.40344,0.0,0.57143,1.0,0.0,1.0,9,10,6,9,0,1,0,0,3,0,0,0,0,0,5,0,0,4,0,0,0,0,10],[8,29,0.2759,0.48213,0.34022,0.21427,0.57143,0.71429,0.0,1.0,8,4,6,8,0,0,0,0,4,0,0,1,0,0,8,0,0,5,0,0,2,0,4],[12,29,0.4138,0.47318,0.32622,0.21427,0.57143,0.71429,0.0,1.0,8,2,4,8,0,0,0,0,4,0,0,1,0,0,7,0,0,7,0,0,3,0,2],[16,29,0.5517,0.36161,0.37794,0.0,0.21429,0.71429,0.0,1.0,16,2,7,16,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,0,2,0,2],[20,29,0.6897,0.41511,0.31,0.0,0.4998,0.71429,0.0,1.0,9,1,0,9,0,1,0,0,3,0,0,3,0,0,6,0,0,8,0,0,1,0,1],[24,29,0.8276,0.4553,0.23536,0.28571,0.4998,0.71429,0.0,0.7143,4,0,0,4,0,1,0,0,5,0,0,6,0,0,7,0,0,9,0,0,0,0,0],[28,29,0.9655,0.6205,0.28033,0.571,0.71429,0.71429,0.0,1.0,2,4,0,2,0,3,0,0,2,0,0,0,0,0,3,0,0,16,0,0,2,0,4],[29,29,1.0,0.66961,0.29545,0.67836,0.71429,0.85704,0.0,1.0,4,6,0,4,0,0,0,0,1,0,0,0,0,0,3,0,0,14,0,0,4,0,6]]},{"b":3,"e":1.0,"k":"volatile","v":0.47319,"x":0.98661,"p":[[0,10,0.0,0.47319,0.3787,0.0,0.57143,0.71429,0.0,1.0,10,6,4,10,0,1,0,0,2,0,0,0,0,0,7,0,0,5,0,0,1,0,6],[4,10,0.4,0.49552,0.37794,0.21427,0.4998,0.89286,0.0,1.0,8,8,4,8,0,0,0,0,6,0,0,2,0,0,4,0,0,3,0,0,1,0,8],[8,10,0.8,0.97767,0.0808,1.0,1.0,1.0,0.571,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,29],[10,10,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]}]},{"i":"e78bb4f09f43bc14","q":"Let  $ x_{1},x_{2},\\cdots,x_{m},y_{1},y_{2},\\cdots,y_{n}$  be positive real numbers. Denote by  $ X \\equal{} \\sum_{i \\equal{} 1}^{m}x,Y \\equal{} \\sum_{j \\equal{} 1}^{n}y.$  Prove that  $ 2XY\\sum_{i \\equal{} 1}^{m}\\sum_{j \\equal{} 1}^{n}|x_{i} \\minus{} y_{j}|\\ge X^2\\sum_{j \\equal{} 1}^{n}\\sum_{l \\equal{} 1}^{n}|y_{i} \\minus{} y_{l}| \\plus{} Y^2\\sum_{i \\equal{} 1}^{m}\\sum_{k \\equal{} 1}^{m}|x_{i} \\minus{} x_{k}|$","t":[{"b":1,"e":0.42857,"k":"rising","v":0.58929,"x":0.85266,"p":[[0,24,0.0,0.58929,0.36025,0.28571,0.64286,1.0,0.0,1.0,4,10,0,4,0,2,0,0,4,0,0,4,0,0,2,0,0,4,0,0,2,0,10],[4,24,0.1667,0.69643,0.32488,0.28571,0.78571,1.0,0.14286,1.0,0,15,0,0,0,2,0,0,7,0,0,2,0,0,2,0,0,3,0,0,1,0,15],[8,24,0.3333,0.73661,0.33524,0.42857,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,18],[12,24,0.5,0.77232,0.2942,0.42857,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,6,0,0,3,0,0,1,0,0,2,0,0,2,0,18],[16,24,0.6667,0.85266,0.20668,0.82132,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,1,0,0,6,0,18],[20,24,0.8333,0.79018,0.2474,0.57142,1.0,1.0,0.2857,1.0,0,17,0,0,0,0,0,0,1,0,0,5,0,0,6,0,0,1,0,0,2,0,17],[24,24,1.0,0.74998,0.29234,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,1,0,0,2,0,0,8,0,0,1,0,0,2,0,0,1,0,17]]},{"b":2,"e":1.0,"k":"rising","v":0.64732,"x":0.89732,"p":[[0,27,0.0,0.74554,0.35307,0.42857,1.0,1.0,0.0,1.0,3,20,0,3,0,0,0,0,2,0,0,6,0,0,0,0,0,1,0,0,0,0,20],[4,27,0.1481,0.64732,0.35533,0.28571,0.78571,1.0,0.0,1.0,2,14,0,2,0,0,0,0,9,0,0,4,0,0,0,0,0,1,0,0,2,0,14],[8,27,0.2963,0.74999,0.32538,0.42859,1.0,1.0,0.0,1.0,1,18,0,1,0,1,0,0,4,0,0,4,0,0,1,0,0,1,0,0,2,0,18],[12,27,0.4444,0.75445,0.31184,0.53539,1.0,1.0,0.0,1.0,1,18,0,1,0,0,0,0,5,0,0,2,0,0,4,0,0,1,0,0,1,0,18],[16,27,0.5926,0.74107,0.32031,0.42857,1.0,1.0,0.14286,1.0,0,19,0,0,0,1,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,19],[20,27,0.7407,0.76784,0.29614,0.53539,1.0,1.0,0.2857,1.0,0,18,0,0,0,0,0,0,7,0,0,1,0,0,2,0,0,3,0,0,1,0,18],[24,27,0.8889,0.67857,0.29451,0.39286,0.71429,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,8,0,0,4,0,0,1,0,0,6,0,0,1,0,12],[27,27,1.0,0.89732,0.17941,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,0,4,0,22]]}]},{"i":"ee4e1dbff0eba298","q":"Let  $ABC$  a triangle and  $O$  his circumcentre.The lines  $OA$  and  $BC$  intersect each other at  $M$  ; the points  $N$  and  $P$  are defined in an analogous way.The tangent line in  $A$  at the circumcircle of triangle  $ABC$  intersect  $NP$  in the point  $X$  ; the points  $Y$  and  $Z$  are defined in an analogous way.Prove that the points  $X$  ,  $Y$  and  $Z$  are collinear.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04017,"p":[[0,20,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.04017,0.12486,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,20,0.6,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.02683,"p":[[0,24,0.0,0.02683,0.09081,0.0,0.0,0.0,0.0,0.43,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5d66a5115d116a44","q":"Let  $ABC$  be a triangle with  $AB<AC<BC$  with circumcircle  $\\Gamma_1$ . The circle  $\\Gamma_2$  has center  $D$  lying on  $\\Gamma_1$  and touches  $BC$  at  $E$  and the extension of  $AB$  at  $F$ . Let  $\\Gamma_1$  and  $\\Gamma_2$  meet at  $K, G$  and the line  $KG$  meets  $EF$  and  $CD$  at  $M, N$ . Show that  $BCNM$  is cyclic.","t":[{"b":1,"e":0.0,"k":"flat","v":0.01339,"x":0.06696,"p":[[0,18,0.0,0.05804,0.13767,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,18,0.2222,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.06696,0.07973,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.08036,"p":[[0,33,0.0,0.04018,0.15251,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,33,0.1212,0.07581,0.13821,0.0,0.0,0.14071,0.0,0.4286,23,0,0,23,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.08036,0.14698,0.0,0.0,0.14286,0.0,0.71429,21,0,1,21,0,7,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,33,0.3636,0.04018,0.09606,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.04018,0.10249,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.03571,0.10102,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"827e14625ef3949a","q":"Let  $BC$  be a chord of a circle  $(O)$  such that  $BC$  is not a diameter. Let  $AE$  be the diameter perpendicular to  $BC$  such that  $A$  belongs to the larger arc  $BC$  of  $(O)$ . Let  $D$  be a point on the larger arc  $BC$  of  $(O)$  which is different from  $A$ . Suppose that  $AD$  intersects  $BC$  at  $S$  and  $DE$  intersects  $BC$  at  $T$ . Let  $F$  be the midpoint of  $ST$  and  $I$  be the second intersection point of the circle  $(ODF)$  with the line  $BC$ .\n1. Let the line passing through  $I$  and parallel to  $OD$  intersect  $AD$  and  $DE$  at  $M$  and  $N$ , respectively. Find the maximum value of the area of the triangle  $MDN$  when  $D$  moves on the larger arc  $BC$  of  $(O)$  (such that  $D \\ne A$ ).\n2. Prove that the perpendicular from  $D$  to  $ST$  passes through the midpoint of  $MN$","t":[{"b":0,"e":0.857,"k":"flat","v":0.73214,"x":0.83035,"p":[[0,17,0.0,0.80357,0.32291,0.82143,1.0,1.0,0.0,1.0,2,19,2,2,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,0,5,0,19],[4,17,0.2353,0.73214,0.37244,0.53571,1.0,1.0,0.0,1.0,4,17,3,4,0,2,0,0,1,0,0,1,0,0,1,0,0,2,0,0,4,0,17],[8,17,0.4706,0.8214,0.15157,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,11,0,9],[12,17,0.7059,0.79462,0.1513,0.71429,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,12,0,0,11,0,6],[16,17,0.9412,0.83035,0.1357,0.71429,0.85714,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,14,0,8],[17,17,1.0,0.80789,0.14995,0.71429,0.857,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0,0,11,0,7]]},{"b":3,"e":0.2857,"k":"falling","v":0.28576,"x":0.88839,"p":[[0,40,0.0,0.88839,0.23887,0.85714,1.0,1.0,0.0,1.0,1,23,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,3,0,23],[4,40,0.1,0.77232,0.346,0.71429,1.0,1.0,0.0,1.0,4,17,1,4,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,5,0,17],[8,40,0.2,0.52678,0.32031,0.2857,0.42857,0.85714,0.0,1.0,2,4,1,2,0,5,0,0,4,0,0,6,0,0,3,0,0,1,0,0,7,0,4],[12,40,0.3,0.62052,0.38067,0.39285,0.71421,1.0,0.0,1.0,5,12,3,5,0,2,0,0,1,0,0,5,0,0,3,0,0,0,0,0,4,0,12],[16,40,0.4,0.41954,0.27888,0.2857,0.42857,0.57111,0.0,1.0,3,4,3,3,0,4,0,0,7,0,0,9,0,0,4,0,0,1,0,0,0,0,4],[20,40,0.5,0.41954,0.28341,0.14286,0.42857,0.57143,0.0,1.0,3,3,2,3,0,6,0,0,4,0,0,9,0,0,5,0,0,0,0,0,2,0,3],[24,40,0.6,0.31696,0.16648,0.14286,0.42857,0.42857,0.0,0.57143,3,0,1,3,0,7,0,0,5,0,0,14,0,0,3,0,0,0,0,0,0,0,0],[28,40,0.7,0.37051,0.15506,0.2857,0.42857,0.4286,0.0,0.57143,1,0,0,1,0,6,0,0,4,0,0,15,0,0,6,0,0,0,0,0,0,0,0],[32,40,0.8,0.34375,0.17445,0.2857,0.42857,0.42857,0.0,0.71429,3,0,2,3,0,4,0,0,7,0,0,15,0,0,1,0,0,2,0,0,0,0,0],[36,40,0.9,0.28576,0.19236,0.10714,0.42857,0.42857,0.0,0.57143,8,0,3,8,0,3,0,0,4,0,0,15,0,0,2,0,0,0,0,0,0,0,0],[40,40,1.0,0.32588,0.18976,0.14286,0.42857,0.42857,0.0,0.85714,3,0,0,3,0,8,0,0,3,0,0,15,0,0,2,0,0,0,0,0,1,0,0]]}]},{"i":"accc226314443098","q":"Given a permutation ( $a_0, a_1, \\ldots, a_n$ ) of the sequence  $0, 1,\\ldots, n$ . A transportation of  $a_i$  with  $a_j$  is called legal if  $a_i=0$  for  $i>0$ , and  $a_{i-1}+1=a_j$ . The permutation ( $a_0, a_1, \\ldots, a_n$ ) is called regular if after a number of legal transportations it becomes ( $1,2, \\ldots, n,0$ ).\nFor which numbers  $n$  is the permutation ( $1, n, n-1, \\ldots, 3, 2, 0$ ) regular?","t":[{"b":2,"e":0.0,"k":"flat","v":0.05357,"x":0.18295,"p":[[0,45,0.0,0.18295,0.16843,0.14286,0.14286,0.17857,0.0,0.71429,6,0,3,6,0,18,0,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[4,45,0.0889,0.14286,0.15568,0.0,0.14286,0.17857,0.0,0.57143,13,0,6,13,0,11,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,45,0.1778,0.13839,0.13592,0.0,0.14286,0.2857,0.0,0.57143,12,0,5,12,0,11,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,45,0.2667,0.15616,0.13998,0.105,0.14286,0.17857,0.0,0.71429,8,0,6,8,0,16,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,45,0.3556,0.08911,0.07772,0.0,0.14286,0.14286,0.0,0.28571,13,0,7,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.08464,0.1376,0.0,0.0,0.14286,0.0,0.57143,20,0,15,20,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,45,0.5333,0.09375,0.13175,0.0,0.0,0.14286,0.0,0.57143,18,0,14,18,0,9,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,45,0.6222,0.08027,0.07929,0.0,0.14143,0.14286,0.0,0.28571,15,0,10,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.06696,0.10092,0.0,0.0,0.14286,0.0,0.42857,20,0,13,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.14286,13,0,10,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,12,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,1,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.08027,0.0708,0.0,0.14286,0.14286,0.0,0.143,14,0,2,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.13828,"x":0.22321,"p":[[0,19,0.0,0.13828,0.15757,0.0,0.14286,0.14286,0.0,0.571,13,0,7,13,0,12,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,19,0.2105,0.14723,0.1803,0.0,0.14286,0.1786,0.0,0.71429,14,0,4,14,0,10,0,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[8,19,0.4211,0.19641,0.20119,0.0,0.14286,0.2857,0.0,0.71429,9,0,4,9,0,13,0,0,5,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[12,19,0.6316,0.14277,0.17496,0.0,0.14286,0.14286,0.0,0.71429,12,0,5,12,0,14,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[16,19,0.8421,0.19643,0.18123,0.10714,0.14286,0.28571,0.0,0.71429,8,0,3,8,0,12,0,0,8,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[19,19,1.0,0.22321,0.18189,0.10714,0.2143,0.28571,0.0,0.71429,8,0,2,8,0,8,0,0,9,0,0,5,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"94e0a683c3490be0","q":"It is known that  $n$  is a positive integer and  $n \\le 144$ . Ten questions of the type \u201cIs  $n$  smaller than  $a$ ?\u201d are allowed. Answers are given with a delay: for  $i = 1, \\ldots , 9$ , the  $i$ -th question is answered only after the  $(i + 1)$ -th question is asked. The answer to the tenth question is given immediately.\nFind a strategy for identifying  $n$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.05803,"x":0.11161,"p":[[0,13,0.0,0.05803,0.22546,0.0,0.0,0.0,0.0,1.0,30,1,22,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,13,0.3077,0.09374,0.2331,0.0,0.0,0.0,0.0,1.0,26,1,11,26,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[8,13,0.6154,0.11161,0.22228,0.0,0.0,0.14286,0.0,0.85714,23,0,9,23,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.14732,"p":[[0,2,0.0,0.14732,0.31841,0.0,0.0,0.0,0.0,1.0,26,2,18,26,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,2],[2,2,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"231b669659467e9f","q":"Let  $A$  denote the set of real numbers  $x$  such that  $0\\le x<1$ . A function  $f:A\\to \\mathbb{R}$  has the properties:\n\n(i)  $f(x)=2f(\\frac{x}{2})$  for all  $x\\in A$ ;\n\n(ii)  $f(x)=1-f(x-\\frac{1}{2})$  if  $\\frac{1}{2}\\le x<1$ .\n\nProve that\n\n(a)  $f(x)+f(1-x)\\ge \\frac{2}{3}$  if  $x$  is rational and  $0<x<1$ .\n\n(b) There are infinitely many odd positive integers  $q$  such that equality holds in (a) when  $x=\\frac{1}{q}$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.08482,"x":0.08482,"p":[[0,3,0.0,0.08482,0.22829,0.0,0.0,0.0,0.0,0.85714,27,0,3,27,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,6,0.0,0.04464,0.11538,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"82ccc76002df93ce","q":"Let  $ABC$  be a triangle with  $AB < BC$ , and let  $E$  and  $F$  be points in  $AC$  and  $AB$  such that  $BF = BC = CE$ , both on the same halfplane as  $A$  with respect to   $BC$ . \n\nLet  $G$  be the intersection of  $BE$  and  $CF$ . Let  $H$  be a point in the parallel through  $G$  to  $AC$  such that  $HG = AF$  (with  $H$  and  $C$  in opposite halfplanes with respect to  $BG$ ). Show that  $\\angle EHG = \\frac{\\angle BAC}{2}$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,27,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,33,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"1a42fa42d8fdcde6","q":"In Lineland there are $n \\geqslant 1$ towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the $2 n$ bulldozers are distinct. Every time when a right and a left bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, the bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes.  Let $A$ and $B$ be two towns, with $B$ being to the right of $A$. We say that town $A$ can sweep town $B$ away if the right bulldozer of $A$ can move over to $B$ pushing off all bulldozers it meets. Similarly, $B$ can sweep $A$ away if the left bulldozer of $B$ can move to $A$ pushing off all bulldozers of all towns on its way.  Prove that there is exactly one town which cannot be swept away by any other one. (Estonia)","t":[{"b":4,"e":0.28571,"k":"flat","v":0.05357,"x":0.12946,"p":[[0,5,0.0,0.05357,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,8,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0625,0.12339,0.0,0.0,0.0,0.0,0.42857,25,0,15,25,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.12946,0.16506,0.0,0.0,0.28571,0.0,0.71429,17,0,0,17,0,4,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.16518,"p":[[0,37,0.0,0.08481,0.16694,0.0,0.0,0.03571,0.0,0.57143,24,0,9,24,0,2,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,37,0.1081,0.04018,0.1197,0.0,0.0,0.0,0.0,0.57143,28,0,10,28,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,37,0.2162,0.08928,0.17768,0.0,0.0,0.03571,0.0,0.71429,24,0,15,24,0,1,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[12,37,0.3243,0.06697,0.12869,0.0,0.0,0.14286,0.0,0.57143,23,0,9,23,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,37,0.4324,0.04911,0.13176,0.0,0.0,0.0,0.0,0.57143,27,0,9,27,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,37,0.5405,0.04455,0.10964,0.0,0.0,0.0,0.0,0.42857,27,0,14,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,24,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,19,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.16063,0.22518,0.0,0.07,0.2857,0.0,1.0,16,1,3,16,0,7,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[36,37,0.973,0.16518,0.16409,0.0,0.14288,0.28571,0.0,0.57143,14,0,2,14,0,3,0,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[37,37,1.0,0.14286,0.17496,0.0,0.0,0.28571,0.0,0.4286,18,0,7,18,0,2,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2a90e5203786df83","q":"In tetrahedron  $ABCS$  no two edges have equal length. Point  $A'$  in plane  $BCS$  is symmetric to  $S$  with respect to the perpendicular bisector of  $BC$ . Points  $B'$  and  $C'$  are defined analagously. Prove that planes  $ABC, AB'C', A'BC'$  abd  $A'B'C$  share a common point.","t":[{"b":4,"e":0.0,"k":"flat","v":0.01339,"x":0.23213,"p":[[0,66,0.0,0.23205,0.20442,0.0,0.21428,0.28571,0.0,0.71429,9,0,9,9,0,7,0,0,9,0,0,2,0,0,4,0,0,1,0,0,0,0,0],[4,66,0.0606,0.23213,0.19147,0.10714,0.21428,0.28571,0.0,0.57143,8,0,4,8,0,8,0,0,9,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[8,66,0.1212,0.18302,0.19957,0.0,0.14286,0.28571,0.0,0.57143,13,0,5,13,0,8,0,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[12,66,0.1818,0.10268,0.15251,0.0,0.0,0.14286,0.0,0.57143,18,0,6,18,0,10,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[16,66,0.2424,0.10259,0.16838,0.0,0.0,0.14286,0.0,0.57143,20,0,6,20,0,7,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[20,66,0.303,0.10268,0.16065,0.0,0.0,0.14286,0.0,0.57143,19,0,4,19,0,8,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[24,66,0.3636,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,66,0.4242,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,66,0.4848,0.23213,0.19147,0.14286,0.14286,0.28571,0.0,0.71429,5,0,4,5,0,15,0,0,5,0,0,2,0,0,4,0,0,1,0,0,0,0,0],[36,66,0.5455,0.15179,0.18189,0.0,0.14286,0.14286,0.0,0.71429,12,0,6,12,0,14,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[40,66,0.6061,0.14723,0.12103,0.105,0.14286,0.14286,0.0,0.42857,8,0,3,8,0,18,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,66,0.6667,0.11143,0.1055,0.0,0.14286,0.14286,0.0,0.42857,12,0,11,12,0,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,66,0.7273,0.13384,0.09406,0.14214,0.14286,0.14286,0.0,0.42857,7,0,6,7,0,21,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,66,0.7879,0.12946,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,5,0,4,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.14723,"x":0.31232,"p":[[0,41,0.0,0.23661,0.16982,0.14286,0.2143,0.42857,0.0,0.57143,6,0,2,6,0,10,0,0,7,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[4,41,0.0976,0.20982,0.20198,0.0,0.14286,0.28571,0.0,0.71429,9,0,3,9,0,11,0,0,6,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[8,41,0.1951,0.25891,0.19701,0.14286,0.28571,0.42857,0.0,0.57143,7,0,3,7,0,7,0,0,9,0,0,3,0,0,6,0,0,0,0,0,0,0,0],[12,41,0.2927,0.2008,0.22831,0.0,0.14286,0.42857,0.0,0.57143,14,0,5,14,0,7,0,0,2,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[16,41,0.3902,0.29017,0.24085,0.14286,0.21431,0.57143,0.0,0.71429,6,0,2,6,0,10,0,0,6,0,0,1,0,0,5,0,0,4,0,0,0,0,0],[20,41,0.4878,0.31232,0.18724,0.14286,0.2857,0.42857,0.14,0.71429,0,0,0,0,0,15,0,0,4,0,0,7,0,0,4,0,0,2,0,0,0,0,0],[24,41,0.5854,0.17839,0.11851,0.14286,0.14286,0.14286,0.0,0.57143,3,0,2,3,0,22,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[28,41,0.6829,0.1517,0.06122,0.14286,0.14286,0.14286,0.0,0.28571,2,0,1,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.16955,0.0558,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.14723,0.04352,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"05797848b02fe954","q":"Let  $ABC$  be acute triangle. Let  $E$  and  $F$  be points on  $BC$ , such that angles  $BAE$  and  $FAC$  are equal. Lines  $AE$  and  $AF$  intersect cirumcircle of  $ABC$  at points  $M$  and  $N$ . On rays  $AB$  and  $AC$  we have points  $P$  and  $R$ , such that angle  $PEA$  is equal to angle  $B$  and angle  $AER$  is equal to angle  $C$ . Let  $L$  be intersection of  $AE$  and  $PR$  and  $D$  be intersection of  $BC$  and  $LN$ . Prove that  $$ \\frac{1}{|MN|}+\\frac{1}{|EF|}=\\frac{1}{|ED|}. $$","t":[{"b":3,"e":0.28571,"k":"flat","v":0.2275,"x":0.40178,"p":[[0,24,0.0,0.2275,0.27867,0.0,0.14143,0.32143,0.0,0.71429,14,0,0,14,0,7,0,0,3,0,0,1,0,0,0,0,0,7,0,0,0,0,0],[4,24,0.1667,0.35712,0.27662,0.14286,0.2857,0.71429,0.0,0.857,4,0,0,4,0,10,0,0,6,0,0,1,0,0,1,0,0,9,0,0,1,0,0],[8,24,0.3333,0.29465,0.3443,0.0,0.14286,0.71429,0.0,1.0,12,2,0,12,0,9,0,0,1,0,0,0,0,0,0,0,0,7,0,0,1,0,2],[12,24,0.5,0.38839,0.24545,0.2857,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,6,0,0,14,0,0,7,0,0,0,0,0,0,0,0,3,0,2],[16,24,0.6667,0.40178,0.21559,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,1,0,0,21,0,0,4,0,0,0,0,0,1,0,0,5,0,0],[20,24,0.8333,0.30798,0.11925,0.2857,0.28571,0.28571,0.14,0.857,0,0,0,0,0,3,0,0,24,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[24,24,1.0,0.30357,0.06917,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,2,0,0,24,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.85714,"k":"rising","v":0.37945,"x":0.83036,"p":[[0,13,0.0,0.38838,0.33164,0.14286,0.21428,0.71429,0.0,0.85714,7,0,0,7,0,9,0,0,1,0,0,1,0,0,2,0,0,7,0,0,5,0,0],[4,13,0.3077,0.37945,0.31865,0.14286,0.14288,0.71429,0.0,0.85714,5,0,0,5,0,12,0,0,1,0,0,1,0,0,1,0,0,8,0,0,4,0,0],[8,13,0.6154,0.83036,0.12078,0.71429,0.85714,0.85714,0.57143,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,14,0,7],[12,13,0.9231,0.79024,0.08746,0.71429,0.857,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,16,0,1],[13,13,1.0,0.7634,0.14984,0.71429,0.85714,0.85714,0.286,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,10,0,0,17,0,1]]}]},{"i":"c246b6cd7a1a0515","q":"In the plane, there are $n \\geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \\ldots, D_{n}$ with radii $R_{1} \\geqslant R_{2} \\geqslant \\ldots \\geqslant R_{n}$. For every $i=1,2, \\ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that  $$ O P_{1}+O P_{2}+\\ldots+O P_{n} \\geqslant R_{6}+R_{7}+\\ldots+R_{n} $$  (A disk is assumed to contain its boundary.) (Iran)","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01784,"p":[[0,19,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"93c5811fa055b78b","q":"Let  $ ABC$  be a triangle with  $ \\measuredangle{BAC} < \\measuredangle{ACB}$ . Let  $ D$ ,  $ E$  be points on the sides  $ AC$  and  $ AB$ , such that the angles  $ ACB$  and  $ BED$  are congruent. If  $ F$  lies in the interior of the quadrilateral  $ BCDE$  such that the circumcircle of triangle  $ BCF$  is tangent to the circumcircle of  $ DEF$  and the circumcircle of  $ BEF$  is tangent to the circumcircle of  $ CDF$ , prove that the points  $ A$ ,  $ C$ ,  $ E$ ,  $ F$  are concyclic.\r\n\r\n*Author: Cosmin Pohoata*","t":[{"b":0,"e":0.28571,"k":"flat","v":0.25892,"x":0.35273,"p":[[0,25,0.0,0.32148,0.13836,0.2857,0.35714,0.42858,0.0,0.43,4,0,1,4,0,0,0,0,12,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.27678,0.15126,0.2857,0.28571,0.42857,0.0,0.42857,6,0,0,6,0,1,0,0,14,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.35273,0.11289,0.2857,0.42857,0.4286,0.0,0.43,2,0,0,2,0,0,0,0,11,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.30357,0.08564,0.2857,0.2857,0.32143,0.14286,0.4286,0,0,0,0,0,4,0,0,20,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.27678,0.03458,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.28125,0.04351,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.28124,0.02485,0.2857,0.28571,0.28571,0.1429,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.25892,0.05575,0.2857,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.32589,"x":0.42858,"p":[[0,34,0.0,0.32589,0.1439,0.2857,0.42857,0.42857,0.0,0.4286,4,0,2,4,0,1,0,0,9,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.33036,0.16918,0.28571,0.42857,0.4286,0.0,0.57143,6,0,0,6,0,0,0,0,5,0,0,20,0,0,1,0,0,0,0,0,0,0,0],[8,34,0.2353,0.38407,0.09069,0.39286,0.42857,0.4286,0.0,0.43,1,0,0,1,0,0,0,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.40179,0.08328,0.42857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.4063,0.0519,0.42857,0.42857,0.42857,0.2857,0.43,0,0,0,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.41965,0.03458,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.42858,1e-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],[32,34,0.9412,0.40184,0.05579,0.42857,0.42857,0.4286,0.2857,0.43,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.41965,0.03458,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63149331513614af","q":"In the acute angle triangle  $ABC$  the point  $O$  is the center of the circumscribed circle and the lines  $OA,OB,OC$  intersect sides  $BC,CA,AB$  respectively in points  $M,N,P$  such that  $\\angle NMP=90^o$ .**(a)** Find the ratios  $\\frac{\\angle AMN}{\\angle NMC}$ , $\\frac{\\angle AMP}{\\angle PMB}$ .**(b)** If any of the angles of the triangle  $ABC$  is  $60^o$ , find the two other angles.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,5,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,23,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.04911,"p":[[0,26,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,23,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.03572,0.12372,0.0,0.0,0.0,0.0,0.57143,29,0,22,29,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,26,0.3077,0.04911,0.14555,0.0,0.0,0.0,0.0,0.71429,28,0,21,28,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,26,0.4615,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,21,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,27,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,31,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e43bad439f617a76","q":"Let  $ABC$  be an acute-angled triangle,  $O$  be its circumcenter,  $BM$  be a median, and  $BH$  be an altitude. Circles  $AOB$  and  $BHC$  meet for the second time at point  $E$ , and circles  $AHB$  and  $BOC$  meet at point  $F$ . Prove that  $ME = MF$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,54,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,54,0.0741,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,54,0.1481,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,54,0.2222,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,54,0.2963,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,54,0.3704,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,54,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,54,0.5185,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,54,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,54,0.6667,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[44,54,0.8148,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,54,0.8889,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,54,0.963,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[54,54,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"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],[4,19,0.2105,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bd2f0e15763f61f8","q":"Let  $ABC$  be an isosceles triangle with base  $AC$ . Points  $D$  and  $E$  are chosen on the sides  $AC$  and  $BC$ , respectively, such that  $CD = DE$ . Let  $H, J,$  and  $K$  be the midpoints of  $DE, AE,$  and  $BD$ , respectively. The circumcircle of triangle  $DHK$  intersects  $AD$  at point  $F$ , whereas the circumcircle of triangle  $HEJ$  intersects  $BE$  at  $G$ . The line through  $K$  parallel to  $AC$  intersects  $AB$  at  $I$ . Let  $IH \\cap GF =$  { $M$ }. Prove that  $J, M,$  and  $K$  are collinear points.","t":[{"b":1,"e":0.71429,"k":"flat","v":0.15178,"x":0.48206,"p":[[0,40,0.0,0.48206,0.38098,0.14286,0.42857,0.89286,0.0,1.0,3,8,2,3,0,12,0,0,0,0,0,3,0,0,0,0,0,5,0,0,1,0,8],[4,40,0.1,0.25447,0.28061,0.14286,0.14286,0.2857,0.0,1.0,5,3,1,5,0,17,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[8,40,0.2,0.32143,0.34442,0.0,0.14286,0.60714,0.0,1.0,9,4,0,9,0,10,0,0,2,0,0,2,0,0,1,0,0,4,0,0,0,0,4],[12,40,0.3,0.34804,0.34255,0.14286,0.14286,0.46429,0.0,1.0,4,5,0,4,0,16,0,0,0,0,0,4,0,0,1,0,0,1,0,0,1,0,5],[16,40,0.4,0.17411,0.198,0.14286,0.14286,0.14286,0.0,1.0,6,1,1,6,0,22,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[20,40,0.5,0.15178,0.18875,0.0,0.14286,0.14286,0.0,0.857,10,0,0,10,0,18,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[24,40,0.6,0.30356,0.24935,0.14286,0.14286,0.4642,0.0,0.71429,4,0,0,4,0,15,0,0,0,0,0,5,0,0,2,0,0,6,0,0,0,0,0],[28,40,0.7,0.37491,0.26912,0.14286,0.35714,0.71429,0.0,0.71429,5,0,2,5,0,7,0,0,4,0,0,5,0,0,1,0,0,10,0,0,0,0,0],[32,40,0.8,0.37054,0.24707,0.14286,0.35714,0.71429,0.0,0.71429,2,0,0,2,0,11,0,0,3,0,0,7,0,0,0,0,0,9,0,0,0,0,0],[36,40,0.9,0.45079,0.25038,0.24999,0.42857,0.71429,0.0,0.71429,2,0,0,2,0,6,0,0,4,0,0,6,0,0,1,0,0,13,0,0,0,0,0],[40,40,1.0,0.48205,0.20453,0.28571,0.42857,0.71429,0.14,0.71429,0,0,0,0,0,4,0,0,5,0,0,10,0,0,1,0,0,12,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"rising","v":0.29893,"x":0.88397,"p":[[0,27,0.0,0.29893,0.31012,0.14286,0.14286,0.42858,0.0,1.0,5,3,1,5,0,16,0,0,2,0,0,2,0,0,1,0,0,2,0,0,1,0,3],[4,27,0.1481,0.40178,0.36672,0.14286,0.14288,0.74996,0.0,1.0,3,7,1,3,0,14,0,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,7],[8,27,0.2963,0.30795,0.33906,0.14286,0.14286,0.39285,0.0,1.0,5,4,2,5,0,18,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,4],[12,27,0.4444,0.37947,0.35285,0.14286,0.14286,0.71429,0.0,1.0,5,4,0,5,0,14,0,0,0,0,0,1,0,0,1,0,0,6,0,0,1,0,4],[16,27,0.5926,0.88397,0.29103,1.0,1.0,1.0,0.0,1.0,2,27,0,2,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,27],[20,27,0.7407,0.78562,0.33899,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,5,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,20],[24,27,0.8889,0.83035,0.27765,0.82132,1.0,1.0,0.14286,1.0,0,20,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,2,0,0,4,0,20],[27,27,1.0,0.79017,0.32338,0.67857,1.0,1.0,0.0,1.0,1,20,0,1,0,4,0,0,0,0,0,0,0,0,3,0,0,3,0,0,1,0,20]]}]},{"i":"c19827ee004462a8","q":"Let  $M_A, M_B, M_C$  be the midpoints of the sides  $BC, CA, AB$  respectively of a non-isosceles triangle  $ABC$ . Points  $H_A, H_B, H_C$  lie on the corresponding sides, different from  $M_A, M_B, M_C$  such that  $M_AH_B=M_AH_C, $   $M_BH_A=M_BH_C,$  and  $M_CH_A=M_CH_B$ . Prove that  $H_A, H_B, H_C$  are the feet of the corresponding altitudes.","t":[{"b":6,"e":0.28571,"k":"flat","v":0.11161,"x":0.14279,"p":[[0,12,0.0,0.11161,0.09932,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.11608,0.08328,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,20,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.14279,0.17129,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,13,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,12,1.0,0.13393,0.10677,0.0,0.14286,0.17868,0.0,0.28571,10,0,0,10,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.12946,"x":0.21428,"p":[[0,19,0.0,0.12947,0.09006,0.10714,0.14286,0.14287,0.0,0.28571,8,0,0,8,0,19,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.21428,0.1923,0.14286,0.14286,0.17868,0.0,0.71429,4,0,0,4,0,20,0,0,3,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[8,19,0.4211,0.12946,0.08268,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.13384,0.09406,0.105,0.14286,0.14287,0.0,0.28571,8,0,0,8,0,18,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.13852,0.10422,0.0,0.14286,0.17857,0.0,0.29,9,0,0,9,0,15,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.16519,0.09522,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,17,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5e1172bd6f935860","q":"Let  $ABC$  be a triangle such that  $AB > BC$  and let  $D$  be a variable point on the line segment  $BC$ . Let  $E$  be the point on the circumcircle of triangle  $ABC$ , lying on the opposite side of  $BC$  from  $A$  such that  $\\angle BAE = \\angle DAC$ . Let  $I$  be the incenter of triangle  $ABD$  and let  $J$  be the incenter of triangle  $ACE$ . Prove that the line  $IJ$  passes through a fixed point, that is independent of  $D$ .\n\n*Proposed by Merlijn Staps*","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.09821,"p":[[0,27,0.0,0.08929,0.16269,0.0,0.0,0.03571,0.0,0.4286,24,0,11,24,0,1,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.09821,0.16146,0.0,0.0,0.2857,0.0,0.42857,23,0,15,23,0,0,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.08036,0.15542,0.0,0.0,0.0,0.0,0.42857,25,0,11,25,0,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.08473,0.14661,0.0,0.0,0.14071,0.0,0.4286,23,0,8,23,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.03125,0.09268,0.0,0.0,0.0,0.0,0.4286,28,0,8,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"rising","v":0.09821,"x":0.26786,"p":[[0,34,0.0,0.09821,0.16146,0.0,0.0,0.2857,0.0,0.42857,23,0,12,23,0,0,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.10268,0.14826,0.0,0.0,0.28571,0.0,0.42857,21,0,12,21,0,1,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.14286,0.15972,0.0,0.0,0.28571,0.0,0.42857,17,0,8,17,0,1,0,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.15625,0.17261,0.0,0.0,0.28571,0.0,0.42857,17,0,16,17,0,0,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.16964,0.16146,0.0,0.21428,0.28571,0.0,0.4286,14,0,13,14,0,2,0,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.14286,0.13832,0.0,0.14286,0.28571,0.0,0.42857,14,0,9,14,0,5,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.20536,0.16342,0.0,0.2857,0.28571,0.0,0.4286,10,0,10,10,0,5,0,0,10,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.24991,0.11853,0.24999,0.28571,0.28571,0.0,0.42857,4,0,4,4,0,4,0,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.24545,0.11433,0.14289,0.28571,0.28571,0.0,0.42857,3,0,1,3,0,7,0,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.26786,0.09279,0.2857,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,6,0,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"729cb54c43507c4c","q":"Given the natural  $n$ . We shall call *word* sequence from  $n$  letters of the alphabet, and *distance*  $\\rho(A, B)$   between *words*  $A=a_1a_2\\dots a_n$  and  $B=b_1b_2\\dots b_n$  , the number of digits in which they differ (that is, the number of such  $i$ , for which  $a_i\\ne b_i$ ). We will say that the *word*  $C$   *lies* between words   $A$  and    $B$  , if  $\\rho (A,B)=\\rho(A,C)+\\rho(C,B)$ . What is the largest number of *words* you can choose so that among any three, there is a *word lying* between the other two?","t":[{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.00446,"p":[[0,7,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,26,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,26,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,18,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,25,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"be407f42bcef5c8a","q":"Let  $ABC$  be a triangle  $D\\in[BC]$  (different than  $A$  and  $B$ ). $E$  is the midpoint of  $[CD]$ .  $F\\in[AC]$  such that  $\\widehat{FEC}=90$  and  $|AF|.|BC|=|AC|.|EC|.$  Circumcircle of  $ADC$  intersect  $[AB]$  at  $G$  different than  $A$ .Prove that tangent to circumcircle of  $AGF$  at  $F$  is touch circumcircle of  $BGE$  too.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,14,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,3,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0267,0.05558,0.0,0.0,0.0,0.0,0.1429,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":4,"e":0.0,"k":"flat","v":0.00446,"x":0.05795,"p":[[0,36,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.05795,0.08636,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"266bedcfaee2e106","q":"Let  $ABC$  be a triangle inscribed in circle  $(O)$  with diamter  $KL$  passes through the midpoint  $M$  of  $AB$  such that  $L, C$  lie on the different sides respect to  $AB$ . A circle passes through  $M, K $ cuts  $LC$  at $ P, Q $ (point  $P$  lies between $ Q, C$ ). The line  $KQ $ cuts  $(LMQ)$  at  $R$ . Prove that  $ARBP$  is cyclic and $ AB$  is the symmedian of triangle  $APR$ .\n\nPlease help :)","t":[{"b":4,"e":0.0,"k":"flat","v":0.07134,"x":0.1517,"p":[[0,24,0.0,0.07581,0.09432,0.0,0.0,0.14286,0.0,0.2857,18,0,1,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.07134,0.09443,0.0,0.0,0.14286,0.0,0.28571,19,0,2,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.08705,0.1345,0.0,0.0,0.14286,0.0,0.5,20,0,1,20,0,7,0,0,3,0,0,1,0,1,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.14278,0.07143,0.14286,0.14286,0.1429,0.0,0.28571,4,0,2,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.1517,0.06122,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.08472,0.11204,0.0,0.07,0.14286,0.0,0.571,16,0,0,16,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.08474,"p":[[0,15,0.0,0.08474,0.11765,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.05804,0.10013,0.0,0.0,0.14286,0.0,0.4286,22,0,1,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.03571,0.10102,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6498643f5329a878","q":"In terms of  $n\\ge2$ , find the largest constant  $c$  such that for all nonnegative  $a_1,a_2,\\ldots,a_n$  satisfying  $a_1+a_2+\\cdots+a_n=n$ , the following inequality holds:\n\\[\\frac1{n+ca_1^2}+\\frac1{n+ca_2^2}+\\cdots+\\frac1{n+ca_n^2}\\le \\frac{n}{n+c}.\\]\n\n*Calvin Deng.*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,22,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,5,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,7,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,7,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a31fa0f7ff856472","q":"Let  $ABC$  be an acute triangle with  $AB\\ne AC$  and circumcircle  $\\omega$ . The angle bisector of  $BAC$  intersects  $BC$  and  $\\omega$  at  $D$  and  $E$  respectively. Circle with diameter  $DE$  intersects  $\\omega$  again at  $F \\ne E$ . Point  $P$  is on  $AF$  such that  $PB = PC$  and  $X$  and  $Y$  are feet of perpendiculars from  $P$  to  $AB$  and  $AC$  respectively. Let  $H$  and  $H'$  be the orthocenters of  $ABC$  and  $AXY$  respectively.  $AH$  meets  $\\omega$  again at  $Q$  . If  $AH'$  and  $HH'$  intersect the circle with diameter  $AH$  again at points  $S$  and  $T$ , respectively, prove that the lines  $AT , HS$  and  $FQ$  are concurrent.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,37,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03572,"p":[[0,77,0.0,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.143,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,77,0.0519,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,77,0.1039,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,77,0.1558,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,77,0.2078,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,77,0.2597,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,77,0.3117,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,77,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,77,0.4156,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,77,0.4675,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,77,0.5195,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,77,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,77,0.6234,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,77,0.6753,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,77,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,77,0.7792,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,77,0.8312,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,77,0.8831,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,77,0.9351,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"ae1b248c61d6906f","q":"Let  $ f(n)$  denote the maximum possible number of right triangles determined by  $ n$  coplanar points. Show that \\[ \\lim_{n\\rightarrow \\infty} \\frac{f(n)}{n^2}\\equal{}\\infty \\;\\textrm{and}\\  \\lim_{n\\rightarrow \\infty}\\frac{f(n)}{n^3}\\equal{}0 .\\] \r\n\r\n*P. Erdos*","t":[{"b":0,"e":1.0,"k":"volatile","v":0.48661,"x":0.99107,"p":[[0,10,0.0,0.55802,0.32214,0.39285,0.57143,0.85714,0.0,1.0,4,6,1,4,0,3,0,0,1,0,0,1,0,0,13,0,0,1,0,0,3,0,6],[4,10,0.4,0.5357,0.30093,0.2857,0.57143,0.71429,0.0,1.0,2,6,1,2,0,5,0,0,2,0,0,2,0,0,12,0,0,3,0,0,0,0,6],[8,10,0.8,0.48661,0.36397,0.14286,0.57143,0.75,0.0,1.0,4,7,2,4,0,9,0,0,1,0,0,0,0,0,7,0,0,3,0,0,1,0,7],[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]]},{"b":5,"e":0.2857,"k":"falling","v":0.11152,"x":0.52213,"p":[[0,25,0.0,0.52213,0.40361,0.14214,0.57143,1.0,0.0,1.0,6,10,2,6,0,7,0,0,0,0,0,1,0,0,5,0,0,1,0,0,2,0,10],[4,25,0.16,0.37033,0.26224,0.14286,0.42857,0.57143,0.0,1.0,3,1,1,3,0,11,0,0,1,0,0,5,0,0,8,0,0,2,0,0,1,0,1],[8,25,0.32,0.42854,0.31338,0.14286,0.35714,0.57143,0.0,1.0,3,5,0,3,0,7,0,0,6,0,0,4,0,0,5,0,0,2,0,0,0,0,5],[12,25,0.48,0.29464,0.26711,0.14286,0.14286,0.57143,0.0,1.0,7,1,4,7,0,11,0,0,2,0,0,2,0,0,7,0,0,2,0,0,0,0,1],[16,25,0.64,0.29015,0.26116,0.14286,0.14286,0.46418,0.0,1.0,5,1,2,5,0,14,0,0,2,0,0,3,0,0,5,0,0,1,0,0,1,0,1],[20,25,0.8,0.38391,0.29328,0.14286,0.35714,0.57143,0.0,1.0,5,2,2,5,0,8,0,0,3,0,0,3,0,0,6,0,0,5,0,0,0,0,2],[24,25,0.96,0.11607,0.10972,0.0,0.14286,0.14286,0.0,0.57143,10,0,4,10,0,20,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[25,25,1.0,0.11152,0.12743,0.0,0.14286,0.14286,0.0,0.57143,13,0,4,13,0,16,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"5518f5e6b52d6222","q":"It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n)=$ $10 n, g(n)=10 n+4$, and $h(2 n)=n$; in other words, one may write 0 or 4 in the end of the number and one may divide an even number by 2. Prove: every positive integer can be constructed starting from 4 and performing a finite number of the operations $f, g$, and $h$ in some order.","t":[{"b":0,"e":0.14286,"k":"rising","v":0.63389,"x":0.80354,"p":[[0,9,0.0,0.63389,0.25238,0.42859,0.71429,0.85714,0.0,1.0,2,2,1,2,0,0,0,0,2,0,0,5,0,0,5,0,0,7,0,0,9,0,2],[4,9,0.4444,0.71427,0.24744,0.57132,0.85714,0.85714,0.0,1.0,1,3,0,1,0,1,0,0,1,0,0,4,0,0,2,0,0,4,0,0,16,0,3],[8,9,0.8889,0.73667,0.22051,0.67857,0.857,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,2,0,0,3,0,0,2,0,0,4,0,0,17,0,3],[9,9,1.0,0.80354,0.15873,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,4,0,0,14,0,7]]},{"b":7,"e":1.0,"k":"falling","v":0.42401,"x":0.75891,"p":[[0,9,0.0,0.66071,0.32093,0.42859,0.857,0.85714,0.0,1.0,3,5,1,3,0,2,0,0,2,0,0,2,0,0,1,0,0,5,0,0,12,0,5],[4,9,0.4444,0.75891,0.2034,0.57143,0.85714,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0,2,0,0,16,0,5],[8,9,0.8889,0.46872,0.24802,0.2857,0.42859,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,9,0,0,6,0,0,6,0,0,0,0,0,7,0,0],[9,9,1.0,0.42401,0.21583,0.28571,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,4,0,0,8,0,0,9,0,0,5,0,0,2,0,0,3,0,0]]}]},{"i":"0c5958f17505e8be","q":"Let  $A$  be the number of ways in which the set  $\\{ 1, 2, . . . , n\\}$  can be partitioned into non-empty subsets. Let  $B$  be the number of ways in which the set  $\\{ 1, 2, . . . , n, n + 1 \\}$  can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that  $A = B$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.08482,"p":[[0,18,0.0,0.08036,0.24468,0.0,0.0,0.0,0.0,1.0,27,2,2,27,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,18,0.2222,0.08482,0.24448,0.0,0.0,0.0,0.0,1.0,26,2,1,26,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,18,0.4444,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.12947,"p":[[0,16,0.0,0.12947,0.29957,0.0,0.0,0.0,0.0,1.0,25,3,0,25,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[4,16,0.25,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,1,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"59b7e554875f696d","q":"Let  $(M, \\cdot)$  be a monoid with identity element  $e.$ a) Prove that if  $M$  is a finite set, then there are no  $a,b \\in M$  such that  $a \\cdot b = e$  and  $b \\cdot a \\neq e.$ b) Provide an example of functions  $f,g: \\mathbb{Z}_{\\ge 0} \\to \\mathbb{Z}_{\\ge 0}$  such that  $f \\circ g = \\mathrm{Id}_{\\mathbb{Z}_{\\ge 0}}$  and  $g \\circ f \\neq \\mathrm{Id}_{\\mathbb{Z}_{\\ge 0}}.$ c) Let  $a,b \\in M$  be such that  $a \\cdot b = e$  and  $b \\cdot a \\neq e.$  Prove that if  $b^n \\cdot a^m = b^q \\cdot a^p$  for some positive integers  $m,n,p,q,$  then  $n=q$  and  $m=p.$","t":[{"b":0,"e":0.857,"k":"flat","v":0.75876,"x":0.82588,"p":[[0,10,0.0,0.75876,0.22995,0.67536,0.85714,0.89286,0.0,1.0,1,8,1,1,0,0,0,0,0,0,0,4,0,0,3,0,0,6,0,0,10,0,8],[4,10,0.4,0.82588,0.23072,0.71429,0.85714,1.0,0.0,1.0,1,14,1,1,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,9,0,14],[8,10,0.8,0.7633,0.2664,0.67536,0.85707,1.0,0.0,1.0,2,10,2,2,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,0,10,0,10],[10,10,1.0,0.80355,0.16268,0.71429,0.857,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,11,0,8]]},{"b":4,"e":1.0,"k":"flat","v":0.70967,"x":0.83005,"p":[[0,18,0.0,0.83005,0.14937,0.71429,0.85714,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,15,0,8],[4,18,0.2222,0.77687,0.23131,0.71429,0.85714,1.0,0.0,1.0,1,10,1,1,0,0,0,0,0,0,0,4,0,0,1,0,0,8,0,0,8,0,10],[8,18,0.4444,0.70967,0.14053,0.57143,0.71429,0.85704,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,6,0,0,13,0,0,9,0,1],[12,18,0.6667,0.79,0.1637,0.71321,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,5,0,0,14,0,6],[16,18,0.8889,0.79465,0.18528,0.71429,0.857,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,3,0,0,4,0,0,13,0,8],[18,18,1.0,0.81237,0.13091,0.71429,0.857,0.85714,0.43,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,13,0,6]]}]},{"i":"ecf6af957ddb5ce8","q":"Let  $F$  be a finite non-empty set of integers and let  $n$  be a positive integer. Suppose that $\\bullet$  Any  $x \\in F$  may be written as  $x=y+z$  for some  $y$ ,  $z \\in F$ ; $\\bullet$  If  $1 \\leq k \\leq n$  and  $x_1$ , ...,  $x_k \\in F$ , then  $x_1+\\cdots+x_k \\neq 0$ . \n\nShow that  $F$  has at least  $2n+2$  elements.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.1383,"x":0.18286,"p":[[0,20,0.0,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.15402,0.05677,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,1,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.14714,0.06668,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.14723,0.02488,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.18286,0.07359,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,21,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.13375,"x":0.16054,"p":[[0,17,0.0,0.16054,0.10568,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,25,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.14638,0.06656,0.14286,0.14286,0.14286,0.0,0.33,3,0,0,3,0,25,0,1,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.14268,0.05051,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.15616,0.05488,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.14679,0.04358,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4df11551ba396c84","q":"Let  $2001$  given points on a circle be coloured either red or green. In one step all points are recoloured simultaneously in the following way: If both direct neighbours of a point  $P$  have the same colour as  $P$ , then the colour of  $P$  remains unchanged, otherwise  $P$  obtains the other colour. Starting with the first colouring  $F_1$ , we obtain the colourings  $F_2,F_3 ,\\ldots .$  after several recolouring steps. Prove that there is a number  $n_0\\le 1000$  such that  $F_{n_0}=F_{n_0 +2}$ . Is the assertion also true if  $1000$  is replaced by  $999$ ?","t":[{"b":1,"e":0.71429,"k":"rising","v":0.51335,"x":0.71872,"p":[[0,9,0.0,0.51335,0.36918,0.14286,0.571,0.85714,0.0,1.0,7,7,3,7,0,2,0,0,3,0,0,2,0,0,5,0,0,4,0,0,2,0,7],[4,9,0.4444,0.67854,0.24224,0.67857,0.71429,0.75,0.0,1.0,2,4,2,2,0,1,0,0,0,0,0,1,0,0,4,0,0,16,0,0,4,0,4],[8,9,0.8889,0.71872,0.18725,0.57143,0.71429,0.78571,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,4,0,0,7,0,0,13,0,0,0,0,8],[9,9,1.0,0.67409,0.13475,0.57143,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,16,0,0,0,0,3]]},{"b":7,"e":1.0,"k":"volatile","v":0.53124,"x":0.98214,"p":[[0,2,0.0,0.53124,0.38338,0.14286,0.64286,0.85714,0.0,1.0,7,6,2,7,0,3,0,0,3,0,0,0,0,0,3,0,0,4,0,0,6,0,6],[2,2,1.0,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28]]}]},{"i":"30ba821309597f34","q":"Let  $ \\Gamma(I,r)$  and  $ \\Gamma(O,R)$  denote the incircle and circumcircle, respectively, of a triangle  $ ABC$ . Consider all the triangels  $ A_iB_iC_i$  which are simultaneously inscribed in  $ \\Gamma(O,R)$  and circumscribed to  $ \\Gamma(I,r)$ . Prove that the centroids of these triangles are concyclic.","t":[{"b":3,"e":0.0,"k":"flat","v":0.02679,"x":0.12054,"p":[[0,44,0.0,0.11831,0.2206,0.0,0.03571,0.14286,0.0,1.0,16,1,0,16,1,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,44,0.0909,0.12054,0.12931,0.0,0.14286,0.1429,0.0,0.4286,14,0,0,14,0,11,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.05804,0.11215,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,44,0.2727,0.1116,0.18202,0.0,0.0,0.16075,0.0,0.64286,21,0,0,21,0,3,0,1,3,0,0,2,0,0,1,1,0,0,0,0,0,0,0],[16,44,0.3636,0.09375,0.14987,0.0,0.0,0.14286,0.0,0.71429,18,0,0,18,0,11,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,44,0.4545,0.0758,0.0873,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.0625,0.12846,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,44,0.6364,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.0558,0.07305,0.0,0.0,0.14286,0.0,0.21429,20,0,0,20,0,11,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[44,44,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.11161,"x":0.20965,"p":[[0,9,0.0,0.11161,0.19144,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,9,0.4444,0.16956,0.23539,0.0,0.14286,0.1429,0.0,1.0,13,1,0,13,0,12,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[8,9,0.8889,0.20088,0.1217,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,2,18,0,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[9,9,1.0,0.20965,0.10104,0.14286,0.1429,0.2857,0.0,0.4286,1,0,0,1,0,18,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4c2715351b2a9e24","q":"Let  $ABCD$  be a circumscribed quadrilateral. Its incircle  $\\omega$  touches the sides  $BC$  and  $DA$  at points  $E$  and  $F$  respectively. It is known that lines  $AB,FE$  and  $CD$  concur. The circumcircles of triangles  $AED$  and  $BFC$  meet  $\\omega$  for the second time at points  $E_1$  and  $F_1$ . Prove that  $EF$  is parallel to  $E_1 F_1$ .","t":[{"b":4,"e":0.14,"k":"flat","v":0.03554,"x":0.13392,"p":[[0,26,0.0,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.125,0.15465,0.0,0.07143,0.1786,0.0,0.57143,16,0,1,16,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,26,0.3077,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.03554,0.07962,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.09366,0.11629,0.0,0.14143,0.14286,0.0,0.57143,15,0,0,15,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,26,0.7692,0.12929,0.11493,0.0,0.14286,0.14286,0.0,0.42857,10,0,0,10,0,17,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.1249,0.12236,0.0,0.14286,0.14286,0.0,0.571,10,0,0,10,0,19,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[26,26,1.0,0.13392,0.15538,0.0,0.14286,0.1429,0.0,0.571,14,0,0,14,0,11,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,13,0.0,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,26,0,1,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"be2529aea396b41e","q":"Let  $H$  be the orthocenter of a given triangle  $ABC$ . Let  $BH$  and  $AC$  meet at a point  $E$ , and  $CH$  and  $AB$  meet at  $F$ . Suppose that  $X$  is a point on the line  $BC$ . Also suppose that the circumcircle of triangle  $BEX$  and the line  $AB$  intersect again at  $Y$ , and the circumcircle of triangle  $CFX$  and the line  $AC$  intersect again at  $Z$ .\nShow that the circumcircle of triangle  $AYZ$  is tangent to the line  $AH$ .\n\n*Proposed by usjl*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.04911,"x":0.17402,"p":[[0,30,0.0,0.17402,0.28735,0.0,0.0,0.14286,0.0,1.0,18,1,4,18,0,8,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,1],[4,30,0.1333,0.07572,0.09426,0.0,0.0,0.14286,0.0,0.42857,17,0,4,17,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.16964,0.27994,0.0,0.07143,0.14286,0.0,1.0,16,2,0,16,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,2],[12,30,0.4,0.14286,0.22868,0.0,0.14286,0.14286,0.0,1.0,15,1,2,15,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[16,30,0.5333,0.11606,0.18704,0.0,0.0,0.14286,0.0,0.85714,17,0,1,17,0,11,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[20,30,0.6667,0.09366,0.12166,0.0,0.0,0.14286,0.0,0.4286,17,0,1,17,0,11,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.07562,0.07951,0.0,0.07,0.14286,0.0,0.28571,16,0,2,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.10269,0.12493,0.0,0.07143,0.1429,0.0,0.4286,16,0,0,16,0,11,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.02232,"x":0.16052,"p":[[0,27,0.0,0.16052,0.25938,0.0,0.07,0.14286,0.0,1.0,16,2,3,16,0,10,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,2],[4,27,0.1481,0.12045,0.20549,0.0,0.0,0.14286,0.0,1.0,17,1,2,17,0,11,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[8,27,0.2963,0.12947,0.19678,0.0,0.14286,0.14287,0.0,1.0,15,1,0,15,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,27,0.4444,0.10704,0.19557,0.0,0.0,0.14286,0.0,0.71429,20,0,1,20,0,8,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[16,27,0.5926,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"18ee5a31b7be8a77","q":"Let  $D$  and  $E$  be arbitrary points on the sides  $BC$  and  $AC$  of triangle  $ABC$ , respectively. The circumcircle of  $\\triangle ADC$  meets for the second time the circumcircle of  $\\triangle BCE$  at point  $F$ . Line  $FE$  meets line  $AD$  at point  $G$ , while line  $FD$  meets line  $BE$  at point  $H$ . Prove that lines  $CF, AH$  and  $BG$  pass through the same point.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,24,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,48,0.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"4345ce983e6ecb57","q":"In a plane the circles  $\\mathcal K_1$  and  $\\mathcal K_2$  with centers  $I_1$  and  $I_2$ , respectively, intersect in two points  $A$  and  $B$ . Assume that  $\\angle I_1AI_2$  is obtuse. The tangent to  $\\mathcal K_1$  in  $A$  intersects  $\\mathcal K_2$  again in  $C$  and the tangent to  $\\mathcal K_2$  in  $A$  intersects  $\\mathcal K_1$  again in  $D$ . Let  $\\mathcal K_3$  be the circumcircle of the triangle  $BCD$ . Let  $E$  be the midpoint of that arc  $CD$  of  $\\mathcal K_3$  that contains  $B$ . The lines  $AC$  and  $AD$  intersect  $\\mathcal K_3$  again in  $K$  and  $L$ , respectively. Prove that the line  $AE$  is perpendicular to  $KL$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.00446,"x":0.07125,"p":[[0,39,0.0,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,3,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,1,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.07125,0.07125,0.0,0.07,0.14286,0.0,0.14286,16,0,3,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,3,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.06697,0.09439,0.0,0.0,0.14286,0.0,0.28571,20,0,1,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01786,"x":0.04465,"p":[[0,47,0.0,0.03116,0.0589,0.0,0.0,0.0,0.0,0.1429,25,0,2,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,3,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,3,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.03554,0.06156,0.0,0.0,0.035,0.0,0.143,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,2,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9aa91f5bfe5717e4","q":"In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible number of flights required to go from one of them to the other. It is known that for any city there are at most 100 cities at distance exactly three from it. Prove that there is no city such that more than 2550 other cities have distance exactly four from it.  (Russia)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04455,"p":[[0,8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,23,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0133,0.05465,0.0,0.0,0.0,0.0,0.2857,30,0,23,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.04455,0.08318,0.0,0.0,0.035,0.0,0.28571,24,0,1,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,18,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,20,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,24,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,23,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"68a0814348669516","q":"Let  $ABCD$  be a cyclic quadrilateral with circumcircle  $\\Omega$  and let diagonals  $AC$  and  $BD$  intersect at  $X$ . Suppose that  $AEFB$  is inscribed in a circumcircle of triangle  $ABX$  such that  $EF$  and  $AB$  are parallel.  $FX$  meets the circumcircle of triangle  $CDX$  again at  $G$ . Let  $EX$  meets  $AB$  at  $P$ , and  $XG$  meets  $CD$  at  $Q$ . Denote by  $S$  the intersection of the perpendicular bisector of  $\\overline{EG}$  and  $\\Omega$  such that  $S$  is closer to  $A$  than  $B$ . Prove that line through  $S$  parallel to  $PQ$  is tangent to  $\\Omega$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,38,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[12,38,0.3158,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,38,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,38,0.7368,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bcffc629ca79bc35","q":"Let  $ABCDEF$  be a convex hexagon whose vertices lie on a circle. Suppose that  $AB\\cdot CD\\cdot EF = BC\\cdot DE\\cdot FA$ . Show that the diagonals  $AD, BE$  and  $CF$  are concurrent.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,14,0.0,0.03125,0.08552,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[14,14,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.03571,"x":0.08036,"p":[[0,7,0.0,0.03571,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0625,0.11812,0.0,0.0,0.03571,0.0,0.4286,24,0,1,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.08036,0.09407,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e04625f5fadad932","q":"Let  $ P\\subseteq \\mathbb{R}^m$  be a non-empty compact convex set and  $ f: P\\rightarrow \\mathbb{R}_{ \\plus{} }$  be a concave function. Prove, that for every  $ \\xi\\in \\mathbb{R}^m$ \r\n\\[ \\int_{P}\\langle \\xi,x \\rangle f(x)dx\\leq \\left[\\frac {m \\plus{} 1}{m \\plus{} 2}\\sup_{x\\in P}{\\langle\\xi,x\\rangle} \\plus{} \\frac {1}{m \\plus{} 2}\\inf_{x\\in P}{\\langle\\xi,x\\rangle}\\right] \\cdot\\int_{P}f(x)dx.\\]","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.08482,"p":[[0,15,0.0,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,2,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.08482,0.1551,0.0,0.0,0.14286,0.0,0.85714,18,0,1,18,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,15,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,7,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.08036,0.08702,0.0,0.07143,0.14286,0.0,0.2857,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"44743e0516f07933","q":"Let  $ABC$  be a triangle, and  $O$  the center of its circumcircle.\r\nLet a line through the point  $O$  intersect the lines  $AB$  and  $AC$  at the points  $M$  and  $N$ , respectively. Denote by  $S$  and  $R$  the midpoints of the segments  $BN$  and  $CM$ , respectively.\r\nProve that  $\\measuredangle ROS=\\measuredangle BAC$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,9,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,24,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"61a853669cfc1f64","q":"Let  $ABC$  be a triangle with circumradius  $R$ , and let  $\\ell_A, \\ell_B, \\ell_C$  be the altitudes through  $A, B, C$  respectively. The altitudes meet at  $H$ . Let  $P$  be an arbitrary point in the same plane as  $ABC$ . The feet of the perpendicular lines through  $P$  onto  $\\ell_A, \\ell_B, \\ell_C$  are  $D, E, F$  respectively. Prove that the areas of  $DEF$  and  $ABC$  satisfy the following equation:  $$   \\operatorname{area}(DEF) = \\frac{{PH}^2}{4R^2} \\cdot \\operatorname{area}(ABC). $$","t":[{"b":2,"e":0.0,"k":"volatile","v":0.01786,"x":0.7008,"p":[[0,11,0.0,0.33036,0.39031,0.0,0.14286,0.85714,0.0,1.0,11,5,0,11,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,5],[4,11,0.3636,0.7008,0.37362,0.39286,0.85714,1.0,0.0,1.0,3,15,0,3,0,4,0,0,1,0,0,1,0,0,2,0,0,1,0,0,5,0,15],[8,11,0.7273,0.09822,0.18707,0.0,0.0,0.14286,0.0,1.0,19,1,0,19,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[11,11,1.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.0892,"x":0.49107,"p":[[0,24,0.0,0.49107,0.4164,0.10714,0.42859,1.0,0.0,1.0,8,10,1,8,0,5,0,0,2,0,0,2,0,0,2,0,0,1,0,0,2,0,10],[4,24,0.1667,0.15624,0.25089,0.0,0.14286,0.1429,0.0,1.0,15,2,1,15,0,11,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[8,24,0.3333,0.0892,0.18469,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,24,0.5,0.14732,0.14054,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,10,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.11607,0.10972,0.0,0.14286,0.14286,0.0,0.4286,12,0,0,12,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.09821,0.12078,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,12,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.11159,0.1274,0.0,0.14286,0.14286,0.0,0.571,14,0,0,14,0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"0e0846dd6e714f7e","q":"Let  $ABC$  be a triangle and let  $P$  be a point in the plane of  $ABC$  that is inside the region of the angle  $BAC$  but outside triangle  $ABC$ .**(a)** Prove that any two of the following statements imply the third.\n\n[list]**(i)** the circumcentre of triangle  $PBC$  lies on the ray  $\\stackrel{\\to}{PA}$ .**(ii)** the circumcentre of triangle  $CPA$  lies on the ray  $\\stackrel{\\to}{PB}$ .**(iii)** the circumcentre of triangle  $APB$  lies on the ray  $\\stackrel{\\to}{PC}$ .[/list]**(b)** Prove that if the conditions in (a) hold, then the circumcentres of triangles  $BPC,CPA$  and  $APB$  lie on the circumcircle of triangle  $ABC$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.03991,"p":[[0,13,0.0,0.03571,0.11845,0.0,0.0,0.0,0.0,0.57143,29,0,3,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,13,0.3077,0.03991,0.11402,0.0,0.0,0.0,0.0,0.57143,27,0,5,27,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,13,0.6154,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,4,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.0,"x":0.04018,"p":[[0,14,0.0,0.04018,0.11425,0.0,0.0,0.0,0.0,0.57143,27,0,4,27,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,14,0.2857,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4e3149180a259a5b","q":"Let  $ABC$  be a triangle with circumcircle  $\\Gamma$ , and points  $E$  and  $F$  are chosen from sides  $CA$ ,  $AB$ , respectively. Let the circumcircle of triangle  $AEF$  and  $\\Gamma$  intersect again at point  $X$ . Let the circumcircles of triangle  $ABE$  and  $ACF$  intersect again at point  $K$ . Line  $AK$  intersect with  $\\Gamma$  again at point  $M$  other than  $A$ , and  $N$  be the reflection point of  $M$  with respect to line  $BC$ . Let  $XN$  intersect with  $\\Gamma$  again at point  $S$  other that  $X$ . \n\nProve that  $SM$  is parallel to  $BC$ . \n\n*Proposed by Ming Hsiao*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.08929,"p":[[0,56,0.0,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,56,0.0714,0.08929,0.10565,0.0,0.07143,0.14286,0.0,0.4286,16,0,0,16,0,13,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,56,0.1429,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,56,0.2143,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.00893,0.0346,0.0,0.0,0.0,0.0,0.143,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,56,0.6429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01786,"x":0.11134,"p":[[0,22,0.0,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.11134,0.12739,0.0,0.14286,0.14286,0.0,0.71429,11,0,0,11,0,20,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,22,0.7273,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"512ca264a988e9e2","q":"Let  $ABC$  be a triangle. Let  $M$  be a variable point interior to the segment  $AB$ , and let  $\\gamma_B$  be the circle through  $M$  and tangent at  $B$  to  $BC$ . Let  $P$  and  $Q$  be the touch points of  $\\gamma_B$  and its tangents from  $A$ , and let  $X$  be the midpoint of the segment  $PQ$ . Similarly, let  $N$  be a variable point interior to the segment  $AC$ , and let  $\\gamma_C$  be the circle through  $M$  and tangent at  $C$  to  $BC$ . Let  $R$  and  $S$  be the touch points of  $\\gamma_C$  and its tangents from  $A$ , and let  $Y$  be the midpoint of the segment  $RS$ . Prove that the line through the centers of the circles  $AMN$  and  $AXY$  passes through a fixed point.","t":[{"b":5,"e":0.0,"k":"flat","v":0.00893,"x":0.0892,"p":[[0,2,0.0,0.0892,0.06909,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.04018,"x":0.09375,"p":[[0,37,0.0,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.08455,0.06994,0.0,0.14143,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.07572,0.07113,0.0,0.14,0.14286,0.0,0.1429,15,0,1,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.07125,0.07125,0.0,0.07,0.14286,0.0,0.14286,16,0,1,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,1,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.14286,20,0,3,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.08018,0.07071,0.0,0.14143,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.08473,0.07009,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9c9ea8f6caca11aa","q":"Let  $AXBY$  be a convex quadrilateral. The incircle of  $\\triangle AXY$  has center  $I_A$  and touches  $\\overline{AX}$  and  $\\overline{AY}$  at  $A_1$  and  $A_2$  respectively. The incircle of  $\\triangle BXY$  has center  $I_B$  and touches  $\\overline{BX}$  and  $\\overline{BY}$  at  $B_1$  and  $B_2$  respectively. Define  $P = \\overline{XI_A} \\cap \\overline{YI_B}$ ,  $Q = \\overline{XI_B} \\cap \\overline{YI_A}$ , and  $R = \\overline{A_1B_1} \\cap \\overline{A_2B_2}$ .\n[list=a]\n[*] Prove that if  $\\angle AXB = \\angle AYB$ , then  $P$ ,  $Q$ ,  $R$  are collinear.\n[*] Prove that if there exists a circle tangent to all four sides of  $AXBY$ , then  $P$ ,  $Q$ ,  $R$  are collinear.\n[/list]","t":[{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.01339,"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],[4,11,0.3636,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.0,"x":0.13393,"p":[[0,25,0.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.13357,0.0345,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.12474,0.04715,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3e123f0ad0e287b8","q":"Let  $A$  be a ring and let  $D$  be the set of its non-invertible elements. If  $a^2=0$  for any  $a \\in D,$  prove that:**a)**  $axa=0$  for all  $a \\in D$  and  $x \\in A$ ;**b)** if  $D$  is a finite set with at least two elements, then there is  $a \\in D,$   $a \\neq 0,$  such that  $ab=ba=0,$  for every  $b \\in D.$ *Ioan B\u0103etu*","t":[{"b":1,"e":0.4286,"k":"rising","v":0.33928,"x":0.65165,"p":[[0,32,0.0,0.49551,0.33689,0.25,0.4998,0.75,0.0,1.0,6,5,4,6,0,2,0,0,2,0,0,6,0,0,6,0,0,2,0,0,3,0,5],[4,32,0.125,0.43754,0.39112,0.0,0.42929,0.85714,0.0,1.0,11,5,6,11,0,2,0,0,2,0,0,2,0,0,3,0,0,3,0,0,4,0,5],[8,32,0.25,0.56231,0.3408,0.39286,0.4998,0.85714,0.0,1.0,4,7,4,4,0,2,0,0,2,0,0,8,0,0,3,0,0,1,0,0,5,0,7],[12,32,0.375,0.33928,0.32878,0.0,0.35714,0.57143,0.0,1.0,13,2,9,13,0,1,0,0,2,0,0,4,0,0,6,0,0,3,0,0,1,0,2],[16,32,0.5,0.45088,0.33523,0.14289,0.42859,0.71429,0.0,1.0,7,4,4,7,0,2,0,0,4,0,0,5,0,0,4,0,0,4,0,0,2,0,4],[20,32,0.625,0.38838,0.32385,0.0,0.42857,0.57143,0.0,1.0,9,3,1,9,0,3,0,0,1,0,0,7,0,0,7,0,0,0,0,0,2,0,3],[24,32,0.75,0.62943,0.31915,0.42857,0.71429,0.89286,0.0,1.0,2,8,0,2,0,3,0,0,1,0,0,6,0,0,3,0,0,4,0,0,5,0,8],[28,32,0.875,0.59821,0.28445,0.42857,0.4286,0.85704,0.0,1.0,2,6,0,2,0,0,0,0,2,0,0,13,0,0,0,0,0,5,0,0,4,0,6],[32,32,1.0,0.65165,0.26226,0.42859,0.71429,0.85714,0.0,1.0,1,5,0,1,0,0,0,0,3,0,0,8,0,0,2,0,0,5,0,0,8,0,5]]},{"b":7,"e":0.71429,"k":"flat","v":0.49104,"x":0.59374,"p":[[0,22,0.0,0.50891,0.34981,0.25,0.57121,0.85704,0.0,1.0,7,5,2,7,0,1,0,0,2,0,0,5,0,0,5,0,0,3,0,0,4,0,5],[4,22,0.1818,0.59374,0.33713,0.39286,0.64286,0.85714,0.0,1.0,3,7,3,3,0,3,0,0,2,0,0,6,0,0,2,0,0,3,0,0,6,0,7],[8,22,0.3636,0.49104,0.33869,0.24999,0.4286,0.71429,0.0,1.0,5,6,2,5,0,3,0,0,4,0,0,5,0,0,4,0,0,4,0,0,1,0,6],[12,22,0.5455,0.5223,0.39545,0.0,0.71429,0.85714,0.0,1.0,9,7,6,9,0,2,0,0,1,0,0,1,0,0,2,0,0,7,0,0,3,0,7],[16,22,0.7273,0.50444,0.34066,0.25,0.4998,0.75,0.0,1.0,6,5,3,6,0,2,0,0,2,0,0,6,0,0,4,0,0,4,0,0,3,0,5]]}]},{"i":"69814e0f73ddb213","q":"Let  $I$  be the incenter of an acute triangle  $\\triangle ABC$ , and let the incircle be  $\\Gamma$ .\nLet the circumcircle of  $\\triangle IBC$  hit  $\\Gamma$  at  $D, E$ , where  $D$  is closer to  $B$  and  $E$  is closer to  $C$ . \nLet  $\\Gamma \\cap BE = K (\\not= E)$ ,  $CD \\cap BI = T$ , and  $CD \\cap \\Gamma = L (\\not= D)$ .\nLet the line passing  $T$  and perpendicular to  $BI$  meet  $\\Gamma$  at  $P$ , where  $P$  is inside  $\\triangle IBC$ . \nProve that the tangent to  $\\Gamma$  at  $P$ ,  $KL$ ,  $BI$  are concurrent.","t":[{"b":1,"e":0.0,"k":"flat","v":0.04447,"x":0.08036,"p":[[0,16,0.0,0.0625,0.11258,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04447,0.06596,0.0,0.0,0.14071,0.0,0.1429,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.07134,0.11288,0.0,0.0,0.14286,0.0,0.42857,21,0,1,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,12,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,16,0.0,0.06696,0.10092,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,2,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"428c4f206457f165","q":"Let  $AK$ ,   $BL$  and  $CM$  be the angle bisectors of a triangle  $ABC$ , with  $K$  on  $BC$ . Let  $P$  and  $Q$  be the points on the lines  $BL$  and  $CM$  respectively such that  $AP = PK$  and  $AQ = QK$ . Prove that  $\\angle PAQ = 90^o -\\frac12  \\angle B AC.$ \t\n\n(I Sharygin)","t":[{"b":1,"e":0.14286,"k":"falling","v":0.18294,"x":0.43741,"p":[[0,15,0.0,0.39731,0.34392,0.14286,0.2143,0.57111,0.0,1.0,1,7,0,1,0,15,0,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,7],[4,15,0.2667,0.27232,0.22689,0.14286,0.14286,0.42857,0.0,1.0,2,2,0,2,0,16,0,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,2],[8,15,0.5333,0.43741,0.38956,0.14286,0.21428,1.0,0.0,1.0,2,10,0,2,0,14,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,10],[12,15,0.8,0.18294,0.08921,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,20,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.20964,0.11295,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,17,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.4286,"k":"flat","v":0.30361,"x":0.37049,"p":[[0,18,0.0,0.32142,0.25999,0.14286,0.28571,0.42857,0.0,1.0,2,3,1,2,0,12,0,0,8,0,0,5,0,0,2,0,0,0,0,0,0,0,3],[4,18,0.2222,0.37049,0.29644,0.14286,0.28571,0.42857,0.0,1.0,1,5,1,1,0,11,0,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,5],[8,18,0.4444,0.30361,0.28739,0.14286,0.14286,0.32143,0.0,1.0,3,3,0,3,0,15,0,0,6,0,0,3,0,0,0,0,0,1,0,0,1,0,3],[12,18,0.6667,0.35277,0.11292,0.2857,0.28571,0.4286,0.14286,0.71429,0,0,0,0,0,2,0,0,16,0,0,12,0,0,1,0,0,1,0,0,0,0,0],[16,18,0.8889,0.33929,0.1948,0.2857,0.28571,0.32143,0.14286,1.0,0,2,0,0,0,5,0,0,19,0,0,5,0,0,1,0,0,0,0,0,0,0,2],[18,18,1.0,0.34821,0.15126,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,17,0,0,10,0,0,1,0,0,0,0,0,0,0,1]]}]},{"i":"079854ac1eb6d508","q":"Let  $ABC$  be a triangle and  $D$  be a point inside triangle  $ABC$ .  $\\Gamma$  is the circumcircle of triangle  $ABC$ , and  $DB$ ,  $DC$  meet  $\\Gamma$  again at  $E$ ,  $F$  , respectively.  $\\Gamma_1$ ,  $\\Gamma_2$  are the circumcircles of triangle  $ADE$  and  $ADF$  respectively. Assume  $X$  is on  $\\Gamma_2$  such that  $BX$  is tangent to  $\\Gamma_2$ . Let  $BX$  meets  $\\Gamma$  again at  $Z$ . Prove that the line  $CZ$  is tangent to  $\\Gamma_1$  .\n\n*Proposed by HakureiReimu*.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,9,0.0,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"73adba4aeea216d5","q":"Let  $(I_b)$ ,  $(I_c)$  are excircles of a triangle  $ABC$ . Given a circle  $ \\omega $  passes through  $A$  and externally tangents to the circles  $(I_b)$  and  $(I_c)$  such that it intersects with  $BC$  at points  $M$ ,  $N$ . \nProve that  $ \\angle BAM=\\angle CAN $ .\n\n\nA. Smirnov","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.10705,"p":[[0,5,0.0,0.10705,0.10099,0.0,0.14286,0.14286,0.0,0.28571,13,0,6,13,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04911,"x":0.12054,"p":[[0,21,0.0,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,6,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.12054,0.13415,0.0,0.07143,0.2857,0.0,0.42857,16,0,7,16,0,6,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,8,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.10268,0.12992,0.0,0.0,0.1786,0.0,0.42857,18,0,9,18,0,6,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.09357,0.11626,0.0,0.0,0.14286,0.0,0.4286,17,0,7,17,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d85f4969d6f202fc","q":"Let  $ABC$  be a scalene triangle whose incircle is tangent to  $BC$ ,  $CA$ ,  $AB$  at  $D$ ,  $E$ ,  $F$  respectively. Lines  $BE$  and  $CF$  meet at  $G$ . Prove that there exists a point  $X$  on the circumcircle of triangle  $EFG$  such that the circumcircles of triangles  $BCX$  and  $EFG$  are tangent, and \\[\\angle BGC = \\angle BXC + \\angle EDF.\\]\n*Kornpholkrit Weraarchakul*","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02223,"p":[[0,36,0.0,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[36,36,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,75,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,75,0.0533,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,75,0.1067,0.01322,0.04109,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,75,0.16,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,75,0.2133,0.01558,0.0427,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,75,0.2667,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,75,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,75,0.3733,0.0133,0.05465,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,75,0.4267,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,75,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,75,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,75,0.5867,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,75,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,75,0.6933,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,75,0.7467,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,75,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,75,0.8533,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,75,0.9067,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,75,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[75,75,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"16679971f365d15d","q":"Let  $ABC$  be a triangle such that  $AB<AC$ . The perpendicular bisector of the side  $BC$  meets the side  $AC$  at the point  $D$ , and the (interior) bisectrix of the angle  $ADB$  meets the circumcircle  $ABC$  at the point  $E$ . Prove that the (interior) bisectrix of the angle  $AEB$  and the line through the incentres of the triangles  $ADE$  and  $BDE$  are perpendicular.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.08009,"x":0.09366,"p":[[0,22,0.0,0.09357,0.10471,0.0,0.07,0.14286,0.0,0.28571,16,0,1,16,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.08929,0.13243,0.0,0.07143,0.14286,0.0,0.71429,16,0,4,16,0,15,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,22,0.3636,0.09366,0.12679,0.0,0.0,0.14286,0.0,0.57143,17,0,1,17,0,11,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,22,0.5455,0.08009,0.07915,0.0,0.14,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.08911,0.07772,0.0,0.14286,0.14286,0.0,0.2857,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"rising","v":0.09822,"x":0.31255,"p":[[0,29,0.0,0.10268,0.11971,0.0,0.14286,0.14286,0.0,0.57143,14,0,2,14,0,15,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,29,0.1379,0.12947,0.14445,0.0,0.14286,0.14286,0.0,0.57143,13,0,1,13,0,13,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,29,0.2759,0.12054,0.12428,0.0,0.14286,0.14286,0.0,0.57143,12,0,1,12,0,15,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,29,0.4138,0.09822,0.0974,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.12054,0.13415,0.0,0.14286,0.14286,0.0,0.4286,14,0,2,14,0,12,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.25884,0.16924,0.14286,0.2857,0.42857,0.0,0.57143,5,0,0,5,0,10,0,0,4,0,0,12,0,0,1,0,0,0,0,0,0,0,0],[24,29,0.8276,0.28126,0.145,0.14297,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,5,0,0,11,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.28569,0.17124,0.14286,0.28571,0.42857,0.0,0.57143,4,0,1,4,0,8,0,0,7,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[29,29,1.0,0.31255,0.14036,0.1429,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,8,0,0,9,0,0,12,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"4f0eb4909f90bf57","q":"In the country of *Sugarland*, there are  $13$  students in the IMO team selection camp. $6$  team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these  $6$  tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation.\nIs it possible that all  $13$  students have a chance of being a team member?\n\n*Proposed by Morteza Saghafian*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0625,"x":0.06696,"p":[[0,17,0.0,0.0625,0.14258,0.0,0.0,0.0,0.0,0.57143,26,0,17,26,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.06696,0.14719,0.0,0.0,0.0,0.0,0.57143,26,0,8,26,0,0,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03563,"x":0.03571,"p":[[0,11,0.0,0.03563,0.09439,0.0,0.0,0.0,0.0,0.42857,27,0,8,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,19,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"0f526be6865ea2ec","q":"Let  $I$  be the center of the circle inscribed in triangle  $ABC$ . The inscribed circle is tangent to side  $BC$  at point  $K$ . Let  $X$  and  $Y$  be points on segments  $BI$  and  $CI$  respectively, such that  $KX \\perp  AB $  and  $KY\\perp  AC$ . The circumscribed circle around triangle  $XYK$  intersects line  $BC$  at point  $D$ . Prove that  $AD \\perp BC$ .\n\n(Matthew Kurskyi)","t":[{"b":6,"e":0.14,"k":"flat","v":0.1741,"x":0.27679,"p":[[0,10,0.0,0.27679,0.19211,0.14286,0.28571,0.42857,0.0,0.857,5,0,1,5,0,7,0,0,10,0,0,7,0,0,2,0,0,0,0,0,1,0,0],[4,10,0.4,0.17411,0.15458,0.0,0.1429,0.28571,0.0,0.71429,10,0,1,10,0,8,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,10,0.8,0.1741,0.15865,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,6,0,0,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[10,10,1.0,0.26337,0.21752,0.0,0.2857,0.42857,0.0,0.71429,9,0,0,9,0,4,0,0,9,0,0,5,0,0,3,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.1874,"x":0.33481,"p":[[0,26,0.0,0.21426,0.17491,0.0,0.28571,0.28571,0.0,0.571,10,0,1,10,0,4,0,0,12,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[4,26,0.1538,0.1874,0.16917,0.0,0.14288,0.28571,0.0,0.57143,10,0,2,10,0,9,0,0,8,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[8,26,0.3077,0.23651,0.19436,0.105,0.2857,0.28571,0.0,0.71429,8,0,0,8,0,6,0,0,12,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[12,26,0.4615,0.19643,0.15046,0.0,0.2143,0.28571,0.0,0.4286,9,0,0,9,0,7,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.31697,0.23619,0.14286,0.42857,0.4286,0.0,0.71429,7,0,0,7,0,6,0,0,2,0,0,11,0,0,2,0,0,4,0,0,0,0,0],[20,26,0.7692,0.3124,0.23271,0.14286,0.28571,0.42857,0.0,0.71429,6,0,0,6,0,5,0,0,10,0,0,4,0,0,2,0,0,5,0,0,0,0,0],[24,26,0.9231,0.33481,0.24381,0.14286,0.28571,0.4642,0.0,1.0,5,1,0,5,0,5,0,0,10,0,0,4,0,0,4,0,0,3,0,0,0,0,1],[26,26,1.0,0.33034,0.20023,0.2857,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,3,0,0,13,0,0,7,0,0,1,0,0,4,0,0,0,0,0]]}]},{"i":"b710e3b02113c9b7","q":"Let  $ABC$  be a triangle with  $AB < AC$  and incentre  $I$ . Let  $E$  be the point on the side  $AC$  such that  $AE = AB$ . Let  $G$  be the point on the line  $EI$  such that  $\\angle IBG = \\angle CBA$  and such that  $E$  and  $G$  lie on opposite sides of  $I$ .\n\nProve that the line  $AI$ , the line perpendicular to  $AE$  at  $E$ , and the bisector of the angle  $\\angle BGI$  are concurrent.","t":[{"b":4,"e":1.0,"k":"rising","v":0.21875,"x":0.55356,"p":[[0,21,0.0,0.21875,0.20511,0.14286,0.14286,0.2857,0.0,1.0,5,1,1,5,0,16,0,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[4,21,0.1905,0.26339,0.20238,0.14286,0.14286,0.32143,0.0,0.71429,2,0,0,2,0,17,0,0,5,0,0,4,0,0,0,0,0,4,0,0,0,0,0],[8,21,0.381,0.55356,0.32291,0.35714,0.42859,0.85714,0.0,1.0,1,7,0,1,0,7,0,0,0,0,0,9,0,0,1,0,0,5,0,0,2,0,7],[12,21,0.5714,0.51339,0.31105,0.1429,0.42857,0.71429,0.14286,1.0,0,6,0,0,0,9,0,0,1,0,0,9,0,0,1,0,0,5,0,0,1,0,6],[16,21,0.7619,0.34366,0.27407,0.14286,0.28571,0.42857,0.0,1.0,3,3,0,3,0,10,0,0,6,0,0,8,0,0,0,0,0,2,0,0,0,0,3],[20,21,0.9524,0.44634,0.30887,0.14286,0.42857,0.71429,0.0,1.0,1,5,0,1,0,9,0,0,4,0,0,9,0,0,0,0,0,3,0,0,1,0,5],[21,21,1.0,0.38837,0.28173,0.14286,0.28571,0.60714,0.0,1.0,2,2,0,2,0,10,0,0,6,0,0,4,0,0,2,0,0,5,0,0,1,0,2]]},{"b":6,"e":0.42857,"k":"flat","v":0.21875,"x":0.28571,"p":[[0,6,0.0,0.21875,0.20511,0.14286,0.14286,0.32143,0.0,0.71429,6,0,1,6,0,17,0,0,1,0,0,5,0,0,0,0,0,3,0,0,0,0,0],[4,6,0.6667,0.26768,0.2962,0.105,0.14286,0.32143,0.0,1.0,8,2,2,8,0,12,0,0,4,0,0,3,0,0,0,0,0,1,0,0,2,0,2],[6,6,1.0,0.28571,0.18898,0.14286,0.2857,0.42857,0.0,0.71429,2,0,0,2,0,13,0,0,6,0,0,8,0,0,0,0,0,3,0,0,0,0,0]]}]},{"i":"3a7d1d57d4a45ac9","q":"Let  $ABCD$  be an inscribed quadrilateral, in  which  $\\angle BAD<90$ . On the rays  $AB$  and  $AD$  are selected points  $K$  and  $L$ , respectively, such that $ KA = KD, LA = LB$ . Let  $N$  - the midpoint of  $AC$ .Prove that if  $\\angle BNC=\\angle DNC $ ,so  $\\angle KNL=\\angle BCD $","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,12,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,18,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2e3cc985cb2d01a3","q":"Let  $N$  be midpoint of the side  $AB$  of a triangle  $ABC$  with  $\\angle A$  greater than  $\\angle B$ . Let  $D$  be a point on the ray  $AC$  such that  $CD=BC$  and  $P$  be a point on the ray  $DN$  which lies on the same side of  $BC$  as  $A$  and satisfies the condition  $\\angle PBC =\\angle A$ . The lines  $PC$  and  $AB$  intersect at  $E$ , and the lines  $BC$  and  $DP$  intersect at  $T$ . Determine the value of   $\\frac{BC}{TC} - \\frac{EA}{EB}$ .","t":[{"b":3,"e":0.42857,"k":"flat","v":0.36161,"x":0.44193,"p":[[0,13,0.0,0.44193,0.16111,0.42857,0.42857,0.571,0.0,0.71429,1,0,0,1,0,3,0,0,1,0,0,17,0,0,7,0,0,3,0,0,0,0,0],[4,13,0.3077,0.40624,0.20237,0.28571,0.42857,0.4642,0.0,0.71429,3,0,0,3,0,3,0,0,3,0,0,15,0,0,3,0,0,5,0,0,0,0,0],[8,13,0.6154,0.40177,0.09738,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,5,0,0,24,0,0,2,0,0,0,0,0,0,0,0],[12,13,0.9231,0.36161,0.07129,0.28571,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.40179,0.06622,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,7,0,0,24,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"rising","v":0.40177,"x":0.83481,"p":[[0,12,0.0,0.40177,0.20651,0.28571,0.42857,0.4642,0.0,0.71429,4,0,0,4,0,1,0,0,5,0,0,14,0,0,3,0,0,5,0,0,0,0,0],[4,12,0.3333,0.41516,0.18335,0.39286,0.42857,0.5711,0.0,0.71429,3,0,1,3,0,1,0,0,4,0,0,15,0,0,6,0,0,3,0,0,0,0,0],[8,12,0.6667,0.41518,0.20316,0.42857,0.42857,0.46429,0.0,0.71429,4,0,1,4,0,1,0,0,2,0,0,17,0,0,3,0,0,5,0,0,0,0,0],[12,12,1.0,0.83481,0.17169,0.71429,0.71429,1.0,0.28571,1.0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,15]]}]},{"i":"f6bc6bdd3823c361","q":"Let  $ABC$  be a triangle inscribed in circle  $(O)$ , with its altitudes  $BH_b, CH_c$  intersect at orthocenter  $H$  ( $H_b \\in AC$ ,  $H_c \\in  AB$ ).  $H_bH_c$  meets  $BC$  at  $P$ . Let  $N$  be the midpoint of  $AH, L$  be the orthogonal projection of  $O$  on the symmedian with respect to angle  $A$  of triangle  $ABC$ . Prove that  $\\angle NLP = 90^o$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,7,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,25,0.0,0.01112,0.03613,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a74c9228316bea1d","q":"Let  $(O_1), (O_2)$  be given two circles intersecting at  $A$  and  $B$ . The tangent lines of  $(O_1)$  at  $A, B$  intersect at   $O$ . Let  $I$  be a point on the circle  $(O_1)$  but outside the circle  $(O_2)$ . The lines  $IA, IB$  intersect circle  $(O_2)$  at  $C, D$ . Denote by  $M$  the midpoint of  $C D$ . Prove that  $I, M, O$  are collinear.","t":[{"b":1,"e":0.2857,"k":"flat","v":0.05357,"x":0.18749,"p":[[0,15,0.0,0.05357,0.17768,0.0,0.0,0.0,0.0,0.71429,29,0,1,29,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[4,15,0.2667,0.11161,0.24675,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[8,15,0.5333,0.18749,0.33774,0.0,0.0,0.2857,0.0,1.0,22,4,1,22,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,4]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.08929,"p":[[0,49,0.0,0.06696,0.20511,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,49,0.0816,0.0625,0.19541,0.0,0.0,0.0,0.0,1.0,28,1,1,28,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,49,0.1633,0.08929,0.26666,0.0,0.0,0.0,0.0,1.0,28,2,4,28,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[12,49,0.2449,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,2,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,49,0.4898,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"42f34cc816b3275c","q":"Let  $I$  be the incenter of a triangle  $ABC$ . The lines passing through  $A$  and parallel to  $BI, CI$  meet the perpendicular bisector to  $AI$  at points  $S, T$  respectively. Let  $Y$  be the common point of  $BT$  and  $CS$ , and  $A^*$  be a point such that  $BICA^*$  is a parallelogram. Prove that the midpoint of segment  $YA^*$  lies on the excircle of the triangle touching the side  $BC$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.01339,"x":0.04018,"p":[[0,17,0.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02455,0.0524,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.04018,0.09606,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00223,"x":0.02232,"p":[[0,8,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"220b2fd537552d49","q":"Let  $ S \\equal{} \\{x_1, x_2, \\ldots, x_{k \\plus{} l}\\}$  be a  $ (k \\plus{} l)$ -element set of real numbers contained in the interval  $ [0, 1]$ ;  $ k$  and  $ l$  are positive integers. A  $ k$ -element subset  $ A\\subset S$  is called *nice* if\r\n\\[ \\left |\\frac {1}{k}\\sum_{x_i\\in A} x_i \\minus{} \\frac {1}{l}\\sum_{x_j\\in S\\setminus A} x_j\\right |\\le \\frac {k \\plus{} l}{2kl}\\]\r\nProve that the number of nice subsets is at least  $ \\dfrac{2}{k \\plus{} l}\\dbinom{k \\plus{} l}{k}$ .\r\n\r\n*Proposed by Andrey Badzyan, Russia*","t":[{"b":5,"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],[4,18,0.2222,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"626c41f8f6bd0ae1","q":"Let  $ABCD$  be a convex quadrilateral with  $AB>AD$  and  $\\angle B=\\angle D=90^{\\circ}$ . Let  $P$  be a point in the side  $AB$  such that  $AP=AD$ . The lines  $PD$  and  $BC$  cut in the point  $Q$ . The perpendicular line to  $AC$  passing by  $Q$  cuts  $AB$  in the point  $R$ . Let  $S$  be the foot of perpendicular of  $D$  to the line  $AC$ . Prove that  $\\angle PSQ=\\angle RCP$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"72b19332752978ec","q":"Let  $ABC$  be a triangle. Circle  $\\Gamma$  passes through point  $A$ , meets segments  $AB$  and  $AC$  again at  $D$  and  $E$  respectively, and intersects segment  $BC$  at  $F$  and  $G$  such that  $F$  lies between  $B$  and  $G$ . The tangent to circle  $(BDF)$  at  $F$  and the tangent to circle  $(CEG)$  at  $G$  meet at  $T$ . Suppose that points  $A$  and  $T$  are distinct. Prove that line  $AT$  is parallel to  $BC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,34,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.04464,0.16146,0.0,0.0,0.0,0.0,0.85714,29,0,7,29,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,34,0.2353,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,10,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0358,"p":[[0,19,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,8,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0358,0.15615,0.0,0.0,0.0,0.0,0.86,30,0,5,30,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"197a30da0708bb34","q":"Let  $AA',BB', CC'$  be the bisectors of the angles of a triangle  $ABC \\ (A' \\in BC, B' \\in CA, C' \\in AB)$ . Prove that each of the lines  $A'B', B'C', C'A'$  intersects the incircle in two points.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[13,13,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,21,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0da3bd34213e1991","q":"Let  $ n$  be an integer, $ n \\geq 3.$  Let  $ x_1, x_2, \\ldots, x_n$  be real numbers such that  $ x_i < x_{i\\plus{}1}$  for  $ 1 \\leq i \\leq n \\minus{} 1$ . Prove that\r\n\r\n\\[ \\frac{n(n\\minus{}1)}{2} \\sum_{i < j} x_ix_j > \\left(\\sum^{n\\minus{}1}_{i\\equal{}1} (n\\minus{}i)\\cdot x_i \\right) \\cdot \\left(\\sum^{n}_{j\\equal{}2} (j\\minus{}1) \\cdot x_j \\right)\\]","t":[{"b":3,"e":0.0,"k":"flat","v":0.01786,"x":0.05357,"p":[[0,11,0.0,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.05357,0.13716,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,11,0.7273,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04902,"p":[[0,25,0.0,0.04902,0.11626,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,25,0.16,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.02455,0.07276,0.0,0.0,0.0,0.0,0.357,28,0,0,28,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a0316ba122e6a9b4","q":"Let  $A$  and  $B$  be two subsets of  $S = \\{1, 2, . . . , 2000\\}$  with  $|A| \\cdot |B| \\geq 3999$ . For a set  $X$  , let  $X-X$  denotes the set  $\\{s-t | s, t \\in X, s \\not = t\\}$ . Prove that  $(A-A) \\cap (B-B)$  is nonempty.","t":[{"b":1,"e":0.0,"k":"volatile","v":0.00893,"x":0.3125,"p":[[0,2,0.0,0.3125,0.411,0.0,0.0,0.71429,0.0,1.0,20,4,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,4],[2,2,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":1.0,"k":"volatile","v":0.30357,"x":0.79017,"p":[[0,13,0.0,0.30357,0.36727,0.0,0.0,0.71429,0.0,1.0,18,2,0,18,0,1,0,0,0,0,0,0,0,0,2,0,0,9,0,0,0,0,2],[4,13,0.3077,0.68303,0.40991,0.53571,0.9285,1.0,0.0,1.0,8,16,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,1,0,16],[8,13,0.6154,0.53125,0.43336,0.0,0.71429,1.0,0.0,1.0,12,10,0,12,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,10],[12,13,0.9231,0.75,0.22303,0.71429,0.71429,0.85714,0.0,1.0,2,5,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,15,0,0,9,0,5],[13,13,1.0,0.79017,0.10092,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,12,0,3]]}]},{"i":"aa579d84ebf998ab","q":"Let  $ABCD$  be a cyclic quadrilateral. The line parallel to  $BD$  passing through  $A$  meets the line parallel to  $AC$  passing through  $B$  at  $E$ . The circumcircle of triangle  $ABE$  meets the lines  $EC$  and  $ED$ , again, at  $F$  and  $G$ , respectively. Prove that the lines  $AB, CD$  and  $FG$  are either parallel or concurrent.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,14,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.01786,"x":0.13839,"p":[[0,40,0.0,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.04464,0.10971,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.10254,0.12498,0.0,0.0,0.14287,0.0,0.43,17,0,0,17,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.13839,0.14934,0.0,0.14286,0.2857,0.0,0.57143,15,0,0,15,0,5,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,40,0.6,0.06696,0.13825,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,4,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,40,0.7,0.10268,0.12992,0.0,0.07143,0.14286,0.0,0.57143,16,0,0,16,0,11,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,40,0.8,0.06696,0.10705,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0758,0.10086,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5d02e6a03c7377e4","q":"Let  $ \\mathcal{F}$  be a family of hexagons  $ H$  satisfying the following properties:\r\n\r\ni)  $ H$  has parallel opposite sides.\r\n\r\nii) Any 3 vertices of  $ H$  can be covered with a strip of width 1.\r\n\r\nDetermine the least  $ \\ell\\in\\mathbb{R}$  such that every hexagon belonging to  $ \\mathcal{F}$  can be covered with a strip of width  $ \\ell$ .\r\n\r\nNote: A strip is the area bounded by two parallel lines separated by a distance  $ \\ell$ . The lines belong to the strip, too.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,10,0.0,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,12,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.11161,"p":[[0,21,0.0,0.08036,0.12846,0.0,0.0,0.2857,0.0,0.28571,23,0,10,23,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.05804,0.11214,0.0,0.0,0.0,0.0,0.28571,25,0,12,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.11161,0.13236,0.0,0.0,0.28571,0.0,0.28571,18,0,12,18,0,3,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"990d75eedddc4fb7","q":"Let  $ABC$  be an acute triangle and  $\\omega$  be its circumcircle. The bisector of  $\\angle BAC$  intersects  $\\omega$  at [another point]  $M$ . Let  $P$  be a point on  $AM$  and inside  $\\triangle ABC$ . Lines passing  $P$  that are parallel to  $AB$  and  $AC$  intersects  $BC$  on  $E, F$  respectively. Lines  $ME, MF$  intersects  $\\omega$  at points  $K, L$  respectively. Prove that  $AM, BL, CK$  are concurrent.","t":[{"b":1,"e":0.0,"k":"flat","v":0.00893,"x":0.04911,"p":[[0,16,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04911,0.12169,0.0,0.0,0.0,0.0,0.57143,26,0,1,26,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,16,0.5,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01786,"x":0.16515,"p":[[0,33,0.0,0.06697,0.16746,0.0,0.0,0.0,0.0,0.57143,26,0,1,26,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[4,33,0.1212,0.0625,0.16728,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[8,33,0.2424,0.07589,0.17122,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,3,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[12,33,0.3636,0.03571,0.11845,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,33,0.4848,0.16515,0.25528,0.0,0.0,0.571,0.0,0.57143,22,0,0,22,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[20,33,0.6061,0.08927,0.20745,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[24,33,0.7273,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.05357,0.09943,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4333aaa50b8303ae","q":"Let  $D$  be the midpoint of the side  $BC$  of a triangle  $ABC$  and  $P$  be a point inside the  $ABD$  satisfying  $\\angle PAD=90^\\circ - \\angle PBD=\\angle CAD$ . Prove that  $\\angle PQB=\\angle BAC$ , where  $Q$  is the intersection point of the lines  $PC$  and  $AD$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.03125,0.12234,0.0,0.0,0.0,0.0,0.57143,30,0,1,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,27,0.2963,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,1,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,45,0.0,0.04018,0.15663,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[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],[8,45,0.1778,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,45,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,45,0.5333,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,45,0.6222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[44,45,0.9778,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"ba34e6b70cde48d5","q":"Let  $ABC$  be a triangle. Points  $D, E, F$  are on segments  $BC$ ,  $CA$ ,  $AB$ , respectively. Suppose that  $AF = 10$ ,  $F B = 10$ ,  $BD = 12$ ,  $DC = 17$ ,  $CE = 11$ , and  $EA = 10$ . Suppose that the circumcircles of  $\\vartriangle BFD$  and  $\\vartriangle CED$  intersect again at  $X$ . Find the circumradius of  $\\vartriangle EXF$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.78571,"x":1.0,"p":[[0,40,0.0,0.79911,0.34784,0.67857,1.0,1.0,0.0,1.0,4,22,2,4,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,22],[4,40,0.1,0.82589,0.31891,0.71429,1.0,1.0,0.0,1.0,3,23,3,3,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,23],[8,40,0.2,0.78571,0.35892,0.57143,1.0,1.0,0.0,1.0,4,22,4,4,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,1,0,22],[12,40,0.3,0.88839,0.27371,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,26],[16,40,0.4,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,40,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":4,"e":1.0,"k":"rising","v":0.82143,"x":1.0,"p":[[0,34,0.0,0.82143,0.29451,0.57143,1.0,1.0,0.0,1.0,1,22,1,1,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,22],[4,34,0.1176,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,34,0.2353,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[20,34,0.5882,0.99777,0.01243,1.0,1.0,1.0,0.92857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,31],[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],[32,34,0.9412,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"062fa001531a39ae","q":"Let  $ABC$  be a triangle and  $n$  a positive integer. Consider on the side  $BC$  the points  $A_1, A_2, ..., A_{2^n-1}$  that divide the side into   $2^n$  equal parts, that is,  $BA_1=A_1A_2=...=A_{2^n-2}A_{2^n-1}=A_{2^n-1}C$ . Set the points  $B_1, B_2, ..., B_{2^n-1}$  and  $C_1, C_2, ..., C_{2^n-1}$  on the sides  $CA$  and  $AB$ , respectively, analogously. Draw the line segments  $AA_1, AA_2, ..., AA_{2^n-1}$ ,  $BB_1, BB_2, ..., BB_{2^n-1}$  and  $CC_1, CC_2, ..., CC_{2^n-1}$ . Find, in terms of  $n$ , the number of regions into which the triangle is divided.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.37946,"x":0.49104,"p":[[0,8,0.0,0.37946,0.20079,0.2857,0.35714,0.42857,0.0,1.0,1,1,0,1,0,5,0,0,10,0,0,9,0,0,4,0,0,2,0,0,0,0,1],[4,8,0.5,0.49104,0.15946,0.42857,0.42859,0.57143,0.0,1.0,1,1,1,1,0,1,0,0,0,0,0,15,0,0,13,0,0,1,0,0,0,0,1],[8,8,1.0,0.38839,0.1439,0.28571,0.42857,0.42858,0.0,0.57143,2,0,0,2,0,1,0,0,7,0,0,16,0,0,6,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.4286,"k":"flat","v":0.28124,"x":0.51789,"p":[[0,60,0.0,0.41063,0.19162,0.28571,0.42857,0.42858,0.0,1.0,1,1,0,1,0,2,0,0,10,0,0,12,0,0,3,0,0,3,0,0,0,0,1],[4,60,0.0667,0.28124,0.19717,0.14286,0.2857,0.42857,0.0,0.57143,5,0,1,5,0,9,0,0,7,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[8,60,0.1333,0.36607,0.2111,0.2857,0.28571,0.57143,0.0,1.0,2,1,0,2,0,5,0,0,11,0,0,5,0,0,7,0,0,1,0,0,0,0,1],[12,60,0.2,0.40625,0.19597,0.2857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,10,0,0,7,0,0,7,0,0,2,0,0,0,0,1],[16,60,0.2667,0.40622,0.19595,0.28571,0.42857,0.57143,0.0,0.71429,3,0,0,3,0,2,0,0,6,0,0,10,0,0,8,0,0,3,0,0,0,0,0],[20,60,0.3333,0.41518,0.17985,0.28571,0.42857,0.46431,0.14286,1.0,0,1,0,0,0,5,0,0,5,0,0,14,0,0,6,0,0,1,0,0,0,0,1],[24,60,0.4,0.45087,0.19918,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,5,0,0,4,0,0,11,0,0,7,0,0,4,0,0,0,0,1],[28,60,0.4667,0.45534,0.19376,0.42857,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,5,0,0,1,0,0,10,0,0,10,0,0,5,0,0,0,0,0],[32,60,0.5333,0.49552,0.18892,0.42857,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,3,0,0,2,0,0,12,0,0,11,0,0,1,0,0,2,0,1],[36,60,0.6,0.51789,0.16265,0.42857,0.57143,0.60714,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,8,0,0,11,0,0,8,0,0,0,0,0],[40,60,0.6667,0.46425,0.1515,0.39286,0.42857,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,8,0,0,12,0,0,10,0,0,1,0,0,0,0,1],[44,60,0.7333,0.41515,0.16112,0.28571,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,5,0,0,5,0,0,12,0,0,8,0,0,2,0,0,0,0,0],[48,60,0.8,0.43749,0.14697,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,8,0,0,10,0,0,10,0,0,2,0,0,0,0,0],[52,60,0.8667,0.41072,0.17034,0.2857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,4,0,0,10,0,0,7,0,0,8,0,0,3,0,0,0,0,0],[56,60,0.9333,0.37499,0.19803,0.28571,0.35714,0.42858,0.0,1.0,2,1,0,2,0,3,0,0,11,0,0,9,0,0,5,0,0,1,0,0,0,0,1],[60,60,1.0,0.43304,0.16164,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,8,0,0,11,0,0,8,0,0,3,0,0,0,0,0]]}]},{"i":"56ba7ee9cceb6cdc","q":"Let  $ABC$  be a triangle,  $I$  its incenter and  $I_a$  its  $A$ -excenter. Let  $\\omega$  be its circuncircle and  $D$  be the intersection of  $AI$  and  $\\omega$ . Let some line  $r$  through  $D$  cut  $BC$  in  $E$  and  $\\omega$  in  $F$ . The lines  $IE$  and  $I_aE$  intersect  $I_aF$  and  $IF$  in  $P$  and  $Q$ , respectively. Furthermore, the circles  $PII_a$  and  $QII_a$  intersect  $I_aE$  and  $IE$  in  $R$  and  $S$ , respectively. Prove that there is a circle passing through  $F,E,R$  and  $S$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.20535,"p":[[0,27,0.0,0.125,0.14174,0.0,0.0,0.28571,0.0,0.42857,17,0,1,17,0,3,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.20535,0.14258,0.0,0.2857,0.28571,0.0,0.42857,10,0,1,10,0,0,0,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.13393,0.14258,0.0,0.0,0.28571,0.0,0.28571,17,0,0,17,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.04455,0.09731,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.11161,0.13709,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":3,"e":0.28571,"k":"rising","v":0.09375,"x":0.28571,"p":[[0,12,0.0,0.09375,0.13651,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,2,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.14286,0.16751,0.0,0.0,0.28571,0.0,0.57143,17,0,2,17,0,2,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,12,0.6667,0.27678,0.04971,0.2857,0.2857,0.28571,0.0,0.28571,1,0,1,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.28571,1e-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]]}]},{"i":"c83a968299724c67","q":"Let  $ABCDE$  be a convex pentagon. Let  $P$  be the intersection of the lines  $CE$  and  $BD$ . Assume that  $\\angle PAD = \\angle ACB$  and  $\\angle CAP = \\angle EDA$ . Prove that the circumcentres of the triangles  $ABC$  and  $ADE$  are collinear with  $P$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,69,0.0,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,69,0.058,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,69,0.1159,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.0267,0.12587,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,69,0.2319,0.03125,0.08552,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,69,0.3478,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,69,0.4058,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,69,0.4638,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,69,0.5217,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,69,0.5797,0.03571,0.11294,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,69,0.6377,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,69,0.6957,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,69,0.7536,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[56,69,0.8116,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,69,0.8696,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,69,0.9275,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,69,0.9855,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[69,69,1.0,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04017,"p":[[0,69,0.0,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,69,0.058,0.03125,0.1504,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,69,0.1159,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,69,0.1739,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,69,0.2319,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,69,0.2899,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,69,0.3478,0.04017,0.12987,0.0,0.0,0.0,0.0,0.571,29,0,0,29,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,69,0.4058,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[40,69,0.5797,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"d739dc99c49531c3","q":"Let  $ABCD$  be a parallelogram and let  $P{}$  be a point inside it such that  $\\angle PDA= \\angle PBA$ . Let  $\\omega_1$  be the excircle of  $PAB$  opposite to the vertex  $A{}$ . Let  $\\omega_2$  be the incircle of the triangle  $PCD$ . Prove that one of the common tangents of  $\\omega_1$  and  $\\omega_2$  is parallel to  $AD$ .\n\n*Ivan Frolov*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,31,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"5ef0a88ca695ff14","q":"Let  $ABC$  be an equilateral triangle such that length of its altitude is  $1$ . Circle with center on the same side of line  $AB$  as point  $C$  and radius  $1$  touches side  $AB$ . Circle rolls on the side  $AB$ . While the circle is rolling, it constantly intersects sides  $AC$  and  $BC$ . Prove that length of an arc of the circle, which lies inside the triangle, is constant","t":[{"b":3,"e":0.4286,"k":"flat","v":0.18295,"x":0.26785,"p":[[0,12,0.0,0.18295,0.15253,0.0,0.21428,0.28571,0.0,0.4286,11,0,1,11,0,5,0,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.23655,0.19093,0.0,0.2857,0.28571,0.0,0.71429,9,0,1,9,0,4,0,0,12,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[8,12,0.6667,0.24554,0.15663,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,10,0,0,11,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[12,12,1.0,0.26785,0.12242,0.14286,0.2857,0.32143,0.0,0.4286,2,0,0,2,0,8,0,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.21426,"x":0.30803,"p":[[0,13,0.0,0.21426,0.1988,0.0,0.14286,0.42857,0.0,0.57143,10,0,1,10,0,9,0,0,4,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[4,13,0.3077,0.24554,0.20897,0.0,0.2857,0.42857,0.0,0.85714,9,0,1,9,0,6,0,0,6,0,0,9,0,0,1,0,0,0,0,0,1,0,0],[8,13,0.6154,0.25,0.17128,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,7,0,0,15,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[12,13,0.9231,0.30803,0.26752,0.14286,0.2857,0.42857,0.0,1.0,5,2,0,5,0,9,0,0,8,0,0,5,0,0,1,0,0,1,0,0,1,0,2],[13,13,1.0,0.23661,0.15815,0.14286,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,9,0,0,12,0,0,5,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"87dc73e3ecb7d57b","q":"Let  $a$ ,  $b$  be positive integers such that  $b^n+n$  is a multiple of  $a^n+n$  for all positive integers  $n$ . Prove that  $a=b$ .\n\n*Proposed by Mohsen Jamali, Iran*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.08482,"p":[[0,16,0.0,0.0625,0.11258,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.08482,0.14223,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.04009,0.07337,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":3,"e":0.0,"k":"flat","v":0.03572,"x":0.14723,"p":[[0,49,0.0,0.03572,0.10102,0.0,0.0,0.0,0.0,0.4286,28,0,1,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.10714,0.16366,0.0,0.0,0.14287,0.0,0.57143,20,0,0,20,0,5,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,49,0.1633,0.06241,0.12841,0.0,0.0,0.035,0.0,0.4286,24,0,0,24,0,5,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.09375,0.14987,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,4,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,49,0.3265,0.14723,0.19393,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,4,0,0,3,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[20,49,0.4082,0.08482,0.15916,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.05804,0.12807,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.04464,0.0974,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.0625,0.13803,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.09366,0.14984,0.0,0.0,0.17857,0.0,0.42857,22,0,0,22,0,2,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.07589,0.13825,0.0,0.0,0.03572,0.0,0.4286,24,0,0,24,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.0625,0.12846,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.12946,0.17627,0.0,0.0,0.2857,0.0,0.4286,20,0,0,20,0,1,0,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.10705,0.1597,0.0,0.0,0.2857,0.0,0.42857,21,0,0,21,0,2,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e378ebcc94ab5862","q":"Let  $a_1$ ,  $a_2$ ,  $a_3$ ,  $\\ldots$  be a sequence of positive integers such that  $a_1=2021$  and   $$ \\sqrt{a_{n+1}-a_n}=\\lfloor \\sqrt{a_n} \\rfloor.  $$  Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence. \n\n*Proposed by Li4, Tsung-Chen Chen, and Ming Hsiao.*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.11161,"x":0.14714,"p":[[0,7,0.0,0.12947,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.14714,0.05631,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.06687,"x":0.125,"p":[[0,14,0.0,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.125,0.05922,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.06687,0.0712,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7751ac59a3aba1d3","q":"Let  $A_1, A_2, ... , A_k$  be the subsets of  $\\left\\{1,2,3,...,n\\right\\}$  such that for all  $1\\leq i,j\\leq k$ : $A_i\\cap A_j \\neq \\varnothing$ . Prove that there are  $n$  distinct positive integers  $x_1,x_2,...,x_n$  such that for each  $1\\leq j\\leq k$ :  $$ lcm_{i \\in A_j}\\left\\{x_i\\right\\}>lcm_{i \\notin A_j}\\left\\{x_i\\right\\} $$  *Proposed by Morteza Saghafian, Mahyar Sefidgaran*","t":[{"b":2,"e":0.0,"k":"falling","v":0.01786,"x":0.48214,"p":[[0,20,0.0,0.45982,0.47074,0.0,0.28571,1.0,0.0,1.0,15,13,1,15,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,13],[4,20,0.2,0.48214,0.48936,0.0,0.21429,1.0,0.0,1.0,15,15,2,15,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[8,20,0.4,0.19643,0.37244,0.0,0.0,0.07143,0.0,1.0,24,5,2,24,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,5],[12,20,0.6,0.20982,0.39927,0.0,0.0,0.0,0.0,1.0,25,6,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,6],[16,20,0.8,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,1,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,20,1.0,0.13393,0.25985,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,1,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,1]]},{"b":3,"e":1.0,"k":"rising","v":0.50892,"x":1.0,"p":[[0,23,0.0,0.50892,0.47774,0.0,0.49979,1.0,0.0,1.0,14,15,2,14,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,15],[4,23,0.1739,0.75893,0.41101,0.67857,1.0,1.0,0.0,1.0,7,23,2,7,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,23],[8,23,0.3478,0.86161,0.31841,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,25],[12,23,0.5217,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[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":"a05ad2b110287a08","q":"Let  $D$ ,  $E$  and  $F$  be points in which incircle of triangle  $ABC$  touches sides  $BC$ ,  $CA$  and  $AB$ , respectively, and let  $I$  be a center of that circle.Furthermore, let  $P$  be a foot of perpendicular from point  $I$  to line  $AD$ , and let  $M$  be midpoint of  $DE$ . If  $\\{N\\}=PM\\cap{AC}$ , prove that  $DN \\parallel EF$","t":[{"b":3,"e":0.14286,"k":"flat","v":0.00893,"x":0.04018,"p":[[0,17,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01339,"x":0.03571,"p":[[0,15,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8c9087ef9819f488","q":"Let  $M$  - be a midpoint of side  $BC$  in triangle  $ABC$ . A cricumcircle of  $ABM$  intersects segment  $AC$  at points  $A$  and  $B_1$  ( $B_1 \\neq A$ ). A circumcircle of  $AMC$  intersects segment  $AB$  at points  $A$  and  $C_1$  ( $C_1 \\neq A$ ). Let  $O$  be a circumcircle of  $AC_1B_1$ . Prove that  $OB=OC$","t":[{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.04464,"p":[[0,11,0.0,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,11,0.3636,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,26,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b99e4e2cbe827287","q":"Let  $I$  be the incenter of triangle  $ABC$ ,  $O_1$  a circle through  $B$  tangent to  $CI$ , and  $O_2$  a circle through  $C$  tangent to  $BI$ . Prove that  $O_1$ , $O_2$  and the circumcircle of  $ABC$  have a common point.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,27,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,16,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7a15809ba11808cd","q":"Let  $ABC$  be an acute triangle such that  $AB<AC$  with orthocenter  $H$ . The altitudes  $BH$  and  $CH$  intersect  $AC$  and  $AB$  at  $B_{1}$  and  $C_{1}$ . Denote by  $M$  the midpoint of  $BC$ . Let  $l$  be the line parallel to  $BC$  passing through  $A$ . The circle around  $ CMC_{1}$  meets the line  $l$  at points  $X$  and  $Y$ , such that  $X$  is on the same side of the line  $AH$  as  $B$  and  $Y$  is on the same side of  $AH$  as  $C$ . The lines  $MX$  and  $MY$  intersect  $CC_{1}$  at  $U$  and  $V$  respectively. Show that the circumcircles of  $ MUV$  and  $ B_{1}C_{1}H$  are tangent.\n\n*Authored by Nikola Velov*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.05357,"x":0.14286,"p":[[0,30,0.0,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,1,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.28571,14,0,2,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.09375,0.09852,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.12929,0.10924,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,19,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.12045,0.0807,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,21,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.09804,0.0661,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.14286,0.09449,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,18,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.10705,0.09446,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,16,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.14286,"k":"flat","v":0.06233,"x":0.10268,"p":[[0,41,0.0,0.0625,0.08703,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.08473,0.09348,0.0,0.07,0.14286,0.0,0.28571,16,0,2,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.06233,0.07067,0.0,0.0,0.14286,0.0,0.1429,18,0,2,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.09813,0.08323,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.0958,0.06575,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,1,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b3e1edba2680d5a2","q":"Let  $ABC$  be a non-isosceles triangle with circumcircle  $k$ , incenter  $I$  and  $C$ -excenter  $I_C$ . Let  $M$  be the midpoint of  $AB$  and  $N$  be the midpoint of arc  $\\widehat{ACB}$  on  $k$ . Prove that  $\\angle IMI_C + \\angle INI_C = 180^{\\circ}$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.00893,"p":[[0,4,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,24,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.00893,"x":0.10268,"p":[[0,11,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,23,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,24,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.08928,0.16656,0.0,0.0,0.07143,0.0,0.57143,24,0,14,24,0,0,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[11,11,1.0,0.10268,0.20277,0.0,0.0,0.2857,0.0,1.0,23,1,11,23,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"85c25f24a178e2ce","q":"Let  $p$  be a prime number for which  $\\frac{p-1}{2}$  is also prime, and let  $a,b,c$  be integers not divisible by  $p$ . Prove that there are at most  $1+\\sqrt {2p}$  positive integers  $n$  such that  $n<p$  and  $p$  divides  $a^n+b^n+c^n$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.00223,"x":0.04464,"p":[[0,22,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.02009,0.04803,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.03125,0.05629,0.0,0.0,0.01786,0.0,0.14286,24,0,0,24,2,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.04464,0.06376,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,2,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.03349,0.0592,0.0,0.0,0.01786,0.0,0.143,24,0,1,24,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02223,"p":[[0,17,0.0,0.00446,0.01729,0.0,0.0,0.0,0.0,0.07143,30,0,0,30,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.01786,0.04374,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.01116,0.0362,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fe549bd622a96dc7","q":"Let  $ABC$  be a scalene triangle,  $M$  be the midpoint of  $BC,P$  be the common point of  $AM$  and the incircle of  $ABC$  closest to  $A$ , and  $Q$  be the common point of the ray  $AM$  and the excircle farthest from  $A$ . The tangent to the incircle at  $P$  meets  $BC$  at point  $X$ , and the tangent to the excircle at  $Q$  meets  $BC$  at  $Y$ . Prove that  $MX=MY$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.00447,"x":0.02679,"p":[[0,10,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.01777,0.04702,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00893,"x":0.02232,"p":[[0,9,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d911b39d634780a0","q":"Let  $a$ ,  $b$  and  $n$  be positive integers, satisfying that  $bn$  divides  $an-a+1$ . Let  $\\alpha=a/b$ . Prove that, when the numbers  $\\lfloor\\alpha\\rfloor,\\lfloor2\\alpha\\rfloor,\\dots,\\lfloor(n-1)\\alpha\\rfloor$  are divided by  $n$ , the residues are  $1,2,\\dots,n-1$ , in some order.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.07589,"p":[[0,36,0.0,0.05804,0.18161,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,36,0.1111,0.07134,0.17493,0.0,0.0,0.0,0.0,0.85714,25,0,1,25,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[8,36,0.2222,0.07589,0.23141,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[12,36,0.3333,0.04688,0.16522,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,36,0.4444,0.04911,0.11495,0.0,0.0,0.01786,0.0,0.57143,24,0,0,24,2,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,36,0.5556,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.01339,0.03762,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04464,"x":0.11607,"p":[[0,8,0.0,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.11607,0.18363,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,8,1.0,0.09143,0.09484,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,13,0,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0906bfbb86f89793","q":"Let  $ABC$  be a triangle with  $AB \\ne AC$ . The incirle of triangle  $ABC$  is tangent to  $BC, CA, AB$  at  $D, E, F$ , respectively. The perpendicular line from  $D$  to  $EF$  intersects  $AB$  at  $X$ . The second intersection point of circumcircles of triangles  $AEF$  and  $ABC$  is  $T$ . Prove that  $TX \\perp T F$","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,23,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,27,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0133,0.05465,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"00b9d929a8146cb4","q":"Let  $D,E,F$  be the points of tangency of the incircle of a given triangle  $ABC$  with sides  $BC, CA, AB,$  respectively. Denote by  $I$  the incenter of  $ABC$ , by  $M$  the midpoint of  $BC$  and by  $G$  the foot of the perpendicular from  $M$  to line  $EF$ . Prove that the line  $ID$  is tangent to the circumcircle of the triangle  $MGI$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.02679,"x":0.1517,"p":[[0,30,0.0,0.08482,0.13296,0.0,0.0,0.14287,0.0,0.42857,21,0,1,21,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.1517,0.17835,0.0,0.14286,0.14287,0.0,0.57143,13,0,1,13,0,12,0,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[8,30,0.2667,0.09822,0.12595,0.0,0.0,0.14287,0.0,0.42857,17,0,0,17,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.04009,0.0816,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.08483,0.11215,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.05134,0.08033,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,8,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"falling","v":0.02679,"x":0.2232,"p":[[0,25,0.0,0.2232,0.19539,0.0,0.14288,0.32143,0.0,0.71429,9,0,0,9,0,8,0,0,7,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[4,25,0.16,0.08473,0.11765,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.10259,0.11968,0.0,0.0,0.1786,0.0,0.28571,17,0,1,17,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.11598,0.1097,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,15,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.06697,0.10092,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.02679,0.05577,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f717d4b445576a7c","q":"Let $A B C D$ be a parallelogram. Point $E$ lies on segment $C D$ such that  $$ 2 \\angle A E B=\\angle A D B+\\angle A C B, $$  and point $F$ lies on segment $B C$ such that  $$ 2 \\angle D F A=\\angle D C A+\\angle D B A . $$  Let $K$ be the circumcenter of triangle $A B D$. Prove that $K E=K F$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"883634c2647bd65f","q":"Let  $ABC$  be an acute triangle with  $\\angle ABC > 45^{\\circ}$  and  $\\angle ACB > 45^{\\circ}$ . Let  $M$  be the midpoint of the side  $BC$ . The circumcircle of triangle  $ABM$  crosses the side  $AC$  again at  $X$  and the circumcircle of triangle  $AMC$  crosses the side  $AB$  again at  $Y$ . The point  $P$  lies on the perpendicular bisector of the segment  $BC$ , so that the points  $A$  and  $P$  lie on the same side of  $XY$ , and  $\\angle YPX = 90^{\\circ} + \\angle BAC$ . Prove that the circumcircles of triangles  $BYP$  and  $CXP$  are tangent.","t":[{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.16962,"p":[[0,20,0.0,0.13829,0.10993,0.14214,0.14286,0.14286,0.0,0.571,7,0,1,7,0,21,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,20,0.2,0.16962,0.18702,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,9,0,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[8,20,0.4,0.12503,0.16273,0.0,0.07143,0.14286,0.0,0.571,16,0,0,16,0,10,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,20,0.6,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.08482,0.14223,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.03572,"x":0.13393,"p":[[0,20,0.0,0.10705,0.08745,0.0,0.14286,0.14286,0.0,0.42857,10,0,1,10,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.13393,0.13333,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,18,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.09821,0.0974,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,14,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.125,0.09942,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,19,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.06696,0.07973,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3a0b0040e443445e","q":"Let  $\\omega$  be a circle in the plane and  $A,B$  two points lying on it. We denote by  $M$  the midpoint of  $AB$  and let  $P \\ne M$  be a new point on  $AB$ . Build circles  $\\gamma$  and  $\\delta$  tangent to  $AB$  at  $P$  and to  $\\omega$  at  $C$ , respectively  $D$ . Consider  $E$  to be the point diametrically opposed to  $D$  in  $\\omega$ . Prove that the circumcenter of  $\\triangle BMC$  lies on the line  $BE$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"8a7398573a0356a3","q":"Let  $ABC$  be a triangle with circumcircle  $w$ . Let  $D$  be the midpoint of arc  $BC$  that contains  $A$ . Define  $E$  and  $F$  similarly. Let the incircle of  $ABC$  touches  $BC,CA,AB$  at  $K,L,M$  respectively. Prove that  $DK,EL,FM$  are concurrent.","t":[{"b":2,"e":0.57143,"k":"rising","v":0.0,"x":0.32139,"p":[[0,18,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.04018,0.15663,0.0,0.0,0.0,0.0,0.71429,30,0,4,30,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.26337,0.26268,0.0,0.2143,0.571,0.0,0.71429,12,0,0,12,0,4,0,0,6,0,0,1,0,0,5,0,0,4,0,0,0,0,0],[16,18,0.8889,0.32139,0.2766,0.10714,0.2143,0.57111,0.0,0.71429,8,0,0,8,0,8,0,0,2,0,0,3,0,0,4,0,0,7,0,0,0,0,0],[18,18,1.0,0.20527,0.23676,0.0,0.14286,0.2857,0.0,0.85714,11,0,0,11,0,11,0,0,4,0,0,1,0,0,2,0,0,2,0,0,1,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,72,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,15,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,72,0.0556,0.02232,0.10171,0.0,0.0,0.0,0.0,0.57143,30,0,14,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,72,0.1111,0.07589,0.22299,0.0,0.0,0.0,0.0,0.85714,28,0,13,28,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[12,72,0.1667,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,20,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,72,0.2222,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,16,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,72,0.2778,0.0625,0.20183,0.0,0.0,0.0,0.0,0.85714,29,0,16,29,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[24,72,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,72,0.3889,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,20,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[32,72,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,72,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,72,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,26,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,72,0.6111,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,29,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,72,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,72,0.7222,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,30,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[56,72,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,72,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,26,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9f38a1c91dc1a416","q":"Let  $ABC$  be an acute triangle with altitudes  $AD$ ,  $BE$ ,  $CF$  where  $D$ ,  $E$ ,  $F$  lie on  $BC$ ,  $AC$ ,  $AB$ , respectively. Let  $M$  be the midpoint of  $BC$ . The circumcircle of triangle  $AEF$  cuts the line  $AM$  at  $A$  and  $X$ . The line  $AM$  cuts the line  $CF$  at  $Y$ . Let  $Z$  be the point of intersection of  $AD$  and  $BX$ . Show that the lines  $YZ$  and  $BC$  are parallel.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"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],[4,26,0.1538,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,1,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,28,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,28,1.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"e217ee89ee5a643e","q":"Let  $p=ab+bc+ac$  be a prime number where  $a,b,c$  are different two by two, show that  $a^3,b^3,c^3$  gives different residues modulo  $p$","t":[{"b":3,"e":0.0,"k":"falling","v":0.00446,"x":0.26339,"p":[[0,16,0.0,0.22768,0.30064,0.0,0.14286,0.28571,0.0,1.0,13,3,0,13,0,8,0,0,5,0,0,0,0,0,3,0,0,0,0,0,0,0,3],[4,16,0.25,0.26339,0.37475,0.0,0.0,0.28571,0.0,1.0,17,5,0,17,0,3,0,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,5],[8,16,0.5,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,16,0.75,0.19642,0.34763,0.0,0.0,0.21429,0.0,1.0,23,3,0,23,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,1,0,3],[16,16,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.00446,"x":0.25891,"p":[[0,12,0.0,0.25891,0.35791,0.0,0.14286,0.28571,0.0,1.0,15,4,0,15,0,6,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2,0,4],[4,12,0.3333,0.24089,0.30818,0.0,0.14286,0.2857,0.0,1.0,12,3,0,12,0,8,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,3],[8,12,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5c9d4eeac027266d","q":"Let  $a$ ,  $b$ ,  $c$  be the lengths of the sides of a triangle  $ABC$ . Prove that  $$ a^2(p-a)(p-b)+b^2(p-b)(p-c)+c^2(p-c)(p-a)\\leqslant\\frac{4}{27}p^4, $$  where  $p$  is the half-perimeter of the triangle  $ABC$ .","t":[{"b":0,"e":0.571,"k":"flat","v":0.53565,"x":0.5982,"p":[[0,10,0.0,0.59374,0.2969,0.28571,0.57143,0.85714,0.0,1.0,1,7,0,1,0,3,0,0,5,0,0,1,0,0,9,0,0,4,0,0,2,0,7],[4,10,0.4,0.5982,0.2299,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,2,0,0,1,0,0,10,0,0,5,0,0,8,0,0,2,0,4],[8,10,0.8,0.56697,0.0977,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,23,0,0,1,0,0,2,0,0],[10,10,1.0,0.53565,0.11292,0.571,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,4,0,0,3,0,0,22,0,0,3,0,0,0,0,0]]},{"b":1,"e":0.2857,"k":"falling","v":0.41515,"x":0.69193,"p":[[0,30,0.0,0.69193,0.19597,0.571,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,7,0,0,7,0,0,6,0,5],[4,30,0.1333,0.68301,0.19801,0.57132,0.64286,0.75,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,8,0,0,1,0,7],[8,30,0.2667,0.65622,0.20159,0.57143,0.57143,0.74996,0.14286,1.0,0,5,0,0,0,1,0,0,0,0,0,4,0,0,14,0,0,5,0,0,3,0,5],[12,30,0.4,0.52679,0.25108,0.28571,0.5005,0.71429,0.14286,1.0,0,4,0,0,0,3,0,0,6,0,0,7,0,0,7,0,0,4,0,0,1,0,4],[16,30,0.5333,0.5134,0.19186,0.42857,0.4286,0.57143,0.0,1.0,1,2,0,1,0,0,0,0,3,0,0,13,0,0,9,0,0,4,0,0,0,0,2],[20,30,0.6667,0.45088,0.15197,0.42857,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,0,0,0,6,0,0,14,0,0,9,0,0,1,0,0,1,0,0],[24,30,0.8,0.54461,0.20957,0.42857,0.4998,0.57143,0.14286,1.0,0,3,0,0,0,1,0,0,3,0,0,12,0,0,9,0,0,2,0,0,2,0,3],[28,30,0.9333,0.45084,0.12926,0.42857,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,14,0,0,11,0,0,1,0,0,0,0,0],[30,30,1.0,0.41515,0.17983,0.28571,0.42857,0.571,0.0,1.0,1,1,0,1,0,3,0,0,6,0,0,13,0,0,8,0,0,0,0,0,0,0,1]]}]},{"i":"9bb525184e3a8285","q":"Let  $x,y$  and  $z$  be positive real numbers such that  $xy+yz+xz=3xyz$ . Prove that \\[ x^2y+y^2z+z^2x \\ge 2(x+y+z)-3 \\] and determine when equality holds.\n\n*UK - David Monk*","t":[{"b":1,"e":0.42857,"k":"flat","v":0.36161,"x":0.49552,"p":[[0,15,0.0,0.49552,0.25249,0.39286,0.42857,0.71429,0.0,1.0,1,2,0,1,0,3,0,0,4,0,0,13,0,0,2,0,0,3,0,0,4,0,2],[4,15,0.2667,0.42857,0.23958,0.28571,0.42857,0.46431,0.0,0.85714,2,0,0,2,0,5,0,0,3,0,0,14,0,0,1,0,0,3,0,0,4,0,0],[8,15,0.5333,0.44641,0.20747,0.28571,0.42857,0.57111,0.0,1.0,1,1,0,1,0,2,0,0,7,0,0,13,0,0,3,0,0,4,0,0,1,0,1],[12,15,0.8,0.4375,0.21998,0.28571,0.42857,0.42858,0.14286,1.0,0,2,0,0,0,3,0,0,10,0,0,12,0,0,1,0,0,3,0,0,1,0,2],[15,15,1.0,0.36161,0.11837,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,4,0,0,4,0,0,23,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.3482,"x":0.51339,"p":[[0,14,0.0,0.42409,0.16163,0.28571,0.42857,0.42858,0.14286,0.85714,0,0,0,0,0,2,0,0,8,0,0,16,0,0,3,0,0,1,0,0,2,0,0],[4,14,0.2857,0.51339,0.28984,0.28571,0.42857,0.75,0.0,1.0,1,5,0,1,0,3,0,0,6,0,0,11,0,0,1,0,0,2,0,0,3,0,5],[8,14,0.5714,0.43302,0.20665,0.39286,0.42857,0.4642,0.0,0.85714,1,0,0,1,0,5,0,0,2,0,0,16,0,0,2,0,0,4,0,0,2,0,0],[12,14,0.8571,0.37933,0.11595,0.28571,0.42857,0.42857,0.14286,0.71,0,0,0,0,0,3,0,0,8,0,0,19,0,0,1,0,0,1,0,0,0,0,0],[14,14,1.0,0.3482,0.14256,0.2857,0.42857,0.42857,0.0,0.5714,2,0,0,2,0,4,0,0,6,0,0,18,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"4382f59f63977c87","q":"Let  $ABC$  be a triangle and let  $D, E$  and  $F$  be the midpoints of sides  $BC, CA$  and  $AB$ , respectively.\nLet  $X\\ne A$  be the intersection of  $AD$  with the circumcircle of  $ABC$ . Let  $\\Omega$  be the circle through  $D$  and  $X$ ,\ntangent to the circumcircle of  $ABC$ . Let  $Y$  and  $Z$  be the intersections of the tangent to  $\\Omega$  at  $D$  with the\nperpendicular bisectors of segments  $DE$  and  $DF$ , respectively. Let  $P$  be the intersection of  $YE$  and  $ZF$  and\nlet  $G$  be the centroid of  $ABC$ . Show that the tangents at  $B$  and  $C$  to the circumcircle of  $ABC$  and the line  $PG$  are concurrent.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.11161,"p":[[0,22,0.0,0.08929,0.16269,0.0,0.0,0.14286,0.0,0.85714,19,0,1,19,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,22,0.1818,0.09821,0.14032,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,12,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,22,0.3636,0.11161,0.1504,0.0,0.0,0.14287,0.0,0.57143,17,0,0,17,0,9,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,22,0.5455,0.06696,0.10092,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.04911,0.11072,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.14286,"k":"flat","v":0.00893,"x":0.14277,"p":[[0,24,0.0,0.11152,0.18807,0.0,0.07,0.14286,0.0,1.0,16,1,0,16,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,24,0.1667,0.08025,0.12837,0.0,0.0,0.14286,0.0,0.571,19,0,0,19,0,11,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,24,0.3333,0.05357,0.1171,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,24,0.5,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.14277,0.08748,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,24,1.0,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4d147ae7fb1cfc3c","q":"Let  $O$  be the circumcenter and  $I$  be the incenter of an acute triangle  $ABC$  with  $m(\\widehat{B}) \\neq m(\\widehat{C})$ . Let  $D$ ,  $E$ ,  $F$  be the midpoints of the sides  $[BC]$ ,  $[CA]$ ,  $[AB]$ , respectively. Let  $T$  be the foot of perpendicular from  $I$  to  $[AB]$ . Let  $P$  be the circumcenter of the triangle  $DEF$  and  $Q$  be the midpoint of  $[OI]$ . If  $A$ ,  $P$ ,  $Q$  are collinear, prove that \\[\\dfrac{|AO|}{|OD|}-\\dfrac{|BC|}{|AT|}=4.\\]","t":[{"b":2,"e":0.42857,"k":"flat","v":0.44643,"x":0.62048,"p":[[0,54,0.0,0.5223,0.27804,0.42857,0.571,0.71429,0.0,1.0,3,2,1,3,0,3,0,0,1,0,0,8,0,0,4,0,0,8,0,0,3,0,2],[4,54,0.0741,0.44643,0.28736,0.24999,0.42857,0.71429,0.0,1.0,2,3,0,2,0,6,0,0,6,0,0,7,0,0,1,0,0,6,0,0,1,0,3],[8,54,0.1481,0.49552,0.28343,0.14286,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,9,0,0,3,0,0,5,0,0,1,0,0,8,0,0,5,0,1],[12,54,0.2222,0.52229,0.3001,0.28571,0.42857,0.71429,0.0,1.0,2,5,0,2,0,4,0,0,3,0,0,8,0,0,4,0,0,4,0,0,2,0,5],[16,54,0.2963,0.52676,0.2511,0.39286,0.57143,0.71429,0.0,1.0,1,1,0,1,0,5,0,0,2,0,0,5,0,0,5,0,0,11,0,0,2,0,1],[20,54,0.3704,0.51786,0.27606,0.28571,0.57143,0.71429,0.0,1.0,1,2,0,1,0,6,0,0,2,0,0,6,0,0,5,0,0,6,0,0,4,0,2],[24,54,0.4444,0.47766,0.21609,0.2857,0.42859,0.71429,0.14286,0.85714,0,0,0,0,0,4,0,0,8,0,0,5,0,0,4,0,0,10,0,0,1,0,0],[28,54,0.5185,0.47765,0.21608,0.42857,0.4998,0.57143,0.0,0.85714,1,0,0,1,0,5,0,0,1,0,0,9,0,0,9,0,0,5,0,0,2,0,0],[32,54,0.5926,0.51781,0.19803,0.42857,0.571,0.71429,0.0,0.85714,1,0,0,1,0,2,0,0,2,0,0,10,0,0,6,0,0,10,0,0,1,0,0],[36,54,0.6667,0.59374,0.16016,0.57132,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,5,0,0,10,0,0,14,0,0,1,0,0],[40,54,0.7407,0.57584,0.15765,0.42857,0.57141,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,9,0,0,11,0,0,6,0,0,4,0,0],[44,54,0.8148,0.54458,0.13569,0.42857,0.571,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,9,0,0,11,0,0,9,0,0,0,0,0],[48,54,0.8889,0.54908,0.13882,0.42857,0.57143,0.60714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,11,0,0,12,0,0,7,0,0,1,0,0],[52,54,0.963,0.62048,0.13177,0.571,0.57143,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,12,0,0,14,0,0,0,0,1],[54,54,1.0,0.58032,0.15542,0.42857,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,7,0,0,11,0,0,10,0,0,2,0,0]]},{"b":3,"e":0.42857,"k":"flat","v":0.40624,"x":0.52677,"p":[[0,11,0.0,0.52677,0.30396,0.28571,0.4998,0.71429,0.0,1.0,3,4,1,3,0,3,0,0,3,0,0,7,0,0,3,0,0,6,0,0,3,0,4],[4,11,0.3636,0.49997,0.27198,0.28571,0.42857,0.71429,0.0,1.0,2,3,1,2,0,3,0,0,4,0,0,9,0,0,4,0,0,5,0,0,2,0,3],[8,11,0.7273,0.42856,0.24484,0.24999,0.42857,0.60714,0.0,0.85714,2,0,1,2,0,6,0,0,3,0,0,11,0,0,2,0,0,5,0,0,3,0,0],[11,11,1.0,0.40624,0.10168,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,2,0,0,24,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"2d1245a9c3bea8df","q":"Let  $ABC$  be a scalene acute-angled triangle with its incenter  $I$  and circumcircle  $\\Gamma$ .\nLine  $AI$  intersects  $\\Gamma$  for the second time at  $M$ . Let  $N$  be the midpoint of  $BC$  and  $T$  be the point\non  $\\Gamma$  such that  $IN \\perp MT$ . Finally, let  $P $  and  $Q$  be the intersection points of  $TB $  and  $TC$ ,\nrespectively, with the line perpendicular to  $AI$  at  $I$ . Show that  $PB = CQ$ .\n*Proposed by Patrik Bak - Slovakia*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,28,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,28,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"741caaecab331bf7","q":"In triangle  $ABC$ , the perpendicular bisectors of sides  $AB$  and  $BC$  intersect side  $AC$  at points  $P$  and  $Q$ , respectively, with point  $P$  lying on the segment  $AQ$ . Prove that the circumscribed circles of the triangles  $PBC$  and  $QBA$  intersect on the bisector of the angle  $PBQ$ .","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.48214,"p":[[0,23,0.0,0.48214,0.45281,0.0,0.28571,1.0,0.0,1.0,12,13,1,12,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,13],[4,23,0.1739,0.22098,0.35596,0.0,0.0,0.2857,0.0,1.0,19,5,1,19,1,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[8,23,0.3478,0.31249,0.39517,0.0,0.14286,0.57111,0.0,1.0,15,7,1,15,0,3,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,7],[12,23,0.5217,0.04911,0.18073,0.0,0.0,0.0,0.0,1.0,28,1,1,28,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,23,0.6957,0.04018,0.12993,0.0,0.0,0.0,0.0,0.71429,27,0,2,27,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[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],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.41072,"p":[[0,25,0.0,0.40625,0.4603,0.0,0.14285,1.0,0.0,1.0,16,11,0,16,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,11],[4,25,0.16,0.40179,0.46898,0.0,0.07143,1.0,0.0,1.0,16,12,1,16,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[8,25,0.32,0.30357,0.41458,0.0,0.0,0.46429,0.0,1.0,17,8,0,17,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,8],[12,25,0.48,0.32589,0.43188,0.0,0.0,1.0,0.0,1.0,17,9,0,17,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,9],[16,25,0.64,0.34375,0.443,0.0,0.0,1.0,0.0,1.0,18,9,1,18,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,9],[20,25,0.8,0.41072,0.45702,0.0,0.21431,1.0,0.0,1.0,15,11,0,15,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,11],[24,25,0.96,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"262d5bc2ab7fc073","q":"Let  $P$  and  $Q$  be arbitrary points on the side  $BC$  of triangle ABC such that  $BP = CQ$ . The common points of segments  $AP$  and  $AQ$  with the incircle form a quadrilateral  $XYZT$ . Find the locus of common points of diagonals of such quadrilaterals.","t":[{"b":3,"e":0.14286,"k":"rising","v":0.03795,"x":0.23656,"p":[[0,15,0.0,0.03795,0.0714,0.0,0.0,0.01786,0.0,0.28571,24,0,2,24,1,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.08017,0.12325,0.0,0.0,0.14286,0.0,0.571,19,0,2,19,0,10,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.23656,0.12183,0.14286,0.14288,0.28571,0.0,0.43,1,0,0,1,0,16,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.19857,0.11118,0.14286,0.14288,0.2857,0.0,0.4286,3,0,0,3,0,16,0,1,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.2008,0.16315,0.14214,0.14286,0.2857,0.0,0.57143,7,0,0,7,0,13,0,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.05125,"x":0.1517,"p":[[0,5,0.0,0.05125,0.07615,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,1,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.1271,0.0906,0.0525,0.14286,0.14286,0.0,0.28571,8,0,0,8,1,18,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.1517,0.10063,0.14214,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,16,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"024db8c4aa3c45e3","q":"Let  $ABCD$  be a non-isosceles trapezoid with  $AB \\parallel CD$ . A circle through  $A$  and  $B$  meets  $AD$ ,  $BC$  at  $E, F$ . The segments  $AF, BE$  meet at  $G$ . The circumcircles of  $\\triangle ADG$  and  $\\triangle BCG$  meet at  $H$ . Show that if  $GD=GC$ ,  $H$  is the orthocenter of  $\\triangle ABG$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,19,0.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,41,0.0,0.06696,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.06696,0.11837,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,4,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[40,41,0.9756,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e8a562570cb21a3f","q":"Let  $\\omega$  be the circumcircle of a triangle  $ABC$ . Let  $P$  be any point on  $\\omega$  different than the verticies of the triangle.\nLine  $AP$  intersects  $BC$  at  $D$ ,  $BP$  intersects  $AC$  at  $E$  and  $CP$  intersects  $AB$  at  $F$ .  Let  $X$  be the projection of  $D$  onto line passing through midpoints of  $AP$  and  $BC$ ,  $Y$  be the projection of  $E$  onto line passing through  $BP$  and  $AC$  and let  $Z$  be the projection of  $F$  onto line passing through midpoints of  $CP$  and  $AB$ . Let  $Q$  be the circumcenter of triangle  $XYZ$ . Prove that all possible points  $Q$ , corresponding to different positions of  $P$  lie on one circle.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.00438,"x":0.1383,"p":[[0,16,0.0,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.1383,0.12103,0.0,0.14286,0.2857,0.0,0.28571,12,0,0,12,0,9,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.13384,0.12845,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04009,"p":[[0,31,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[31,31,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d6e260733d807a22","q":"Let  $AA_1 , BB_1$ , and  $CC_1$  be the altitudes of the non-isosceles acute-angled triangle  $ABC$ . The circles circumscibred around the triangles  $ABC$  and  $A_1 B_1 C$  intersect again at the point  $P , Z$  is the intersection point of the tangents to the circumscribed circle of the triangle  $ABC$  conducted at points  $A$  and  $B$  . Prove that lines  $AP , BC$  and  $ZC_1$  are concurrent.","t":[{"b":1,"e":0.0,"k":"flat","v":0.02679,"x":0.03571,"p":[[0,8,0.0,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.03571,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,2,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.03116,0.08541,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.05339,"p":[[0,7,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.05339,0.09262,0.0,0.0,0.14071,0.0,0.42857,22,0,12,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"630e2ec47172fdb5","q":"Given a natural number  $n{}$  find the smallest  $\\lambda$  such that\\[\\gcd(x(x + 1)\\cdots(x + n - 1), y(y + 1)\\cdots(y + n - 1)) \\leqslant (x-y)^\\lambda,\\] for any positive integers  $y{}$  and  $x \\geqslant y + n$ .","t":[{"b":2,"e":0.42857,"k":"rising","v":0.08928,"x":0.34822,"p":[[0,13,0.0,0.08928,0.15872,0.0,0.0,0.14286,0.0,0.57143,23,0,5,23,0,2,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,13,0.3077,0.11607,0.2126,0.0,0.0,0.14286,0.0,0.71429,23,0,4,23,0,2,0,0,2,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[8,13,0.6154,0.30357,0.08564,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,23,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[12,13,0.9231,0.34822,0.09407,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,12,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.33929,0.09279,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,14,0,0,15,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"volatile","v":0.08929,"x":0.33022,"p":[[0,15,0.0,0.08929,0.16269,0.0,0.0,0.07143,0.0,0.57143,24,0,2,24,0,0,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.33022,0.16921,0.2857,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,4,0,0,11,0,0,8,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"0f3c2b890c3a02e1","q":"Let\n\\[M=\\{1, 2, 3, \\ldots, 2022\\}\\]\nDetermine the least positive integer  $k$ , such that for every  $k$  subsets of  $M$  with the cardinality of each subset equal to  $3$ , there are two of these subsets with exactly one common element.","t":[{"b":2,"e":0.42857,"k":"flat","v":0.29464,"x":0.4375,"p":[[0,23,0.0,0.39732,0.20434,0.28571,0.42857,0.57143,0.0,0.71429,3,0,3,3,0,4,0,0,4,0,0,10,0,0,8,0,0,3,0,0,0,0,0],[4,23,0.1739,0.4375,0.18877,0.42857,0.42857,0.57143,0.0,0.71429,3,0,2,3,0,1,0,0,3,0,0,12,0,0,10,0,0,3,0,0,0,0,0],[8,23,0.3478,0.29464,0.20183,0.14286,0.28571,0.42857,0.0,0.57143,7,0,3,7,0,3,0,0,10,0,0,5,0,0,7,0,0,0,0,0,0,0,0],[12,23,0.5217,0.40625,0.22047,0.28571,0.42857,0.57143,0.0,0.85714,4,0,2,4,0,2,0,0,5,0,0,10,0,0,7,0,0,3,0,0,1,0,0],[16,23,0.6957,0.39732,0.20743,0.28571,0.42857,0.57143,0.0,0.85714,4,0,4,4,0,2,0,0,5,0,0,9,0,0,11,0,0,0,0,0,1,0,0],[20,23,0.8696,0.38391,0.19376,0.2857,0.42857,0.57143,0.0,0.71429,3,0,3,3,0,3,0,0,7,0,0,9,0,0,8,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.37945,"x":0.41964,"p":[[0,12,0.0,0.41517,0.21828,0.28571,0.42857,0.57143,0.0,0.71429,4,0,4,4,0,2,0,0,5,0,0,7,0,0,10,0,0,4,0,0,0,0,0],[4,12,0.3333,0.37945,0.2219,0.14286,0.42857,0.57143,0.0,0.85714,3,0,2,3,0,6,0,0,5,0,0,7,0,0,8,0,0,2,0,0,1,0,0],[8,12,0.6667,0.39284,0.17127,0.28571,0.42857,0.57111,0.14286,0.71429,0,0,0,0,0,5,0,0,10,0,0,8,0,0,6,0,0,3,0,0,0,0,0],[12,12,1.0,0.41964,0.12846,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,12,0,0,8,0,0,1,0,0,0,0,0]]}]},{"i":"99085c0e7f44b2d3","q":"Let  $ABCD$  be a cyclic quadrilateral. Point  $P$  is on line  $CB$  such that  $CP=CA$ and  $B$  lies between  $C$  and  $P$ . Point  $Q$  is on line  $CD$  such that  $CQ=CA$  and  $D$  lies between  $C$  and  $Q$ . Prove that the incentre of triangle  $ABD$  lies on line  $PQ.$","t":[{"b":5,"e":0.14286,"k":"flat","v":0.02679,"x":0.07143,"p":[[0,21,0.0,0.07143,0.15972,0.0,0.0,0.0,0.0,0.71429,25,0,1,25,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,21,0.1905,0.05804,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0625,0.11259,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.02679,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.06697,0.12364,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0625,0.12846,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04464,"x":0.07589,"p":[[0,30,0.0,0.07589,0.15146,0.0,0.0,0.0,0.0,0.42857,25,0,1,25,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.04464,0.10971,0.0,0.0,0.0,0.0,0.42857,27,0,1,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.06696,0.12364,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.07589,0.13355,0.0,0.0,0.07143,0.0,0.42857,24,0,0,24,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0625,0.11811,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ab9d2f47cedea971","q":"Let  $ABC$  be an acute triangle such that  $\\angle B > \\angle C$ . Let  $D$  and  $E$  be the points on the segments  $BC$  and  $CA$ , respectively, such that  $AD$  bisects  $\\angle A$  and  $BE\\perp AC$ . Finally, let  $M$  be the midpoint of the side  $BC$ . Suppose that the circumcircle of  $\\triangle CDE$  intersects  $AD$  again at a point  $X$  different from  $D$ . Prove that  $\\angle XME = 90^{\\circ} - \\angle BAC$ .","t":[{"b":0,"e":0.2857,"k":"flat","v":0.10714,"x":0.22321,"p":[[0,39,0.0,0.16071,0.17035,0.0,0.14286,0.28571,0.0,0.71429,13,0,0,13,0,7,0,0,9,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,39,0.1026,0.13393,0.14698,0.0,0.14286,0.14287,0.0,0.57143,13,0,0,13,0,12,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,39,0.2051,0.2054,0.16734,0.0,0.14286,0.42857,0.0,0.43,9,0,1,9,0,9,0,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.19643,0.1915,0.0,0.14286,0.32143,0.0,0.71429,12,0,2,12,0,6,0,0,6,0,0,7,0,0,0,0,0,1,0,0,0,0,0],[16,39,0.4103,0.10714,0.14725,0.0,0.0,0.17857,0.0,0.4286,19,0,1,19,0,5,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.12054,0.15199,0.0,0.0,0.1786,0.0,0.4286,17,0,0,17,0,7,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.13384,0.18188,0.0,0.0,0.2857,0.0,0.71429,18,0,1,18,0,4,0,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[28,39,0.7179,0.22321,0.15947,0.10714,0.2857,0.32143,0.0,0.4286,8,0,2,8,0,6,0,0,10,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.20982,0.14718,0.14286,0.2857,0.28571,0.0,0.71429,6,0,1,6,0,9,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[36,39,0.9231,0.22321,0.11258,0.24999,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,2,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.20973,0.10711,0.14286,0.2857,0.28571,0.0,0.28571,5,0,0,5,0,7,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.28571,"k":"flat","v":0.06696,"x":0.25446,"p":[[0,33,0.0,0.125,0.17035,0.0,0.0,0.2857,0.0,0.71429,18,0,1,18,0,4,0,0,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,33,0.1212,0.16964,0.14914,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,5,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.11607,0.15746,0.0,0.0,0.2857,0.0,0.4286,19,0,0,19,0,4,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.14732,0.19061,0.0,0.07143,0.2857,0.0,0.85714,16,0,0,16,0,5,0,0,8,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[16,33,0.4848,0.06696,0.12364,0.0,0.0,0.03571,0.0,0.42857,24,0,3,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.08929,0.10564,0.0,0.0,0.14286,0.0,0.28571,17,0,1,17,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.23661,0.11071,0.14286,0.2857,0.28571,0.0,0.4286,2,0,0,2,0,11,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.2366,0.10479,0.24999,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,4,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.25446,0.07771,0.2857,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"47201d36fe7c62fc","q":"Let  $ABC$  be a triangle with  $AB < AC$ . Its incircle with centre  $I$  touches the sides  $BC, CA,$  and  $AB$  in the points  $D, E,$  and  $F$  respectively. The angle bisector  $AI$  intersects the lines  $DE$  and  $DF$  in the points  $X$  and  $Y$  respectively. Let  $Z$  be the foot of the altitude through  $A$  with respect to  $BC$ .\n\nProve that  $D$  is the incentre of the triangle  $XYZ$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,5,0.0,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,7,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.02679,0.08328,0.0,0.0,0.0,0.0,0.4286,28,0,4,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e7514534ba2e5526","q":"Let  $\\{a_n\\}_{n\\geq 0}$  be a non-decreasing, unbounded sequence of non-negative integers with  $a_0=0$ . Let the number of members of the sequence not exceeding  $n$  be  $b_n$ . Prove that \\[ (a_0 + a_1 + \\cdots + a_m)( b_0 + b_1 + \\cdots + b_n ) \\geq (m + 1)(n + 1). \\]","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,7,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"56d1e396592ef0a8","q":"Let  $ABC$  be an acute scalene triangle with incenter  $I$ , circumcircle  $w_1$ , and denote the circumcircle of  $BIC$  as  $w_2$ . Suppose point  $P$  lies on  $w_2$  and is inside  $w_1$ . Let  $X,Y$  lie on  $BC$  with  $XP \\perp BP, YP \\perp PC$ . Circles  $O_1, O_2$  are drawn tangent to  $w_1$  at points on the same side of  $BC$  as  $A$  and tangent to  $BC$  at  $X,Y$  respectively. Let the centers of those two circles be  $Z_1, Z_2$ . Let  $D$  be the point on  $w_2$  opposite to  $P$  and let  $E$  be the foot of the altitude from  $P$  to  $BC$ . Show that  $DE \\perp Z_1Z_2$","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"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],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,4,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[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],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"6c9e46bf145734df","q":"Let  $B$  and  $D$  be points on segments  $[AE]$  and  $[AF]$  respectively. Excircles of triangles  $ABF$  and  $ADE$  touching sides  $BF$  and  $DE$  is the same, and its center is  $I$ .  $BF$  and  $DE$  intersects at  $C$ . Let  $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$  be the circumcenters of triangles  $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$  respectively.**a)** Show that points  $P_1, P_2, P_3, P_4$  concylic and points  $Q_1, Q_2, Q_3, Q_4$  concylic.**b)** Denote centers of theese circles as  $O_1$  and  $O_2$ . Prove that  $O_1, O_2$  and  $I$  are collinear.","t":[{"b":6,"e":0.14286,"k":"flat","v":0.04018,"x":0.13375,"p":[[0,18,0.0,0.04018,0.12492,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,18,0.2222,0.0625,0.17105,0.0,0.0,0.0,0.0,0.71429,28,0,1,28,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[8,18,0.4444,0.09804,0.19041,0.0,0.0,0.14071,0.0,0.71429,22,0,1,22,0,5,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[12,18,0.6667,0.0892,0.1812,0.0,0.0,0.035,0.0,0.71429,24,0,2,24,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,18,0.8889,0.12936,0.23241,0.0,0.0,0.14287,0.0,0.71429,22,0,0,22,0,3,0,0,2,0,0,1,0,0,1,0,0,3,0,0,0,0,0],[18,18,1.0,0.13375,0.21109,0.0,0.0,0.2857,0.0,0.71429,20,0,1,20,0,3,0,0,5,0,0,1,0,0,1,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,17,0.0,0.06696,0.11837,0.0,0.0,0.03571,0.0,0.28571,24,0,1,24,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"2fcd8595a5c93cd2","q":"Let  $ABCD$  be a cyclic quadrilateral with circumcircle  $\\omega$  centered at  $O$ , whose diagonals intersect at  $H$ . Let  $O_1$  and  $O_2$  be the circumcenters of triangles  $AHD$  and  $BHC$ . A line through  $H$  intersects  $\\omega$  at  $M_1$  and  $M_2$  and intersects the circumcircles of triangles  $O_1HO$  and  $O_2HO$  at  $N_1$  and  $N_2$ , respectively, so that  $N_1$  and  $N_2$  lie inside  $\\omega$ . Prove that  $M_1N_1 = M_2N_2$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,39,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,35,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9654e0ff494e62b8","q":"Let  $H$  and  $I$  be the orthocenter and incenter, respectively, of an acute-angled triangle  $ABC$ . The circumcircle of the triangle  $BCI$  intersects the segment  $AB$  at the point  $P$  different from  $B$ . Let  $K$  be the projection of  $H$  onto  $AI$  and  $Q$  the reflection of  $P$  in  $K$ . Show that  $B$ ,  $H$  and  $Q$  are collinear.\n\n*Proposed by Mads Christensen, Denmark*","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,31,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.00893,"x":0.04009,"p":[[0,8,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.14286,23,0,6,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c8f456a645c8cbdd","q":"Let  $\\lfloor \\bullet \\rfloor$  denote the floor function. For nonnegative integers  $a$  and  $b$ , their *bitwise xor*, denoted  $a \\oplus b$ , is the unique nonnegative integer such that   $$  \\left \\lfloor \\frac{a}{2^k}  \\right \\rfloor+ \\left\\lfloor\\frac{b}{2^k} \\right\\rfloor - \\left\\lfloor \\frac{a\\oplus b}{2^k}\\right\\rfloor $$   is even for every  $k \\ge 0$ . Find all positive integers  $a$  such that for any integers  $x>y\\ge 0$ , we have \\[ x\\oplus ax \\neq y \\oplus ay. \\]\n\n*Carl Schildkraut*","t":[{"b":3,"e":0.14286,"k":"flat","v":0.04018,"x":0.08929,"p":[[0,13,0.0,0.07589,0.11837,0.0,0.0,0.14286,0.0,0.42857,21,0,1,21,0,6,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.08929,0.12242,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,12,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,13,0.6154,0.08929,0.11152,0.0,0.07143,0.14286,0.0,0.42857,16,0,1,16,0,14,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04018,"x":0.23661,"p":[[0,31,0.0,0.07589,0.11837,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,31,0.129,0.11161,0.17399,0.0,0.0,0.14286,0.0,0.71429,19,0,1,19,0,6,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,31,0.2581,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.23661,0.15815,0.14286,0.2857,0.28571,0.0,0.85714,3,0,0,3,0,12,0,0,13,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[16,31,0.5161,0.16509,0.13417,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,12,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.125,0.15465,0.0,0.14286,0.14286,0.0,0.85714,11,0,0,11,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,31,0.7742,0.09822,0.09062,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.11161,0.10555,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,16,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.12053,0.1017,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,15,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a436e5c30d7041cf","q":"Let  $ABC$  be an acute triangle and  $D$ ,  $E$ ,  $F$  be the feet of the altitudes through  $A$ ,  $B$ ,  $C$  respectively.  Call  $Y$  and  $Z$  the feet of the perpendicular lines from  $B$  and  $C$  to  $FD$  and  $DE$ , respectively. Let  $F_1$  be the symmetric of  $F$  with respect to  $E$  and  $E_1$  be the symmetric of  $E$  with respect to  $F$ . If  $3EF=FD+DE$ , prove that  $\\angle BZF_1=\\angle CYE_1$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.03125,"x":0.20089,"p":[[0,21,0.0,0.15625,0.15303,0.0,0.21428,0.28571,0.0,0.42857,15,0,1,15,0,1,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.20089,0.16311,0.0,0.28571,0.28571,0.0,0.42857,12,0,0,12,0,0,0,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.125,0.14617,0.0,0.0,0.28571,0.0,0.42857,18,0,0,18,0,1,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.15625,0.17261,0.0,0.0,0.28571,0.0,0.4286,17,0,0,17,0,0,0,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.07143,0.12371,0.0,0.0,0.07143,0.0,0.28571,24,0,1,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.28571,"k":"flat","v":0.15625,"x":0.26786,"p":[[0,4,0.0,0.15625,0.13997,0.0,0.2857,0.28571,0.0,0.28571,14,0,1,14,0,1,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5515af1bdec3aee4","q":"Let  $p$  be a prime number. Find all subsets  $S\\subseteq\\mathbb Z/p\\mathbb Z$  such that\n1. if  $a,b\\in S$ , then  $ab\\in S$ , and\n2. there exists an  $r\\in S$  such that for all  $a\\in S$ , we have  $r-a\\in S\\cup\\{0\\}$ .\n\n*Proposed by Harun Khan*","t":[{"b":3,"e":0.42857,"k":"flat","v":0.4375,"x":0.53569,"p":[[0,13,0.0,0.53569,0.18557,0.42857,0.42859,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,1,0,0,15,0,0,9,0,0,2,0,0,2,0,2],[4,13,0.3077,0.49999,0.16751,0.42857,0.42857,0.57111,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,21,0,0,5,0,0,1,0,0,1,0,2],[8,13,0.6154,0.50445,0.14718,0.42857,0.42857,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,19,0,0,10,0,0,0,0,0,0,0,2],[12,13,0.9231,0.48663,0.13764,0.42857,0.42857,0.46525,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,23,0,0,5,0,0,1,0,0,1,0,1],[13,13,1.0,0.4375,0.06121,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,26,0,0,4,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.571,"k":"flat","v":0.45981,"x":0.55355,"p":[[0,12,0.0,0.54464,0.1729,0.42857,0.42857,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,19,0,0,6,0,0,3,0,0,2,0,2],[4,12,0.3333,0.55355,0.21354,0.42857,0.42857,0.57143,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,19,0,0,6,0,0,0,0,0,1,0,5],[8,12,0.6667,0.5,0.14724,0.42857,0.42857,0.57143,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,19,0,0,7,0,0,2,0,0,1,0,1],[12,12,1.0,0.45981,0.07769,0.42857,0.42857,0.42858,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,24,0,0,6,0,0,1,0,0,0,0,0]]}]},{"i":"578232919e74213a","q":"Let  $n \\ge 2$  be a positive integer. Suppose  $a_1, a_2, \\dots, a_n$  are distinct integers. For  $k = 1, 2, \\dots, n$ , let\n\\[ s_k := \\prod_{\\substack{i \\not= k,  1 \\le i \\le n}} |a_k - a_i|, \\]\ni.e.  $s_k$  is the product of all terms of the form  $|a_k - a_i|$ , where  $i \\in \\{ 1, 2, \\dots, n \\}$  and  $i \\not= k$ . \nFind the largest positive integer  $M$  such that  $M$  divides the least common multiple of  $s_1, s_2, \\dots, s_n$  for any choices of  $a_1, a_2, \\dots, a_n$ .","t":[{"b":0,"e":0.571,"k":"flat","v":0.46872,"x":0.70535,"p":[[0,47,0.0,0.46872,0.17213,0.42857,0.42857,0.4642,0.1429,1.0,0,1,0,0,0,1,0,0,5,0,0,18,0,0,3,0,0,3,0,0,1,0,1],[4,47,0.0851,0.53572,0.20516,0.42857,0.42857,0.57143,0.14286,1.0,0,4,0,0,0,1,0,0,1,0,0,17,0,0,7,0,0,2,0,0,0,0,4],[8,47,0.1702,0.56249,0.22851,0.42857,0.57143,0.71429,0.14286,1.0,0,2,0,0,0,4,0,0,0,0,0,9,0,0,7,0,0,7,0,0,3,0,2],[12,47,0.2553,0.69643,0.26905,0.42857,0.71429,1.0,0.14286,1.0,0,11,0,0,0,2,0,0,1,0,0,6,0,0,4,0,0,7,0,0,1,0,11],[16,47,0.3404,0.70535,0.24985,0.42859,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,0,0,0,9,0,0,3,0,0,6,0,0,3,0,10],[20,47,0.4255,0.53123,0.20588,0.42857,0.57121,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,4,0,0,9,0,0,8,0,0,5,0,0,3,0,1],[24,47,0.5106,0.5625,0.23941,0.42857,0.50001,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,4,0,0,10,0,0,3,0,0,7,0,0,3,0,3],[28,47,0.5957,0.57134,0.25015,0.42857,0.57143,0.71429,0.14,1.0,0,4,0,0,0,3,0,0,2,0,0,10,0,0,5,0,0,5,0,0,3,0,4],[32,47,0.6809,0.46875,0.24804,0.14286,0.50001,0.60714,0.14286,1.0,0,1,0,0,0,9,0,0,1,0,0,6,0,0,8,0,0,5,0,0,2,0,1],[36,47,0.766,0.58478,0.21536,0.42857,0.57143,0.71429,0.14286,1.0,0,4,0,0,0,1,0,0,3,0,0,7,0,0,11,0,0,5,0,0,1,0,4],[40,47,0.8511,0.5357,0.20516,0.42857,0.57143,0.60714,0.14286,1.0,0,2,0,0,0,3,0,0,1,0,0,10,0,0,10,0,0,5,0,0,1,0,2],[44,47,0.9362,0.59812,0.28008,0.42857,0.57143,0.89286,0.14,1.0,0,8,0,0,0,2,0,0,5,0,0,7,0,0,6,0,0,3,0,0,1,0,8],[47,47,1.0,0.61603,0.21853,0.42857,0.57143,0.75,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,10,0,0,9,0,0,3,0,0,3,0,5]]},{"b":2,"e":0.42857,"k":"flat","v":0.42411,"x":0.54018,"p":[[0,28,0.0,0.47321,0.21261,0.42857,0.42857,0.46431,0.0,1.0,1,2,0,1,0,2,0,0,2,0,0,19,0,0,2,0,0,3,0,0,1,0,2],[4,28,0.1429,0.45088,0.17895,0.42857,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,20,0,0,3,0,0,1,0,0,0,0,2],[8,28,0.2857,0.54018,0.19475,0.42857,0.42857,0.60714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,18,0,0,4,0,0,4,0,0,1,0,3],[12,28,0.4286,0.43304,0.12619,0.42857,0.42857,0.4286,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,21,0,0,6,0,0,1,0,0,0,0,0],[16,28,0.5714,0.44642,0.09277,0.42857,0.42857,0.4286,0.2857,0.71429,0,0,0,0,0,0,0,0,3,0,0,24,0,0,3,0,0,2,0,0,0,0,0],[20,28,0.7143,0.42411,0.09771,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,23,0,0,3,0,0,1,0,0,0,0,0],[24,28,0.8571,0.44643,0.08564,0.42857,0.42857,0.42857,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,28,0,0,2,0,0,0,0,0,1,0,0],[28,28,1.0,0.4375,0.03458,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"036f57a84e3fd87a","q":"Let  $L$  be the foot of the internal bisector of  $\\angle B$  in an acute-angled triangle  $ABC.$  The points  $D$  and  $E$  are the midpoints of the smaller arcs  $AB$  and  $BC$  respectively in the circumcircle  $\\omega$  of  $\\triangle ABC.$  Points  $P$  and  $Q$  are marked on the extensions of the segments  $BD$  and  $BE$  beyond  $D$  and  $E$  respectively so that  $\\measuredangle APB=\\measuredangle CQB=90^{\\circ}.$  Prove that the midpoint of  $BL$  lies on the line  $PQ.$","t":[{"b":2,"e":0.28571,"k":"flat","v":0.01786,"x":0.07143,"p":[[0,26,0.0,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.0625,0.11259,0.0,0.0,0.14286,0.0,0.57143,21,0,1,21,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,26,0.3077,0.06696,0.09438,0.0,0.0,0.14286,0.0,0.28571,20,0,1,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.08482,"p":[[0,29,0.0,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.08482,0.09354,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.05357,0.1171,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[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],[29,29,1.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"12511febaf43f0b0","q":"Let  $ABC$  be an acute triangle with orthocenter  $ H $  and  $AB<AC.$  Let  $\\Omega_1$  be a circle with diameter  $AC$  and  $\\Omega_2$  a circle with diameter  $ AB.$  Line  $BH$  intersects  $\\Omega_1$  in points  $ D $  and  $E$  such that  $E$  is not on segment  $BH.$   Line  $ CH $  intersects  $\\Omega_2$  in points  $ F $  and  $G$  such that  $G$  is not on segment  $CH.$  Prove that the lines  $EG, DF$  and  $BC$  are concurrent.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.01786,"x":0.06241,"p":[[0,23,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.04437,0.06582,0.0,0.0,0.14,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02679,"x":0.11143,"p":[[0,16,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,2,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.08019,0.07072,0.0,0.14143,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.11143,0.05897,0.14,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0bf30b71dba4431c","q":"Let  $ABC$  be an acute triangle with  $AC > AB$  and  $O$  its circumcenter. Let  $D$  be a point on segment  $BC$  such that  $O$  lies inside triangle  $ADC$  and  $\\angle DAO + \\angle ADB = \\angle ADC$ . Let  $P$  and  $Q$  be the circumcenters of triangles  $ABD$  and  $ACD$  respectively, and let  $M$  be the intersection of lines  $BP$  and  $CQ$ . Show that lines  $AM, PQ$  and  $BC$  are concurrent.\n\n*Pablo Ja\u00e9n, Panama*","t":[{"b":2,"e":0.0,"k":"falling","v":0.03571,"x":0.30339,"p":[[0,38,0.0,0.30339,0.27385,0.14286,0.14286,0.42857,0.0,1.0,4,2,0,4,0,14,0,0,3,0,0,5,0,0,2,0,0,1,0,0,1,0,2],[4,38,0.1053,0.25884,0.27071,0.0,0.14286,0.42857,0.0,0.85714,10,0,0,10,0,9,0,0,3,0,0,4,0,0,1,0,0,3,0,0,2,0,0],[8,38,0.2105,0.17857,0.21128,0.0,0.14286,0.1786,0.0,0.857,12,0,0,12,0,12,0,0,1,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[12,38,0.3158,0.08911,0.11702,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,12,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.06697,0.07974,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.2857,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.06463,0.07848,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,1,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"falling","v":0.06696,"x":0.27232,"p":[[0,16,0.0,0.27232,0.34875,0.0,0.14286,0.57143,0.0,1.0,15,2,0,15,0,6,0,0,1,0,0,1,0,0,2,0,0,2,0,0,3,0,2],[4,16,0.25,0.25445,0.27369,0.0,0.14286,0.42858,0.0,0.85714,10,0,0,10,0,10,0,0,3,0,0,2,0,0,2,0,0,3,0,0,2,0,0],[8,16,0.5,0.19197,0.22192,0.0,0.14286,0.28571,0.0,0.85714,12,0,2,12,0,10,0,0,3,0,0,4,0,0,1,0,0,1,0,0,1,0,0],[12,16,0.75,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.10259,0.07344,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d5a2601ae8c0d899","q":"Let  $k$  be an integer and  $k > 1$ . Define a sequence  $\\{a_n\\}$  as follows: $a_0 = 0$ , $a_1 = 1$ , and $a_{n+1} = ka_n + a_{n-1}$  for  $n = 1,2,...$ .\n\nDetermine, with proof, all possible  $k$  for which there exist non-negative integers  $l,m (l \\not= m)$  and positive integers  $p,q$  such that  $a_l + ka_p = a_m + ka_q$ .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.32589,"x":0.42857,"p":[[0,38,0.0,0.35266,0.16744,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,6,0,0,11,0,0,12,0,0,2,0,0,0,0,0,0,0,1],[4,38,0.1053,0.375,0.17765,0.2857,0.28571,0.4286,0.14286,1.0,0,1,0,0,0,4,0,0,14,0,0,8,0,0,4,0,0,1,0,0,0,0,1],[8,38,0.2105,0.3348,0.1163,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,0,0,0,21,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[12,38,0.3158,0.37945,0.14984,0.2857,0.28571,0.4286,0.1429,0.857,0,0,0,0,0,1,0,0,18,0,0,7,0,0,4,0,0,1,0,0,1,0,0],[16,38,0.4211,0.41073,0.15462,0.28571,0.42857,0.4286,0.14286,0.85714,0,0,0,0,0,2,0,0,10,0,0,14,0,0,3,0,0,2,0,0,1,0,0],[20,38,0.5263,0.42857,0.19885,0.28571,0.42857,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,12,0,0,14,0,0,1,0,0,1,0,0,1,0,2],[24,38,0.6316,0.34149,0.10211,0.2857,0.28571,0.42858,0.14286,0.571,0,0,0,0,0,2,0,0,18,0,0,9,0,1,2,0,0,0,0,0,0,0,0],[28,38,0.7368,0.36604,0.15942,0.2857,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,2,0,0,18,0,0,8,0,0,2,0,0,0,0,0,2,0,0],[32,38,0.8421,0.34373,0.11764,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,19,0,0,8,0,0,2,0,0,1,0,0,0,0,0],[36,38,0.9474,0.32589,0.10853,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,19,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[38,38,1.0,0.33928,0.12243,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,20,0,0,8,0,0,0,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.27231,"x":0.31696,"p":[[0,6,0.0,0.31696,0.09932,0.2857,0.28571,0.42857,0.14286,0.5714,0,0,0,0,0,4,0,0,18,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[4,6,0.6667,0.28125,0.11564,0.1429,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,8,0,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.27231,0.12679,0.14286,0.2857,0.28571,0.07143,0.57143,0,0,0,0,1,10,0,1,13,0,0,5,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"ef1b89174acc65f0","q":"Let $A, B, C, D$ be four points on a circle $\\omega$ in that order. Let $P$ and $Q$ be two points on the line $(A B)$ such that the points $Q, A, B, P$ are collinear in that order, the circumcircle of $A D Q$ is tangent to the line $(A C)$, and the circumcircle of $B C P$ is tangent to the line $(B D)$. Let $M$ and $N$ be the midpoints of the segments $[B C]$ and $[A D]$, respectively.\nProve that the tangent to the circumcircle of $A N Q$ at $A$, the tangent to the circumcircle of $B M P$ at $B$, and the line $(C D)$ are concurrent.","t":[{"b":3,"e":0.0,"k":"falling","v":0.00447,"x":0.16062,"p":[[0,14,0.0,0.16062,0.14175,0.0,0.14286,0.28571,0.0,0.42857,12,0,0,12,0,6,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0625,0.1234,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"rising","v":0.09374,"x":0.38393,"p":[[0,39,0.0,0.16518,0.17536,0.0,0.14286,0.28571,0.0,0.57143,14,0,0,14,0,6,0,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[4,39,0.1026,0.09374,0.14983,0.0,0.0,0.14286,0.0,0.571,21,0,0,21,0,4,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,39,0.2051,0.33033,0.13087,0.2857,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,4,0,0,14,0,0,10,0,0,3,0,0,0,0,0,0,0,0],[12,39,0.3077,0.3125,0.09062,0.2857,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,3,0,0,21,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[16,39,0.4103,0.33034,0.07519,0.28571,0.28571,0.42857,0.2857,0.571,0,0,0,0,0,0,0,0,23,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[20,39,0.5128,0.32589,0.09606,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,18,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[24,39,0.6154,0.29464,0.06121,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,2,0,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.38393,0.07524,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.36612,0.0794,0.28571,0.42857,0.42857,0.14286,0.43,0,0,0,0,0,1,0,0,12,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.36607,0.07936,0.28571,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,12,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.34811,0.10078,0.2857,0.35714,0.42857,0.14,0.571,0,0,0,0,0,3,0,0,13,0,0,15,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"87336ffcd9739c28","q":"Let  $p$  and  $q$  be prime numbers of the form  $4k+3$ . Suppose that there exist integers  $x$  and  $y$  such that  $x^2-pqy^2=1$ . Prove that there exist positive integers  $a$  and  $b$  such that  $|pa^2-qb^2|=1$ .","t":[{"b":5,"e":0.42857,"k":"flat","v":0.14719,"x":0.39729,"p":[[0,34,0.0,0.39729,0.3775,0.0,0.42857,0.71429,0.0,1.0,13,5,0,13,0,0,0,0,2,0,0,3,0,0,5,0,0,3,0,0,1,0,5],[4,34,0.1176,0.27679,0.34058,0.0,0.0,0.71429,0.0,1.0,17,2,1,17,0,0,0,0,5,0,0,1,0,0,0,0,0,7,0,0,0,0,2],[8,34,0.2353,0.15177,0.25236,0.0,0.0,0.42857,0.0,0.71429,23,0,3,23,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,0,0,0],[12,34,0.3529,0.14719,0.24055,0.0,0.0,0.28571,0.0,0.71429,22,0,0,22,0,0,0,0,4,0,0,2,0,0,1,0,0,3,0,0,0,0,0],[16,34,0.4706,0.22768,0.18851,0.0,0.28571,0.28571,0.0,0.71429,10,0,0,10,0,1,0,0,17,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[20,34,0.5882,0.34822,0.16728,0.2857,0.42857,0.4286,0.0,0.71429,3,0,1,3,0,2,0,0,10,0,0,14,0,0,1,0,0,2,0,0,0,0,0],[24,34,0.7059,0.3258,0.13952,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,6,0,0,10,0,0,14,0,0,0,0,0,1,0,0,0,0,0],[28,34,0.8235,0.30357,0.14174,0.2857,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,2,0,0,17,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[32,34,0.9412,0.27232,0.10926,0.14286,0.28571,0.32143,0.14286,0.42857,0,0,0,0,0,11,0,0,13,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.25,0.11845,0.14286,0.2857,0.28571,0.0,0.42857,1,0,0,1,0,13,0,0,11,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"volatile","v":0.51338,"x":0.98661,"p":[[0,22,0.0,0.51338,0.39425,0.0,0.42857,1.0,0.0,1.0,9,10,2,9,0,0,0,0,1,0,0,8,0,0,1,0,0,3,0,0,0,0,10],[4,22,0.1818,0.76784,0.29397,0.57132,1.0,1.0,0.0,1.0,2,17,2,2,0,0,0,0,0,0,0,5,0,0,2,0,0,6,0,0,0,0,17],[8,22,0.3636,0.61606,0.37532,0.39286,0.71429,1.0,0.0,1.0,7,11,4,7,0,0,0,0,1,0,0,1,0,0,4,0,0,8,0,0,0,0,11],[12,22,0.5455,0.52232,0.34738,0.2857,0.64286,0.71429,0.0,1.0,7,6,2,7,0,0,0,0,3,0,0,5,0,0,1,0,0,10,0,0,0,0,6],[16,22,0.7273,0.81249,0.2933,0.71429,1.0,1.0,0.0,1.0,2,19,2,2,0,1,0,0,0,0,0,1,0,0,2,0,0,5,0,0,2,0,19],[20,22,0.9091,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[22,22,1.0,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30]]}]},{"i":"917bc0c72e0188a7","q":"In a scalene triangle  $ABC$ ,  $H$  is the orthocenter, and  $G$  is the centroid. Let  $A_b$  and  $A_c$  be points on  $AB$  and  $AC$ , respectively, such that  $B$ ,  $C$ ,  $A_b$ ,  $A_c$  are cyclic, and the points  $A_b$ ,  $A_c$ ,  $H$  are collinear.  $O_a$  is the circumcenter of the triangle  $AA_bA_c$ .  $O_b$  and  $O_c$  are defined similarly. Prove that the centroid of the triangle  $O_aO_bO_c$  lies on the line  $HG$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.04464,"x":0.16072,"p":[[0,24,0.0,0.12947,0.13054,0.0,0.14286,0.1786,0.0,0.4286,13,0,3,13,0,11,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.42857,22,0,5,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.12499,0.16265,0.0,0.0,0.17868,0.0,0.571,17,0,4,17,0,7,0,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[12,24,0.5,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,1,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.28571,22,0,11,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.16072,0.12752,0.0,0.14286,0.28571,0.0,0.42857,9,0,0,9,0,12,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.125,0.15047,0.0,0.07143,0.17857,0.0,0.4286,16,0,0,16,0,8,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.09821,"p":[[0,15,0.0,0.0958,0.14236,0.0,0.0,0.14286,0.0,0.57143,19,0,4,19,0,7,0,1,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.09821,0.13092,0.0,0.0,0.14286,0.0,0.57143,17,0,7,17,0,10,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.03571,0.07986,0.0,0.0,0.0,0.0,0.2857,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4ad0d58ee49236ab","q":"Let  $ABC$  be an acuted-angle triangle and let  $H$  be it's orthocenter. Let  $PQ$  be a segment through  $H$  such that  $P$  lies on  $AB$  and  $Q$  lies on  $AC$  and such that  $ \\angle PHB= \\angle CHQ$ . Finally, in the circumcircle of  $\\triangle ABC$ , consider  $M$  such that  $M$  is the mid point of the arc  $BC$  that doesn't contain  $A$ . Prove that  $MP=MQ$ Proposed by Eduardo Velasco/Marco Figueroa","t":[{"b":2,"e":0.57143,"k":"rising","v":0.42847,"x":0.61161,"p":[[0,14,0.0,0.42847,0.25009,0.25,0.57143,0.57143,0.0,0.71429,6,0,2,6,0,2,0,0,2,0,0,3,0,0,14,0,0,5,0,0,0,0,0],[4,14,0.2857,0.61161,0.1394,0.57143,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,0,0,0,0,0,0,15,0,0,15,0,0,0,0,0],[8,14,0.5714,0.61157,0.10854,0.57143,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,0,0,0,19,0,0,12,0,0,0,0,0],[12,14,0.8571,0.60709,0.10716,0.57143,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,0,0,0,20,0,0,11,0,0,0,0,0],[14,14,1.0,0.59367,0.11905,0.57132,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,0,0,0,20,0,0,10,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"rising","v":0.28567,"x":0.66067,"p":[[0,21,0.0,0.42856,0.24484,0.14286,0.57143,0.57143,0.0,0.71429,4,0,3,4,0,6,0,0,0,0,0,3,0,0,14,0,0,5,0,0,0,0,0],[4,21,0.1905,0.48659,0.19185,0.42857,0.57143,0.57143,0.0,0.71429,2,0,1,2,0,2,0,0,3,0,0,3,0,0,18,0,0,4,0,0,0,0,0],[8,21,0.381,0.51786,0.21943,0.42857,0.57143,0.57143,0.0,1.0,1,2,0,1,0,3,0,0,2,0,0,6,0,0,13,0,0,5,0,0,0,0,2],[12,21,0.5714,0.28567,0.2448,0.14286,0.14288,0.571,0.0,0.71429,7,0,5,7,0,10,0,0,4,0,0,1,0,0,7,0,0,3,0,0,0,0,0],[16,21,0.7619,0.6428,0.13835,0.57143,0.57143,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,18,0,0,9,0,0,0,0,3],[20,21,0.9524,0.65177,0.07088,0.57143,0.71429,0.71429,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,18,0,0,0,0,0],[21,21,1.0,0.66067,0.11713,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,15,0,0,0,0,2]]}]},{"i":"a4aa7e5f50cc36c8","q":"Let $ABC$ be a right triangle at $C$. Let $D$ be the foot of the altitude from $C$, and $Z$ the point on $[AB]$ such that $AC = AZ$. The bisector of $\\widehat{BAC}$ intersects $(CB)$ and $(CZ)$ at $X$ and $Y$ respectively. Show that the four points $B, X, Y, D$ lie on the same circle.","t":[{"b":1,"e":1.0,"k":"rising","v":0.62051,"x":0.96429,"p":[[0,21,0.0,0.62051,0.26872,0.53539,0.71429,0.71429,0.0,1.0,2,5,0,2,0,2,0,0,1,0,0,3,0,0,4,0,0,15,0,0,0,0,5],[4,21,0.1905,0.6384,0.26482,0.42857,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,3,0,0,0,0,0,10,0,0,1,0,0,9,0,0,2,0,7],[8,21,0.381,0.79463,0.22852,0.71429,0.78571,1.0,0.14286,1.0,0,13,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,12,0,0,3,0,13],[12,21,0.5714,0.90625,0.23585,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,26],[16,21,0.7619,0.9375,0.11259,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,2,0,24],[20,21,0.9524,0.96429,0.08748,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,27],[21,21,1.0,0.91508,0.18547,0.85714,1.0,1.0,0.14,1.0,0,23,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,5,0,23]]},{"b":7,"e":1.0,"k":"rising","v":0.55353,"x":0.97321,"p":[[0,10,0.0,0.55353,0.31298,0.28571,0.50071,0.75,0.0,1.0,1,6,0,1,0,6,0,0,2,0,0,7,0,0,1,0,0,7,0,0,2,0,6],[4,10,0.4,0.91071,0.20438,1.0,1.0,1.0,0.14286,1.0,0,25,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,4,0,0,1,0,25],[8,10,0.8,0.92411,0.20198,1.0,1.0,1.0,0.14286,1.0,0,27,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,27],[10,10,1.0,0.97321,0.12595,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,30]]}]},{"i":"298b17599f65fe53","q":"Let  $ABC$  be an acute-angled scalene triangle and  $\\Gamma$  be its circumcircle.  $K$  is the foot of the internal bisector of  $\\angle BAC$  on  $BC$ . Let  $M$  be the midpoint of the arc  $BC$  containing  $A$ .  $MK$  intersect  $\\Gamma$  again at  $A'$ .  $T$  is the intersection of the tangents at  $A$  and  $A'$ .  $R$  is the intersection of the perpendicular to  $AK$  at  $A$  and perpendicular to  $A'K$  at  $A'$ . Show that  $T, R$  and  $K$  are collinear.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,23,0.0,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,7,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,16,0.0,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.03125,0.08552,0.0,0.0,0.0,0.0,0.2857,28,0,1,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"91b55f2cd8233564","q":"Let  $ABC$  be triangle with  $A$ -excenter  $J$ . The reflection of  $J$  in  $BC$  is  $K$ . The points  $E$  and  $F$  are on  $BJ, CJ$  such that  $\\angle EAB=\\angle CAF=90^{\\circ}$ . Prove that  $\\angle FKE+\\angle FJE=180^{\\circ}$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.11152,"x":0.18304,"p":[[0,38,0.0,0.11152,0.06898,0.105,0.14286,0.14286,0.0,0.2857,8,0,3,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.11161,0.09932,0.0,0.14286,0.14286,0.0,0.42857,11,0,7,11,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.12946,0.05486,0.14286,0.14286,0.14286,0.0,0.2857,4,0,2,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.13393,0.08702,0.14286,0.14286,0.14286,0.0,0.42857,6,0,5,6,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.14732,0.10403,0.14286,0.14286,0.14287,0.0,0.42857,6,0,4,6,0,21,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.13831,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,1,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.18304,0.09606,0.14286,0.14286,0.14287,0.14286,0.42857,0,0,0,0,0,27,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.17402,0.07774,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,27,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.16054,0.05928,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,29,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,2,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.14286,0.03571,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14,"k":"flat","v":0.1159,"x":0.24106,"p":[[0,30,0.0,0.12054,0.1017,0.0,0.14286,0.14286,0.0,0.42857,9,0,7,9,0,21,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.1159,0.08323,0.105,0.14286,0.14286,0.0,0.42857,8,0,5,8,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.16956,0.08331,0.14286,0.14286,0.1429,0.0,0.42857,2,0,2,2,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.16063,0.11153,0.14286,0.14286,0.14286,0.0,0.42857,5,0,4,5,0,21,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.14277,0.07143,0.14286,0.14286,0.14287,0.0,0.28571,4,0,3,4,0,24,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.24106,0.14477,0.14286,0.14286,0.32142,0.0,0.57143,1,0,0,1,0,18,0,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[24,30,0.8,0.18295,0.08922,0.14286,0.14286,0.1786,0.0,0.4286,1,0,1,1,0,23,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.1917,0.11083,0.14286,0.14286,0.2857,0.0,0.42857,2,0,0,2,0,21,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.16965,0.07523,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,28,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fbb65027d190061c","q":"Let  $n\\geq 2$  be an integer. For each natural  $m$  and each integer sequence  $0<k_1<k_2<\\cdots <k_m$  for which  $k_1+\\cdots+k_m=n$ , Michael wrote down the number  $\\frac{1}{k_1\\cdot k_2\\cdots k_m} $  on the board. Prove that the sum of the numbers on the board is less than  $1$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.04455,"x":0.13839,"p":[[0,46,0.0,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.09366,0.14552,0.0,0.0,0.14286,0.0,0.71429,17,0,0,17,0,13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,46,0.3478,0.11598,0.20651,0.0,0.07,0.14286,0.0,1.0,16,1,0,16,0,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[20,46,0.4348,0.13839,0.16935,0.10714,0.14286,0.14286,0.0,1.0,8,1,0,8,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,46,0.5217,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.01786,"x":0.15179,"p":[[0,47,0.0,0.11161,0.22227,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,47,0.0851,0.09822,0.11538,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,47,0.1702,0.12946,0.2,0.0,0.14286,0.14286,0.0,0.71429,15,0,0,15,0,14,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[12,47,0.2553,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.14732,0.22724,0.0,0.14286,0.14286,0.0,1.0,13,1,0,13,0,16,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,1],[20,47,0.4255,0.15179,0.21998,0.0,0.14286,0.14286,0.0,0.71429,15,0,1,15,0,12,0,0,0,0,0,1,0,0,1,0,0,3,0,0,0,0,0],[24,47,0.5106,0.10259,0.1758,0.0,0.14143,0.14286,0.0,1.0,15,1,0,15,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,47,0.5957,0.14732,0.25626,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,9,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,1],[32,47,0.6809,0.07143,0.17857,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[36,47,0.766,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6efa70c2827d533c","q":"Let  $X$  and  $Y$  be two sets of points in the plane and  $M$  be a set of segments connecting points from  $X$  and  $Y$  . Let  $k$  be a natural number. Prove that the segments from  $M$  can be painted using  $k$  colors in such a way that for any point  $x \\in X \\cup Y$  and two colors  $\\alpha$  and  $\\beta$   $(\\alpha \\neq \\beta)$ , the difference between the number of  $\\alpha$ -colored segments and the number of  $\\beta$ -colored segments originating in  $X$  is less than or equal to  $1$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.04465,"x":0.14285,"p":[[0,14,0.0,0.04465,0.11539,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,14,0.2857,0.05804,0.16311,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[8,14,0.5714,0.11159,0.15454,0.0,0.0,0.2857,0.0,0.571,19,0,0,19,0,4,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,14,0.8571,0.14277,0.10714,0.0,0.14286,0.2857,0.0,0.28571,9,0,0,9,0,14,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.14285,0.12372,0.0,0.14286,0.2857,0.0,0.28571,12,0,0,12,0,8,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.02205,"x":0.07142,"p":[[0,3,0.0,0.07142,0.17124,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[3,3,1.0,0.02205,0.05125,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f6e0779d397dcea5","q":"Let  $ABC$  be an acute scalene triangle with  $O$  as its circumcenter. Point  $P$  lies inside triangle  $ABC$  with  $\\angle PAB = \\angle PBC$  and  $\\angle PAC = \\angle PCB$ . Point  $Q$  lies on line  $BC$  with  $QA = QP$ . Prove that  $\\angle AQP = 2\\angle OQB$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.01339,"x":0.11607,"p":[[0,12,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,5,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.04018,0.10249,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.11607,0.09061,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,18,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,35,0.0,0.01777,0.0777,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,2,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4b64009cbf1e6de7","q":"For an  ${n\\times n}$  matrix  $A$ , let  $X_{i}$  be the set of entries in row  $i$ , and  $Y_{j}$  the set of entries in column  $j$ ,  ${1\\leq i,j\\leq n}$ . We say that  $A$  is *golden* if  ${X_{1},\\dots ,X_{n},Y_{1},\\dots ,Y_{n}}$  are distinct sets. Find the least integer  $n$  such that there exists a  ${2004\\times 2004}$  golden matrix with entries in the set  ${\\{1,2,\\dots ,n\\}}$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.0625,"x":0.14286,"p":[[0,17,0.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,11,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.12045,0.13414,0.0,0.14143,0.17857,0.0,0.42857,15,0,12,15,0,9,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.14286,0.10714,0.10714,0.14286,0.1429,0.0,0.42857,8,0,6,8,0,17,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.10268,0.09606,0.0,0.14286,0.14286,0.0,0.28571,13,0,10,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.07589,"x":0.18303,"p":[[0,18,0.0,0.13822,0.13592,0.0,0.14286,0.28571,0.0,0.4286,13,0,6,13,0,9,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.08483,0.10632,0.0,0.0,0.14286,0.0,0.286,18,0,12,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.07589,0.10705,0.0,0.0,0.14286,0.0,0.28571,20,0,12,20,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.18303,0.11425,0.14286,0.14286,0.28571,0.0,0.42857,6,0,5,6,0,12,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ed357bfe096d67a4","q":"Let  $ABC$  be a triagle inscribed in a circle  $(O)$ . A variable line through the orthocenter  $H$  of the triangle meets the circle  $(O)$  at two points  $P , Q$ . Two lines through  $P, Q$  that are perpendicular to  $AP , AQ$  respectively meet  $BC$  at  $M, N$  respectively. Prove that the line through  $P$  perpendicular to  $OM$  and the line through  $Q$  perpendicular to  $ON$  meet each other at a point on the circle   $(O)$ .\n\nNguy\u1ec5n V\u0103n Linh","t":[{"b":0,"e":0.71429,"k":"volatile","v":0.20536,"x":0.6205,"p":[[0,12,0.0,0.36161,0.33117,0.14286,0.2857,0.71429,0.0,1.0,7,2,0,7,0,8,0,0,5,0,0,2,0,0,0,0,0,5,0,0,3,0,2],[4,12,0.3333,0.20536,0.21998,0.0,0.14286,0.28571,0.0,0.85714,11,0,1,11,0,9,0,0,5,0,0,4,0,0,1,0,0,1,0,0,1,0,0],[8,12,0.6667,0.25893,0.30396,0.0,0.14286,0.32143,0.0,1.0,12,2,0,12,0,6,0,0,6,0,0,2,0,0,2,0,0,0,0,0,2,0,2],[12,12,1.0,0.6205,0.16214,0.57143,0.57143,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,2,0,0,13,0,0,10,0,0,3,0,1]]},{"b":3,"e":0.28571,"k":"flat","v":0.21875,"x":0.34374,"p":[[0,17,0.0,0.25893,0.24338,0.0,0.14286,0.42857,0.0,0.85714,9,0,1,9,0,8,0,0,5,0,0,4,0,0,3,0,0,2,0,0,1,0,0],[4,17,0.2353,0.34374,0.26451,0.14286,0.28571,0.46418,0.0,1.0,7,2,0,7,0,2,0,0,9,0,0,6,0,0,5,0,0,1,0,0,0,0,2],[8,17,0.4706,0.31696,0.15866,0.28571,0.28571,0.32143,0.0,1.0,2,1,0,2,0,1,0,0,21,0,0,7,0,0,0,0,0,0,0,0,0,0,1],[12,17,0.7059,0.21875,0.14719,0.10714,0.28571,0.28571,0.0,0.42857,8,0,0,8,0,4,0,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.25893,0.10972,0.2857,0.28571,0.28571,0.0,0.4286,3,0,0,3,0,4,0,0,21,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.28125,0.09771,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,2,0,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ee16db3f365eadab","q":"Let  $\\Omega$  be the circumscribed circle of the acute triangle  $ABC$  and  $ D $  a point the small arc  $BC$  of  $\\Omega$ . Points  $E$  and  $ F $  are on the sides  $ AB$  and  $AC$ , respectively, such that the quadrilateral  $CDEF$  is a parallelogram. Point  $G$  is on the small arc  $AC$  such that lines  $DC$  and  $BG$  are parallel. Prove that the angles  $GFC$  and  $BAC$  are equal.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.01339,"x":0.09821,"p":[[0,18,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,9,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,4,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.08482,0.12299,0.0,0.0,0.17857,0.0,0.28571,21,0,8,21,0,3,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,6,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.05803,0.11214,0.0,0.0,0.03571,0.0,0.42857,24,0,7,24,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.09821,0.12595,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.15625,"p":[[0,44,0.0,0.06696,0.12364,0.0,0.0,0.03571,0.0,0.42857,24,0,4,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.08036,0.11258,0.0,0.0,0.14286,0.0,0.28571,20,0,5,20,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.0892,0.11706,0.0,0.0,0.14286,0.0,0.28571,19,0,1,19,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.14285,0.13363,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,4,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.12054,0.13415,0.0,0.0,0.2857,0.0,0.28571,17,0,0,17,0,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.15625,0.13533,0.0,0.21428,0.28571,0.0,0.28571,13,0,0,13,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.11608,0.13092,0.0,0.0,0.2857,0.0,0.28571,17,0,0,17,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.15616,0.13534,0.0,0.21428,0.2857,0.0,0.28571,13,0,0,13,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2e291f825c1912d8","q":"Let  $ABC$  be an acute-angled triangle such that  $AB<AC$ . Let  $D$  be the point of intersection of the perpendicular bisector of the side  $BC$  with the side  $AC$ . Let  $P$  be a point on the shorter arc  $AC$  of the circumcircle of the triangle  $ABC$  such that  $DP \\parallel BC$ . Finally, let  $M$  be the midpoint of the side  $AB$ . Prove that  $\\angle APD=\\angle MPB$ .\n\n*Proposed by Dominik Burek, Poland*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,9,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,12,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,32,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,8,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"586faaee157fb112","q":"Let  $A,B,C,D$  be four distinct points on a circle such that the lines  $(AC)$  and  $(BD)$  intersect at  $E$ , the lines  $(AD)$  and  $(BC)$  intersect at  $F$  and such that  $(AB)$  and  $(CD)$  are not parallel.\r\n\r\nProve that  $C,D,E,F$  are on the same circle if, and only if,  $(EF)\\bot(AB)$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.06688,"x":0.24999,"p":[[0,12,0.0,0.19193,0.25899,0.0,0.0,0.32143,0.0,1.0,17,1,9,17,0,4,0,0,3,0,0,2,0,0,5,0,0,0,0,0,0,0,1],[4,12,0.3333,0.1607,0.25188,0.0,0.0,0.1786,0.0,1.0,19,1,6,19,0,5,0,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,1],[8,12,0.6667,0.24999,0.22866,0.0,0.14288,0.4286,0.0,0.71429,10,0,6,10,0,7,0,0,4,0,0,4,0,0,6,0,0,1,0,0,0,0,0],[12,12,1.0,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,4,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.04455,"x":0.19633,"p":[[0,17,0.0,0.19633,0.19805,0.0,0.14286,0.28571,0.0,0.57143,12,0,9,12,0,7,0,0,6,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[4,17,0.2353,0.10714,0.15972,0.0,0.0,0.1786,0.0,0.57143,20,0,17,20,0,4,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.12491,0.21941,0.0,0.0,0.14286,0.0,0.85714,17,0,14,17,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0],[12,17,0.7059,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.2857,23,0,20,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e533d6cbc38ada9c","q":"Let  $ABC$  be a triangle and  $\\Gamma$  its circumcircle. Let  $A_1$  be the midpoint of arc  $BC$  containing  $A$  and  $A_2$  be the touchpoint of the incircle of  $ABC$  with  $BC$ . Line  $A_1A_2$  intersects  $\\Gamma$    at $ A_3$ . Similarly, are defined  $B_3$  and  $C_3$  . Let us take the points  $P, Q$  and  $R$  so that  $QR, PR$  and  $PQ$  they are tangent to  $\\Gamma$  at  $A_3$ ,  $B_3$  and  $C_3$  respectively. Prove that  $PA$ ,  $QB$ , and  $RC$  concur.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,37,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,35,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c29b3fc8c68cac5e","q":"Let  $M$  and  $N$  be two regular polygonic area.Define  $K(M,N)$  as the midpoints of segments  $[AB]$  such that  $A$  belong to  $M$  and  $B$  belong to  $N$ . Find all situations of  $M$  and  $N$  such that  $K(M,N)$  is a regualr polygonic area too.","t":[{"b":2,"e":0.2857,"k":"falling","v":0.50885,"x":0.6741,"p":[[0,10,0.0,0.6741,0.22653,0.57143,0.71429,0.85714,0.14286,1.0,0,2,0,0,0,2,0,0,1,0,0,4,0,0,6,0,0,5,0,0,12,0,2],[4,10,0.4,0.65623,0.22263,0.5713,0.71429,0.85714,0.0,1.0,1,1,0,1,0,1,0,0,0,0,0,5,0,0,7,0,0,6,0,0,11,0,1],[8,10,0.8,0.58485,0.17981,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,9,0,0,8,0,0,6,0,0,6,0,0],[10,10,1.0,0.50885,0.14695,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,10,0,0,15,0,0,1,0,0,2,0,0]]},{"b":6,"e":0.71429,"k":"falling","v":0.40625,"x":0.70532,"p":[[0,16,0.0,0.70532,0.1819,0.57143,0.71429,0.85714,0.2857,1.0,0,1,0,0,0,0,0,0,3,0,0,1,0,0,5,0,0,10,0,0,12,0,1],[4,16,0.25,0.54003,0.24922,0.42857,0.57143,0.71429,0.0,0.85714,3,0,2,3,0,0,0,0,4,0,0,5,0,0,8,0,0,6,0,0,6,0,0],[8,16,0.5,0.40625,0.12931,0.28571,0.42857,0.46431,0.0,0.57143,1,0,0,1,0,0,0,0,10,0,0,13,0,0,8,0,0,0,0,0,0,0,0],[12,16,0.75,0.43305,0.19391,0.28571,0.42857,0.57111,0.0,0.85714,1,0,0,1,0,2,0,0,9,0,0,10,0,0,4,0,0,5,0,0,1,0,0],[16,16,1.0,0.44192,0.18333,0.28571,0.42857,0.571,0.0,0.85714,1,0,0,1,0,2,0,0,6,0,0,13,0,0,5,0,0,4,0,0,1,0,0]]}]},{"i":"3dcde10c4d914b43","q":"Let  $(X, <)$  be an arbitrary ordered set. Show that the elements of  $X$  can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour.\n\n(translated by L. Erd\u0151s)","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.15177,"p":[[0,14,0.0,0.15177,0.20494,0.0,0.0,0.17857,0.0,0.57143,17,0,8,17,0,7,0,0,1,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[4,14,0.2857,0.13381,0.18529,0.0,0.0,0.14287,0.0,0.57143,17,0,5,17,0,8,0,0,2,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[8,14,0.5714,0.07133,0.12361,0.0,0.0,0.14286,0.0,0.571,21,0,4,21,0,8,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.04447,0.075,0.0,0.0,0.14,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.05357,"x":0.16964,"p":[[0,24,0.0,0.15624,0.19676,0.0,0.07143,0.28571,0.0,0.57143,16,0,5,16,0,7,0,0,2,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[4,24,0.1667,0.12277,0.15282,0.0,0.14286,0.14286,0.0,0.57143,15,0,3,15,0,11,0,1,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,24,0.3333,0.13838,0.19058,0.0,0.0,0.2857,0.0,0.57143,18,0,11,18,0,5,0,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[12,24,0.5,0.10268,0.17215,0.0,0.0,0.14286,0.0,0.71429,19,0,6,19,0,9,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,24,0.6667,0.16964,0.18707,0.0,0.14286,0.17857,0.0,0.71429,11,0,5,11,0,13,0,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[20,24,0.8333,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.06697,0.13825,0.0,0.0,0.14286,0.0,0.71429,22,0,2,22,0,8,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"6a360aa44a3aa6a6","q":"Let  $ABC$  be an acute-angled triangle. Point  $P$  is such that  $AP = AB$  and  $PB \\parallel AC$ . Point  $Q$  is such that  $AQ = AC$  and  $CQ \\parallel AB$ . Segments  $CP$  and  $BQ$  meet at point  $X$ . Prove that the circumcenter of triangle  $ABC$  lies on the circumcircle of triangle  $PXQ$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,14,0.0,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.04455,"p":[[0,27,0.0,0.04455,0.08318,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.01777,0.04701,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"432f6e6ef775f755","q":"Let  $\\bigtriangleup ABC$  be an acute triangle with a point  $D$  on side  $BC$ . Let  $J$  be a point on side  $AC$  such that  $\\angle BAD = 2\\angle ADJ$ , and  $\\omega$  be the circumcircle of triangle  $\\bigtriangleup CDJ$ . The line  $AD$  intersects  $\\omega$  again at a point  $P$ , and  $Q$  is the feet of the altitude from  $J$  to  $AB$ .\nProve that if  $JP = JQ$ , then the line perpendicular to  $DJ$  through  $A$  is tangent to  $\\omega$ .\n\n*Proposed by Ivan Chan - Malaysia*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,57,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,57,0.0702,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,57,0.1404,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,57,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,57,0.7018,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,13,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,2,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"3886a4dc1817d4dc","q":"Let  $\\mathbb{N}$  be the set of positive integers. Across all function  $f:\\mathbb{N}\\to\\mathbb{N}$  such that   $$ mn+1\\text{ divides } f(m)f(n)+1 $$   for all positive integers  $m$  and  $n$ , determine all possible values of  $f(101).$","t":[{"b":5,"e":0.0,"k":"flat","v":0.04464,"x":0.3482,"p":[[0,28,0.0,0.12491,0.12751,0.0,0.14286,0.14286,0.0,0.57143,12,0,2,12,0,14,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,28,0.1429,0.21875,0.18205,0.14286,0.14286,0.28571,0.0,0.57143,5,0,2,5,0,17,0,0,3,0,0,2,0,0,5,0,0,0,0,0,0,0,0],[8,28,0.2857,0.19642,0.20436,0.0,0.14286,0.42857,0.0,0.57143,14,0,2,14,0,5,0,0,2,0,0,9,0,0,2,0,0,0,0,0,0,0,0],[12,28,0.4286,0.3482,0.33869,0.0,0.14286,0.57143,0.0,1.0,9,4,2,9,0,8,0,0,1,0,0,1,0,0,8,0,0,1,0,0,0,0,4],[16,28,0.5714,0.2633,0.28599,0.0,0.14286,0.42857,0.0,1.0,10,2,0,10,0,8,0,0,5,0,0,3,0,0,3,0,0,0,0,0,1,0,2],[20,28,0.7143,0.22766,0.19513,0.14286,0.14286,0.28571,0.0,0.57143,7,0,1,7,0,12,0,0,6,0,0,1,0,0,6,0,0,0,0,0,0,0,0],[24,28,0.8571,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.13393,"x":0.46874,"p":[[0,57,0.0,0.16063,0.21651,0.0,0.14286,0.14286,0.0,0.85714,13,0,1,13,0,14,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0],[4,57,0.0702,0.19643,0.22232,0.0,0.14286,0.28571,0.0,0.85714,11,0,2,11,0,11,0,0,4,0,0,2,0,0,2,0,0,1,0,0,1,0,0],[8,57,0.1404,0.13393,0.13803,0.0,0.14286,0.14287,0.0,0.57143,12,0,0,12,0,13,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,57,0.2105,0.3125,0.33775,0.0,0.14288,0.4286,0.0,1.0,10,4,1,10,0,7,0,0,4,0,0,4,0,0,1,0,0,1,0,0,1,0,4],[16,57,0.2807,0.46874,0.33165,0.14286,0.42857,0.75,0.14286,1.0,0,7,0,0,0,11,0,0,4,0,0,6,0,0,2,0,0,1,0,0,1,0,7],[20,57,0.3509,0.2634,0.21461,0.14286,0.14286,0.28571,0.14286,1.0,0,1,0,0,0,21,0,0,4,0,0,3,0,0,2,0,0,0,0,0,1,0,1],[24,57,0.4211,0.20536,0.11811,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,23,0,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[28,57,0.4912,0.26338,0.15196,0.14286,0.14286,0.42857,0.14286,0.57143,0,0,0,0,0,18,0,0,4,0,0,7,0,0,3,0,0,0,0,0,0,0,0],[32,57,0.5614,0.24999,0.15565,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,20,0,0,4,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[36,57,0.6316,0.21866,0.13361,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,21,0,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[40,57,0.7018,0.22768,0.12807,0.14286,0.14286,0.2857,0.14286,0.57143,0,0,0,0,0,21,0,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[44,57,0.7719,0.24107,0.14032,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,20,0,0,4,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[48,57,0.8421,0.23206,0.1547,0.14286,0.14286,0.32143,0.0,0.57143,1,0,0,1,0,21,0,0,2,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[52,57,0.9123,0.22321,0.11258,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,20,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.24107,0.14032,0.14286,0.14286,0.32143,0.14286,0.57143,0,0,0,0,0,20,0,0,4,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[57,57,1.0,0.1875,0.10374,0.14286,0.14286,0.17857,0.0,0.57143,1,0,0,1,0,23,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"946a1a278663662f","q":"Let $A B C$ be an acute-angled triangle with circumcenter $O$. Let $\\Gamma$ be a circle with centre on the altitude from $A$ in $A B C$, passing through vertex $A$ and points $P$ and $Q$ on sides $A B$ and $A C$. Assume that $B P \\cdot C Q=A P \\cdot A Q$. Prove that $\\Gamma$ is tangent to the circumcircle of triangle BOC.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.04465,"p":[[0,38,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.02232,0.08073,0.0,0.0,0.0,0.0,0.4286,29,0,2,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,2,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,38,0.7368,0.04465,0.10972,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,20,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"6bb195551b5cfec0","q":"Let  $n,k,m$  be positive integers, where  $k\\ge 2$  and  $n\\le m < \\frac{2k-1}{k}n$ . Let  $A$  be a subset of  $\\{1,2,\\ldots ,m\\}$  with  $n$  elements. Prove that every integer in the range  $\\left(0,\\frac{n}{k-1}\\right)$  can be expressed as  $a-b$ , where  $a,b\\in A$ .","t":[{"b":3,"e":1.0,"k":"flat","v":0.47321,"x":0.61606,"p":[[0,6,0.0,0.47321,0.21261,0.28571,0.42857,0.46429,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,15,0,0,2,0,0,3,0,0,0,0,3],[4,6,0.6667,0.50897,0.19538,0.42857,0.42859,0.57143,0.2857,1.0,0,3,0,0,0,0,0,0,4,0,0,18,0,0,5,0,0,1,0,0,1,0,3],[6,6,1.0,0.61606,0.20652,0.42857,0.64286,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,9,0,0,4,0,0,10,0,0,3,0,3]]},{"b":4,"e":0.42857,"k":"rising","v":0.50445,"x":0.79911,"p":[[0,34,0.0,0.50445,0.1636,0.42857,0.42857,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,13,0,0,4,0,0,8,0,0,1,0,0],[4,34,0.1176,0.69643,0.23351,0.42857,0.71429,1.0,0.28571,1.0,0,9,0,0,0,0,0,0,1,0,0,9,0,0,3,0,0,8,0,0,2,0,9],[8,34,0.2353,0.67408,0.20898,0.53539,0.71429,0.71429,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,6,0,0,4,0,0,13,0,0,1,0,6],[12,34,0.3529,0.79911,0.21387,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,9,0,0,3,0,14],[16,34,0.4706,0.68302,0.22795,0.42857,0.71429,0.89286,0.28571,1.0,0,8,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,10,0,0,1,0,8],[20,34,0.5882,0.71874,0.23004,0.42857,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,9,0,0,3,0,0,9,0,0,0,0,11],[24,34,0.7059,0.76337,0.21013,0.57143,0.71429,1.0,0.28571,1.0,0,11,0,0,0,0,0,0,1,0,0,3,0,0,5,0,0,9,0,0,3,0,11],[28,34,0.8235,0.70978,0.23553,0.42859,0.71429,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,9,0,0,6,0,0,5,0,0,1,0,11],[32,34,0.9412,0.74999,0.22017,0.57143,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,6,0,0,5,0,0,8,0,0,1,0,12],[34,34,1.0,0.69196,0.22619,0.42857,0.71429,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,10,0,0,4,0,0,8,0,0,1,0,9]]}]},{"i":"2429bbb368dda15d","q":"Let  $E$  and  $F$  are distinct points on the diagonal  $AC$  of a parallelogram  $ABCD$  . Prove that , if there exists a cricle through  $E,F$  tangent to rays  $BA,BC$  then there also exists a cricle through  $E,F$  tangent to rays  $DA,DC$ .","t":[{"b":1,"e":0.0,"k":"falling","v":0.02223,"x":0.29915,"p":[[0,17,0.0,0.29915,0.29314,0.0,0.28571,0.42895,0.0,1.0,11,1,4,11,0,4,0,0,4,0,0,6,0,0,1,0,0,4,0,0,1,0,1],[4,17,0.2353,0.29232,0.33333,0.0,0.1429,0.42857,0.0,1.0,11,4,5,11,0,6,0,1,5,0,0,2,0,0,1,0,0,2,0,0,0,0,4],[8,17,0.4706,0.20981,0.27194,0.0,0.14286,0.42857,0.0,1.0,15,1,4,15,0,6,0,0,2,0,0,5,0,0,1,0,0,1,0,0,1,0,1],[12,17,0.7059,0.1874,0.26107,0.0,0.0,0.2857,0.0,1.0,17,1,2,17,0,4,0,0,4,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[16,17,0.9412,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.04009,0.09597,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.08465,"x":0.52465,"p":[[0,19,0.0,0.52465,0.33778,0.28571,0.4998,0.857,0.0,1.0,6,4,3,6,0,1,0,0,2,0,0,7,0,0,2,0,0,4,1,0,5,0,4],[4,19,0.2105,0.2232,0.25737,0.0,0.14286,0.42857,0.0,1.0,13,1,3,13,0,6,0,0,4,0,0,5,0,0,1,0,0,2,0,0,0,0,1],[8,19,0.4211,0.15149,0.17923,0.0,0.14143,0.1429,0.0,0.71429,12,0,2,12,1,12,0,0,3,0,0,1,0,1,1,0,0,1,0,0,0,0,0],[12,19,0.6316,0.09822,0.10374,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.16517,0.22332,0.0,0.14286,0.1786,0.0,1.0,13,1,0,13,0,11,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[19,19,1.0,0.08465,0.12291,0.0,0.0,0.14287,0.0,0.4286,19,0,0,19,0,9,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"45e94254a3911e97","q":"Given an inscribed quadrangle  $ ABCD $ , whose sides intersect at points  $ K $  and  $ M $ . The bisectors of the angles  $ K $  and  $ M $  intersect the sides of  $ ABCD $  at four points. Prove that these four points form a rhombus.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.09375,"x":0.23205,"p":[[0,11,0.0,0.21428,0.28571,0.0,0.07143,0.42857,0.0,1.0,16,2,1,16,0,4,0,0,3,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[4,11,0.3636,0.23205,0.2983,0.0,0.07,0.42857,0.0,1.0,16,2,1,16,0,4,0,0,1,0,0,5,0,0,3,0,0,1,0,0,0,0,2],[8,11,0.7273,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.10259,0.1085,0.0,0.14286,0.14286,0.0,0.4286,13,0,0,13,0,17,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.04018,"x":0.23214,"p":[[0,7,0.0,0.21874,0.26721,0.0,0.07143,0.42857,0.0,1.0,16,1,0,16,0,3,0,0,2,0,0,6,0,0,3,0,0,1,0,0,0,0,1],[4,7,0.5714,0.23214,0.3004,0.0,0.07143,0.32143,0.0,1.0,16,2,0,16,0,2,0,0,6,0,0,3,0,0,0,0,0,3,0,0,0,0,2],[7,7,1.0,0.04018,0.0735,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"327d45832d9edc2c","q":"Let $A B C D$ be a convex quadrilateral with $\\angle A B C>90^{\\circ}, \\angle C D A>90^{\\circ}$, and  $\\angle D A B=\\angle B C D$. Denote by $E$ and $F$ the reflections of $A$ in lines $B C$ and $C D$, respectively. Suppose that the segments $A E$ and $A F$ meet the line $B D$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $B E K$ and $D F L$ are tangent to each other. (Slovakia)","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,35,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[12,35,0.3429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,39,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"649364cd9619d44c","q":"Let  $\\alpha$  be a real number. Determine all polynomials  $P$  with real coefficients such that   $$ P(2x+\\alpha)\\leq (x^{20}+x^{19})P(x) $$   holds for all real numbers  $x$ .\n\n*Proposed by Walther Janous, Austria*","t":[{"b":0,"e":0.28571,"k":"falling","v":0.16518,"x":0.54018,"p":[[0,40,0.0,0.54018,0.29393,0.28571,0.42857,0.85714,0.14286,1.0,0,7,0,0,0,2,0,0,8,0,0,12,0,0,0,0,0,0,0,0,3,0,7],[4,40,0.1,0.5,0.29014,0.28571,0.42857,0.75,0.14286,1.0,0,6,0,0,0,3,0,0,10,0,0,10,0,0,0,0,0,1,0,0,2,0,6],[8,40,0.2,0.47322,0.24856,0.39286,0.42857,0.4286,0.0,1.0,1,4,1,1,0,2,0,0,5,0,0,17,0,0,1,0,0,1,0,0,1,0,4],[12,40,0.3,0.3973,0.23617,0.24999,0.42857,0.4286,0.0,1.0,1,2,1,1,0,7,0,0,6,0,0,11,0,0,2,0,0,3,0,0,0,0,2],[16,40,0.4,0.38393,0.1729,0.28571,0.42857,0.42857,0.14286,1.0,0,1,0,0,0,5,0,0,8,0,0,15,0,0,2,0,0,1,0,0,0,0,1],[20,40,0.5,0.38838,0.18976,0.28571,0.35714,0.42857,0.14286,1.0,0,1,0,0,0,4,0,0,12,0,0,11,0,0,2,0,0,1,0,0,1,0,1],[24,40,0.6,0.31696,0.1461,0.14286,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,10,0,0,8,0,0,12,0,0,1,0,0,1,0,0,0,0,0],[28,40,0.7,0.2008,0.11219,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,22,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.16518,0.08827,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,22,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.20982,0.10092,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,21,0,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.22321,0.12339,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,21,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.42857,"k":"flat","v":0.44197,"x":0.7232,"p":[[0,76,0.0,0.52232,0.33619,0.28571,0.42857,1.0,0.0,1.0,1,9,1,1,0,4,0,0,10,0,0,5,0,0,0,0,0,3,0,0,0,0,9],[4,76,0.0526,0.51339,0.29851,0.28571,0.42857,0.75,0.14286,1.0,0,7,0,0,0,5,0,0,5,0,0,12,0,0,1,0,0,1,0,0,1,0,7],[8,76,0.1053,0.44197,0.28873,0.1429,0.42857,0.46431,0.0,1.0,1,4,0,1,0,8,0,0,3,0,0,12,0,0,1,0,0,1,0,0,2,0,4],[12,76,0.1579,0.59374,0.28146,0.42857,0.4998,0.89286,0.14286,1.0,0,8,0,0,0,3,0,0,2,0,0,11,0,0,4,0,0,3,0,0,1,0,8],[16,76,0.2105,0.48652,0.2446,0.28571,0.42857,0.46431,0.14,1.0,0,4,0,0,0,2,0,0,7,0,0,15,0,0,2,0,0,0,0,0,2,0,4],[20,76,0.2632,0.61606,0.33586,0.28571,0.57143,1.0,0.14286,1.0,0,10,0,0,0,4,0,0,8,0,0,2,0,0,3,0,0,0,0,0,5,0,10],[24,76,0.3158,0.66071,0.31084,0.28571,0.78564,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,10,0,0,4,0,0,1,0,0,1,0,0,5,0,11],[28,76,0.3684,0.61158,0.28175,0.28571,0.57121,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,8,0,0,5,0,0,4,0,0,0,0,0,9,0,5],[32,76,0.4211,0.59821,0.2822,0.28571,0.57143,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,12,0,0,2,0,0,3,0,0,3,0,0,7,0,5],[36,76,0.4737,0.59373,0.29257,0.28571,0.4998,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,10,0,0,5,0,0,2,0,0,0,0,0,9,0,5],[40,76,0.5263,0.64732,0.25996,0.28571,0.78564,0.85714,0.14286,1.0,0,1,0,0,0,1,0,0,8,0,0,1,0,0,2,0,0,4,0,0,15,0,1],[44,76,0.5789,0.69641,0.27375,0.28571,0.85714,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,9,0,0,0,0,0,1,0,0,4,0,0,12,0,6],[48,76,0.6316,0.70982,0.28456,0.39286,0.85714,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,7,0,0,2,0,0,0,0,0,0,0,0,16,0,6],[52,76,0.6842,0.64732,0.26482,0.42857,0.78564,0.85714,0.14286,1.0,0,3,0,0,0,1,0,0,6,0,0,5,0,0,2,0,0,2,0,0,13,0,3],[56,76,0.7368,0.7232,0.28334,0.42857,0.85714,1.0,0.2857,1.0,0,13,0,0,0,0,0,0,5,0,0,6,0,0,1,0,0,3,0,0,4,0,13],[60,76,0.7895,0.62946,0.30275,0.39286,0.50001,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,7,0,0,8,0,0,1,0,0,2,0,0,3,0,10],[64,76,0.8421,0.70535,0.28333,0.42857,0.78564,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,5,0,0,7,0,0,1,0,0,3,0,0,4,0,12],[68,76,0.8947,0.60713,0.2988,0.28571,0.57143,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,10,0,0,5,0,0,0,0,0,2,0,0,8,0,6],[72,76,0.9474,0.60714,0.28793,0.42857,0.42857,0.89286,0.2857,1.0,0,8,0,0,0,0,0,0,7,0,0,12,0,0,0,0,0,0,0,0,5,0,8],[76,76,1.0,0.65625,0.32115,0.28571,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,8,0,0,5,0,0,0,0,0,4,0,0,2,0,12]]}]},{"i":"272a75b3039eb411","q":"Let  $n$  be a positive integer. Let  $B_n$  be the set of all binary strings of length  $n$ . For a binary string  $s_1\\hdots s_n$ , we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by  $b$ . Then, we replace  $s_b$  with  $1-s_b$ . A string  $a$  is said to be a *descendant* of  $b$  if  $a$  can be obtained from  $b$  through a finite number of twists. A subset of  $B_n$  is called *divided* if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of  $B_n$ .\n\n*Remark.* Here is an example of a twist:  $101100 \\rightarrow 101000$  because  $1\\mid 0\\mid 11\\mid 00$  has  $4$  blocks of consecutive digits. \n\n*Viktor Simjanoski*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.125,"x":0.15616,"p":[[0,66,0.0,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.14286,2,0,2,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.15616,0.06548,0.14286,0.14286,0.14286,0.0,0.42857,1,0,1,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,66,0.1212,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,4,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,66,0.1818,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,3,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,66,0.2424,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,3,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,66,0.303,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,2,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,66,0.3636,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,66,0.4242,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,66,0.4848,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,2,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,66,0.5455,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,2,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,66,0.6061,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,66,0.6667,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,66,0.7273,0.14286,1e-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],[52,66,0.7879,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,66,0.8485,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,66,0.9091,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.12946,"x":0.1517,"p":[[0,41,0.0,0.1383,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,2,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.42857,1,0,1,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,2,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.12946,0.04164,0.14286,0.14286,0.14286,0.0,0.14286,3,0,3,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8d93e3d65a3f353d","q":"Let  $M$  be the set of the integer numbers from the range  $[-n, n]$ . The subset  $P$  of  $M$  is called a *base subset* if every number from  $M$  can be expressed as a sum of some different numbers from  $P$ . Find the smallest natural number  $k$  such that every  $k$  numbers that belongs to  $M$  form a base subset.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.08018,"x":0.33482,"p":[[0,43,0.0,0.33482,0.15407,0.2857,0.28571,0.42857,0.0,0.71429,2,0,1,2,0,3,0,0,14,0,0,9,0,0,3,0,0,1,0,0,0,0,0],[4,43,0.093,0.31241,0.15755,0.2857,0.28571,0.42857,0.0,0.71429,2,0,1,2,0,4,0,0,17,0,0,6,0,0,1,0,0,2,0,0,0,0,0],[8,43,0.186,0.33481,0.15404,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,3,0,0,19,0,0,4,0,0,3,0,0,2,0,0,0,0,0],[12,43,0.2791,0.23214,0.18814,0.0,0.2857,0.42857,0.0,0.57143,9,0,1,9,0,6,0,0,8,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[16,43,0.3721,0.16955,0.15337,0.0,0.14286,0.28571,0.0,0.57143,11,0,5,11,0,8,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,43,0.4651,0.13393,0.17105,0.0,0.0,0.2857,0.0,0.57143,17,0,4,17,0,6,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,43,0.5581,0.19196,0.17717,0.0,0.14286,0.28571,0.0,0.71429,11,0,2,11,0,6,0,0,10,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[28,43,0.6512,0.16517,0.17533,0.0,0.14286,0.28571,0.0,0.57143,14,0,3,14,0,5,0,0,9,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[32,43,0.7442,0.08018,0.10051,0.0,0.0,0.14286,0.0,0.28571,18,0,4,18,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.15177,0.21407,0.0,0.07143,0.2857,0.0,0.85714,16,0,7,16,0,7,0,0,6,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[40,43,0.9302,0.19195,0.21607,0.0,0.14286,0.28571,0.0,0.85714,11,0,3,11,0,11,0,0,4,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[43,43,1.0,0.14286,0.17128,0.0,0.14286,0.14287,0.0,0.71429,13,0,5,13,0,12,0,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.33482,"p":[[0,23,0.0,0.33482,0.16982,0.2857,0.28571,0.42857,0.0,0.85714,2,0,1,2,0,3,0,0,16,0,0,6,0,0,4,0,0,0,0,0,1,0,0],[4,23,0.1739,0.30801,0.16405,0.2857,0.28571,0.42857,0.0,0.57143,4,0,1,4,0,2,0,0,16,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[8,23,0.3478,0.3258,0.19968,0.24999,0.28571,0.42857,0.0,0.71429,5,0,1,5,0,3,0,0,10,0,0,8,0,0,4,0,0,2,0,0,0,0,0],[12,23,0.5217,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,21,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,22,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,22,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"eb7fcebe7053d6c0","q":"Let $A B C$ be an acute triangle such that $A B<A C$. Let $\\omega$ be the circumcircle of $A B C$ and assume that the tangent to $\\omega$ at $A$ intersects the line $B C$ at $D$. Let $\\Omega$ be the circle with center $D$ and radius $A D$. Denote by $E$ the second intersection point of $\\omega$ and $\\Omega$. Let $M$ be the midpoint of $B C$. If the line $B E$ meets $\\Omega$ again at $X$, and the line $C X$ meets $\\Omega$ for the second time at $Y$, show that $A, Y$ and $M$ are collinear.\n\n## Proposed by North Macedonia","t":[{"b":0,"e":0.0,"k":"flat","v":0.03572,"x":0.05348,"p":[[0,7,0.0,0.05348,0.0927,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.02679,"x":0.08473,"p":[[0,32,0.0,0.08473,0.12295,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.07589,0.10091,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.07143,0.09449,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"88394bce1e2ce4d2","q":"Let  $ABC$  be a scalene and acute triangle with circumcenter  $O$ . Let  $\\omega$  be the circle with center  $A$ , tangent to  $BC$  at  $D$ . Suppose there are two points  $F$  and  $G$  on  $\\omega$  such that  $FG \\perp AO$ ,  $\\angle BFD = \\angle DGC$  and the couples of points  $(B,F)$  and  $(C,G)$  are in different halfplanes with respect to the line  $AD$ . Show that the tangents to  $\\omega$  at  $F$  and  $G$  meet on the circumcircle of  $ABC$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.0,"x":0.04455,"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],[4,23,0.1739,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.01777,0.04701,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.02232,"x":0.07143,"p":[[0,45,0.0,0.03571,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.07134,0.07978,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"91a6e6db13246da5","q":"Let  $ABC$  be an acute triangle. Construct a point  $X$  on the different side of  $C$  with respect to the line  $AB$  and construct a point  $Y$  on the different side of  $B$  with respect to the line  $AC$  such that  $BX=AC$ ,  $CY=AB$ , and  $AX=AY$ .  Let  $A'$  be the reflection of  $A$  across the perpendicular bisector of  $BC$ . Suppose that  $X$  and  $Y$  lie on different sides of the line  $AA'$ , prove that points  $A$ ,  $A'$ ,  $X$ , and  $Y$  lie on a circle.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,24,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9028f2347d45f12d","q":"Let  $p$  be a prime number and  $k,r$  are positive integers such that  $p>r$ . If  $pk+r$  divides  $p^p+1$  then prove that  $r$  divides  $k$ .","t":[{"b":1,"e":0.28571,"k":"flat","v":0.14286,"x":0.27232,"p":[[0,26,0.0,0.24107,0.3719,0.0,0.0,0.32142,0.0,1.0,18,5,2,18,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[4,26,0.1538,0.27232,0.37348,0.0,0.0,0.42857,0.0,1.0,17,5,5,17,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,0,1,0,5],[8,26,0.3077,0.19643,0.26184,0.0,0.07143,0.2857,0.0,1.0,16,1,4,16,0,4,0,0,6,0,0,1,0,0,2,0,0,2,0,0,0,0,1],[12,26,0.4615,0.14286,0.30514,0.0,0.0,0.03571,0.0,1.0,24,3,6,24,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[16,26,0.6154,0.22321,0.3008,0.0,0.0,0.32143,0.0,1.0,17,2,11,17,0,1,0,0,6,0,0,4,0,0,0,0,0,1,0,0,1,0,2],[20,26,0.7692,0.24554,0.2402,0.0,0.21428,0.42857,0.0,0.85714,11,0,3,11,0,5,0,0,7,0,0,3,0,0,4,0,0,1,0,0,1,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.10714,"x":0.35714,"p":[[0,14,0.0,0.25445,0.29823,0.0,0.21429,0.42857,0.0,1.0,15,2,1,15,0,1,0,0,6,0,0,4,0,0,2,0,0,2,0,0,0,0,2],[4,14,0.2857,0.19196,0.29147,0.0,0.0,0.21429,0.0,1.0,18,1,3,18,0,6,0,0,0,0,0,2,0,0,2,0,0,2,0,0,1,0,1],[8,14,0.5714,0.14286,0.25754,0.0,0.0,0.14286,0.0,1.0,20,1,2,20,0,5,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,1],[12,14,0.8571,0.35714,0.36246,0.0,0.14286,0.75,0.0,1.0,9,3,0,9,0,8,0,0,3,0,0,2,0,0,1,0,0,1,0,0,5,0,3],[14,14,1.0,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"57920cdfa800ce05","q":"Let  $ABC$  be an acute triangle with circumcircle  $\\Gamma$  and let  $D$  be the midpoint of minor arc  $BC$ . Let  $E, F$  be on  $\\Gamma$  such that  $DE \\bot AC$  and  $DF \\bot AB$ . Lines  $BE$  and  $DF$  meet at  $G$ , and lines  $CF$  and  $DE$  meet at  $H$ . Show that  $BCHG$  is a parallelogram.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0982,"x":0.20982,"p":[[0,10,0.0,0.20982,0.23415,0.0,0.14286,0.35714,0.0,0.57143,15,0,3,15,0,3,0,0,6,0,0,0,0,0,8,0,0,0,0,0,0,0,0],[4,10,0.4,0.12052,0.20855,0.0,0.0,0.17857,0.0,0.57143,23,0,11,23,0,1,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[8,10,0.8,0.0982,0.19374,0.0,0.0,0.03571,0.0,0.57143,24,0,7,24,0,2,0,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.17411,"x":0.31696,"p":[[0,14,0.0,0.17411,0.19475,0.0,0.14286,0.2857,0.0,0.57143,15,0,7,15,0,4,0,0,7,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[4,14,0.2857,0.21427,0.24742,0.0,0.14286,0.32144,0.0,0.85714,15,0,4,15,0,3,0,0,6,0,0,1,0,0,6,0,0,0,0,0,1,0,0],[8,14,0.5714,0.30355,0.10561,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,1,0,0,26,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,14,0.8571,0.31696,0.12234,0.28571,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,1,0,0,25,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[14,14,1.0,0.2857,0.10098,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,0,0,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"b85d132f30c029d3","q":"Let  $p > 2$  be a prime number. Prove that there is a permutation  $k_1, k_2, ..., k_{p-1}$  of numbers  $1,2,...,p-1$  such that the number  $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$   is divisible by  $p$ .\n\nNote: The numbers  $k_1, k_2, ..., k_{p-1}$  are a permutation of the numbers  $1,2,...,p-1$  if each of of numbers  $1,2,...,p-1$  appears exactly once among the numbers  $k_1, k_2, ..., k_{p-1}$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.02679,"x":0.15178,"p":[[0,35,0.0,0.15178,0.31121,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,2],[4,35,0.1143,0.09822,0.23808,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[8,35,0.2286,0.03571,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,35,0.3429,0.12045,0.22334,0.0,0.0,0.14286,0.0,0.85714,20,0,0,20,0,7,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[16,35,0.4571,0.13393,0.25985,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,3,0,0],[20,35,0.5714,0.04464,0.13092,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,35,0.6857,0.05804,0.14223,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,5,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[28,35,0.8,0.06241,0.18185,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,35,0.9143,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0625,0.16728,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.14286,"p":[[0,52,0.0,0.09375,0.2131,0.0,0.0,0.14286,0.0,0.85714,23,0,1,23,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[4,52,0.0769,0.14286,0.23419,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,0,8,0,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0],[8,52,0.1538,0.13839,0.28679,0.0,0.0,0.14286,0.0,1.0,23,2,0,23,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,2],[12,52,0.2308,0.07143,0.20825,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[16,52,0.3077,0.08482,0.23652,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[20,52,0.3846,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,52,0.5385,0.08258,0.19488,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,3,0,0,0,0,1,1,0,0,1,0,0,0,0,0,1,0,0],[32,52,0.6154,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[48,52,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"d982b4688db08b03","q":"Let  $I$  be an open interval of length  $\\frac{1}{n}$ , where  $n$  is a positive integer. Find the maximum possible number of rational numbers of the form  $\\frac{a}{b}$  where  $1 \\le b \\le n$  that lie in  $I$ .","t":[{"b":2,"e":0.2857,"k":"flat","v":0.25893,"x":0.30803,"p":[[0,24,0.0,0.29019,0.02486,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.25893,0.11538,0.2857,0.28571,0.28571,0.0,0.57143,4,0,2,4,0,1,0,0,25,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,24,0.3333,0.28111,0.12055,0.28571,0.28571,0.28571,0.0,0.71,3,0,1,3,0,0,0,0,26,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[12,24,0.5,0.29019,0.09094,0.28571,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,0,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,24,0.6667,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.30803,0.06298,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[24,24,1.0,0.28571,1e-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]]},{"b":4,"e":0.2857,"k":"flat","v":0.24107,"x":0.29464,"p":[[0,28,0.0,0.26339,0.10779,0.2857,0.28571,0.28571,0.0,0.4286,4,0,1,4,0,0,0,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.28571,5,0,1,5,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.27232,0.17261,0.2857,0.28571,0.28571,0.0,1.0,5,1,2,5,0,0,0,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,28,0.4286,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.29464,0.07936,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,28,0.7143,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[28,28,1.0,0.28571,1e-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":"bda54e7ab222915a","q":"Let $A B C$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\\angle P A Q, \\angle P B Q$, and $\\angle P C Q$ are the altitudes of triangle $A B C$. Prove that the midpoint of $\\overline{P Q}$ lies on the Euler line of $A B C$.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.06697,"x":0.11152,"p":[[0,12,0.0,0.06697,0.10705,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.05804,"x":0.08482,"p":[[0,11,0.0,0.07143,0.15568,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,11,0.3636,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.4286,18,0,1,18,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.08009,0.07064,0.0,0.14,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9e11fd44b7df0839","q":"In the coordinate plane are finitely many walls, which are disjoint line segments, none of which are parallel to either axis. A bulldozer starts at an arbitrary point and moves in the $+x$ direction. Every time it hits a wall, it turns at a right angle to its path, away from the wall, and continues moving. (Thus the bulldozer always moves parallel to the axes.) Prove that it is impossible for the bulldozer to hit both sides of every wall.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,9,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,33,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,33,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"108ff812e5c4201c","q":"Let  $ABC$  be a triangle such that  $AC\\not= BC,AB<AC$  and let  $K$  be it's circumcircle. The tangent to  $K$  at the point  $A$  intersects the line  $BC$  at the point  $D$ . Let  $K_1$  be the circle tangent to  $K$  and to the segments  $(AD),(BD)$ . We denote by  $M$  the point where  $K_1$  touches  $(BD)$ . Show that  $AC=MC$  if and only if  $AM$  is the bisector of the  $\\angle DAB$ .\n\n*Neculai Roman*","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.00438,"p":[[0,22,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,11,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"13b65e8131a03afa","q":"Let $A B C D$ be a cyclic convex quadrilateral and $\\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals $A C$ and $B D$, let $L$ be the center of the circle tangent to sides $A B, B C$, and $C D$, and let $M$ be the midpoint of the arc $B C$ of $\\Gamma$ not containing $A$ and $D$. Prove that the excenter of triangle $B C E$ opposite $E$ lies on the line $L M$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,27,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"73859fe14dddb0a9","q":"Let  $ABCD$  be a convex quadrilateral for which \\[ AB+CD=\\sqrt{2}\\cdot AC\\qquad\\text{and}\\qquad BC+DA=\\sqrt{2}\\cdot BD.\\] Prove that  $ABCD$  is a parallelogram.","t":[{"b":0,"e":0.0,"k":"flat","v":0.05357,"x":0.13839,"p":[[0,7,0.0,0.08473,0.18507,0.0,0.0,0.14286,0.0,1.0,21,1,1,21,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,7,0.5714,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.13839,0.14934,0.0,0.07143,0.28571,0.0,0.4286,16,0,0,16,0,3,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.04464,"x":0.1115,"p":[[0,19,0.0,0.09375,0.13175,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.1115,0.1416,0.0,0.07,0.14286,0.0,0.571,16,0,0,16,0,10,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,19,0.4211,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.2857,24,0,1,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.05795,0.08637,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"83153801a0380373","q":"Let $A B C$ be a scalene triangle with circumcircle $\\Omega$, and suppose the incircle of $A B C$ touches $B C$ at $D$. The angle bisector of $\\angle A$ meets $B C$ and $\\Omega$ at $K$ and $M$. The circumcircle of $\\triangle D K M$ intersects the $A$-excircle at $S_{1}, S_{2}$, and $\\Omega$ at $T \\neq M$. Prove that line $A T$ passes through either $S_{1}$ or $S_{2}$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,35,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"caf14f5c843bc408","q":"Let $ABC$ be a triangle with all angles acute, and $H$ its orthocenter. The bisectors of $\\widehat{ABH}$ and $\\widehat{ACH}$ intersect at a point $I$. Show that $I$ is collinear with the midpoints of $[BC]$ and $[AH]$.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.09804,"x":0.54908,"p":[[0,52,0.0,0.48661,0.28763,0.2857,0.42857,0.85714,0.0,0.85714,2,0,0,2,0,3,0,0,10,0,0,3,0,0,2,0,0,3,0,0,9,0,0],[4,52,0.0769,0.54908,0.25532,0.28571,0.571,0.85714,0.14286,0.85714,0,0,0,0,0,3,0,0,7,0,0,5,0,0,4,0,0,3,0,0,10,0,0],[8,52,0.1538,0.38384,0.28001,0.14286,0.28571,0.71429,0.0,0.85714,4,0,2,4,0,6,0,0,9,0,0,3,0,0,1,0,0,5,0,0,4,0,0],[12,52,0.2308,0.18741,0.20027,0.14286,0.14286,0.14287,0.0,0.85714,6,0,1,6,0,20,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[16,52,0.3077,0.14723,0.14501,0.14286,0.14286,0.14286,0.0,0.85714,6,0,0,6,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,52,0.3846,0.10268,0.07349,0.0,0.14286,0.14286,0.0,0.2857,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,52,0.4615,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,52,0.5385,0.12946,0.05486,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,52,0.6154,0.10715,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,52,0.6923,0.12938,0.05484,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,52,0.7692,0.09804,0.0661,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,52,0.8462,0.10697,0.06176,0.105,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,52,0.9231,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,52,1.0,0.12027,0.05176,0.14214,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.24554,"x":0.50892,"p":[[0,5,0.0,0.50892,0.30916,0.28571,0.42857,0.85704,0.0,1.0,3,1,0,3,0,4,0,0,5,0,0,5,0,0,0,0,0,6,0,0,8,0,1],[4,5,0.8,0.24554,0.2237,0.14286,0.2857,0.28571,0.0,0.85714,7,0,0,7,0,8,0,0,13,0,0,0,0,0,1,0,0,1,0,0,2,0,0],[5,5,1.0,0.29464,0.07936,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"b2121a312a9d289b","q":"Let  $p, q, r$  be prime numbers and  $t, n$  be natural numbers such that  $p^2 +qt =(p + t)^n$  and  $p^2 + qr = t^4$  .\na) Show that  $n < 3$ .\nb) Determine all the numbers  $p, q, r, t, n$  that satisfy the given conditions.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.45085,"x":0.53115,"p":[[0,10,0.0,0.51338,0.24184,0.28571,0.42859,0.71429,0.14286,1.0,0,1,0,0,0,4,0,0,6,0,0,7,0,0,2,0,0,9,0,0,3,0,1],[4,10,0.4,0.47321,0.23538,0.28571,0.42857,0.57143,0.14286,1.0,0,3,0,0,0,2,0,0,10,0,0,10,0,0,3,0,0,3,0,0,1,0,3],[8,10,0.8,0.53115,0.23498,0.42857,0.4286,0.71429,0.14,1.0,0,3,0,0,0,2,0,0,5,0,0,11,0,0,4,0,0,5,0,0,2,0,3],[10,10,1.0,0.45085,0.1679,0.28571,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,7,0,0,8,0,0,10,0,0,4,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"flat","v":0.50445,"x":0.6473,"p":[[0,3,0.0,0.50445,0.19227,0.28571,0.57121,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,6,0,0,7,0,0,9,0,0,1,0,0],[3,3,1.0,0.6473,0.0944,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,20,0,0,0,0,0]]}]},{"i":"e3cf93c22c206b7a","q":"Let  $P(z)=a_nz^n+a_{n-1}z^{n-1}+\\ldots+a_mz^m$  be a polynomial with complex coefficients such that  $a_m\\neq 0, a_n\\neq 0$  and  $n>m$ . Prove that\n\\[\\text{max}_{|z|=1}\\{|P(z)|\\}\\ge\\sqrt{2|a_ma_n|+\\sum_{k=m}^{n} |a_k|^2}\\]","t":[{"b":0,"e":0.14286,"k":"flat","v":0.09813,"x":0.22322,"p":[[0,7,0.0,0.22322,0.26711,0.0,0.14286,0.2857,0.0,1.0,10,2,0,10,0,12,0,0,3,0,0,3,0,0,1,0,0,1,0,0,0,0,2],[4,7,0.5714,0.16518,0.21756,0.0,0.14286,0.2857,0.0,1.0,13,1,0,13,0,10,0,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[7,7,1.0,0.09813,0.13089,0.0,0.0,0.14286,0.0,0.4286,17,0,0,17,0,11,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.57143,"k":"flat","v":0.15179,"x":0.24098,"p":[[0,18,0.0,0.24098,0.24341,0.0,0.14286,0.32143,0.0,1.0,10,1,0,10,0,7,0,0,7,0,0,3,0,0,3,0,0,1,0,0,0,0,1],[4,18,0.2222,0.17857,0.23145,0.0,0.14286,0.17857,0.0,1.0,12,1,0,12,0,12,0,0,4,0,0,0,0,0,2,0,0,1,0,0,0,0,1],[8,18,0.4444,0.15179,0.21998,0.0,0.14286,0.14287,0.0,1.0,15,1,0,15,0,10,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[12,18,0.6667,0.23219,0.29182,0.0,0.14286,0.42857,0.0,1.0,13,2,0,13,0,8,0,0,2,0,0,4,0,0,2,0,0,0,0,0,1,0,2],[16,18,0.8889,0.16072,0.1171,0.14286,0.14286,0.1429,0.0,0.4286,6,0,0,6,0,19,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.16964,0.1448,0.10714,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,15,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"373558fcb6d26eed","q":"Let $A B C D$ be a convex quadrilateral and let $P$ and $Q$ be points in $A B C D$ such that $P Q D A$ and $Q P B C$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $P Q$ such that $\\angle P A E=\\angle Q D E$ and $\\angle P B E=\\angle Q C E$. Show that the quadrilateral $A B C D$ is cyclic.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04909,"p":[[0,34,0.0,0.04909,0.1317,0.0,0.0,0.0,0.0,0.571,27,0,1,27,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,34,0.1176,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.03572,0.08749,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.02679,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,49,0.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.03571,0.15152,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,49,0.4898,0.03125,0.12234,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,49,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0b60b4126fc8cb7a","q":"Let  $ABCD$  be a convex quadrilateral with  $\\angle A+\\angle D=90^\\circ$  and  $E$  the point of intersection of its diagonals. The line  $\\ell$  cuts the segments  $AB$ ,  $CD$ ,  $AE$  and  $ED$  in points  $X,Y,Z,T$ , respectively. Suppose that  $AZ=CE$  and  $BE=DT$ . Prove that the length of the segment  $XY$  is not larger than the diameter of the the circumcircle of  $ETZ$ .\n*Proposed by A. Kuznetsov, I. Frolov*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,14,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,2,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[14,14,1.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,23,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9020f6f3ab61f99a","q":"Let  $ABCD$  be a convex quadrilateral. The common external tangents to circles  $(ABC)$  and  $(ACD)$  meet at point  $E$ , the common external tangents to circles  $(ABD)$  and  $(BCD)$  meet at point  $F$ . Let  $F$  lie on  $AC$ , prove that  $E$  lies on  $BD$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.125,"x":0.29018,"p":[[0,21,0.0,0.25,0.10101,0.2857,0.28571,0.28571,0.0,0.42857,4,0,0,4,0,1,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.25893,0.11538,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,4,0,0,22,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,21,0.381,0.2366,0.09851,0.14289,0.2857,0.28571,0.0,0.42857,3,0,0,3,0,6,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.29018,0.08364,0.2857,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,2,0,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.125,0.13243,0.0,0.07143,0.28571,0.0,0.28571,16,0,0,16,0,4,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.18303,0.13474,0.0,0.2857,0.28571,0.0,0.42857,10,0,0,10,0,4,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.14286,0.13363,0.0,0.14286,0.28571,0.0,0.28571,14,0,0,14,0,4,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.25,"x":0.37049,"p":[[0,34,0.0,0.25,0.10101,0.2857,0.2857,0.28571,0.0,0.42857,3,0,1,3,0,4,0,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.28123,0.13112,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,2,0,0,23,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[8,34,0.2353,0.35262,0.15138,0.28571,0.28571,0.46418,0.0,0.71429,1,0,0,1,0,1,0,0,21,0,0,1,0,0,7,0,0,1,0,0,0,0,0],[12,34,0.3529,0.29015,0.14048,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,2,0,0,22,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[16,34,0.4706,0.30803,0.1017,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,3,0,0,24,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[20,34,0.5882,0.37049,0.14217,0.28571,0.28571,0.571,0.14286,0.71429,0,0,0,0,0,1,0,0,21,0,0,1,0,0,8,0,0,1,0,0,0,0,0],[24,34,0.7059,0.29016,0.14498,0.2857,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,3,0,0,20,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[28,34,0.8235,0.30357,0.07784,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,28,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,34,0.9412,0.35701,0.13813,0.2857,0.28571,0.57025,0.14286,0.57143,0,0,0,0,0,2,0,0,21,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[34,34,1.0,0.32585,0.10239,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,25,0,0,2,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"b2cbd690e4013c16","q":"Let  $PRUS$  be a trapezium such that  $\\angle PSR = 2\\angle QSU$  and  $\\angle SPU = 2 \\angle UPR$ . Let  $Q$  and  $T$  be on  $PR$  and  $SU$  respectively such that  $SQ$  and  $PU$  bisect  $\\angle PSR$  and  $\\angle SPU$  respectively. Let  $PT$  meet  $SQ$  at  $E$ . The line through  $E$  parallel to  $SR$  meets  $PU$  in  $F$  and the line through  $E$  parallel to  $PU$  meets  $SR$  in  $G$ . Let  $FG$  meet  $PR$  and  $SU$  in  $K$  and  $L$  respectively. Prove that  $KF$  =  $FG$  =  $GL$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.11152,"p":[[0,41,0.0,0.08036,0.11811,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.11152,0.17761,0.0,0.0,0.14286,0.0,0.71429,19,0,3,19,0,7,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,41,0.1951,0.07589,0.11285,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.02223,0.06281,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.14286,"k":"flat","v":0.03125,"x":0.13384,"p":[[0,49,0.0,0.04009,0.0816,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.05339,0.06893,0.0,0.0,0.14286,0.0,0.14286,20,0,1,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.08027,0.11253,0.0,0.0,0.14286,0.0,0.42857,19,0,1,19,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.10259,0.1299,0.0,0.14143,0.14286,0.0,0.57143,15,0,0,15,0,14,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,49,0.3265,0.09375,0.09182,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.11152,0.08549,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.09813,0.09057,0.0,0.14286,0.14287,0.0,0.28571,13,0,0,13,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.08036,0.08702,0.0,0.07143,0.14286,0.0,0.28571,16,0,1,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.06688,0.07965,0.0,0.0,0.14286,0.0,0.2857,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.05804,0.07873,0.0,0.0,0.14286,0.0,0.2857,20,0,0,20,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.10259,0.07345,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.13384,0.08702,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,20,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.12036,0.07235,0.14,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"07f217db03e18e4d","q":"Let  $ABCD$  be a quadrilateral inscribed in a circle with center  $O$  and  $E$  be the intersection of segments  $AC$  and  $BD$ . Let  $\\omega_1$  be the circumcircle of  $ADE$  and  $\\omega_2$  be the circumcircle of  $BCE$ . The tangent to  $\\omega_1$  at  $A$  and the tangent to  $\\omega_2$  at  $C$  meet at  $P$ . The tangent to  $\\omega_1$  at  $D$  and the tangent to  $\\omega_2$  at  $B$  meet at  $Q$ . Show that  $OP=OQ$ .\n\n*Merlijn Staps*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,14,0.0,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,1,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.01344,"p":[[0,33,0.0,0.01344,0.07482,0.0,0.0,0.0,0.0,0.43,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,33,0.4848,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a75c010e2eaa605b","q":"Let  $\\{a_n\\}$  be a bounded real sequence.\n(a) Prove that if X is a positive-measure subset of  $\\mathbb R$ , then for almost all  $x\\in X$ , there exist a subsequence  $\\{y_n\\}$  of X such that   $$ \\sum_{n=1}^\\infty (n(y_n-x)-a_n)=1 $$  (b) construct an unbounded sequence  $\\{a_n\\}$  for which the above equation is also true.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.14277,"x":0.28121,"p":[[0,12,0.0,0.15161,0.07089,0.14286,0.14286,0.14286,0.0,0.42857,2,0,1,2,0,27,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.14277,0.05051,0.14286,0.14286,0.14286,0.0,0.2857,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.28121,0.17318,0.14286,0.14286,0.42857,0.14,0.71429,0,0,0,0,0,17,0,0,5,0,0,5,0,0,4,0,0,1,0,0,0,0,0],[12,12,1.0,0.22545,0.16279,0.14286,0.14286,0.2857,0.0,0.71429,2,0,0,2,0,19,0,1,4,0,0,3,0,0,2,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.14268,"x":0.16295,"p":[[0,6,0.0,0.14509,0.0725,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,23,0,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.16295,0.0697,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,26,0,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0ff15a128220b635","q":"Let  $ ABC $  be a triangle, and let  $M$  be midpoint of  $BC$ . Let  $ I_b $  and  $ I_c $  be incenters of  $ AMB $  and  $ AMC $ . Prove that the second intersection of circumcircles of  $ ABI_b $  and  $ ACI_c $  distinct from  $A$  lies on line  $AM$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,27,0.0,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02233,"p":[[0,33,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.02233,0.07243,0.0,0.0,0.0,0.0,0.286,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,33,0.9697,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"2375f831ec168c0d","q":"Let $P$ be an odd-degree integer-coefficient polynomial. 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":5,"e":0.14286,"k":"flat","v":0.17857,"x":0.31469,"p":[[0,9,0.0,0.24329,0.21931,0.125,0.14286,0.32143,0.0,0.71429,7,0,0,7,1,10,0,0,6,0,0,3,0,0,2,0,0,3,0,0,0,0,0],[4,9,0.4444,0.31469,0.21264,0.14286,0.28571,0.46418,0.0,0.71429,5,0,1,5,0,6,0,0,8,0,1,4,0,0,6,0,0,2,0,0,0,0,0],[8,9,0.8889,0.17857,0.12372,0.14286,0.14286,0.2857,0.0,0.4286,5,0,0,5,0,18,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.21865,0.1881,0.14286,0.14286,0.2857,0.0,0.71429,4,0,0,4,2,16,0,0,4,0,0,1,0,0,4,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.4286,"k":"flat","v":0.0692,"x":0.27006,"p":[[0,12,0.0,0.27006,0.23803,0.0,0.28571,0.4642,0.0,0.71429,9,0,0,9,0,5,0,1,8,0,0,1,0,0,5,0,0,3,0,0,0,0,0],[4,12,0.3333,0.10937,0.16169,0.0,0.07141,0.14286,0.0,0.71429,15,0,0,15,3,10,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,12,0.6667,0.0692,0.11221,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,1,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,12,1.0,0.16731,0.14128,0.14214,0.14286,0.14286,0.0,0.571,6,0,0,6,1,19,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"16aa778e5fc49690","q":"Let $A$ be a set of real numbers such that $A$ has at least four elements. Suppose $A$ has the property that $a^{2}+b c$ is a rational number for all distinct numbers $a, b, c$ in $A$. Prove that there exists a positive integer $M$ such that $a \\sqrt{M}$ is a rational number for every $a$ in $A$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.35711,"x":0.5982,"p":[[0,6,0.0,0.59374,0.29474,0.28571,0.71429,0.85714,0.0,1.0,2,1,0,2,0,1,0,0,8,0,0,0,0,0,4,0,0,3,0,0,13,0,1],[4,6,0.6667,0.5982,0.31831,0.39286,0.64286,0.85714,0.0,1.0,3,4,0,3,0,1,0,0,4,0,0,6,0,0,2,0,0,1,0,0,11,0,4],[6,6,1.0,0.35711,0.17854,0.2857,0.28571,0.42858,0.0,0.85714,2,0,0,2,0,1,0,0,18,0,0,4,0,0,5,0,0,1,0,0,1,0,0]]},{"b":5,"e":0.57143,"k":"flat","v":0.61155,"x":0.7053,"p":[[0,24,0.0,0.61156,0.30143,0.39285,0.71429,0.85714,0.0,1.0,1,5,0,1,0,4,0,0,3,0,0,3,0,0,4,0,0,5,0,0,7,0,5],[4,24,0.1667,0.61604,0.29111,0.28571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,1,0,0,7,0,0,4,0,0,2,0,0,3,0,0,10,0,4],[8,24,0.3333,0.68747,0.20653,0.571,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,6,0,0,8,0,0,5,0,0,7,0,5],[12,24,0.5,0.6964,0.21355,0.57132,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,3,0,0,9,0,5],[16,24,0.6667,0.61155,0.26057,0.4286,0.57143,0.857,0.0,1.0,2,2,0,2,0,0,0,0,4,0,0,3,0,0,8,0,0,4,0,0,9,0,2],[20,24,0.8333,0.7053,0.23676,0.5354,0.71429,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,3,0,0,5,0,0,5,0,0,4,0,0,8,0,7],[24,24,1.0,0.66069,0.22233,0.5354,0.71429,0.85714,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,7,0,0,7,0,0,7,0,0,6,0,4]]}]},{"i":"ace8647cff8f6f9c","q":"Let  $ABCD$  be a cyclic quadrilateral (In the same order) inscribed into the circle  $\\odot (O)$ . Let  $\\overline{AC}$   $\\cap$   $\\overline{BD}$   $=$   $E$ . A randome line  $\\ell$  through  $E$  intersects  $\\overline{AB}$  at  $P$  and  $BC$  at  $Q$ .  A circle  $\\omega$  touches  $\\ell$  at  $E$  and passes through  $D$ . Given,  $\\omega$   $\\cap$   $\\odot (O)$   $=$   $R$ . Prove,  Points  $B,Q,R,P$  are concyclic.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,30,0.0,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.07589,0.15146,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.04018,0.12492,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.05357,0.13243,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.04464,0.12078,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.04464,"x":0.33915,"p":[[0,32,0.0,0.05804,0.13296,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.04464,0.12078,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.33915,0.15013,0.2857,0.42857,0.42857,0.0,0.71,3,0,0,3,0,2,0,0,9,0,0,17,0,0,0,0,0,1,0,0,0,0,0],[12,32,0.375,0.15607,0.0655,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.1875,0.08328,0.14286,0.14286,0.2857,0.0,0.42857,1,0,1,1,0,21,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.19187,0.07673,0.14286,0.14286,0.2857,0.14,0.42857,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.20518,0.09418,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,18,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.2008,0.09359,0.14286,0.14286,0.28571,0.0,0.42857,1,0,0,1,0,19,0,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.19187,0.08464,0.14286,0.14286,0.2857,0.0,0.42857,1,0,0,1,0,20,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1ce2e7b703f7f1fc","q":"Let  $ABC$  be an acute triangle with  $AB<AC<BC$   and let  $D$  be a point on it's extension of  $BC$  towards  $C$ . Circle  $c_1$ , with center  $A$  and radius  $AD$ , intersects lines  $AC,AB$  and  $CB$  at points  $E,F$ , and  $G$  respectively. Circumscribed circle  $c_2$  of triangle  $AFG$  intersects again lines  $FE,BC,GE$  and  $DF$  at points  $J,H,H' $  and  $J'$   respectively. Circumscribed circle  $c_3$  of triangle  $ADE$  intersects again lines  $FE,BC,GE$  and  $DF$  at points  $I,K,K' $  and  $I' $   respectively. Prove that the quadrilaterals  $HIJK$  and  $H'I'J'K '$  are cyclic and the centers of their circumscribed circles coincide.\n\nby Evangelos Psychas, Greece","t":[{"b":0,"e":0.4286,"k":"flat","v":0.23206,"x":0.42411,"p":[[0,30,0.0,0.26777,0.16275,0.14286,0.28571,0.42857,0.0,0.4286,4,0,0,4,0,11,0,0,2,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.23206,0.19154,0.0,0.14286,0.42857,0.0,0.4286,10,0,0,10,0,7,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.30786,0.16427,0.14286,0.42857,0.42857,0.0,0.4286,4,0,0,4,0,7,0,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.40625,0.08828,0.42857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,1,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.39723,0.08579,0.42857,0.42857,0.42857,0.14,0.4286,0,0,0,0,0,3,0,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.39286,0.10102,0.42857,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,2,0,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.41518,0.05486,0.42857,0.42857,0.42857,0.14286,0.4286,0,0,0,0,0,1,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.42411,0.02486,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.41518,0.04164,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.42857,"k":"rising","v":0.25437,"x":0.42866,"p":[[0,33,0.0,0.25437,0.16656,0.14286,0.21428,0.42857,0.0,0.42857,5,0,0,5,0,11,0,0,2,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.26786,0.14617,0.14286,0.1429,0.42857,0.0,0.4286,1,0,0,1,0,16,0,0,1,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.32143,0.16367,0.14286,0.42857,0.42857,0.0,0.4286,4,0,0,4,0,6,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.3213,0.16774,0.14286,0.42857,0.42857,0.0,0.43,5,0,0,5,0,4,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.28563,0.17504,0.14286,0.42857,0.42857,0.0,0.4286,6,0,0,6,0,6,0,0,2,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.28572,0.15972,0.14286,0.42857,0.42857,0.0,0.4286,3,0,0,3,0,11,0,0,1,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.42857,1e-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],[28,33,0.8485,0.42415,0.02487,0.42857,0.42857,0.42857,0.28571,0.43,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.42857,1e-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],[33,33,1.0,0.42866,0.00035,0.42857,0.42857,0.42857,0.42857,0.43,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"25559fe565c6fb15","q":"In Greifswald there are three schools called  $A,B$  and  $C$ , each of which is attended by at least one student. Among any three students, one from  $A$ , one from  $B$  and one from  $C$ , there are two knowing each other and two not knowing each other. Prove that at least one of the following holds:\n\n\n- Some student from  $A$  knows all students from  $B$ .\n- Some student from  $B$  knows all students from  $C$ .\n- Some student from  $C$  knows all students from  $A$ .","t":[{"b":3,"e":0.14286,"k":"flat","v":0.16509,"x":0.19625,"p":[[0,10,0.0,0.19625,0.10574,0.14286,0.14286,0.28571,0.0,0.4286,2,0,0,2,0,19,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.16964,0.06621,0.14286,0.14286,0.14287,0.0,0.28571,1,0,1,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.16964,0.05576,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.16509,0.05191,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.14259,"x":0.17848,"p":[[0,18,0.0,0.16072,0.13243,0.10714,0.14286,0.1786,0.0,0.42857,8,0,3,8,0,16,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.15179,0.04971,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.17848,0.08752,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,27,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f6dcc9ac7c0fc53d","q":"Let  $ABC$  be a non-right triangle and let  $M$  be the midpoint of  $BC$ . Let  $D$  be a point on  $AM$  (D\u2260A, D\u2260M). Let \u03c91 be a circle through  $D$  that intersects  $BC$  at  $B$  and let \u03c92 be a circle through  $D$  that intersects  $BC$  at  $C$ . Let  $AB$  intersect \u03c91 at  $B$  and  $E$ , and let  $AC$  intersect \u03c92 at  $C$  and  $F$ .\nProve, that the tangent on \u03c91 at  $E$  and the tangent on \u03c92 at  $F$  intersect on  $AM$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.05356,"p":[[0,31,0.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.05356,0.12747,0.0,0.0,0.0,0.0,0.571,26,0,1,26,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,31,0.5161,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.0,"k":"flat","v":0.0,"x":0.04009,"p":[[0,8,0.0,0.02683,0.0835,0.0,0.0,0.0,0.0,0.43,28,0,1,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.04009,0.0816,0.0,0.0,0.0,0.0,0.28571,25,0,2,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"64f4018566568eb9","q":"Let  ${ABC}$  be an acute angled triangle, let  ${O}$  be its circumcentre, and let  ${D,E,F}$  be points on the sides  ${BC,CA,AB}$ , respectively. The circle  ${(c_1)}$  of radius  ${FA}$ , centered at  ${F}$ , crosses the segment  ${OA}$  at  ${A'}$  and the circumcircle  ${(c)}$  of the triangle  ${ABC}$ again at  ${K}$ . Similarly, the circle  ${(c_2)}$  of radius  $DB$ , centered at  $D$ , crosses the segment  $\\left( OB \\right)$  at  ${B}'$  and the circle  ${(c)}$  again at  ${L}$ . Finally, the circle  ${(c_3)}$  of radius  $EC$ , centered at  $E$ , crosses the segment  $\\left( OC \\right)$ at  ${C}'$  and the circle  ${(c)}$  again at  ${M}$ . Prove that the quadrilaterals  $BKF{A}',CLD{B}'$  and  $AME{C}'$  are all cyclic, and their circumcircles share a common point.\n\n Evangelos Psychas (Greece)","t":[{"b":4,"e":0.0,"k":"falling","v":0.00446,"x":0.32134,"p":[[0,7,0.0,0.32134,0.3031,0.0,0.2857,0.60714,0.0,0.85714,12,0,0,12,0,3,0,0,2,0,0,4,0,0,3,0,0,7,0,0,1,0,0],[4,7,0.5714,0.15625,0.27976,0.0,0.0,0.17857,0.0,0.85714,23,0,0,23,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,2,0,0],[7,7,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.04464,"x":0.32143,"p":[[0,31,0.0,0.32143,0.35892,0.0,0.21428,0.71429,0.0,1.0,15,2,0,15,0,1,0,0,4,0,0,1,0,0,2,0,0,4,0,0,3,0,2],[4,31,0.129,0.20979,0.26479,0.0,0.07143,0.28571,0.0,1.0,16,1,0,16,0,2,0,0,7,0,0,2,0,0,2,0,0,2,0,0,0,0,1],[8,31,0.2581,0.11151,0.23615,0.0,0.0,0.035,0.0,0.85714,24,0,0,24,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,2,0,0],[12,31,0.3871,0.25445,0.33451,0.0,0.0,0.46418,0.0,1.0,18,1,1,18,0,0,0,0,5,0,0,1,0,0,1,0,0,3,0,0,3,0,1],[16,31,0.5161,0.07588,0.20195,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[20,31,0.6452,0.16515,0.29254,0.0,0.0,0.17857,0.0,1.0,23,1,0,23,0,1,0,0,1,0,0,0,0,0,3,0,0,3,0,0,0,0,1],[24,31,0.7742,0.06696,0.18893,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[28,31,0.9032,0.16069,0.32288,0.0,0.0,0.03571,0.0,1.0,24,2,1,24,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,2],[31,31,1.0,0.04464,0.14914,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"f5e3c1a6d57391f2","q":"In triangle  $ABC$ ,  $D$  is a variable point on line  $BC$ . Points  $E,F$  are on segments  $AC, AB$  respectively such that  $BF=BD$  and  $CD=CE$ . Circles  $(AEF)$  and  $(ABC)$  meet again at  $S$ . Lines  $EF$  and  $BC$  meet at  $P$  and circles  $(PDS)$  and  $(AEF)$  meet again at  $Q$ . Prove that, as  $D$  varies, isogonal conjugate of  $Q$  with respect to triangle  $ ABC$  lies on a fixed circle.\n\n*Proposed by Serdar Bozdag*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,44,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,44,0.6364,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":2,"e":0.0,"k":"flat","v":0.0,"x":0.01777,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"8898ce8501b7799a","q":"Let $ABC$ be a triangle such that $\\widehat{BAC}=60^{\\circ}$. Let $O$ be the center of the circumcircle of triangle $ABC$, $H$ its orthocenter, and $I$ the point of intersection of the angle bisectors. Show that the points $B$, $O$, $I$, $H$, and $C$ are concyclic.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.80357,"x":0.94643,"p":[[0,23,0.0,0.84821,0.23673,0.85714,0.85714,1.0,0.0,1.0,1,15,1,1,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,12,0,15],[4,23,0.1739,0.91517,0.13296,0.85714,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,14,0,17],[8,23,0.3478,0.89284,0.19885,0.85714,1.0,1.0,0.0,1.0,1,19,1,1,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,9,0,19],[12,23,0.5217,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21],[16,23,0.6957,0.89731,0.12492,0.857,1.0,1.0,0.57143,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,8,0,17],[20,23,0.8696,0.80357,0.13243,0.71429,0.71429,0.89286,0.57143,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,6,0,8],[23,23,1.0,0.83481,0.12428,0.71429,0.85707,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,10,0,9]]},{"b":2,"e":0.85714,"k":"flat","v":0.80802,"x":0.96428,"p":[[0,24,0.0,0.96428,0.07144,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,25],[4,24,0.1667,0.91517,0.10013,0.85714,0.92857,1.0,0.57143,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,14,0,16],[8,24,0.3333,0.80802,0.31055,0.857,1.0,1.0,0.0,1.0,2,18,2,2,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,7,0,18],[12,24,0.5,0.8125,0.29974,0.85711,1.0,1.0,0.0,1.0,2,17,2,2,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,8,0,17],[16,24,0.6667,0.83035,0.30813,0.85714,1.0,1.0,0.0,1.0,3,17,3,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,17],[20,24,0.8333,0.93301,0.07132,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[24,24,1.0,0.94643,0.07784,0.85714,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,21]]}]},{"i":"f3daebcbdefaa3b8","q":"Let $A B C$ be an acute, scalene triangle, and let $M, N$, and $P$ be the midpoints of $\\overline{B C}, \\overline{C A}$, and $\\overline{A B}$, respectively. Let the perpendicular bisectors of $\\overline{A B}$ and $\\overline{A C}$ intersect ray $A M$ in points $D$ and $E$ respectively, and let lines $B D$ and $C E$ intersect in point $F$, inside triangle $A B C$. Prove that points $A, N, F$, and $P$ all lie on one circle.","t":[{"b":0,"e":0.57143,"k":"rising","v":0.37499,"x":0.75893,"p":[[0,20,0.0,0.37499,0.30877,0.14286,0.28571,0.57111,0.0,1.0,4,3,0,4,0,9,0,0,6,0,0,4,0,0,2,0,0,2,0,0,2,0,3],[4,20,0.2,0.40159,0.29989,0.14286,0.28571,0.57111,0.0,1.0,2,4,0,2,0,8,0,0,9,0,0,4,0,0,2,0,0,2,0,0,1,0,4],[8,20,0.4,0.47753,0.34544,0.14286,0.42857,0.85704,0.0,1.0,2,6,0,2,0,9,0,0,4,0,0,4,0,0,2,0,0,2,0,0,3,0,6],[12,20,0.6,0.44622,0.30269,0.24999,0.35714,0.60682,0.0,1.0,1,5,0,1,0,7,0,0,8,0,0,6,0,0,2,0,0,2,0,0,1,0,5],[16,20,0.8,0.75893,0.27067,0.57143,0.85714,1.0,0.1429,1.0,0,14,0,0,0,1,0,0,4,0,0,1,0,0,3,0,0,6,0,0,3,0,14],[20,20,1.0,0.69196,0.23175,0.57143,0.71429,0.85714,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,4,0,0,6,0,0,5,0,0,9,0,5]]},{"b":1,"e":1.0,"k":"volatile","v":0.44862,"x":0.97321,"p":[[0,9,0.0,0.49553,0.36593,0.14289,0.35714,0.85714,0.0,1.0,4,7,0,4,0,5,0,0,7,0,0,2,0,0,2,0,0,1,0,0,4,0,7],[4,9,0.4444,0.44862,0.31401,0.2857,0.35714,0.66082,0.0,1.0,4,5,0,4,0,3,0,0,9,0,0,3,0,0,4,1,0,3,0,0,0,0,5],[8,9,0.8889,0.88613,0.19554,0.83929,1.0,1.0,0.14286,1.0,0,21,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,3,1,0,3,0,21],[9,9,1.0,0.97321,0.07523,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,2,0,28]]}]},{"i":"362298464fb5cca7","q":"Let  $S$  be a set of positive integers such that if  $a$  and  $b$  are elements of  $S$  such that  $a<b$ , then  $b-a$  divides the least common multiple of  $a$  and  $b$ , and the quotient is an element of  $S$ . Prove that the cardinality of  $S$  is less than or equal to  $2$ .","t":[{"b":3,"e":0.28571,"k":"flat","v":0.375,"x":0.47307,"p":[[0,15,0.0,0.47307,0.13067,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,14,0,0,10,0,0,3,0,0,0,0,0],[4,15,0.2667,0.41962,0.1921,0.42857,0.42857,0.57111,0.0,0.71429,4,0,0,4,0,0,0,0,3,0,0,15,0,0,7,0,0,3,0,0,0,0,0],[8,15,0.5333,0.375,0.16269,0.28571,0.42857,0.4286,0.0,0.71429,2,0,0,2,0,2,0,0,10,0,0,11,0,0,6,0,0,1,0,0,0,0,0],[12,15,0.8,0.38835,0.20273,0.2857,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,1,0,0,9,0,0,6,0,0,10,0,0,2,0,0,0,0,0],[15,15,1.0,0.40625,0.10171,0.28571,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,11,0,0,15,0,0,6,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.4286,"k":"flat","v":0.42408,"x":0.50443,"p":[[0,38,0.0,0.46426,0.14284,0.42857,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,16,0,0,7,0,0,4,0,0,0,0,0],[4,38,0.1053,0.43742,0.16335,0.39286,0.42857,0.57111,0.0,0.71429,2,0,0,2,0,0,0,0,6,0,0,12,0,0,10,0,0,2,0,0,0,0,0],[8,38,0.2105,0.44191,0.13049,0.39286,0.42857,0.571,0.14286,0.71429,0,0,0,0,0,1,0,0,7,0,0,14,0,0,8,0,0,2,0,0,0,0,0],[12,38,0.3158,0.49106,0.17473,0.42857,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,12,0,0,6,0,0,5,0,0,2,0,0],[16,38,0.4211,0.47766,0.14553,0.28593,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,9,0,0,6,0,0,15,0,0,1,0,0,1,0,0],[20,38,0.5263,0.49103,0.17833,0.39286,0.4998,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,8,0,0,9,0,0,6,0,0,1,0,0],[24,38,0.6316,0.50443,0.14716,0.42857,0.571,0.57143,0.2857,0.857,0,0,0,0,0,0,0,0,6,0,0,9,0,0,12,0,0,4,0,0,1,0,0],[28,38,0.7368,0.42408,0.14498,0.28571,0.42857,0.57141,0.14286,0.71429,0,0,0,0,0,1,0,0,12,0,0,8,0,0,9,0,0,2,0,0,0,0,0],[32,38,0.8421,0.4955,0.14277,0.42857,0.4998,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,4,0,0,11,0,0,11,0,0,5,0,0,0,0,0],[36,38,0.9474,0.42856,0.12371,0.28571,0.42857,0.4642,0.2857,0.71429,0,0,0,0,0,0,0,0,10,0,0,14,0,0,6,0,0,2,0,0,0,0,0],[38,38,1.0,0.43744,0.11252,0.28571,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,12,0,0,11,0,0,0,0,0,0,0,0]]}]},{"i":"d80fc9e7946b38a4","q":"Let $A B C$ be a triangle with $A B>A C$. The internal angle bisector of $\\angle B A C$ intersects the side $B C$ at $D$. The circles with diameters $B D$ and $C D$ intersect the circumcircle of $\\triangle A B C$ a second time at $P \\neq B$ and $Q \\neq C$, respectively. The lines $P Q$ and $B C$ intersect at $X$. Prove that $A X$ is tangent to the circumcircle of $\\triangle A B C$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.02232,"x":0.04455,"p":[[0,19,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,3,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.04455,0.13564,0.0,0.0,0.0,0.0,0.71429,27,0,1,27,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,19,0.4211,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.06697,"p":[[0,44,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.05357,0.14174,0.0,0.0,0.0,0.0,0.71429,26,0,1,26,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,44,0.1818,0.04911,0.13175,0.0,0.0,0.0,0.0,0.71429,25,0,1,25,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,44,0.2727,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.04465,0.15335,0.0,0.0,0.0,0.0,0.85714,27,0,5,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[20,44,0.4545,0.06697,0.18893,0.0,0.0,0.0,0.0,0.85714,27,0,3,27,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[24,44,0.5455,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.04464,0.10972,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.04455,0.08318,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.05804,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e10170cbd1c35f32","q":"Let  ${ABC}$  be a triangle inscribed in a circle. Point  ${P}$  is the center of the  arc  ${BAC}$ . The circle with the diameter  ${CP}$  intersects the angle bisector of angle  ${\\angle BAC}$   at points  ${K, L}$    ${(|AK| <|AL|)}$ . Point  ${M}$  is the reflection of  ${L}$  with respect to line  ${BC}$ . Prove that the circumcircle of the triangle  ${BKM}$  passes through the center of the segment  ${BC}$  .","t":[{"b":5,"e":0.0,"k":"flat","v":0.04911,"x":0.06696,"p":[[0,5,0.0,0.06696,0.12364,0.0,0.0,0.03571,0.0,0.42857,24,0,3,24,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.04911,0.09181,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.03572,"x":0.04464,"p":[[0,2,0.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,4,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,1.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cf76b2cde3b96dee","q":"The points of a circle are colored by three colors. Prove that there exist infinitely many isosceles triangles inscribed in the circle whose vertices are of the same color.","t":[{"b":4,"e":0.0,"k":"falling","v":0.09366,"x":0.39271,"p":[[0,34,0.0,0.3704,0.24429,0.14286,0.28571,0.57143,0.0,0.85714,3,0,0,3,0,8,0,0,6,0,0,4,0,0,5,0,0,5,0,0,1,0,0],[4,34,0.1176,0.39271,0.31729,0.14286,0.35714,0.71429,0.0,0.85714,6,0,0,6,0,9,0,0,1,0,0,2,0,0,4,0,0,5,0,0,5,0,0],[8,34,0.2353,0.23658,0.24897,0.10714,0.14286,0.42858,0.0,0.85714,8,0,0,8,0,15,0,0,0,0,0,2,0,0,5,0,0,0,0,0,2,0,0],[12,34,0.3529,0.2187,0.23133,0.0,0.14286,0.46418,0.0,0.71429,10,0,0,10,0,13,0,0,0,0,0,1,0,0,7,0,0,1,0,0,0,0,0],[16,34,0.4706,0.23213,0.27139,0.0,0.14286,0.2857,0.0,0.85714,11,0,0,11,0,11,0,0,3,0,0,0,0,0,1,0,0,5,0,0,1,0,0],[20,34,0.5882,0.19196,0.23038,0.0,0.14286,0.14286,0.0,0.85714,9,0,0,9,0,16,0,0,3,0,0,0,0,0,1,0,0,1,0,0,2,0,0],[24,34,0.7059,0.10706,0.09446,0.0,0.14286,0.14286,0.0,0.42857,11,0,0,11,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.09366,0.06779,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.09813,0.07519,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.571,"k":"rising","v":0.32589,"x":0.56245,"p":[[0,26,0.0,0.32589,0.25313,0.14286,0.21428,0.57143,0.0,0.85714,3,0,0,3,0,13,0,0,4,0,0,3,0,0,4,0,0,3,0,0,2,0,0],[4,26,0.1538,0.39705,0.28502,0.14286,0.28571,0.711,0.0,0.85714,2,0,0,2,0,12,0,0,3,0,0,2,0,0,4,0,0,5,0,0,4,0,0],[8,26,0.3077,0.56245,0.14698,0.571,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,2,0,0,1,0,0,3,0,0,17,0,0,9,0,0,0,0,0],[12,26,0.4615,0.52676,0.15334,0.4286,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,3,0,0,4,0,0,17,0,0,6,0,0,0,0,0],[16,26,0.6154,0.51771,0.15891,0.42857,0.571,0.60714,0.14,0.71429,0,0,0,0,0,1,0,0,5,0,0,7,0,0,11,0,0,8,0,0,0,0,0],[20,26,0.7692,0.55794,0.07455,0.571,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,3,0,0,26,0,0,2,0,0,0,0,0],[24,26,0.9231,0.50432,0.13348,0.42857,0.571,0.57143,0.1429,0.71429,0,0,0,0,0,1,0,0,4,0,0,7,0,0,17,0,0,3,0,0,0,0,0],[26,26,1.0,0.5535,0.11709,0.53465,0.57143,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,7,0,0,18,0,0,6,0,0,0,0,0]]}]},{"i":"c85810e4a06a8da1","q":"Let  $k$  be a positive integer.  Some of the  $2k$ -element subsets of a given set are marked.  Suppose that for any subset of cardinality less than or equal to  $(k+1)^2$  all the marked subsets contained in it (if any) have a common element.  Show that all the marked subsets have a common element.","t":[{"b":1,"e":0.571,"k":"flat","v":0.53123,"x":0.74107,"p":[[0,33,0.0,0.74107,0.38372,0.39285,1.0,1.0,0.0,1.0,2,21,0,2,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,21],[4,33,0.1212,0.63837,0.32535,0.42857,0.57121,1.0,0.0,1.0,1,13,0,1,0,3,0,0,1,0,0,9,0,0,5,0,0,0,0,0,0,0,13],[8,33,0.2424,0.54015,0.36022,0.35714,0.4998,1.0,0.0,1.0,5,10,0,5,0,3,0,0,0,0,0,8,0,0,6,0,0,0,0,0,0,0,10],[12,33,0.3636,0.72322,0.32525,0.42857,1.0,1.0,0.14286,1.0,0,17,0,0,0,4,0,0,0,0,0,8,0,0,1,0,0,1,0,0,1,0,17],[16,33,0.4848,0.70533,0.27651,0.42857,0.64286,1.0,0.0,1.0,1,13,1,1,0,0,0,0,0,0,0,9,0,0,6,0,0,2,0,0,1,0,13],[20,33,0.6061,0.55801,0.24317,0.42857,0.57121,0.57143,0.0,1.0,2,5,0,2,0,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,1,0,5],[24,33,0.7273,0.61605,0.20959,0.42857,0.57143,0.71429,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,13,0,0,8,0,0,5,0,0,0,0,6],[28,33,0.8485,0.5625,0.18535,0.42857,0.4286,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,1,0,0,16,0,0,6,0,0,4,0,0,3,0,2],[32,33,0.9697,0.53123,0.1525,0.42857,0.4286,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,18,0,0,9,0,0,3,0,0,0,0,2],[33,33,1.0,0.61603,0.20341,0.42857,0.57143,0.71429,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,12,0,0,10,0,0,3,0,0,2,0,5]]},{"b":5,"e":0.42857,"k":"falling","v":0.44195,"x":0.69195,"p":[[0,14,0.0,0.6875,0.41563,0.14286,1.0,1.0,0.0,1.0,2,19,0,2,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,19],[4,14,0.2857,0.65178,0.333,0.42857,0.71429,1.0,0.0,1.0,3,11,0,3,0,1,0,0,1,0,0,8,0,0,1,0,0,4,0,0,3,0,11],[8,14,0.5714,0.69195,0.35734,0.42857,0.9285,1.0,0.0,1.0,3,16,0,3,0,2,0,0,0,0,0,7,0,0,1,0,0,2,0,0,1,0,16],[12,14,0.8571,0.48661,0.15093,0.42857,0.42857,0.42858,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,25,0,0,2,0,0,1,0,0,2,0,1],[14,14,1.0,0.44195,0.0416,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0]]}]},{"i":"2225a276b710c934","q":"Let $A$ be a set of $n \\geq 2$ positive integers, and let $f(x)=\\sum_{a \\in A} x^{a}$. Prove that there exists a complex number $z$ with $|z|=1$ and $|f(z)|=\\sqrt{n-2}$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.07589,"x":0.08706,"p":[[0,12,0.0,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.08706,0.08881,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,14,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.06241,"x":0.10036,"p":[[0,6,0.0,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.10036,0.10233,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,1,15,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.08482,0.13296,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a0710332023d26b3","q":"Let  $ABC$  an acute triangle and  $H$  its orthocenter. Let  $E$  and  $F$  be the intersection of lines  $BH$  and  $CH$  with  $AC$  and  $AB$  respectively, and let  $D$  be the intersection of lines  $EF$  and  $BC$ . Let  $\\Gamma_1$  be the circumcircle of  $AEF$ , and  $\\Gamma_2$  the circumcircle of  $BHC$ . The line  $AD$  intersects  $\\Gamma_1$  at point  $I \\neq A$ . Let  $J$  be the feet of the internal bisector of  $\\angle{BHC}$  and  $M$  the midpoint of the arc  $\\stackrel{\\frown}{BC}$  from  $\\Gamma_2$  that contains the point  $H$ . The line  $MJ$  intersects  $\\Gamma_2$  at point  $N \\neq M$ . Show that the triangles  $EIF$  and  $CNB$  are similar.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,34,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,15,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,9,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,13,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,7,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,11,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"877421d75a4d982f","q":"Let  ${A}$  be a finite set and  ${\\rightarrow}$  be a binary relation on it such that for any  ${a,b,c \\in A}$ , if  ${a\\neq b}, {a \\rightarrow c}$  and  ${b \\rightarrow c}$  then either  ${a \\rightarrow b}$  or  ${b \\rightarrow a}$  (or possibly both). Let  ${B,\\,B \\subset A}$  be minimal with the property: for any  ${a \\in A \\setminus B}$  there exists  ${b \\in B}$ , such that either  ${a \\rightarrow b}$  or  ${b \\rightarrow a}$  (or possibly both). \nSupposing that  ${A}$  has at most  ${k}$  elements that are pairwise not in relation  ${\\rightarrow}$ , prove that  ${B}$  has at most  ${k}$  elements.","t":[{"b":5,"e":0.71429,"k":"flat","v":0.31686,"x":0.43293,"p":[[0,7,0.0,0.33928,0.33071,0.0,0.2857,0.60714,0.0,0.85714,11,0,2,11,0,4,0,0,4,0,0,2,0,0,3,0,0,2,0,0,6,0,0],[4,7,0.5714,0.43293,0.3398,0.14286,0.28571,0.75,0.0,1.0,3,3,1,3,0,11,0,0,3,0,0,2,0,0,2,0,0,3,0,0,5,0,3],[7,7,1.0,0.31686,0.15873,0.14286,0.28571,0.42857,0.14,0.71429,0,0,0,0,0,11,0,0,8,0,0,9,0,0,3,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.4286,"k":"flat","v":0.27228,"x":0.32579,"p":[[0,11,0.0,0.29007,0.29986,0.0,0.14286,0.4642,0.0,1.0,10,2,1,10,0,7,0,0,4,0,0,3,0,0,4,0,0,1,0,0,1,0,2],[4,11,0.3636,0.27228,0.24061,0.105,0.21428,0.42857,0.0,0.85714,8,0,3,8,0,8,0,0,4,0,0,7,0,0,3,0,0,0,0,0,2,0,0],[8,11,0.7273,0.32579,0.28627,0.14286,0.2143,0.57111,0.0,1.0,5,1,4,5,0,11,0,0,5,0,0,2,0,0,4,0,0,1,0,0,3,0,1],[11,11,1.0,0.32132,0.17504,0.14286,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,9,0,0,6,0,0,9,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"cf50a3fca2ce79d5","q":"Let  $ABCD$  be a cyclic quadrilateral whose diagonals  $AC$  and  $BD$  meet at  $E$ . The extensions of the sides  $AD$  and  $BC$  beyond  $A$  and  $B$  meet at  $F$ . Let  $G$  be the point such that  $ECGD$  is a parallelogram, and let  $H$  be the image of  $E$  under reflection in  $AD$ . Prove that  $D,H,F,G$  are concyclic.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0267,"p":[[0,37,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.0267,0.08316,0.0,0.0,0.0,0.0,0.42857,28,0,1,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.0067,0.02743,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02223,"p":[[0,7,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6acf621011421267","q":"Let  $ABC$  be an acute triangle with circumcircle  $\\omega$  such that  $AB<AC$ . Let  $M$  be the midpoint of the arc  $BC$  of~ $\\omega$  containing the point~ $A$ , and let  $X\\neq M$  be the other point on  $\\omega$  such that  $AX=AM$ . Points  $E$  and  $F$  are chosen on sides  $AC$  and  $AB$  of the triangle  $ABC$  such that  $EX=EC$  and  $FX=FB$ . Prove that  $AE=AF$ .","t":[{"b":2,"e":0.1429,"k":"flat","v":0.13838,"x":0.43301,"p":[[0,18,0.0,0.29464,0.36234,0.0,0.07143,0.57143,0.0,1.0,16,2,9,16,0,2,0,0,3,0,0,2,0,0,2,0,0,0,0,0,5,0,2],[4,18,0.2222,0.43301,0.37538,0.0,0.42857,0.74996,0.0,1.0,10,6,7,10,0,1,0,0,4,0,0,3,0,0,5,0,0,1,0,0,2,0,6],[8,18,0.4444,0.24991,0.16757,0.14286,0.2857,0.28571,0.0,0.85714,4,0,0,4,0,9,0,0,13,0,0,5,0,0,0,0,0,0,0,0,1,0,0],[12,18,0.6667,0.13838,0.15351,0.0,0.14286,0.2857,0.0,0.571,14,0,0,14,0,9,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,18,0.8889,0.15179,0.17835,0.0,0.07143,0.28571,0.0,0.57143,16,0,0,16,0,5,0,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[18,18,1.0,0.1875,0.18013,0.0,0.14288,0.28571,0.0,0.57143,13,0,0,13,0,4,0,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"volatile","v":0.20982,"x":0.5803,"p":[[0,17,0.0,0.48658,0.36744,0.0,0.49979,0.857,0.0,1.0,9,4,5,9,0,0,0,0,3,0,0,4,0,0,3,0,0,3,0,0,6,0,4],[4,17,0.2353,0.20982,0.28344,0.0,0.0,0.32143,0.0,1.0,17,2,8,17,0,2,0,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,2],[8,17,0.4706,0.5803,0.33107,0.28571,0.57121,0.85714,0.0,1.0,3,7,0,3,0,3,0,0,3,0,0,3,0,0,6,0,0,3,0,0,4,0,7],[12,17,0.7059,0.42855,0.30928,0.14289,0.42857,0.71429,0.0,1.0,5,2,0,5,0,4,0,0,6,0,0,5,0,0,3,0,0,3,0,0,4,0,2],[16,17,0.9412,0.43747,0.3173,0.14286,0.42857,0.71429,0.0,1.0,6,4,0,6,0,3,0,0,4,0,0,6,0,0,4,0,0,5,0,0,0,0,4],[17,17,1.0,0.48436,0.35207,0.24999,0.42857,0.85704,0.0,1.0,5,6,0,5,1,2,0,0,6,0,0,4,0,0,3,0,0,2,0,0,3,0,6]]}]},{"i":"90db5195910cbbad","q":"Let $A B C D$ be a convex quadrilateral and let $P$ be the point of intersection of $A C$ and $B D$. Suppose that $A C+A D=B C+B D$. Prove that the internal angle bisectors of $\\angle A C B, \\angle A D B$, and $\\angle A P B$ meet at a common point.","t":[{"b":1,"e":0.0,"k":"flat","v":0.03571,"x":0.09821,"p":[[0,9,0.0,0.09821,0.21556,0.0,0.0,0.03571,0.0,0.85714,24,0,2,24,0,3,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[4,9,0.4444,0.04464,0.13092,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,9,0.8889,0.03571,0.08748,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.05804,0.1063,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"falling","v":0.03125,"x":0.18308,"p":[[0,23,0.0,0.18308,0.27491,0.0,0.0,0.28571,0.0,0.85714,19,0,2,19,0,3,0,0,3,0,0,2,0,0,1,0,0,2,0,0,2,0,0],[4,23,0.1739,0.14287,0.24744,0.0,0.0,0.28571,0.0,0.85714,22,0,2,22,0,1,0,0,4,0,0,1,0,0,1,0,0,2,0,0,1,0,0],[8,23,0.3478,0.08036,0.18189,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,23,0.5217,0.12054,0.22899,0.0,0.0,0.14286,0.0,0.85714,22,0,1,22,0,4,0,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,0],[16,23,0.6957,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.04902,0.08449,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.03125,0.08553,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a62286236635e303","q":"Let  $k\\ge2$  be an integer. Find the smallest integer  $n \\ge k+1$  with the property that there exists a set of  $n$  distinct real numbers such that each of its elements can be written as a sum of  $k$  other distinct elements of the set.","t":[{"b":0,"e":0.2857,"k":"rising","v":0.06697,"x":0.28571,"p":[[0,57,0.0,0.06697,0.10092,0.0,0.0,0.14286,0.0,0.28571,21,0,8,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,57,0.0702,0.14732,0.12103,0.0,0.14286,0.28571,0.0,0.28571,11,0,7,11,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,57,0.1404,0.18304,0.12492,0.0,0.2857,0.28571,0.0,0.28571,9,0,8,9,0,5,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,57,0.2105,0.25,0.07986,0.2857,0.28571,0.28571,0.0,0.28571,2,0,2,2,0,4,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.22321,0.15126,0.0,0.2857,0.28571,0.0,0.57143,9,0,8,9,0,0,0,0,20,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,57,0.3509,0.25,0.08748,0.2857,0.28571,0.28571,0.0,0.28571,3,0,2,3,0,2,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.26339,0.0724,0.2857,0.28571,0.28571,0.0,0.28571,2,0,1,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.25892,0.08328,0.2857,0.28571,0.28571,0.0,0.28571,3,0,1,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.28571,0.03571,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.28571,1e-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],[40,57,0.7018,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.28571,1e-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],[48,57,0.8421,0.28125,0.02486,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.28571,1e-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],[57,57,1.0,0.28571,1e-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]]},{"b":4,"e":0.14286,"k":"rising","v":0.08482,"x":0.28571,"p":[[0,39,0.0,0.09375,0.11071,0.0,0.0,0.14286,0.0,0.28571,17,0,11,17,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,10,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.19196,0.13651,0.10714,0.2857,0.28571,0.0,0.57143,8,0,5,8,0,7,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,39,0.3077,0.21875,0.13356,0.14286,0.2857,0.28571,0.0,0.4286,6,0,2,6,0,7,0,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.20982,0.12869,0.10714,0.28571,0.28571,0.0,0.4286,8,0,8,8,0,2,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.27678,0.03458,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.27232,0.04163,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.27232,0.05486,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.27232,0.04164,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.28571,1e-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],[39,39,1.0,0.25446,0.05906,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b94157699f7db24b","q":"Let  $n$  be a positive integer. Prove that the numbers   $$ 1^1, 3^3, 5^5, ..., (2n-1)^{2n-1} $$   all give different remainders when divided by  $2^n$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.07143,"p":[[0,30,0.0,0.07134,0.16362,0.0,0.0,0.035,0.0,0.71429,24,0,0,24,0,5,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[4,30,0.1333,0.07143,0.16751,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,3,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,30,0.2667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.06695,0.19877,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.16071,"p":[[0,8,0.0,0.16071,0.28065,0.0,0.0,0.2857,0.0,0.85714,22,0,0,22,0,1,0,0,4,0,0,0,0,0,0,0,0,3,0,0,2,0,0],[4,8,0.5,0.0625,0.19212,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"34821e42ab6d6816","q":"Let  $ABC$  be a scalene triangle. Given the center  $I$  of the inscribe circle and the points  $K_1$ ,  $K_2$  and  $K_3$  where the inscribed circle is tangent to the sides  $BC$ ,  $AC$  and  $AB$ . Using only a ruler, construct the center of the circumscribed circle of triangle  $ABC$ .\n\n (Hryhorii Filippovskyi)","t":[{"b":2,"e":0.14286,"k":"rising","v":0.02232,"x":0.31248,"p":[[0,31,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,20,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.12945,0.20625,0.0,0.0,0.1429,0.0,0.71429,19,0,0,19,0,6,0,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[8,31,0.2581,0.14723,0.18724,0.0,0.14143,0.2857,0.0,0.71429,15,0,0,15,0,7,0,0,8,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[12,31,0.3871,0.31248,0.26827,0.14286,0.2143,0.57111,0.0,0.857,6,0,0,6,0,10,0,0,5,0,0,2,0,0,2,0,0,6,0,0,1,0,0],[16,31,0.5161,0.22322,0.18877,0.14286,0.14286,0.28571,0.0,0.71429,6,0,0,6,0,13,0,0,7,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[20,31,0.6452,0.28561,0.20825,0.14286,0.2857,0.42895,0.0,0.71429,4,0,0,4,0,11,0,0,7,0,0,3,0,0,5,0,0,2,0,0,0,0,0],[24,31,0.7742,0.2098,0.21715,0.10714,0.14286,0.2857,0.0,0.71429,8,0,0,8,0,15,0,0,3,0,0,1,0,0,2,0,0,3,0,0,0,0,0],[28,31,0.9032,0.21866,0.17855,0.14286,0.1429,0.2857,0.0,0.71429,5,0,0,5,0,14,0,0,9,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[31,31,1.0,0.20963,0.20515,0.0,0.14286,0.32143,0.0,0.71429,9,0,0,9,0,12,0,0,3,0,0,5,0,0,1,0,0,2,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02237,"p":[[0,40,0.0,0.02237,0.07274,0.0,0.0,0.0,0.0,0.29,29,0,23,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,23,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,21,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,25,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,12,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,11,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,8,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,25,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0171d2fa03c621ac","q":"Given natural numbers  $a$  and  $b$ , such that  $a<b<2a$ . Some cells on a graph are colored such that in every rectangle with dimensions  $A \\times B$  or  $B \\times A$ , at least one cell is colored. For which greatest  $\\alpha$  can you say that for every natural number  $N$  you can find a square  $N \\times N$  in which at least  $\\alpha \\cdot N^2$  cells are colored?","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,90,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,19,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,90,0.0444,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,90,0.0889,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,90,0.1333,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,90,0.1778,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,90,0.2222,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,90,0.2667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,90,0.3111,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,90,0.3556,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,90,0.4,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,90,0.4444,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,90,0.4889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,90,0.5333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,90,0.5778,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,90,0.6222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,90,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,90,0.7111,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,90,0.7556,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,90,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,90,0.8444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,90,0.8889,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,90,0.9333,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,90,0.9778,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[90,90,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.03571,"p":[[0,36,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,14,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,9,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5c9bdd8ede95b3b1","q":"Let $A B C D$ be an isosceles trapezoid with $A B \\| C D$. The inscribed circle $\\omega$ of triangle $B C D$ meets $C D$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\\angle D A C$ such that $E F \\perp C D$. Let the circumscribed circle of triangle $A C F$ meet line $C D$ at $C$ and $G$. Prove that the triangle $A F G$ is isosceles.","t":[{"b":0,"e":1.0,"k":"rising","v":0.17411,"x":0.82143,"p":[[0,27,0.0,0.29911,0.38525,0.0,0.14286,0.60714,0.0,1.0,13,6,1,13,0,9,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,6],[4,27,0.1481,0.20089,0.33855,0.0,0.0,0.17857,0.0,1.0,19,4,1,19,0,5,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,4],[8,27,0.2963,0.17411,0.31285,0.0,0.0,0.14286,0.0,1.0,18,3,3,18,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[12,27,0.4444,0.28125,0.39687,0.0,0.07143,0.5,0.0,1.0,16,6,3,16,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,6],[16,27,0.5926,0.61607,0.46625,0.10714,1.0,1.0,0.0,1.0,8,19,1,8,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19],[20,27,0.7407,0.77232,0.40067,0.85714,1.0,1.0,0.0,1.0,5,24,1,5,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24],[24,27,0.8889,0.75893,0.41869,0.78571,1.0,1.0,0.0,1.0,6,24,0,6,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24],[27,27,1.0,0.82143,0.37287,1.0,1.0,1.0,0.0,1.0,4,26,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26]]},{"b":2,"e":0.0,"k":"falling","v":0.02232,"x":0.29902,"p":[[0,38,0.0,0.29902,0.39348,0.0,0.14286,0.42857,0.0,1.0,13,6,0,13,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,6],[4,38,0.1053,0.29018,0.39687,0.0,0.14286,0.28582,0.0,1.0,13,7,1,13,0,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,7],[8,38,0.2105,0.15625,0.31208,0.0,0.0,0.14286,0.0,1.0,20,3,0,20,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3],[12,38,0.3158,0.12947,0.26088,0.0,0.0,0.14286,0.0,1.0,19,2,0,19,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[16,38,0.4211,0.11598,0.24855,0.0,0.0,0.14286,0.0,1.0,21,2,0,21,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[20,38,0.5263,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.03116,0.08541,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[36,38,0.9474,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0620ca9d4005aaa8","q":"Let $\\left(a_{n}\\right)_{n \\geqslant 1}$ be a sequence of non-zero natural numbers such that $a_{n+2 m}$ divides $a_{n}+a_{n+m}$ for all integers $m \\geqslant 1$ and $n \\geqslant 1$. Prove that this sequence is ultimately periodic, that is, there exist two integers $N \\geqslant 1$ and $d \\geqslant 1$ such that $a_{n}=a_{n+d}$ for all $n \\geqslant N$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.06688,"p":[[0,33,0.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.0625,0.13333,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.06688,0.11279,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.04464,0.0974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.00438,"x":0.08928,"p":[[0,20,0.0,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.08482,0.12299,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.08928,0.12242,0.0,0.0,0.17857,0.0,0.28571,20,0,0,20,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"754651aa2cc0ebc0","q":"Let  $ABC$  be an acute triangle with  $AB < AC$ , and let  $H$  be its orthocenter. Let  $D$ ,  $E$ ,  $F$  and  $M$  be the midpoints of  $BC$ ,  $AC$ , and  $AH$ , respectively. Prove that the circumcircles of triangles  $AHD$ ,  $BMC$ , and  $DEF$  pass through a common point.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,94,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,94,0.0426,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,94,0.0851,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,94,0.1277,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,94,0.1702,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,94,0.2128,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,94,0.2553,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,94,0.2979,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,11,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,94,0.3404,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,15,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,94,0.383,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,22,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,94,0.4255,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,17,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,94,0.4681,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,20,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,94,0.5106,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,23,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,94,0.5532,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,26,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,94,0.5957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,94,0.6383,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,94,0.6809,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,94,0.7234,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,94,0.766,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,94,0.8085,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,94,0.8511,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,94,0.8936,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,80,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,80,0.05,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,80,0.1,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,80,0.15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,80,0.2,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,80,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,80,0.3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,80,0.35,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,80,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,80,0.45,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,80,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,80,0.55,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,80,0.6,0.00447,0.02488,0.0,0.0,0.0,0.0,0.143,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,80,0.65,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,80,0.7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,80,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,80,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,80,0.85,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,26,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3d65acb62b6abdcd","q":"Let  $ABC$  be an acute scalene triangle with incenter  $I$ , and let  $M$  be the circumcenter of triangle  $BIC$ .  Points  $D$ ,  $B'$ , and  $C'$  lie on side  $BC$  so that  $ \\angle BIB' = \\angle CIC' = \\angle IDB = \\angle IDC = 90^{\\circ} $ .  Define  $P = \\overline{AB} \\cap \\overline{MC'}$ ,  $Q = \\overline{AC} \\cap \\overline{MB'}$ ,  $S = \\overline{MD} \\cap \\overline{PQ}$ , and  $K = \\overline{SI} \\cap \\overline{DF}$ , where segment  $EF$  is a diameter of the incircle selected so that  $S$  lies in the interior of segment  $AE$ .  It is known that  $KI=15x$ ,  $SI=20x+15$ ,  $BC=20x^{5/2}$ , and  $DI=20x^{3/2}$ , where  $x = \\tfrac ab(n+\\sqrt p)$  for some positive integers  $a$ ,  $b$ ,  $n$ ,  $p$ , with  $p$  prime and  $\\gcd(a,b)=1$ . Compute  $a+b+n+p$ .\n\n*Proposed by Evan Chen*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,37,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,13,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,17,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,23,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,21,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,26,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,13,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f966eb861d0912b5","q":"Let  $ABC$  be an acute angled triangle $.$  Let  $D$  be the foot of the internal angle bisector of  $\\angle BAC$  and let  $M$  be the midpoint of  $AD.$  Let  $X$  be a point on segment  $BM$  such that  $\\angle MXA=\\angle DAC.$  Prove that  $AX$  is perpendicular to  $XC.$","t":[{"b":1,"e":0.0,"k":"flat","v":0.03125,"x":0.09821,"p":[[0,35,0.0,0.08911,0.11702,0.0,0.0,0.14286,0.0,0.42857,18,0,1,18,0,9,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,5,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.07133,0.14717,0.0,0.0,0.14071,0.0,0.57143,23,0,4,23,0,6,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,35,0.3429,0.09821,0.10374,0.0,0.14286,0.14286,0.0,0.28571,15,0,1,15,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,3,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.04018,0.08172,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.08473,0.10007,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.09598,0.11238,0.0,0.0,0.1429,0.0,0.28571,17,0,0,17,0,8,0,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.1429,"k":"flat","v":0.04464,"x":0.12054,"p":[[0,33,0.0,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.42857,22,0,2,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.10268,0.1197,0.0,0.0,0.17857,0.0,0.28571,17,0,2,17,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.08037,0.12341,0.0,0.0,0.1429,0.0,0.42857,21,0,2,21,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.08482,0.11214,0.0,0.0,0.14286,0.0,0.28571,19,0,2,19,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.05804,0.12807,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,33,0.6061,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.11161,0.11143,0.0,0.14286,0.1429,0.0,0.28571,14,0,0,14,0,11,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.09821,0.12595,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.12045,0.12426,0.0,0.14143,0.2857,0.0,0.28571,15,0,0,15,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.12054,0.12428,0.0,0.14286,0.2857,0.0,0.28571,15,0,0,15,0,7,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"19eca3781bd0a42e","q":"The sequence  $ \\{x_{n}\\}$  is defined by  $ x_{1} \\equal{} 2,x_{2} \\equal{} 12$ , and  $ x_{n \\plus{} 2} \\equal{} 6x_{n \\plus{} 1} \\minus{} x_{n}$ ,  $ (n \\equal{} 1,2,\\ldots)$ . Let  $ p$  be an odd prime number, let  $ q$  be a prime divisor of  $ x_{p}$ . Prove that if  $ q\\neq2,3,$  then  $ q\\geq 2p \\minus{} 1$ .","t":[{"b":1,"e":0.42857,"k":"falling","v":0.29464,"x":0.5178,"p":[[0,26,0.0,0.5178,0.17033,0.42859,0.57143,0.57143,0.0,0.71429,1,0,1,1,0,1,0,0,4,0,0,3,0,0,17,0,0,6,0,0,0,0,0],[4,26,0.1538,0.48661,0.18161,0.28571,0.50001,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,7,0,0,7,0,0,9,0,0,6,0,0,1,0,0],[8,26,0.3077,0.46426,0.19883,0.28571,0.42857,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,8,0,0,8,0,0,6,0,0,5,0,0,2,0,0],[12,26,0.4615,0.49107,0.24206,0.28571,0.42857,0.71429,0.14286,1.0,0,2,0,0,0,2,0,0,11,0,0,7,0,0,1,0,0,7,0,0,2,0,2],[16,26,0.6154,0.3125,0.08328,0.28571,0.28571,0.42857,0.14286,0.4286,0,0,0,0,0,3,0,0,20,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.29464,0.08703,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,4,0,0,23,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[24,26,0.9231,0.32588,0.08914,0.28571,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,1,0,0,23,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[26,26,1.0,0.33926,0.12748,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,14,0,0,9,0,0,4,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.57143,"k":"flat","v":0.41518,"x":0.66501,"p":[[0,50,0.0,0.59373,0.16408,0.5354,0.71429,0.71429,0.2857,0.857,0,0,0,0,0,0,0,0,5,0,0,3,0,0,7,0,0,16,0,0,1,0,0],[4,50,0.08,0.66501,0.09847,0.57143,0.71214,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,13,0,0,4,0,0],[8,50,0.16,0.53566,0.1515,0.42857,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,4,0,0,9,0,0,12,0,0,5,0,0,2,0,0],[12,50,0.24,0.57137,0.18558,0.42857,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,6,0,0,4,0,0,11,0,0,6,0,0,5,0,0],[16,50,0.32,0.54017,0.18808,0.42857,0.57143,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,6,0,0,8,0,0,10,0,0,4,0,0,3,0,1],[20,50,0.4,0.52229,0.18422,0.28571,0.57143,0.60714,0.2857,1.0,0,1,0,0,0,0,0,0,10,0,0,1,0,0,13,0,0,7,0,0,0,0,1],[24,50,0.48,0.53121,0.21792,0.28571,0.4998,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,11,0,0,5,0,0,3,0,0,8,0,0,5,0,0],[28,50,0.56,0.48655,0.18507,0.28571,0.49979,0.57143,0.0,0.85714,1,0,1,1,0,0,0,0,8,0,0,7,0,0,9,0,0,6,0,0,1,0,0],[32,50,0.64,0.49996,0.16365,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,6,0,0,8,0,0,11,0,0,5,0,0,1,0,0],[36,50,0.72,0.49548,0.18203,0.39286,0.4998,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,7,0,0,8,0,0,11,0,0,3,0,0,1,0,1],[40,50,0.8,0.50443,0.22298,0.28571,0.42859,0.60714,0.2857,1.0,0,2,0,0,0,0,0,0,11,0,0,8,0,0,5,0,0,3,0,0,3,0,2],[44,50,0.88,0.41518,0.13534,0.28571,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,14,0,0,9,0,0,7,0,0,2,0,0,0,0,0],[48,50,0.96,0.45533,0.21258,0.28571,0.42859,0.57143,0.14286,0.85714,0,0,0,0,0,3,0,0,11,0,0,5,0,0,6,0,0,4,0,0,3,0,0],[50,50,1.0,0.52671,0.16534,0.42857,0.571,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,5,0,0,10,0,0,10,0,0,4,0,0,3,0,0]]}]},{"i":"4a093a4ce8747744","q":"Let  $ABC$  a triangle and let  $\\omega$  be its circumcircle. Let  $E{}$  be the midpoint of the minor arc  $BC$  of  $\\omega$ , and  $M{}$  the midpoint of  $BC$ . Let  $V$  be the other point of intersection of  $AM$  with  $\\omega$ ,  $F{}$  the point of intersection of  $AE$  with  $BC$ ,  $X{}$  the other point of intersection of the circumcircle of  $FEM$  with  $\\omega$ ,  $X'$  the reflection of  $V{}$  with respect to  $M{}$ ,  $A'{}$  the foot of the perpendicular from  $A{}$  to  $BC$  and  $S{}$  the other point of intersection of  $XA'$  with  $\\omega$ . If  $Z \\in \\omega$  with  $Z\\neq X$  is such that  $AX = AZ$ , then prove that  $S, X'$  and  $Z{}$  are collinear.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,73,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,73,0.0548,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[12,73,0.1644,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,73,0.2192,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,73,0.3288,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[40,73,0.5479,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,73,0.6027,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,50,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"233da602dd3f1701","q":"Let  $m$  be an integer where  $|m|\\ge 2$ . Let  $a_1,a_2,\\cdots$  be a sequence of integers such that  $a_1,a_2$  are not both zero, and for any positive integer  $n$ ,  $a_{n+2}=a_{n+1}-ma_n$ .\n\nProve that if positive integers  $r>s\\ge 2$  satisfy  $a_r=a_s=a_1$ , then  $r-s\\ge |m|$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.03571,"x":0.20536,"p":[[0,15,0.0,0.05804,0.13296,0.0,0.0,0.0,0.0,0.57143,26,0,2,26,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.20536,0.22286,0.0,0.2857,0.28571,0.0,1.0,13,1,0,13,0,1,0,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,15,0.5333,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,15,0.8,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.12053,"p":[[0,14,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.12053,0.2055,0.0,0.0,0.17857,0.0,0.71429,22,0,2,22,0,2,0,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[8,14,0.5714,0.08929,0.16269,0.0,0.0,0.17857,0.0,0.71429,23,0,0,23,0,1,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,14,0.8571,0.05357,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4f10387fdcb88c15","q":"Let  $ABC$  be an acute-angled triangle with  $AB < AC$ . Tangent to its circumcircle  $\\Omega$  at  $A$  intersects the line  $BC$  at  $D$ . Let  $G$  be the centroid of  $\\triangle ABC$  and let  $AG$  meet  $\\Omega$  again at  $H \\neq A$ . Suppose the line  $DG$  intersects the lines  $AB$  and  $AC$  at  $E$  and  $F$ , respectively. Prove that  $\\angle EHG = \\angle GHF$ .(Slovakia)","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01768,"p":[[0,33,0.0,0.01768,0.04678,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[12,33,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01116,"x":0.03571,"p":[[0,24,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.03348,0.05918,0.0,0.0,0.01786,0.0,0.14286,24,0,2,24,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,2,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,2,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7d862eb6a1ad59bd","q":"Let  $ABC$  be a triangle with  $AB = 3, AC = 4,$  and  $BC = 5$ , let  $P$  be a point on  $BC$ , and let  $Q$  be the point (other than  $A$ ) where the line through  $A$  and  $P$  intersects the circumcircle of  $ABC$ . Prove that\n\\[PQ\\le \\frac{25}{4\\sqrt{6}}.\\]","t":[{"b":5,"e":0.71429,"k":"flat","v":0.70085,"x":0.78125,"p":[[0,9,0.0,0.78125,0.14279,0.71429,0.78571,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,0,11,0,5],[4,9,0.4444,0.74551,0.1461,0.67857,0.78564,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,8,0,0,15,0,1],[8,9,0.8889,0.70085,0.13058,0.57143,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,11,0,0,10,0,0],[9,9,1.0,0.71427,0.12373,0.67857,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,14,0,0,10,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.76783,"x":0.83928,"p":[[0,18,0.0,0.76783,0.16657,0.71429,0.78564,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,10,0,0,11,0,5],[4,18,0.2222,0.77231,0.12809,0.71429,0.78571,0.85714,0.571,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,10,0,0,13,0,3],[8,18,0.4444,0.81249,0.09743,0.82143,0.85714,0.85714,0.571,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,23,0,1],[12,18,0.6667,0.83035,0.06621,0.85714,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,27,0,0],[16,18,0.8889,0.83928,0.08564,0.85714,0.85714,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,23,0,3],[18,18,1.0,0.82588,0.08552,0.82132,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,22,0,2]]}]},{"i":"84a8db3fb6075edc","q":"Let  $N,K,L$  be points on  $AB,BC,CA$  such that  $CN$  bisector of angle  $\\angle ACB$  and  $AL=BK$ .Let  $BL\\cap AK=P$ .If  $I,J$  be incenters of triangles  $\\triangle BPK$  and  $\\triangle ALP$  and  $IJ\\cap CN=Q$  prove that  $IQ=JP$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"03025a2c18d9d0a0","q":"Let  $p$  be a prime number in the form  $p=4k+3$ . Prove that if the numbers  $x_0,y_0,z_0,t_0$  are solutions of the equation  $x^{2p}+y^{2p}+z^{2p}=t^{2p}$ , then at least one of them is divisible by  $p$ . *(Plamen Koshlukov)*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0067,"p":[[0,31,0.0,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,31,0.2581,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.00222,0.01235,0.0,0.0,0.0,0.0,0.071,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,0,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"551156dd324207cb","q":"Let $A B C$ be an acute triangle with circumcircle $\\omega$. Let $t$ be a tangent line to $\\omega$. Let $t_{a}, t_{b}$, and $t_{c}$ be the lines obtained by reflecting $t$ in the lines $B C, C A$, and $A B$, respectively. Show that the circumcircle of the triangle determined by the lines $t_{a}, t_{b}$, and $t_{c}$ is tangent to the circle $\\omega$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,41,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,35,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"726291b7f95bc908","q":"In the land of wonders, there are $n$ cities. Each pair of cities is connected by a one-way road, which starts from one of the two cities and arrives at the other. To find her way, Alice questions the King of Hearts: with each question, Alice chooses a pair of cities, and the King of Hearts tells her which is the starting city of the road connecting these two cities.\nProve that, in $5 n$ questions or fewer, Alice can determine whether there exists a city from which at most one road starts.","t":[{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.03125,"p":[[0,17,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,21,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,22,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,24,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,14,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,16,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.04911,"p":[[0,9,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,25,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.04911,0.13651,0.0,0.0,0.0,0.0,0.57143,28,0,27,28,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,30,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7ab133e38ca4979c","q":"Let  $\\mathcal{P}$  be a convex polygon and  $\\textbf{T}$  be a triangle with vertices among the vertices of  $\\mathcal{P}$ . By removing  $\\textbf{T}$  from  $\\mathcal{P}$ , we end up with  $0, 1, 2,$  or  $3$  smaller polygons (possibly with shared vertices) which we call the effect of  $\\textbf{T}$ . A triangulation of  $P$  is a way of dissecting it into some triangles using some non-intersecting diagonals. We call a triangulation of  $\\mathcal{P}$   $\\underline{\\text{beautiful}}$ , if for each of its triangles, the effect of this triangle contains exactly one polygon with an odd number of vertices. Prove that a triangulation of  $\\mathcal{P}$  is beautiful if and only if we can remove some of its diagonals and end up with all regions as quadrilaterals.","t":[{"b":3,"e":0.14286,"k":"volatile","v":0.16054,"x":0.40179,"p":[[0,4,0.0,0.40179,0.26591,0.14286,0.42857,0.71429,0.0,0.71429,2,0,2,2,0,12,0,0,1,0,0,2,0,0,5,0,0,10,0,0,0,0,0],[4,4,1.0,0.16054,0.05928,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,26,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.16955,"x":0.45973,"p":[[0,11,0.0,0.45973,0.319,0.14286,0.42857,0.71429,0.0,1.0,1,4,0,1,0,12,0,0,2,0,0,2,0,0,2,0,0,9,0,0,0,0,4],[4,11,0.3636,0.28106,0.21582,0.14286,0.14286,0.42857,0.14,0.71429,0,0,0,0,0,21,0,0,2,0,0,3,0,0,1,0,0,5,0,0,0,0,0],[8,11,0.7273,0.16955,0.08331,0.14286,0.14286,0.1429,0.0,0.42857,2,0,1,2,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.16955,0.06625,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b71ab6887486952f","q":"Let  $n,k$  be arbitrary positive integers. We fill the entries of an  $n\\times k$  array with integers such that all the  $n$  rows contain the integers  $1,2,\\dots,k$  in some order. Add up the numbers in all  $k$  columns \u2013 let  $S$  be the largest of these sums. What is the minimal value of  $S$ ?","t":[{"b":2,"e":0.57143,"k":"flat","v":0.34821,"x":0.38393,"p":[[0,12,0.0,0.35266,0.16744,0.28571,0.28571,0.46418,0.0,0.57143,3,0,3,3,0,1,0,0,14,0,0,6,0,0,8,0,0,0,0,0,0,0,0],[4,12,0.3333,0.38393,0.13092,0.28571,0.42857,0.42857,0.0,0.71429,1,0,1,1,0,0,0,0,13,0,0,13,0,0,4,0,0,1,0,0,0,0,0],[8,12,0.6667,0.34821,0.15542,0.2857,0.28571,0.46429,0.14286,0.57143,0,0,0,0,0,7,0,0,12,0,0,5,0,0,8,0,0,0,0,0,0,0,0],[12,12,1.0,0.375,0.14174,0.28571,0.42857,0.42857,0.0,0.57143,1,0,1,1,0,2,0,0,12,0,0,10,0,0,7,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"rising","v":0.34821,"x":0.54466,"p":[[0,32,0.0,0.34821,0.12846,0.28571,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,1,0,0,18,0,0,7,0,0,5,0,0,0,0,0,0,0,0],[4,32,0.125,0.36159,0.13823,0.28571,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,1,0,0,17,0,0,6,0,0,7,0,0,0,0,0,0,0,0],[8,32,0.25,0.35714,0.15972,0.28571,0.35714,0.42857,0.0,0.57143,2,0,2,2,0,3,0,0,11,0,0,9,0,0,7,0,0,0,0,0,0,0,0],[12,32,0.375,0.38393,0.1448,0.28571,0.28571,0.46431,0.0,0.71429,1,0,1,1,0,0,0,0,16,0,0,7,0,0,7,0,0,1,0,0,0,0,0],[16,32,0.5,0.45089,0.08828,0.42857,0.42857,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,4,0,0,19,0,0,9,0,0,0,0,0,0,0,0],[20,32,0.625,0.54466,0.05565,0.57132,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0],[24,32,0.75,0.54018,0.09932,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,0,2,0,0,1,0,0],[28,32,0.875,0.52229,0.07666,0.42857,0.57143,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,9,0,0,22,0,0,0,0,0,0,0,0],[32,32,1.0,0.54017,0.06901,0.53539,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,23,0,0,1,0,0,0,0,0]]}]},{"i":"fce5ffababcc8b10","q":"Let  $\\Omega$  be the circumcircle of an acute triangle  $ABC$ . Points  $D$ ,  $E$ ,  $F$  are the midpoints of the inferior arcs  $BC$ ,  $CA$ ,  $AB$ , respectively, on  $\\Omega$ . Let  $G$  be the antipode of  $D$  in  $\\Omega$ . Let  $X$  be the intersection of lines  $GE$  and  $AB$ , while  $Y$  the intersection of lines  $FG$  and  $CA$ . Let the circumcenters of triangles  $BEX$  and  $CFY$  be points  $S$  and  $T$ , respectively. Prove that  $D$ ,  $S$ ,  $T$  are collinear. \n    \n*Proposed by kyou46 and Li4.*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"70dd0b3cf1f71e2c","q":"Let  $ABC$  be an acute non-isosceles triangle with circumcircle  $\\omega$ , circumcenter  $O$  and orthocenter  $H$ . We draw a line perpendicular to  $AH$  through  $O$  and a line perpendicular to  $AO$  through  $H$ . Prove that the points of intersection of these lines with sides  $AB$  and  $AC$  lie on a circle, which is tangent to  $\\omega$ .\n*Proposed by A. Kuznetsov*","t":[{"b":3,"e":0.0,"k":"flat","v":0.0625,"x":0.12045,"p":[[0,16,0.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.2857,19,0,3,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.09804,0.1259,0.0,0.07,0.14286,0.0,0.4286,16,0,0,16,0,13,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.12045,0.14333,0.0,0.14286,0.14286,0.0,0.71429,13,0,0,13,0,14,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,16,0.75,0.06669,0.11261,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,16,1.0,0.06697,0.07974,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.02661,"x":0.08482,"p":[[0,7,0.0,0.08482,0.13296,0.0,0.0,0.14286,0.0,0.57143,19,0,6,19,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02661,0.05539,0.0,0.0,0.0,0.0,0.1429,26,0,6,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5b9c8958b598ba80","q":"Let $A B C D$ be a cyclic quadrilateral satisfying $A D^{2}+B C^{2}=A B^{2}$. The diagonals of $A B C D$ intersect at $E$. Let $P$ be a point on side $\\overline{A B}$ satisfying $\\angle A P D=\\angle B P C$. Show that line $P E$ bisects $\\overline{C D}$.","t":[{"b":5,"e":0.0,"k":"falling","v":0.00446,"x":0.19186,"p":[[0,16,0.0,0.19186,0.28707,0.0,0.0,0.32143,0.0,1.0,20,1,0,20,0,1,0,0,3,0,0,2,0,0,2,0,0,3,0,0,0,0,1],[4,16,0.25,0.12052,0.26511,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[8,16,0.5,0.14277,0.28347,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[12,16,0.75,0.05357,0.1915,0.0,0.0,0.0,0.0,0.85714,29,0,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,16,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.21429,"p":[[0,29,0.0,0.10714,0.18898,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,6,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[4,29,0.1379,0.10714,0.21724,0.0,0.0,0.03571,0.0,0.85714,24,0,0,24,0,2,0,0,1,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[8,29,0.2759,0.13839,0.20666,0.0,0.0,0.28571,0.0,0.57143,21,0,0,21,0,1,0,0,3,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[12,29,0.4138,0.21429,0.30514,0.0,0.0,0.42857,0.0,1.0,18,2,0,18,0,3,0,0,2,0,0,3,0,0,2,0,0,2,0,0,0,0,2],[16,29,0.5517,0.09822,0.14914,0.0,0.0,0.14286,0.0,0.57143,19,0,0,19,0,8,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,29,0.6897,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"12585d4ea98f4739","q":"Let  $ABC$  be an acute triangle with circumcenter  $O$ , orthocenter  $H$ , and circumcircle  $\\Omega$ . Let  $M$  be the midpoint of  $AH$  and  $N$  the midpoint of  $BH$ . Assume the points  $M$ ,  $N$ ,  $O$ ,  $H$  are distinct and lie on a circle  $\\omega$ . Prove that the circles  $\\omega$  and  $\\Omega$  are internally tangent to each other.\n\n*Dhroova Aiylam and Evan Chen*","t":[{"b":0,"e":0.2857,"k":"falling","v":0.26785,"x":0.56696,"p":[[0,15,0.0,0.56696,0.34716,0.28571,0.42857,1.0,0.0,1.0,2,11,2,2,0,0,0,0,14,0,0,0,0,0,3,0,0,2,0,0,0,0,11],[4,15,0.2667,0.41964,0.36932,0.24999,0.28571,0.78571,0.0,1.0,7,8,1,7,0,1,0,0,14,0,0,0,0,0,1,0,0,1,0,0,0,0,8],[8,15,0.5333,0.27679,0.04971,0.28571,0.28571,0.28571,0.0,0.286,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.28571,1e-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],[15,15,1.0,0.26785,0.05922,0.2857,0.28571,0.28571,0.0,0.28571,1,0,1,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"falling","v":0.28572,"x":0.53571,"p":[[0,13,0.0,0.53571,0.36246,0.28571,0.35714,1.0,0.0,1.0,4,10,1,4,0,0,0,0,12,0,0,1,0,0,3,0,0,1,0,0,1,0,10],[4,13,0.3077,0.37946,0.32461,0.2857,0.28571,0.57143,0.0,1.0,7,5,3,7,0,0,0,0,16,0,0,0,0,0,2,0,0,2,0,0,0,0,5],[8,13,0.6154,0.28572,5e-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],[12,13,0.9231,0.32143,0.09449,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[13,13,1.0,0.32142,0.09449,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,0,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"a795416949d865ea","q":"Let  $(G, *)$  a group of  $n > 1$  elements, and let  $g \\in G$  be an element distinct from the identity. \nAna and Bob play with the group  $G$  on the following way: \nStarting with Ana and playing alternately, each player selects an element of  $G$  that has not been selected before, until each element of  $G$  have been selected or a player have selected the elements  $a$  and  $a *\n g$  for some  $a \\in G$ . \nIn that case it is said that the player loses and his opponent wins. $a)$  If  $n$  is odd, show that, independent of element  $g$ , one of the two players has\na winning strategy and determines which player\npossesses such a strategy. $b)$  If  $n$  is even, show that there exists an element  $g \\in G$  for which none of the players\nhas a winning strategy.\n\nNote: A group  $(G, *)$  es a set  $G$  together with a binary operation  $* : G\\times G \\to G$  that satisfy the following properties $(i)$   $*$  is asociative:  $\\forall a, b, c \\in G (a * b) * c = a * (b * c)$ ; $(ii)$  there exists an identity element  $e \\in G$  such that  $\\forall a \\in G, a *e = e * a = a;$  $(iii)$  there exists inverse elements:  $\\forall a \\in G \\exists a^{-1} \\in G$  such that  $a*a^{-1} = a^{-1}\n *a = e.$","t":[{"b":2,"e":0.85714,"k":"rising","v":0.44194,"x":0.79472,"p":[[0,15,0.0,0.48661,0.329,0.10714,0.57143,0.71429,0.0,1.0,8,2,8,8,0,1,0,0,0,0,0,4,0,0,7,0,0,6,0,0,4,0,2],[4,15,0.2667,0.44194,0.27746,0.28571,0.42859,0.60714,0.0,1.0,7,1,7,7,0,0,0,0,2,0,0,8,0,0,7,0,0,6,0,0,1,0,1],[8,15,0.5333,0.79472,0.2081,0.71429,0.85714,0.895,0.0,1.0,1,8,1,1,0,0,0,0,1,0,0,0,0,0,1,0,0,10,0,0,11,0,8],[12,15,0.8,0.77679,0.10062,0.71429,0.71429,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,6,0,4],[15,15,1.0,0.77232,0.09354,0.71429,0.71429,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,22,0,0,7,0,3]]},{"b":7,"e":0.57143,"k":"flat","v":0.49999,"x":0.57128,"p":[[0,13,0.0,0.49999,0.31743,0.2857,0.57143,0.71429,0.0,1.0,7,2,6,7,0,0,0,0,3,0,0,2,0,0,8,0,0,6,0,0,4,0,2],[4,13,0.3077,0.57128,0.35887,0.32145,0.71429,0.85714,0.0,1.0,8,3,8,8,0,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,10,0,3],[8,13,0.6154,0.558,0.04163,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0],[12,13,0.9231,0.54018,0.10555,0.57143,0.57143,0.57143,0.0,0.57143,1,0,1,1,0,0,0,0,0,0,0,3,0,0,28,0,0,0,0,0,0,0,0],[13,13,1.0,0.52231,0.14987,0.57143,0.57143,0.57143,0.0,0.71429,2,0,2,2,0,0,0,0,1,0,0,2,0,0,26,0,0,1,0,0,0,0,0]]}]},{"i":"112f800c12cebde5","q":"Let  $ABCD$  be cyclic quadrilateral. Let  $AC$  and  $BD$  intersect at  $R$ , and let  $AB$  and  $CD$  intersect at  $K$ .  Let  $M$  and  $N$  are points on  $AB$  and  $CD$  such that  $\\frac{AM}{MB}=\\frac{CN}{ND}$ . Let  $P$  and  $Q$  be the intersections of  $MN$  with the diagonals of  $ABCD$ .  Prove that circumcircles of triangles  $KMN$  and  $PQR$  are tangent at a fixed point.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,5,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,3,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.06241,"p":[[0,14,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,3,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,3,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.06241,0.10055,0.0,0.0,0.14286,0.0,0.28571,22,0,4,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[14,14,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"737eb79fe804232d","q":"Let  $P_n$  be the number of permutations  $\\pi$  of  $\\{1,2,\\dots,n\\}$  such that \\[|i-j|=1\\text{ implies }|\\pi(i)-\\pi(j)|\\le 2\\] for all  $i,j$  in  $\\{1,2,\\dots,n\\}.$  Show that for  $n\\ge 2,$  the quantity \\[P_{n+5}-P_{n+4}-P_{n+3}+P_n\\] does not depend on  $n,$  and find its value.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,17,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,49,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,14,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,15,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,10,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,25,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"942e1a6260b8ded3","q":"Let  $I$  and  $I_a$  be the incenter and excenter (opposite vertex  $A$ ) of a triangle  $ABC$ , respectively. Let  $A'$  be the point on its circumcircle opposite to  $A$ , and  $A_1$  be the foot of the altitude from  $A$ . Prove that  $\\angle IA_1I_a=\\angle IA'I_a$ .\n\n*(Proposed by Pavel Kozhevnikov)*","t":[{"b":2,"e":0.57143,"k":"volatile","v":0.0357,"x":0.57136,"p":[[0,7,0.0,0.0357,0.11839,0.0,0.0,0.0,0.0,0.571,29,0,3,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.10268,0.23753,0.0,0.0,0.0,0.0,1.0,26,1,1,26,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[7,7,1.0,0.57136,0.00016,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01786,"x":0.01786,"p":[[0,8,0.0,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,3,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,8,0.5,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,3,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3a946cea57ad2d36","q":"Let  $\\alpha$  be a real number in the interval  $(0,1).$  Prove that there exists a sequence  $(\\varepsilon_n)_{n\\geq 1}$  where each term is either  $0$  or  $1$  such that the sequence  $(s_n)_{n\\geq 1}$  \\[s_n=\\frac{\\varepsilon_1}{n(n+1)}+\\frac{\\varepsilon_2}{(n+1)(n+2)}+...+\\frac{\\varepsilon_n}{(2n-1)2n}\\]verifies the inequality \\[0\\leq \\alpha-2ns_n\\leq\\frac{2}{n+1}\\] for any  $n\\geq 2.$","t":[{"b":4,"e":0.0,"k":"flat","v":0.01786,"x":0.11607,"p":[[0,4,0.0,0.11607,0.20652,0.0,0.0,0.14286,0.0,1.0,18,1,1,18,0,10,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[4,4,1.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04911,"x":0.20536,"p":[[0,13,0.0,0.14731,0.27543,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,6,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,2],[4,13,0.3077,0.20536,0.33681,0.0,0.0,0.2857,0.0,1.0,19,3,1,19,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,2,0,3],[8,13,0.6154,0.12054,0.26513,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[12,13,0.9231,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.04911,0.09852,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d336bc49281562e0","q":"Let  $K,L, M$  be the midpoints of  $BC,CA,AB$  repectively on a given triangle  $ABC$ . Let  $\\Gamma$  be a circle passing through  $B$  and tangent to the circumcircle of  $KLM$ , say at  $X$ . Suppose that  $LX$  and  $BC$  meet at  $\\Gamma$  . Show that  $CX$  is perpendicular to  $AB$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.09375,"x":0.17857,"p":[[0,8,0.0,0.17857,0.28347,0.0,0.0,0.2857,0.0,1.0,18,1,0,18,0,5,0,0,4,0,0,1,0,0,0,0,0,1,0,0,2,0,1],[4,8,0.5,0.17857,0.32341,0.0,0.0,0.14286,0.0,1.0,21,3,0,21,0,4,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,3],[8,8,1.0,0.09375,0.12682,0.0,0.0,0.14286,0.0,0.57143,17,0,0,17,0,11,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.01339,"x":0.18741,"p":[[0,29,0.0,0.18741,0.29762,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,3,0,0,5,0,0,0,0,0,0,0,0,2,0,0,2,0,1],[4,29,0.1379,0.08482,0.19516,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,29,0.2759,0.15625,0.29312,0.0,0.0,0.14286,0.0,1.0,20,2,0,20,0,6,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[12,29,0.4138,0.10268,0.2402,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0],[16,29,0.5517,0.07589,0.24218,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,29,0.6897,0.08482,0.19516,0.0,0.0,0.14286,0.0,0.85714,23,0,0,23,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[24,29,0.8276,0.05357,0.13716,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,29,0.9655,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"93bd599e1ba2be4e","q":"Let  $ABC$  be an acute triangle and  $D, E, F$  are the midpoints of  $BC, CA, AB$ , respectively. The circle with diameter  $AD$  intersects the lines  $AB$  and  $AC$  at points  $P$  and  $Q$  , respectively. The lines through  $P$  and  $Q$  parallel to  $BC$  intersect  $DE$  at point  $R$  and  $DF$  at point  $S$ , respectively. The circumcircle of  $DPR$  intersects  $AB$  at  $X$ , the circumcircle of  $DQS$  intersects  $AC$  in  $Y$ , and these two circles intersect again point  $Z$ . Prove that  $Z$  is the midpoint of  $XY$ .","t":[{"b":1,"e":0.28571,"k":"flat","v":0.25446,"x":0.33018,"p":[[0,7,0.0,0.33018,0.28232,0.14,0.21431,0.60714,0.0,0.85714,7,0,1,7,0,9,0,0,2,0,0,4,0,0,2,0,0,7,0,0,1,0,0],[4,7,0.5714,0.25446,0.26422,0.0,0.14286,0.32143,0.0,0.71429,9,0,2,9,0,11,0,0,4,0,0,1,0,0,0,0,0,7,0,0,0,0,0],[7,7,1.0,0.30803,0.09523,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,1,0,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.24107,"x":0.24107,"p":[[0,4,0.0,0.24107,0.22142,0.0,0.14288,0.42857,0.0,0.71429,9,0,2,9,0,8,0,0,6,0,0,5,0,0,1,0,0,3,0,0,0,0,0]]}]},{"i":"4d76e25510245c8a","q":"Let  $ABC$  be an acute triangle with orthocenter  $H$  and altitudes  $AA_1$ ,  $BB_1$ ,  $CC_1$ . The lines  $AB$  and  $A_1B_1$  intersect at  $C_2$  and  $\\ell_C$  is the line through the midpoint of  $CH$ , perpendicular to  $CC_2$ . The lines  $\\ell_A$  and  $\\ell_B$  are defined analogously. Prove that the lines  $\\ell_A$ ,  $\\ell_B$  and  $\\ell_C$  are concurrent.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.11598,"p":[[0,36,0.0,0.11161,0.12745,0.0,0.14286,0.1429,0.0,0.4286,15,0,7,15,0,11,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.11598,0.16916,0.0,0.0,0.14286,0.0,0.71429,17,0,11,17,0,9,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,36,0.2222,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,23,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,24,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.07581,"x":0.27678,"p":[[0,22,0.0,0.15616,0.22407,0.0,0.0,0.28571,0.0,1.0,17,1,11,17,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[4,22,0.1818,0.11607,0.19377,0.0,0.0,0.14286,0.0,0.71429,20,0,19,20,0,5,0,0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[8,22,0.3636,0.12045,0.11902,0.0,0.14286,0.14286,0.0,0.4286,13,0,5,13,0,12,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.07581,0.09433,0.0,0.0,0.14286,0.0,0.42857,17,0,12,17,0,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.23213,0.1324,0.14286,0.14286,0.28571,0.14286,0.57143,0,0,0,0,0,20,0,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[20,22,0.9091,0.2366,0.1621,0.14286,0.14288,0.2857,0.0,0.71429,2,0,0,2,0,17,0,0,7,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[22,22,1.0,0.27678,0.12843,0.14286,0.28571,0.42857,0.14286,0.571,0,0,0,0,0,13,0,0,9,0,0,9,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"faa86dadff40cef8","q":"Let  $S = \\{(x, y) \\in Z^2 | 0 \\le x \\le 11, 0\\le y \\le 9\\}$ . Compute the number of sequences  $(s_0, s_1, . . . , s_n)$  of elements in  $S$  (for any positive integer  $n \\ge 2$ ) that satisfy the following conditions: $\\bullet$   $s_0 = (0, 0)$  and  $s_1 = (1, 0)$ , $\\bullet$   $s_0, s_1, . . . , s_n$  are distinct, $\\bullet$  for all integers  $2  \\le  i  \\le  n$ ,  $s_i$  is obtained by rotating  $s_{i-2}$  about  $s_{i-1}$  by either  $90^o$  or  $180^o$  in the\nclockwise direction.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.03562,"x":0.14268,"p":[[0,9,0.0,0.03571,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,25,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,9,0.4444,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,17,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.1292,0.04156,0.14286,0.14286,0.14286,0.0,0.14286,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04018,"x":0.07125,"p":[[0,5,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,22,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.07125,0.1785,0.0,0.0,0.14071,0.0,1.0,22,1,20,22,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"a047ba2ae0270fbc","q":"Let  $G$  be a finite group with the following property:\n\nIf  $f$  is an automorphism of  $G$ , then there exists  $m\\in\\mathbb{N^\\star}$ , so that  $f(x)=x^{m} $  for all  $x\\in G$ .\n\nProve that G is commutative.\n\n*Marian Andronache*","t":[{"b":2,"e":0.42857,"k":"volatile","v":0.14286,"x":0.91518,"p":[[0,21,0.0,0.14286,0.32537,0.0,0.0,0.0,0.0,1.0,26,3,0,26,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[4,21,0.1905,0.54018,0.41609,0.0,0.71429,1.0,0.0,1.0,11,9,0,11,0,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,2,0,9],[8,21,0.381,0.46427,0.40406,0.0,0.42859,0.85714,0.0,1.0,11,7,0,11,0,2,0,0,0,0,0,4,0,0,1,0,0,5,0,0,2,0,7],[12,21,0.5714,0.85714,0.14725,0.71429,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,6,0,14],[16,21,0.7619,0.91518,0.13296,0.85714,1.0,1.0,0.42857,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,7,0,20],[20,21,0.9524,0.87054,0.14445,0.71429,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,7,0,15],[21,21,1.0,0.87946,0.15198,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,11,0,15]]},{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.29909,"p":[[0,31,0.0,0.29909,0.42311,0.0,0.0,0.71429,0.0,1.0,21,6,0,21,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,6],[4,31,0.129,0.17411,0.35667,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,4],[8,31,0.2581,0.09821,0.24598,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[12,31,0.3871,0.1875,0.34707,0.0,0.0,0.17857,0.0,1.0,23,4,0,23,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,4],[16,31,0.5161,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"0c5ddea451632558","q":"Let  $ABC$  be a triangle with circumcircle  $\\Gamma$ .  $D$  is the midpoint of arc  $BC$  (this arc does not contain  $A$ ).  $E$  is the common point of  $BC$  and the perpendicular bisector of  $BD$ .  $F$  is the common point of  $AC$  and the parallel to  $AB$  containing  $D$ .  $G$  is the common point of  $EF$  and  $AB$ .   $H$  is the common point of  $GD$  and  $AC$ . Show that  $GAH$  is isosceles.","t":[{"b":1,"e":0.28571,"k":"flat","v":0.09804,"x":0.1875,"p":[[0,7,0.0,0.09804,0.09732,0.0,0.14286,0.14286,0.0,0.42857,13,0,1,13,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.14286,0.14286,0.0,0.14286,0.2857,0.0,0.42857,13,0,5,13,0,9,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.1875,0.1448,0.0,0.2857,0.28571,0.0,0.57143,10,0,0,10,0,4,0,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.10259,"x":0.18972,"p":[[0,11,0.0,0.10714,0.10714,0.0,0.14286,0.14286,0.0,0.4286,13,0,4,13,0,15,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.10259,0.09602,0.0,0.14286,0.14286,0.0,0.28571,13,0,2,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.16964,0.1357,0.0,0.21428,0.28571,0.0,0.42857,11,0,0,11,0,5,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.18972,0.1124,0.14286,0.2857,0.28571,0.0,0.28571,6,0,0,6,1,8,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f11ba7f87d1adbf0","q":"Let  $ABC$  be an acute triangle, and let  $AA_1, BB_1$ , and  $CC_1$  be its altitudes. Segments  $AA_1$  and  $B_1C_1$  meet at point  $K$ . The perpendicular bisector of segment  $A_1K$  intersects sides  $AB$  and  $AC$  at  $L$  and  $M$ , respectively. Prove that points  $A,A_1, L$ , and  $M$  lie on a circle.","t":[{"b":1,"e":0.0,"k":"flat","v":0.02232,"x":0.05358,"p":[[0,7,0.0,0.05358,0.06917,0.0,0.0,0.14286,0.0,0.143,20,0,7,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,4,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.04465,"x":0.06241,"p":[[0,9,0.0,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,4,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,5,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,11,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9926502a112d78ed","q":"Let $ABC$ be a triangle with $\\widehat{BAC}=60^{\\circ}$ and let $\\Gamma$ be its circumcircle. Let $H$ be the orthocenter of $ABC$ and $S$ the midpoint of the arc $\\widehat{BC}$ not containing $A$. Let $P$ be the point on $\\Gamma$ such that $\\widehat{SPH}=90^{\\circ}$. Show that there exists a circle passing through $P, S$ and which is tangent to $(AB)$ and $(AC)$.","t":[{"b":4,"e":0.28571,"k":"flat","v":0.31699,"x":0.45536,"p":[[0,25,0.0,0.31699,0.20744,0.24999,0.28571,0.42893,0.0,0.71429,6,0,0,6,0,2,0,0,12,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[4,25,0.16,0.45536,0.1448,0.42857,0.42857,0.46429,0.14286,0.85714,0,0,0,0,0,1,0,0,5,0,0,18,0,0,4,0,0,3,0,0,1,0,0],[8,25,0.32,0.43304,0.10999,0.42857,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,6,0,0,16,0,0,9,0,0,0,0,0,0,0,0],[12,25,0.48,0.39286,0.13832,0.28571,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,2,0,0,12,0,0,12,0,0,4,0,0,2,0,0,0,0,0],[16,25,0.64,0.38393,0.12079,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,1,0,0,9,0,0,17,0,0,4,0,0,0,0,0,0,0,0],[20,25,0.8,0.38391,0.14478,0.28571,0.42857,0.42857,0.0,0.71429,1,0,0,1,0,2,0,0,10,0,0,13,0,0,5,0,0,1,0,0,0,0,0],[24,25,0.96,0.375,0.14617,0.28571,0.42857,0.42858,0.0,0.57143,1,0,0,1,0,3,0,0,10,0,0,11,0,0,7,0,0,0,0,0,0,0,0],[25,25,1.0,0.36158,0.14715,0.28571,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,2,0,0,15,0,0,8,0,0,5,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.24554,"x":0.41516,"p":[[0,24,0.0,0.24554,0.16458,0.14286,0.28571,0.42857,0.0,0.4286,7,0,0,7,0,6,0,0,8,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.31241,0.11552,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,4,0,0,16,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[8,24,0.3333,0.41516,0.13532,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,12,0,0,9,0,0,9,0,0,1,0,0,0,0,0],[12,24,0.5,0.35713,0.11291,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,2,0,0,16,0,0,10,0,0,4,0,0,0,0,0,0,0,0],[16,24,0.6667,0.35713,0.12875,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,15,0,0,10,0,0,3,0,0,1,0,0,0,0,0],[20,24,0.8333,0.33034,0.10371,0.28571,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,0,0,0,21,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[24,24,1.0,0.37946,0.14987,0.28571,0.28571,0.46428,0.14286,0.71429,0,0,0,0,0,2,0,0,17,0,0,5,0,0,6,0,0,2,0,0,0,0,0]]}]},{"i":"c8ea033827f8d964","q":"Let $A$ be a finite set with more than one element. Prove that the number of nonequivalent sets $S$ which tile $A$ is always even.","t":[{"b":1,"e":1.0,"k":"rising","v":0.5491,"x":1.0,"p":[[0,26,0.0,0.5491,0.43464,0.10714,0.64286,1.0,0.0,1.0,8,13,3,8,0,3,0,0,4,0,0,0,0,0,1,0,0,1,0,0,2,0,13],[4,26,0.1538,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[8,26,0.3077,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,26,0.4615,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[16,26,0.6154,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,26,0.7692,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[24,26,0.9231,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[26,26,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":4,"e":1.0,"k":"volatile","v":0.44633,"x":0.97768,"p":[[0,16,0.0,0.57588,0.3838,0.24999,0.64286,1.0,0.0,1.0,4,11,0,4,0,4,0,0,5,0,0,1,0,0,2,0,0,3,0,0,2,0,11],[4,16,0.25,0.89275,0.22048,0.96425,1.0,1.0,0.14,1.0,0,24,0,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,24],[8,16,0.5,0.77232,0.3131,0.57142,1.0,1.0,0.0,1.0,2,18,1,2,0,0,0,0,2,0,0,3,0,0,4,0,0,0,0,0,3,0,18],[12,16,0.75,0.44633,0.3793,0.14286,0.28571,0.89275,0.0,1.0,5,8,0,5,0,8,0,0,5,0,0,1,0,0,3,0,0,1,0,0,1,0,8],[16,16,1.0,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28]]}]},{"i":"eb0faf888adbb7c6","q":"Let  circles  $ \\Gamma $  and  $ \\omega $  are  circumcircle and incircle  of the  triangle  $ABC$ ,   the incircle  touches  sides   $BC,CA,AB$  at the points   $A_1,B_1,C_1$ .   Let  $A_2$   and   $B_2$   lies the lines  $A_1I$  and  $B_1I$  ( $A_1$  and  $A_2$  lies  different sides  from  $I$ ,  $B_1$  and  $B_2$  lies  different sides from  $I$ ) such that    $IA_2=IB_2=R$ .  Prove that : \n(a)  $AA_2=BB_2=IO$ ;\n(b)   The lines  $AA_2$   and   $BB_2$    intersect   on the circle   $ \\Gamma ;$","t":[{"b":3,"e":0.14286,"k":"flat","v":0.09357,"x":0.1607,"p":[[0,25,0.0,0.13393,0.11258,0.0,0.14286,0.1429,0.0,0.4286,10,0,1,10,0,15,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.1607,0.17402,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,9,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[8,25,0.32,0.09357,0.08448,0.0,0.14286,0.14286,0.0,0.2857,13,0,7,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.13839,0.14054,0.0,0.14286,0.2857,0.0,0.42857,13,0,2,13,0,10,0,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.2857,10,0,3,10,0,19,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.12054,0.12931,0.0,0.14286,0.14286,0.0,0.4286,13,0,8,13,0,14,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.13384,0.10062,0.14214,0.14286,0.14286,0.0,0.4286,7,0,0,7,0,22,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.10706,0.08745,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.13823,"x":0.165,"p":[[0,6,0.0,0.14714,0.1262,0.105,0.14286,0.14287,0.0,0.57143,8,0,0,8,0,18,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,6,0.6667,0.13823,0.10999,0.0,0.14286,0.1429,0.0,0.42857,9,0,0,9,0,16,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.165,0.13885,0.14214,0.14286,0.14287,0.0,0.71429,6,0,0,6,0,19,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"637bf2e457bb1440","q":"Let  $ABCD$  be a quadrilateral with a incircle  $\\omega$ . Let  $I$  be the center of  $\\omega$ , suppose that the lines  $AD$  and  $BC$  intersect at  $Q$  and the lines  $AB$  and  $CD$  intersect at  $P$  with  $B$  is in the segment  $AP$  and  $D$  is in the segment  $AQ$ . Let  $X$  and  $Y$  the incenters of  $\\triangle PBD$  and  $\\triangle QBD$  respectively. Let  $R$  be the intersection of  $PY$  and  $QX$ . Prove that the line  $IR$  is perpendicular to  $BD$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,38,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"7199e8ece18d1b26","q":"Let $\\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\\ell \\in \\mathcal{L}$ a point $f(\\ell)$ on $\\ell$. Suppose that for any point $X$, and for any three lines $\\ell_{1}, \\ell_{2}, \\ell_{3}$ passing through $X$, the points $f\\left(\\ell_{1}\\right), f\\left(\\ell_{2}\\right), f\\left(\\ell_{3}\\right)$ and $X$ lie on a circle.  Prove that there is a unique point $P$ such that $f(\\ell)=P$ for any line $\\ell$ passing through $P$. (Australia)","t":[{"b":3,"e":0.71429,"k":"flat","v":0.22098,"x":0.35715,"p":[[0,6,0.0,0.35715,0.27664,0.14286,0.42857,0.42858,0.0,1.0,5,3,1,5,0,7,0,0,2,0,0,13,0,0,1,0,0,1,0,0,0,0,3],[4,6,0.6667,0.22098,0.22257,0.0,0.14286,0.32143,0.0,0.85714,9,0,0,9,0,10,0,1,4,0,0,6,0,0,0,0,0,0,0,0,2,0,0],[6,6,1.0,0.30355,0.18119,0.14286,0.28571,0.42857,0.0,0.71429,2,0,0,2,0,12,0,0,3,0,0,11,0,0,3,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.17857,"x":0.38834,"p":[[0,17,0.0,0.24545,0.22088,0.14286,0.14286,0.42857,0.0,1.0,5,1,0,5,0,15,0,0,3,0,0,6,0,0,1,0,0,1,0,0,0,0,1],[4,17,0.2353,0.28779,0.21285,0.14286,0.2857,0.42857,0.0,0.85714,5,0,0,5,0,9,0,0,6,0,0,7,0,1,2,0,0,1,0,0,1,0,0],[8,17,0.4706,0.23652,0.14992,0.14286,0.14286,0.32143,0.0,0.57143,2,0,0,2,0,17,0,0,5,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[12,17,0.7059,0.38834,0.29724,0.14286,0.28571,0.46536,0.0,1.0,3,2,0,3,0,9,0,0,5,0,0,7,0,0,1,0,0,1,0,0,4,0,2],[16,17,0.9412,0.19857,0.15233,0.14286,0.14286,0.42857,0.0,0.42857,5,0,0,5,1,17,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.17857,0.21053,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,2,9,0,0,2,0,0,4,0,0,1,0,0,2,0,0,0,0,0]]}]},{"i":"cc7b7ff6e1a46c80","q":"Let  $ABC$  be a triangle, let  $I$  be its incenter, let  $\\Omega$  be its circumcircle, and let  $\\omega$  be the  $A$ - mixtilinear incircle. Let  $D,E$  and  $T$  be the intersections of  $\\omega$  and  $AB,AC$  and  $\\Omega$ , respectively, let the line  $IT$  cross  $\\omega$  again at  $P$ , and let lines  $PD$  and  $PE$  cross the line  $BC$  at  $M$  and  $N$  respectively. Prove that points  $D,E,M,N$  are concyclic. What is the center of this circle?","t":[{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,43,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,6,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,9,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,10,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,5,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.02232,0.05188,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.02232,"x":0.03125,"p":[[0,7,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,5,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"21a754d29bd4c505","q":"Let  $P$  be a cyclic polygon with circumcenter  $O$  that does not lie on any diagonal, and let  $S$  be the set of points on 2D plane containing  $P$  and  $O$ . \n\nThe  $\\textit{Matcha Sweep Game}$  is a game between two players  $A$  and  $B$ , with  $A$  going first, such that each choosing a nonempty subset  $T$  of points in  $S$  that has not been previously chosen, and such that if  $T$  has at least  $3$  vertices then  $T$  forms a convex polygon. The game ends with all points have been chosen, with the player picking the last point wins. \n\nFor which polygons  $P$  can  $A$  guarantee a win? \n\n*Proposed by Anzo Teh Zhao Yang*","t":[{"b":2,"e":1.0,"k":"flat","v":0.51786,"x":0.99554,"p":[[0,47,0.0,0.8125,0.32427,0.85714,1.0,1.0,0.0,1.0,3,20,3,3,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,5,0,20],[4,47,0.0851,0.70536,0.34981,0.42857,0.85714,1.0,0.0,1.0,4,15,2,4,0,0,0,0,0,0,0,7,0,0,1,0,0,2,0,0,3,0,15],[8,47,0.1702,0.68304,0.40364,0.35714,0.85714,1.0,0.0,1.0,6,15,6,6,0,2,0,0,0,0,0,2,0,0,1,0,0,0,0,0,6,0,15],[12,47,0.2553,0.51786,0.42521,0.0,0.57143,1.0,0.0,1.0,11,9,11,11,0,1,0,0,0,0,0,2,0,0,4,0,0,0,0,0,5,0,9],[16,47,0.3404,0.87946,0.24251,0.85714,1.0,1.0,0.0,1.0,2,19,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,9,0,19],[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],[24,47,0.5106,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[28,47,0.5957,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[32,47,0.6809,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,47,0.766,0.98214,0.04726,1.0,1.0,1.0,0.857,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[40,47,0.8511,0.95535,0.17655,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,28],[44,47,0.9362,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[47,47,1.0,0.90179,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28]]},{"b":6,"e":0.85714,"k":"flat","v":0.5357,"x":0.875,"p":[[0,28,0.0,0.74999,0.32928,0.57143,0.92857,1.0,0.0,1.0,3,16,1,3,0,1,0,0,0,0,0,2,0,0,5,0,0,1,0,0,4,0,16],[4,28,0.1429,0.625,0.40049,0.21429,0.85707,1.0,0.0,1.0,8,11,7,8,0,0,0,0,1,0,0,0,0,0,5,0,0,1,0,0,6,0,11],[8,28,0.2857,0.5357,0.36596,0.2857,0.57121,1.0,0.0,1.0,5,9,4,5,0,2,0,0,6,0,0,2,0,0,4,0,0,3,0,0,1,0,9],[12,28,0.4286,0.875,0.15047,0.85714,0.85714,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,0,0,12,0,14],[16,28,0.5714,0.67411,0.39967,0.42857,0.85714,1.0,0.0,1.0,8,10,8,8,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,10],[20,28,0.7143,0.75,0.32143,0.71429,0.85714,1.0,0.0,1.0,4,11,4,4,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,0,11,0,11],[24,28,0.8571,0.64726,0.39607,0.42825,0.85714,1.0,0.0,1.0,8,10,8,8,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,9,0,10]]}]},{"i":"e190dd1ea4ba6a2c","q":"The positive integers  $1,2,...,121$  are arranged in the squares of a  $11 \\times 11$  table. Dima found the product of numbers in each row and Sasha found the product of the numbers in each column. Could they get the same set of  $11$  numbers?\n\n*Proposed by S. Berlov*","t":[{"b":1,"e":1.0,"k":"rising","v":0.32143,"x":1.0,"p":[[0,17,0.0,0.32143,0.44607,0.0,0.0,1.0,0.0,1.0,20,9,0,20,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,9],[4,17,0.2353,0.42857,0.47515,0.0,0.0,1.0,0.0,1.0,17,12,0,17,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,12],[8,17,0.4706,0.61161,0.4549,0.0,1.0,1.0,0.0,1.0,10,17,0,10,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,17],[12,17,0.7059,0.56249,0.44741,0.0,0.71429,1.0,0.0,1.0,11,14,0,11,0,0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,1,0,14],[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":0.0,"k":"volatile","v":0.0,"x":0.76339,"p":[[0,29,0.0,0.42411,0.48377,0.0,0.0,1.0,0.0,1.0,18,12,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,12],[4,29,0.1379,0.38393,0.45518,0.0,0.0,1.0,0.0,1.0,18,9,0,18,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,9],[8,29,0.2759,0.49107,0.45728,0.0,0.5,1.0,0.0,1.0,14,11,1,14,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,3,0,11],[12,29,0.4138,0.62946,0.45577,0.0,1.0,1.0,0.0,1.0,10,18,0,10,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,18],[16,29,0.5517,0.51786,0.48936,0.0,0.78571,1.0,0.0,1.0,15,15,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,15],[20,29,0.6897,0.71875,0.43519,0.21429,1.0,1.0,0.0,1.0,8,22,0,8,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,22],[24,29,0.8276,0.76339,0.39385,0.64286,1.0,1.0,0.0,1.0,6,22,0,6,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,22],[28,29,0.9655,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"73e519c2f45a59a4","q":"Let  $x_1,\\ldots ,x_n$  be positive real numbers. Show that there exist  $a_1,\\ldots ,a_n\\in\\{-1,1\\}$  such that:\n\\[a_1x_1^2+a_2x_2^2+\\ldots +a_nx_n^2\\ge (a_1x_1+a_2x_2+\\ldots + a_n x_n)^2\\]","t":[{"b":2,"e":0.85714,"k":"flat","v":0.42856,"x":0.63391,"p":[[0,20,0.0,0.57139,0.38631,0.25,0.71429,1.0,0.0,1.0,7,10,2,7,0,1,0,0,3,0,0,1,0,0,3,0,0,6,0,0,1,0,10],[4,20,0.2,0.43301,0.37708,0.0,0.4998,0.71429,0.0,1.0,11,5,0,11,0,1,0,0,3,0,0,1,0,0,4,0,0,6,0,0,1,0,5],[8,20,0.4,0.63391,0.35703,0.28571,0.71429,1.0,0.0,1.0,4,11,0,4,0,2,0,0,3,0,0,1,0,0,2,0,0,8,0,0,1,0,11],[12,20,0.6,0.55798,0.37688,0.25,0.57143,1.0,0.0,1.0,6,10,0,6,0,2,0,0,3,0,0,2,0,0,4,0,0,5,0,0,0,0,10],[16,20,0.8,0.51332,0.31103,0.2857,0.571,0.857,0.0,0.85714,6,0,0,6,0,0,0,0,4,0,0,3,0,0,7,0,0,2,0,0,10,0,0],[20,20,1.0,0.42856,0.30092,0.2857,0.35714,0.60714,0.0,1.0,6,1,0,6,0,1,0,0,9,0,0,2,0,0,6,0,0,2,0,0,5,0,1]]},{"b":4,"e":0.0,"k":"falling","v":0.08929,"x":0.48212,"p":[[0,17,0.0,0.48212,0.37922,0.10714,0.42859,0.85714,0.0,1.0,8,7,1,8,0,1,0,0,6,0,0,2,0,0,2,0,0,4,0,0,2,0,7],[4,17,0.2353,0.38391,0.3415,0.10714,0.28571,0.71429,0.0,1.0,8,3,0,8,0,5,0,0,5,0,0,3,0,0,2,0,0,3,0,0,3,0,3],[8,17,0.4706,0.39732,0.36375,0.0,0.28571,0.71429,0.0,1.0,10,5,2,10,0,2,0,0,5,0,0,4,0,0,1,0,0,4,0,0,1,0,5],[12,17,0.7059,0.08929,0.12242,0.0,0.0,0.1786,0.0,0.28571,20,0,0,20,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.15625,0.14445,0.0,0.21428,0.28571,0.0,0.42857,14,0,1,14,0,2,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.13391,0.15943,0.0,0.0,0.2857,0.0,0.571,17,0,2,17,0,3,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"aa8eaeab65878158","q":"Let  $ABC$  be an non-isosceles triangle with incenter  $I$ , circumcenter  $O$  and a point  $D$  on segment  $BC $ such that  $(BID) $ cut segments  $AB $  at $ E $ and  $(CID) $ cuts segment  $AC $ at  $F$  Circle  $(DEF)$  cuts segments  $AB$ , $AC $ again at  $M,N$ . Let  $P$  The intersection of  $IB$   and   $DE $  ,  $Q$  The intersection of  $IC$ and  $DF$  . Prove that  $EN,FM,PQ $ are parallel and the median of vertex  $I$ in triangle  $IPQ$  bisects the arc  $BAC$  of  $(O)$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.03571,"x":0.13821,"p":[[0,29,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,3,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.13821,0.06666,0.14286,0.14286,0.14286,0.0,0.42857,3,0,3,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.13161,0.06212,0.14286,0.14286,0.14286,0.0,0.2857,4,0,3,4,1,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.11589,0.05568,0.14214,0.14286,0.14286,0.0,0.1429,6,0,6,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.12938,0.04161,0.14286,0.14286,0.14286,0.0,0.1429,3,0,2,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.11589,0.05568,0.14214,0.14286,0.14286,0.0,0.1429,6,0,5,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.02679,"x":0.14277,"p":[[0,11,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,3,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ac06c71798f6c6a1","q":"Let $ABC$ be a triangle and let $A_{1}, B_{1}$, and $C_{1}$ be the points of contact of the $A$-excircle, denoted by $\\omega_{A}$, with the side $BC$ and the rays $[AC)$ and $[AB)$, respectively. Let $P$ be the midpoint of the segment $\\left[B_{1} C_{1}\\right]$. The line $\\left(A_{1} P\\right)$ intersects the circle $\\omega_{A}$ again at point $X$. The tangent to the circumcircle of triangle $ABC$ at point $A$ and the tangent to the circle $\\omega_{A}$ at point $X$ intersect at $R$. Show that $RX = RP$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.00446,"x":0.15616,"p":[[0,21,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.15616,0.09007,0.14286,0.14286,0.1786,0.0,0.28571,5,0,0,5,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.13822,0.07562,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,23,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.10697,0.08741,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,10,0.0,0.05357,0.13716,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,10,0.4,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"11c94778667c0383","q":"Let $\\mathbb{Q}_{>0}$ be the set of positive rational numbers. Let $f: \\mathbb{Q}_{>0} \\rightarrow \\mathbb{R}$ be a function satisfying the conditions  $$ f(x) f(y) \\geqslant f(x y) \\text { and } f(x+y) \\geqslant f(x)+f(y) $$  for all $x, y \\in \\mathbb{Q}_{>0}$. Given that $f(a)=a$ for some rational $a>1$, prove that $f(x)=x$ for all $x \\in \\mathbb{Q}_{>0}$.  (Bulgaria)","t":[{"b":0,"e":0.0,"k":"flat","v":0.05357,"x":0.08482,"p":[[0,8,0.0,0.08482,0.17075,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,2,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,8,0.5,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14,"k":"flat","v":0.10491,"x":0.26339,"p":[[0,12,0.0,0.10491,0.18209,0.0,0.0,0.14286,0.0,0.85714,18,0,0,18,1,9,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[4,12,0.3333,0.14287,0.22304,0.0,0.0,0.14287,0.0,0.8571,17,0,0,17,0,8,0,0,4,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[8,12,0.6667,0.26339,0.32559,0.0,0.14286,0.42857,0.0,1.0,14,2,1,14,0,5,0,0,4,0,0,2,0,0,1,0,0,2,0,0,2,0,2],[12,12,1.0,0.17857,0.20825,0.0,0.14286,0.2857,0.0,0.85714,10,0,0,10,0,13,0,0,6,0,0,1,0,0,0,0,0,0,0,0,2,0,0]]}]},{"i":"dd8dd201470f0a8c","q":"Let  $I$  be the incentre of triangle  $ABC$  with  $AB>AC$  and let the line  $AI$  intersect the side  $BC$  at  $D$ . Suppose that point  $P$  lies on the segment  $BC$  and satisfies  $PI=PD$ . Further, let  $J$  be the point obtained by reflecting  $I$  over the perpendicular bisector of  $BC$ , and let  $Q$  be the other intersection of the circumcircles of the triangles  $ABC$  and  $APD$ . Prove that  $\\angle BAQ=\\angle CAJ$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"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],[16,32,0.5,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"3f3f73fc3a398c64","q":"The function  $ F$  is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every  $ n \\geq 0,$ \r\n\r\n(i)  $ F(4n) \\equal{} F(2n) \\plus{} F(n),$ \r\n(ii)  $ F(4n \\plus{} 2) \\equal{} F(4n) \\plus{} 1,$ \r\n(iii)  $ F(2n \\plus{} 1) \\equal{} F(2n) \\plus{} 1.$ \r\n\r\nProve that for each positive integer  $ m,$  the number of integers  $ n$  with  $ 0 \\leq n < 2^m$  and  $ F(4n) \\equal{} F(3n)$  is  $ F(2^{m \\plus{} 1}).$","t":[{"b":4,"e":0.85714,"k":"flat","v":0.59371,"x":0.74997,"p":[[0,70,0.0,0.6607,0.24936,0.42857,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,7,0,0,2,0,0,4,0,0,7,0,0,7,0,5],[4,70,0.0571,0.59371,0.25781,0.42857,0.57143,0.74996,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,7,0,0,8,0,0,4,0,0,3,0,5],[8,70,0.1143,0.67406,0.19311,0.57143,0.64271,0.857,0.14286,1.0,0,4,0,0,0,1,0,0,0,0,0,3,0,0,12,0,0,7,0,0,5,0,4],[12,70,0.1714,0.60265,0.25688,0.42859,0.57143,0.71429,0.0,1.0,2,4,0,2,0,0,0,0,3,0,0,5,0,0,7,0,0,8,0,0,3,0,4],[16,70,0.2286,0.7232,0.21111,0.57142,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,3,0,0,7,0,0,5,0,0,9,0,6],[20,70,0.2857,0.71875,0.15765,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,11,0,0,6,0,4],[24,70,0.3429,0.74104,0.16149,0.57143,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,11,0,0,7,0,5],[28,70,0.4,0.74996,0.15157,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,12,0,0,9,0,4],[32,70,0.4571,0.70534,0.16343,0.57143,0.71429,0.85704,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,13,0,0,6,0,3],[36,70,0.5143,0.70975,0.17312,0.571,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,9,0,0,7,0,4],[40,70,0.5714,0.65621,0.20474,0.5354,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,5,0,0,6,0,0,9,0,0,6,0,3],[44,70,0.6286,0.68741,0.1766,0.571,0.71429,0.85704,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,3,0,0,10,0,0,8,0,0,7,0,3],[48,70,0.6857,0.66514,0.19105,0.571,0.57143,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,1,0,0,5,0,0,11,0,0,6,0,0,5,0,4],[52,70,0.7429,0.69194,0.16411,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,9,0,0,8,0,2],[56,70,0.8,0.65171,0.17837,0.571,0.57143,0.85704,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,5,0,0,12,0,0,5,0,0,7,0,2],[60,70,0.8571,0.63834,0.14281,0.571,0.64286,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,11,0,0,11,0,0,5,0,0],[64,70,0.9143,0.74997,0.15975,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,7,0,0,8,0,0,11,0,4],[68,70,0.9714,0.69192,0.16411,0.57143,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,7,0,0,9,0,2],[70,70,1.0,0.7098,0.18725,0.57143,0.71429,0.85714,0.2857,1.0,0,5,0,0,0,0,0,0,1,0,0,3,0,0,8,0,0,9,0,0,6,0,5]]},{"b":7,"e":1.0,"k":"rising","v":0.60261,"x":0.83036,"p":[[0,25,0.0,0.60261,0.23072,0.42857,0.57143,0.74996,0.14286,1.0,0,2,0,0,0,2,0,0,4,0,0,3,0,0,9,0,0,6,0,0,6,0,2],[4,25,0.16,0.6562,0.18853,0.571,0.57143,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,3,0,0,13,0,0,5,0,0,6,0,3],[8,25,0.32,0.62497,0.24679,0.42857,0.57143,0.74996,0.0,1.0,1,6,0,1,0,0,0,0,3,0,0,5,0,0,10,0,0,5,0,0,2,0,6],[12,25,0.48,0.62943,0.25471,0.4286,0.57143,0.75,0.0,1.0,1,6,0,1,0,1,0,0,2,0,0,5,0,0,8,0,0,7,0,0,2,0,6],[16,25,0.64,0.63387,0.18879,0.571,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,2,0,0,5,0,0,12,0,0,6,0,0,4,0,3],[20,25,0.8,0.683,0.21051,0.571,0.71429,0.85714,0.2857,1.0,0,6,0,0,0,0,0,0,2,0,0,4,0,0,9,0,0,7,0,0,4,0,6],[24,25,0.96,0.82585,0.14617,0.71429,0.85714,1.0,0.571,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,6,0,0,12,0,9],[25,25,1.0,0.83036,0.14913,0.71429,0.85714,1.0,0.4286,1.0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,7,0,11]]}]},{"i":"3d577441ca3a37bc","q":"Let  $a,b,c,d$  be pairwise distinct positive integers such that   $$ \\frac{a}{a+b}+\\frac{b}{b+c}+\\frac{c}{c+d}+\\frac{d}{d+a} $$   is an integer. Prove that  $a+b+c+d$  is **not** a prime number.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.12492,"x":0.45534,"p":[[0,10,0.0,0.45534,0.29973,0.14286,0.42859,0.71429,0.0,1.0,3,3,0,3,0,7,0,0,3,0,0,4,0,0,4,0,0,8,0,0,0,0,3],[4,10,0.4,0.35266,0.29662,0.10714,0.28571,0.57143,0.0,1.0,8,2,1,8,0,3,0,0,8,0,0,2,0,0,6,0,0,2,0,0,1,0,2],[8,10,0.8,0.36601,0.27656,0.14286,0.28571,0.57111,0.0,0.85714,4,0,1,4,0,9,0,0,4,0,0,6,0,0,2,0,0,3,0,0,4,0,0],[10,10,1.0,0.12492,0.04722,0.14286,0.14286,0.14286,0.0,0.143,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.03125,"x":0.43302,"p":[[0,32,0.0,0.43302,0.27544,0.2857,0.42859,0.71429,0.0,1.0,4,1,1,4,0,3,0,0,7,0,0,6,0,0,2,0,0,7,0,0,2,0,1],[4,32,0.125,0.28563,0.26491,0.0,0.14286,0.4286,0.0,1.0,9,1,2,9,0,8,0,0,1,0,0,7,0,0,4,0,0,2,0,0,0,0,1],[8,32,0.25,0.24088,0.25867,0.0,0.14286,0.32143,0.0,0.85714,11,0,0,11,0,8,0,0,5,0,0,1,0,0,3,0,0,3,0,0,1,0,0],[12,32,0.375,0.22322,0.23402,0.0,0.14286,0.32143,0.0,1.0,10,1,0,10,0,9,0,0,5,0,0,5,0,0,1,0,0,1,0,0,0,0,1],[16,32,0.5,0.28554,0.28323,0.0,0.21428,0.57111,0.0,0.71429,13,0,0,13,0,3,0,0,3,0,0,2,0,0,6,0,0,5,0,0,0,0,0],[20,32,0.625,0.14286,0.21429,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,13,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[24,32,0.75,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.05349,0.06905,0.0,0.0,0.14286,0.0,0.143,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8d217a4f3e0bb294","q":"Let  $n$  be a positive integer and let  $V$  be a  $(2n-1)$ -dimensional vector space over the two-element field. Prove that for arbitrary vectors  $v_1,\\dots,v_{4n-1} \\in V,$  there exists a sequence  $1\\leq i_1<\\dots<i_{2n}\\leq 4n-1$  of indices such that  $v_{i_1}+\\dots+v_{i_{2n}}=0.$","t":[{"b":5,"e":0.0,"k":"flat","v":0.01786,"x":0.09821,"p":[[0,37,0.0,0.05804,0.12301,0.0,0.0,0.0,0.0,0.42857,26,0,1,26,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,3,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.0625,0.13333,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,0,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.07143,0.12372,0.0,0.0,0.07143,0.0,0.28571,24,0,0,24,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0625,0.11259,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.05804,0.11214,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.09821,0.1357,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02232,"x":0.11607,"p":[[0,55,0.0,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.05794,0.11207,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.11607,0.1357,0.0,0.0,0.28571,0.0,0.28571,18,0,0,18,0,2,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.05357,0.10564,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.07589,0.12363,0.0,0.0,0.17857,0.0,0.28571,23,0,0,23,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[55,55,1.0,0.08036,0.12846,0.0,0.0,0.2857,0.0,0.28571,23,0,0,23,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8cf4587983570481","q":"Let  $ABCDE$  be an equiangular pentagon. Line  $CD$  intersects  $AE$  extended and  $AB$  extended at  $F$  and  $H$ , respectively.  $ED$  and  $BC$  extended meet at  $G$ . Segments  $CE$  and  $BD$  meet at  $I$ . If a circle can be drawn passing through  $C,D,G,$  and  $I$ , what is the measure of  $\\angle {FGH}$ ?","t":[{"b":2,"e":0.14286,"k":"flat","v":0.02232,"x":0.2366,"p":[[0,33,0.0,0.14714,0.12104,0.105,0.14286,0.14286,0.0,0.42857,8,0,4,8,0,18,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.16518,0.14773,0.0,0.14286,0.2857,0.0,0.42857,10,0,4,10,0,12,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.21875,0.12869,0.14286,0.21428,0.28571,0.0,0.4286,4,0,3,4,0,12,0,0,11,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.2366,0.14107,0.14286,0.21435,0.28571,0.0,0.714,2,0,1,2,0,14,0,0,11,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[16,33,0.4848,0.22321,0.11259,0.14286,0.21428,0.28571,0.0,0.4286,2,0,2,2,0,14,0,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.12946,0.14445,0.0,0.14286,0.14286,0.0,0.42857,14,0,8,14,0,11,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,18,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,17,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.28571,14,0,2,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.0892,0.08559,0.0,0.14286,0.14286,0.0,0.28571,14,0,2,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.10704,"x":0.25893,"p":[[0,16,0.0,0.10704,0.14719,0.0,0.0,0.14286,0.0,0.571,17,0,11,17,0,10,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,16,0.25,0.14732,0.11564,0.0,0.14286,0.2857,0.0,0.42857,9,0,8,9,0,14,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.25893,0.13092,0.14286,0.28571,0.28571,0.0,0.4286,4,0,3,4,0,5,0,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,5,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5fe817201715628a","q":"Let  $p,q$  be primes, where  $p>q$ . Define  $t=\\gcd(p!-1,q!-1)$ . Prove that  $t\\le p^{\\frac{p}{3}}$ .","t":[{"b":1,"e":0.2857,"k":"flat","v":0.25,"x":0.29017,"p":[[0,19,0.0,0.2633,0.05208,0.2857,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.25,0.06186,0.25,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.28571,1e-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],[12,19,0.6316,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.25446,0.08552,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,1,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.22321,"x":0.25884,"p":[[0,12,0.0,0.23214,0.09278,0.14289,0.2857,0.28571,0.0,0.28571,3,0,2,3,0,6,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.25884,0.05595,0.2857,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.23661,0.07668,0.14286,0.2857,0.28571,0.0,0.286,1,0,1,1,0,9,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.22321,0.07087,0.14286,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,14,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bbf0d9154abcc4e5","q":"Let  $M$  be a subset of  $\\{1,2,..., 1998\\}$  with  $1000$  elements. Prove that it is always possible to find two elements  $a$  and  $b$  in  $M$ , not necessarily distinct, such that  $a + b$  is a power of  $2$ .","t":[{"b":4,"e":0.0,"k":"volatile","v":0.39284,"x":0.57142,"p":[[0,7,0.0,0.57142,0.3896,0.10714,0.71429,0.85714,0.0,1.0,8,7,7,8,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,0,7,0,7],[4,7,0.5714,0.39284,0.33502,0.0,0.35714,0.71429,0.0,1.0,9,1,8,9,0,4,0,0,3,0,0,3,0,0,2,0,0,6,0,0,4,0,1]]},{"b":6,"e":0.57143,"k":"flat","v":0.21429,"x":0.51783,"p":[[0,13,0.0,0.45089,0.32948,0.10714,0.5,0.71429,0.0,0.85714,8,0,7,8,0,1,0,0,5,0,0,2,0,0,3,0,0,6,0,0,7,0,0],[4,13,0.3077,0.51783,0.2714,0.39286,0.57121,0.71429,0.0,1.0,3,2,3,3,0,2,0,0,3,0,0,6,0,0,7,0,0,6,0,0,3,0,2],[8,13,0.6154,0.46427,0.30722,0.24999,0.42859,0.71429,0.0,1.0,6,1,5,6,0,2,0,0,3,0,0,6,0,0,4,0,0,5,0,0,5,0,1],[12,13,0.9231,0.21429,0.28121,0.0,0.14286,0.32143,0.0,1.0,15,1,15,15,0,6,0,0,3,0,0,3,0,0,1,0,0,2,0,0,1,0,1],[13,13,1.0,0.45538,0.24336,0.42857,0.4286,0.57143,0.0,0.85714,5,0,5,5,0,0,0,0,2,0,0,11,0,0,8,0,0,3,0,0,3,0,0]]}]},{"i":"578e666005221191","q":"Let $P$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $X, Y$, and $Z$ be points on the interior of $AB, BC$, and $CD$ respectively, such that\n\n$$\n\\frac{|AX|}{|XB|}=\\frac{|BY|}{|YC|}=\\frac{|CZ|}{|ZD|}=2\n$$\n\nSuppose moreover that $XY$ is tangent to the circumcircle of $\\triangle CYZ$ and that $YZ$ is tangent to the circumcircle of $\\triangle BXY$. Prove that $\\angle APD = \\angle XYZ$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.01339,"x":0.14732,"p":[[0,10,0.0,0.10268,0.22934,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,1,0,0,3,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[4,10,0.4,0.14732,0.21572,0.0,0.0,0.1786,0.0,0.71429,18,0,0,18,0,6,0,0,2,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[8,10,0.8,0.01563,0.04277,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00893,"x":0.13382,"p":[[0,34,0.0,0.13382,0.25981,0.0,0.0,0.14071,0.0,0.857,23,0,1,23,0,3,0,0,1,0,0,0,0,0,1,0,0,3,0,0,1,0,0],[4,34,0.1176,0.09812,0.22709,0.0,0.0,0.14071,0.0,1.0,23,1,0,23,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[8,34,0.2353,0.06696,0.12364,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.04464,0.0974,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.05348,0.0927,0.0,0.0,0.14071,0.0,0.2857,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7f0669efcf0b207a","q":"Three pairwise distinct positive integers $a, b, c$, with $\\operatorname{gcd}(a, b, c)=1$, satisfy\n\n$$\na\\left|(b-c)^{2}, \\quad b\\right|(c-a)^{2} \\quad \\text { and } \\quad c \\mid(a-b)^{2} \\text {. }\n$$\n\nProve that there does not exist a non-degenerate triangle with side lengths $a, b, c$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.09822,"p":[[0,15,0.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.09366,0.08454,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.02232,0.0487,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,2,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,43,0.0,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,1,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.07589,0.07973,0.0,0.07143,0.14286,0.0,0.2857,16,0,1,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[40,43,0.9302,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"dc30b370b00b0b44","q":"Let  $S$  denote the set of rational numbers in the interval  $(0,1)$ . Determine, with proof, if there exists a subset  $T$  of  $S$  such that every element in  $S$  can be uniquely written as the sum of finitely many distinct elements in  $T$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"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],[4,41,0.0976,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.04018,0.15251,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,41,0.3902,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,41,0.5854,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.03571,0.12372,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a51ed4b7633a7fe2","q":"Karl starts with $n$ cards labeled $1,2, \\ldots, n$ lined up in random order on his desk. He calls a pair $(a, b)$ of cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. Karl picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ cards to its left, then it now has $i$ cards to its right. He then picks up the card labeled 2 and reinserts it in the same manner, and so on, until he has picked up and put back each of the cards $1, \\ldots, n$ exactly once in that order. For example, if $n=4$, then one example of a process is  $$ 3142 \\longrightarrow 3412 \\longrightarrow 2341 \\longrightarrow 2431 \\longrightarrow 2341 $$  which has three swapped pairs both before and after. Show that, no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,34,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,5,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,40,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"806e5a04de6f89a7","q":"Let  $\\mathbb{N}$  be the set of positive integers. Determine all positive integers  $k$  for which there exist functions  $f:\\mathbb{N} \\to \\mathbb{N}$  and  $g: \\mathbb{N}\\to \\mathbb{N}$  such that  $g$  assumes infinitely many values  and such that   $$  f^{g(n)}(n)=f(n)+k $$   holds for every positive integer  $n$ .\n\n(*Remark.* Here,  $f^{i}$  denotes the function  $f$  applied  $i$  times i.e  $f^{i}(j)=f(f(\\dots f(j)\\dots ))$ .)","t":[{"b":2,"e":0.0,"k":"flat","v":0.04464,"x":0.0625,"p":[[0,7,0.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,9,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0625,0.1234,0.0,0.0,0.03571,0.0,0.4286,24,0,9,24,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02223,"x":0.07143,"p":[[0,18,0.0,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,6,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,10,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.04902,0.10472,0.0,0.0,0.0,0.0,0.4286,25,0,8,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,22,0,6,22,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,6,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.02223,0.06281,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"461a76d17f5a2927","q":"Let  $n$  be a positive integer, and let  $p(x)$  be a polynomial of\ndegree  $n$  with integer coefficients.  Prove that  $$ \\max_{0\\le x\\le1} \\big|p(x)\\big| &gt; \\frac1{e^n}. $$  Proposed by G\u00e9za K\u00f3s, E\u00f6tv\u00f6s University, Budapest","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"da22da2eda027f83","q":"Let  $V$  be a convex polygon.\r\n(a) Show that if  $V$  has  $3k$  vertices, then  $V$  can be triangulated such that each vertex is in an odd number of triangles.\r\n(b) Show that if the number of vertices is not divisible with 3, then  $V$  can be triangulated such that exactly 2 vertices have an even number of triangles.","t":[{"b":5,"e":0.1429,"k":"flat","v":0.0,"x":0.18303,"p":[[0,18,0.0,0.14732,0.2382,0.0,0.0,0.28571,0.0,1.0,21,1,14,21,0,1,0,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[4,18,0.2222,0.1383,0.23003,0.0,0.0,0.2857,0.0,0.85714,22,0,9,22,0,1,0,0,2,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[8,18,0.4444,0.18303,0.28846,0.0,0.0,0.42857,0.0,1.0,21,1,5,21,0,1,0,0,1,0,0,4,0,0,2,0,0,1,0,0,1,0,1],[12,18,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.08027,"x":0.19197,"p":[[0,27,0.0,0.10714,0.22868,0.0,0.0,0.0,0.0,0.85714,25,0,8,25,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,0,0],[4,27,0.1481,0.12509,0.23088,0.0,0.0,0.14286,0.0,1.0,22,1,6,22,0,3,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,1],[8,27,0.2963,0.14286,0.22304,0.0,0.0,0.21432,0.0,0.85714,20,0,5,20,0,4,0,0,0,0,0,6,0,0,1,0,0,0,0,0,1,0,0],[12,27,0.4444,0.09821,0.24338,0.0,0.0,0.0,0.0,1.0,26,1,6,26,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[16,27,0.5926,0.08929,0.16656,0.0,0.0,0.03571,0.0,0.57143,24,0,2,24,0,1,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[20,27,0.7407,0.19197,0.23038,0.0,0.14286,0.2857,0.0,1.0,13,1,0,13,0,8,0,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,1],[24,27,0.8889,0.08027,0.1333,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,4,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.11161,0.1461,0.0,0.0,0.1786,0.0,0.4286,18,0,0,18,0,6,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"34c9d8e443584194","q":"The incircle of the triangle  $ABC$  touches the sides  $BC, CA$  and  $AB{}$  at  $D,E$  and  $F{}$  respectively. Let the circle  $\\omega$  touch the segments  $CA{}$  and  $AB{}$  at  $Q{}$  and  $R{}$  respectively, and the points  $M{}$  and  $N{}$  are selected on the segments  $AB{}$  and  $AC{}$  respectively, so that the segments  $CM{}$  and  $BN{}$  touch  $\\omega$ . The bisectors of  $\\angle NBC$  and  $\\angle MCB$  intersect the segments  $DE{}$  and  $DF{}$  at  $K{}$  and  $L{}$  respectively. Prove that the lines  $RK{}$  and  $QL{}$  intersect on  $\\omega$ .\n\n*Proposed by Tran Quang Hung*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"d8b41efc8fa55fcd","q":"Let  $N$  be a positive integer. Determine all positive integers  $n$  that satisfy the following condition:\n\nFor any list  $d_1, d_2, \\ldots, d_k$  of divisors of  $n$  (not necessarily distinct) such that\n\\[\n\\frac{1}{d_1} + \\frac{1}{d_2} + \\cdots + \\frac{1}{d_k} > N,\n\\]\nthere exists a subset of the fractions  $\\frac{1}{d_1}, \\frac{1}{d_2}, \\ldots, \\frac{1}{d_k}$  whose sum is exactly  $N$ .","t":[{"b":2,"e":0.42857,"k":"flat","v":0.49103,"x":0.59364,"p":[[0,32,0.0,0.4955,0.255,0.28571,0.4998,0.71429,0.0,1.0,1,1,0,1,0,4,0,0,6,0,0,5,0,0,6,0,0,5,0,0,4,0,1],[4,32,0.125,0.53567,0.23144,0.28593,0.571,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,8,0,0,6,0,0,8,0,0,4,0,0,2,0,3],[8,32,0.25,0.59364,0.27005,0.42857,0.571,0.85714,0.0,1.0,1,5,0,1,0,1,0,0,3,0,0,10,0,0,4,0,0,3,0,0,5,0,5],[12,32,0.375,0.57585,0.20973,0.42857,0.57143,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,5,0,0,7,0,0,11,0,0,3,0,0,3,0,3],[16,32,0.5,0.5178,0.21649,0.42857,0.57121,0.57143,0.0,1.0,1,1,1,1,0,2,0,0,4,0,0,5,0,0,14,0,0,2,0,0,3,0,1],[20,32,0.625,0.49103,0.21703,0.39286,0.4998,0.60714,0.0,0.85714,1,0,1,1,0,3,0,0,4,0,0,8,0,0,8,0,0,5,0,0,3,0,0],[24,32,0.75,0.55352,0.2165,0.42857,0.571,0.60714,0.14286,1.0,0,2,0,0,0,2,0,0,3,0,0,8,0,0,11,0,0,2,0,0,4,0,2],[28,32,0.875,0.57139,0.13363,0.571,0.57143,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,4,0,0,16,0,0,8,0,0,1,0,0],[32,32,1.0,0.49546,0.21121,0.42857,0.571,0.57143,0.0,1.0,2,1,1,2,0,1,0,0,4,0,0,7,0,0,11,0,0,6,0,0,0,0,1]]},{"b":5,"e":1.0,"k":"rising","v":0.49553,"x":0.72306,"p":[[0,15,0.0,0.55799,0.2509,0.42857,0.57143,0.71429,0.0,1.0,1,2,1,1,0,2,0,0,4,0,0,7,0,0,4,0,0,8,0,0,4,0,2],[4,15,0.2667,0.59812,0.27779,0.28571,0.71429,0.85714,0.14,1.0,0,1,0,0,0,4,0,0,5,0,0,4,0,0,1,0,0,5,0,0,12,0,1],[8,15,0.5333,0.49553,0.23952,0.28571,0.4286,0.71429,0.0,1.0,1,1,0,1,0,2,0,0,7,0,0,8,0,0,5,0,0,4,0,0,4,0,1],[12,15,0.8,0.66964,0.25862,0.42857,0.78564,0.85714,0.0,1.0,1,2,1,1,0,0,0,0,5,0,0,4,0,0,0,0,0,6,0,0,14,0,2],[15,15,1.0,0.72306,0.20807,0.57143,0.857,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,3,0,0,3,0,0,3,0,0,6,0,0,14,0,3]]}]},{"i":"3abec84f0b47ba89","q":"Two circles $O_{1}$ and $O_{2}$ intersect at $M$ and $N$. The common tangent to the two circles closest to $M$ touches $O_{1}$ and $O_{2}$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflections of $A$ and $B$ with respect to $M$ respectively. The circumcircle of triangle $D C M$ intersects the circles $O_{1}$ and $O_{2}$ at points $E$ and $F$ distinct from $M$. Show that the circumcircles of triangles $M E F$ and $N E F$ have the same radius.","t":[{"b":1,"e":0.42857,"k":"rising","v":0.06687,"x":0.29464,"p":[[0,13,0.0,0.06687,0.11831,0.0,0.0,0.14286,0.0,0.42857,22,0,2,22,0,7,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.21875,0.11285,0.14286,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,3,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.28125,0.12103,0.28571,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,0,0,0,26,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[12,13,0.9231,0.26785,0.13243,0.2857,0.28571,0.28571,0.0,0.71429,4,0,0,4,0,1,0,0,24,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[13,13,1.0,0.29464,0.1234,0.28571,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,1,0,0,25,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.04911,"x":0.07134,"p":[[0,7,0.0,0.07134,0.08741,0.0,0.0,0.14286,0.0,0.28571,18,0,3,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,6,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.06696,0.08737,0.0,0.0,0.14286,0.0,0.2857,19,0,1,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4b73bd60003dcdcd","q":"Three nonnegative real numbers $r_{1}, r_{2}, r_{3}$ are written on a blackboard. These numbers have the property that there exist integers $a_{1}, a_{2}, a_{3}$, not all zero, satisfying $a_{1} r_{1}+a_{2} r_{2}+a_{3} r_{3}=0$. We are permitted to perform the following operation: find two numbers $x, y$ on the blackboard with $x \\leq y$, then erase $y$ and write $y-x$ in its place. Prove that after a finite number of such operations, we can end up with at least one 0 on the blackboard.","t":[{"b":2,"e":0.0,"k":"flat","v":0.04464,"x":0.15179,"p":[[0,11,0.0,0.15179,0.25489,0.0,0.0,0.14286,0.0,0.85714,17,0,1,17,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0],[4,11,0.3636,0.08482,0.18161,0.0,0.0,0.14286,0.0,1.0,20,1,0,20,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,11,0.7273,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.01786,"x":0.05357,"p":[[0,11,0.0,0.05357,0.11152,0.0,0.0,0.14286,0.0,0.57143,23,0,1,23,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,11,0.3636,0.04893,0.06761,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"203a785de9c5d06e","q":"Let  $\\mathbb{Z}_{>0}$  denote the set of positive integers. For any positive integer  $k$ , a function  $f: \\mathbb{Z}_{>0} \\to \\mathbb{Z}_{>0}$  is called *$k$ -good* if  $\\gcd(f(m) + n, f(n) + m) \\le k$  for all  $m \\neq n$ . Find all  $k$  such that there exists a  $k$ -good function.\n\n*Proposed by James Rickards, Canada*","t":[{"b":0,"e":0.0,"k":"flat","v":0.13839,"x":0.24107,"p":[[0,43,0.0,0.19643,0.18123,0.0,0.2857,0.28571,0.0,0.85714,11,0,6,11,0,3,0,0,16,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,43,0.093,0.1741,0.13235,0.0,0.2857,0.28571,0.0,0.28571,11,0,6,11,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.21429,0.16751,0.0,0.28571,0.28571,0.0,0.71429,9,0,5,9,0,3,0,0,18,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[12,43,0.2791,0.19643,0.17405,0.0,0.2857,0.28571,0.0,0.85714,10,0,7,10,0,4,0,0,17,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[16,43,0.3721,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.42857,4,0,4,4,0,3,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.20535,0.11258,0.14286,0.2857,0.28571,0.0,0.28571,6,0,4,6,0,6,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.24107,0.10374,0.2857,0.28571,0.28571,0.0,0.42857,4,0,4,4,0,3,0,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.21875,0.12364,0.14286,0.28571,0.28571,0.0,0.4286,7,0,7,7,0,2,0,0,22,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.1875,0.1357,0.0,0.2857,0.28571,0.0,0.28571,11,0,8,11,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.16741,0.14129,0.0,0.2857,0.28571,0.0,0.42857,12,0,12,12,1,2,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.13839,0.14054,0.0,0.07143,0.28571,0.0,0.28571,16,0,15,16,0,1,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.19634,0.11714,0.14214,0.2857,0.28571,0.0,0.28571,7,0,4,7,0,6,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.28571,"k":"flat","v":0.14955,"x":0.19196,"p":[[0,14,0.0,0.15625,0.19019,0.0,0.07143,0.28571,0.0,0.85714,16,0,10,16,0,2,0,0,12,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,14,0.2857,0.19196,0.18766,0.0,0.14286,0.28571,0.0,1.0,9,1,7,9,0,8,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,14,0.5714,0.18304,0.1197,0.10714,0.2857,0.28571,0.0,0.28571,8,0,6,8,0,7,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.14955,0.14101,0.0,0.14286,0.2857,0.0,0.4286,13,0,3,13,0,6,0,1,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.17402,0.11145,0.14214,0.14286,0.28571,0.0,0.28571,7,0,0,7,0,11,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"95c0ebf10e5a5170","q":"Let  $M$  be a positive integer.  $f(x):=x^3+ax^2+bx+c\\in\\mathbb Z[x]$  satisfy  $|a|,|b|,|c|\\le M.$   $x_1,x_2$  are different roots of  $f.$  Prove that   $$ |x_1-x_2|>\\frac 1{M^2+3M+1}. $$  *Created by Jingjun Han*","t":[{"b":3,"e":0.0,"k":"flat","v":0.06696,"x":0.08929,"p":[[0,5,0.0,0.06696,0.09438,0.0,0.0,0.14286,0.0,0.28571,20,0,1,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.07589,0.15146,0.0,0.0,0.14286,0.0,0.71429,22,0,5,22,0,7,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[5,5,1.0,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.4286,14,0,0,14,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.10705,"x":0.12053,"p":[[0,14,0.0,0.12053,0.18249,0.0,0.0,0.14286,0.0,0.71429,17,0,5,17,0,9,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[4,14,0.2857,0.10705,0.15566,0.0,0.0,0.14286,0.0,0.71429,17,0,1,17,0,10,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,14,0.5714,0.11608,0.16536,0.0,0.07143,0.14286,0.0,0.71429,16,0,0,16,0,11,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"d24fa565d56a8462","q":"Let  $R$  be a noncommutative finite ring with multiplicative identity element  $1$ . Show that if the subring generated by  $I \\cup \\{1\\}$  is  $R$  for each nonzero ideal  $I$  then  $R$  is simple.","t":[{"b":0,"e":0.0,"k":"flat","v":0.04902,"x":0.16509,"p":[[0,16,0.0,0.16509,0.18936,0.0,0.14286,0.2857,0.0,0.71429,14,0,1,14,0,7,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[4,16,0.25,0.13829,0.1655,0.0,0.07,0.2857,0.0,0.571,16,0,1,16,0,6,0,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[8,16,0.5,0.12501,0.15466,0.0,0.0,0.2857,0.0,0.4286,18,0,3,18,0,3,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.42857,22,0,4,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.04902,0.09173,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.42857,"k":"rising","v":0.11152,"x":0.48657,"p":[[0,26,0.0,0.16955,0.16919,0.0,0.14286,0.28571,0.0,0.4286,13,0,5,13,0,7,0,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.11152,0.14165,0.0,0.0,0.1786,0.0,0.57143,17,0,2,17,0,7,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,26,0.3077,0.21427,0.1923,0.0,0.14288,0.42857,0.0,0.57143,10,0,0,10,0,8,0,0,5,0,0,6,0,0,3,0,0,0,0,0,0,0,0],[12,26,0.4615,0.4375,0.21998,0.39286,0.42857,0.57143,0.0,0.71429,4,0,0,4,0,1,0,0,3,0,0,12,0,0,5,0,0,7,0,0,0,0,0],[16,26,0.6154,0.31249,0.18361,0.14286,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,7,0,0,9,0,0,9,0,0,2,0,0,2,0,0,0,0,0],[20,26,0.7692,0.44641,0.20123,0.2857,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,3,0,0,8,0,0,6,0,0,7,0,0,7,0,0,0,0,0],[24,26,0.9231,0.48657,0.19838,0.39286,0.4998,0.60714,0.0,0.85714,1,0,0,1,0,2,0,0,5,0,0,8,0,0,8,0,0,7,0,0,1,0,0],[26,26,1.0,0.45067,0.22921,0.2857,0.42859,0.60714,0.0,0.85714,2,0,0,2,0,4,0,0,5,0,0,6,0,0,7,0,0,7,0,0,1,0,0]]}]},{"i":"4040e2834f61b178","q":"Let $A B C$ be a triangle. Prove that there is a line $\\ell$ (in the plane of triangle $A B C$ ) such that the intersection of the interior of triangle $A B C$ and the interior of its reflection $A^{\\prime} B^{\\prime} C^{\\prime}$ in $\\ell$ has area more than $\\frac{2}{3}$ the area of triangle $A B C$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.09821,"p":[[0,14,0.0,0.05357,0.16269,0.0,0.0,0.0,0.0,0.85714,27,0,15,27,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,14,0.2857,0.09821,0.2683,0.0,0.0,0.0,0.0,1.0,27,2,14,27,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.02232,0.08828,0.0,0.0,0.0,0.0,0.4286,30,0,25,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.13839,"p":[[0,20,0.0,0.13839,0.32632,0.0,0.0,0.0,0.0,1.0,27,3,8,27,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,3],[4,20,0.2,0.11161,0.29824,0.0,0.0,0.0,0.0,1.0,28,2,16,28,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[8,20,0.4,0.03572,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,6,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,20,0.6,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e8f64d8546aa885a","q":"Let  $P$  and  $Q$  be points taken inside of triangle  $ABC$  such that  $\\angle APB=\\angle AQC$  and  $\\angle APC=\\angle AQB$ . Circumcircle of  $APQ$  intersects  $AB$  and  $AC$  second time at  $K$  and  $L$  respectively. Prove that  $B,C,L,K$  are concyclic.","t":[{"b":2,"e":0.14,"k":"flat","v":0.06687,"x":0.07589,"p":[[0,8,0.0,0.07589,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.06687,0.10085,0.0,0.0,0.14286,0.0,0.28571,21,0,2,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.0892,"p":[[0,23,0.0,0.0892,0.09938,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,12,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.06025,0.08584,0.0,0.0,0.14286,0.0,0.2857,20,0,1,20,1,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,2,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.143,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,2,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.03116,0.05889,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"0c11248e637e4d36","q":"Let  $ABC$  be a triangle with incentre  $I$ . A line  $r$  that passes through  $I$  intersects the circumcircles of triangles  $AIB$  and  $AIC$  at points  $P$  and  $Q$ , respectively. Prove that the circumcentre of triangle  $APQ$  is on the circumcircle of  $ABC$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,22,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"77629cb438c88dc6","q":"The sequence $a_{n}$ is defined as follows: $a_{1}=56, a_{n+1}=a_{n}-1 / a_{n}$. Show that $a_{n}<0$ for some $n$ such that $0<\\mathrm{n}<2002$.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.3482,"x":0.44187,"p":[[0,12,0.0,0.44187,0.31823,0.14286,0.28571,0.71429,0.0,1.0,1,4,0,1,0,10,0,0,7,0,0,2,0,0,1,0,0,5,0,0,2,0,4],[4,12,0.3333,0.40177,0.29973,0.14286,0.28571,0.71429,0.0,1.0,4,2,0,4,0,8,0,0,5,0,0,2,0,0,4,0,0,6,0,0,1,0,2],[8,12,0.6667,0.3482,0.24205,0.14286,0.28571,0.4286,0.0,1.0,2,1,0,2,0,9,0,0,9,0,0,5,0,0,2,0,0,3,0,0,1,0,1],[12,12,1.0,0.37944,0.17714,0.2857,0.42857,0.571,0.14286,0.85714,0,0,0,0,0,7,0,0,8,0,0,8,0,0,8,0,0,0,0,0,1,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.15179,"x":0.35268,"p":[[0,11,0.0,0.33482,0.28928,0.14286,0.21428,0.46429,0.0,1.0,4,1,0,4,0,12,0,0,5,0,0,3,0,0,1,0,0,3,0,0,3,0,1],[4,11,0.3636,0.15179,0.10677,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,25,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,11,0.7273,0.35268,0.1636,0.2857,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,4,0,0,13,0,0,9,0,0,4,0,0,0,0,0,1,0,0],[11,11,1.0,0.29465,0.16728,0.14286,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,7,0,0,6,0,0,13,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"c95dee32f9a8f167","q":"The set  $ \\{a_0, a_1, \\ldots, a_n\\}$  of real numbers satisfies the following conditions:\r\n\r**(i)**  $ a_0 \\equal{} a_n \\equal{} 0,$ \r**(ii)** for  $ 1 \\leq k \\leq n \\minus{} 1,$  \\[ a_k \\equal{} c \\plus{} \\sum^{n\\minus{}1}_{i\\equal{}k} a_{i\\minus{}k} \\cdot \\left(a_i \\plus{} a_{i\\plus{}1} \\right)\\]\r\n\r\nProve that  $ c \\leq \\frac{1}{4n}.$","t":[{"b":1,"e":0.57143,"k":"rising","v":0.23661,"x":0.50446,"p":[[0,11,0.0,0.23661,0.21312,0.0,0.21428,0.32144,0.0,0.57143,10,0,2,10,0,6,0,0,8,0,0,1,0,0,7,0,0,0,0,0,0,0,0],[4,11,0.3636,0.31697,0.23887,0.0,0.42857,0.57143,0.0,0.57143,10,0,0,10,0,1,0,0,4,0,0,6,0,0,11,0,0,0,0,0,0,0,0],[8,11,0.7273,0.42856,0.22015,0.28571,0.57143,0.57143,0.0,0.57143,5,0,0,5,0,2,0,0,2,0,0,2,0,0,21,0,0,0,0,0,0,0,0],[11,11,1.0,0.50446,0.13356,0.53571,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,4,0,0,3,0,0,24,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.22768,"x":0.50446,"p":[[0,106,0.0,0.41071,0.23077,0.28571,0.5,0.57143,0.0,1.0,5,1,0,5,0,0,0,0,8,0,0,3,0,0,15,0,0,0,0,0,0,0,1],[4,106,0.0377,0.31249,0.22709,0.10714,0.28571,0.57143,0.0,0.57143,8,0,0,8,0,2,0,0,10,0,0,0,0,0,12,0,0,0,0,0,0,0,0],[8,106,0.0755,0.32143,0.17496,0.2857,0.28571,0.42857,0.0,0.57143,4,0,0,4,0,2,0,0,15,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[12,106,0.1132,0.35268,0.17852,0.2857,0.28571,0.57143,0.0,0.57143,2,0,0,2,0,4,0,0,14,0,0,1,0,0,11,0,0,0,0,0,0,0,0],[16,106,0.1509,0.2857,0.17126,0.2857,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,2,0,0,18,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[20,106,0.1887,0.31696,0.21049,0.14286,0.28571,0.57143,0.0,0.57143,6,0,0,6,0,3,0,0,12,0,0,0,0,0,11,0,0,0,0,0,0,0,0],[24,106,0.2264,0.25446,0.15866,0.14286,0.28571,0.32143,0.0,0.57143,5,0,1,5,0,7,0,0,12,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[28,106,0.2642,0.27231,0.1935,0.10714,0.28571,0.32143,0.0,0.57143,8,0,0,8,0,1,0,0,15,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[32,106,0.3019,0.32588,0.22082,0.14286,0.28571,0.57143,0.0,0.57143,7,0,1,7,0,2,0,0,10,0,0,1,0,0,12,0,0,0,0,0,0,0,0],[36,106,0.3396,0.22768,0.19516,0.0,0.28571,0.28571,0.0,0.57143,10,0,0,10,0,4,0,0,12,0,0,1,0,0,5,0,0,0,0,0,0,0,0],[40,106,0.3774,0.28571,0.19885,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,3,0,0,12,0,0,3,0,0,7,0,0,0,0,0,0,0,0],[44,106,0.4151,0.33479,0.17714,0.28571,0.28571,0.571,0.0,0.57143,3,0,0,3,0,3,0,0,15,0,0,2,0,0,9,0,0,0,0,0,0,0,0],[48,106,0.4528,0.27679,0.20806,0.0,0.28571,0.42857,0.0,0.57143,9,0,1,9,0,1,0,0,12,0,0,3,0,0,7,0,0,0,0,0,0,0,0],[52,106,0.4906,0.32589,0.1996,0.14289,0.28571,0.57143,0.0,0.57143,5,0,0,5,0,4,0,0,9,0,0,5,0,0,9,0,0,0,0,0,0,0,0],[56,106,0.5283,0.25447,0.17762,0.14286,0.28571,0.32143,0.0,0.57143,6,0,0,6,0,7,0,0,11,0,0,4,0,0,4,0,0,0,0,0,0,0,0],[60,106,0.566,0.31241,0.20659,0.14286,0.28571,0.46429,0.0,0.57143,6,0,0,6,0,5,0,0,6,0,0,7,0,0,8,0,0,0,0,0,0,0,0],[64,106,0.6038,0.32143,0.18211,0.14289,0.28571,0.46431,0.0,0.57143,3,0,0,3,0,6,0,0,11,0,0,4,0,0,8,0,0,0,0,0,0,0,0],[68,106,0.6415,0.37947,0.20705,0.28571,0.42859,0.57143,0.0,0.57143,4,0,0,4,0,3,0,0,8,0,0,2,0,0,15,0,0,0,0,0,0,0,0],[72,106,0.6792,0.30357,0.21354,0.10714,0.28571,0.57143,0.0,0.57143,8,0,0,8,0,1,0,0,11,0,0,3,0,0,9,0,0,0,0,0,0,0,0],[76,106,0.717,0.29464,0.20497,0.14286,0.28571,0.57143,0.0,0.57143,6,0,0,6,0,5,0,0,11,0,0,1,0,0,9,0,0,0,0,0,0,0,0],[80,106,0.7547,0.26339,0.20858,0.0,0.28571,0.42857,0.0,0.57143,9,0,1,9,0,3,0,0,11,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[84,106,0.7925,0.27676,0.20493,0.10714,0.28571,0.42857,0.0,0.57143,8,0,1,8,0,3,0,0,11,0,0,3,0,0,7,0,0,0,0,0,0,0,0],[88,106,0.8302,0.37945,0.1698,0.28571,0.35714,0.57143,0.0,0.57143,2,0,0,2,0,2,0,0,12,0,0,5,0,0,11,0,0,0,0,0,0,0,0],[92,106,0.8679,0.48219,0.14615,0.42857,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,1,0,0,4,0,0,5,0,0,21,0,0,0,0,0,0,0,0],[96,106,0.9057,0.50446,0.13356,0.53572,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,4,0,0,3,0,0,24,0,0,0,0,0,0,0,0],[100,106,0.9434,0.45978,0.12231,0.28571,0.4998,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,7,0,0,16,0,0,0,0,0,0,0,0],[104,106,0.9811,0.47768,0.14987,0.42857,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,1,0,0,5,0,0,4,0,0,21,0,0,0,0,0,0,0,0],[106,106,1.0,0.45535,0.12077,0.28571,0.4286,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,9,0,0,8,0,0,15,0,0,0,0,0,0,0,0]]}]},{"i":"634aa0fbe5e6fc6e","q":"Twenty-\ufb01ve points are given on the plane.  Among any three of them, one can choose two less than one inch apart.  Prove that there are 13 points among them which lie in a circle of radius 1.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.16518,"x":0.33034,"p":[[0,30,0.0,0.27214,0.24587,0.14286,0.14286,0.28571,0.0,1.0,2,1,0,2,0,18,0,0,6,0,0,0,0,0,3,0,0,0,0,0,2,0,1],[4,30,0.1333,0.25447,0.27833,0.0,0.14286,0.42857,0.0,1.0,10,1,2,10,0,10,0,0,3,0,0,2,0,0,3,0,0,2,0,0,1,0,1],[8,30,0.2667,0.31695,0.29823,0.14286,0.28571,0.32143,0.0,1.0,7,2,2,7,0,6,0,0,11,0,0,1,0,0,2,0,0,0,0,0,3,0,2],[12,30,0.4,0.33034,0.28889,0.14286,0.14286,0.57143,0.0,1.0,3,2,1,3,0,15,0,0,3,0,0,1,0,0,6,0,0,0,0,0,2,0,2],[16,30,0.5333,0.28561,0.22307,0.14286,0.14288,0.42857,0.0,1.0,1,1,0,1,0,17,0,0,5,0,0,4,0,0,3,0,0,0,0,0,1,0,1],[20,30,0.6667,0.3257,0.29295,0.14286,0.21428,0.46418,0.0,1.0,4,3,2,4,0,12,0,0,6,0,0,2,0,0,4,0,0,0,0,0,1,0,3],[24,30,0.8,0.17411,0.20119,0.10714,0.14286,0.14286,0.0,1.0,8,1,5,8,0,18,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[28,30,0.9333,0.24554,0.18638,0.14286,0.14286,0.28571,0.0,0.85714,1,0,1,1,0,20,0,0,5,0,0,1,0,0,4,0,0,0,0,0,1,0,0],[30,30,1.0,0.16518,0.11355,0.14286,0.14286,0.14286,0.0,0.57143,3,0,3,3,0,25,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.19643,"x":0.41963,"p":[[0,49,0.0,0.29017,0.2823,0.14286,0.14288,0.42857,0.0,1.0,7,1,2,7,0,10,0,0,6,0,0,2,0,0,3,0,0,0,0,0,3,0,1],[4,49,0.0816,0.41963,0.3368,0.14286,0.2857,0.60714,0.0,1.0,2,6,1,2,0,12,0,0,4,0,0,2,0,0,4,0,0,2,0,0,0,0,6],[8,49,0.1633,0.19643,0.22517,0.10714,0.14286,0.2857,0.0,1.0,8,1,0,8,0,15,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[12,49,0.2449,0.24554,0.22934,0.14286,0.14286,0.28571,0.0,1.0,5,1,1,5,0,14,0,0,7,0,0,2,0,0,2,0,0,0,0,0,1,0,1],[16,49,0.3265,0.27679,0.20806,0.14286,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,10,0,0,11,0,0,1,0,0,4,0,0,1,0,0,1,0,0],[20,49,0.4082,0.20534,0.12335,0.14286,0.14286,0.28571,0.0,0.571,3,0,0,3,0,16,0,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,49,0.4898,0.20089,0.1063,0.14286,0.21428,0.28571,0.0,0.42857,4,0,0,4,0,12,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.21429,0.18211,0.14286,0.14288,0.28571,0.0,1.0,5,1,0,5,0,13,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[32,49,0.6531,0.25893,0.12078,0.14286,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,7,0,0,20,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[36,49,0.7347,0.22322,0.08702,0.14286,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,10,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.26338,0.10775,0.24999,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,22,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[44,49,0.898,0.25893,0.12078,0.14286,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,10,0,0,17,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[48,49,0.9796,0.25446,0.09268,0.25,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,23,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[49,49,1.0,0.22312,0.12345,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,12,0,0,14,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"a03ddee042014539","q":"Let  $\\mathcal{C}_1$  and  $\\mathcal{C}_2$  be two circumferences intersecting at points  $A$  and  $B$ . Let  $C$  be a point on line  $AB$  such that  $B$  lies between  $A$  and  $C$ . Let  $P$  and  $Q$  be points on  $\\mathcal{C}_1$  and  $\\mathcal{C}_2$  respectively such that  $CP$  and  $CQ$  are tangent to  $\\mathcal{C}_1$  and  $\\mathcal{C}_2$  respectively,  $P$  is not inside  $\\mathcal{C}_2$  and  $Q$  is not inside  $\\mathcal{C}_1$ . Line  $PQ$  cuts  $\\mathcal{C}_1$  at  $R$  and  $\\mathcal{C}_2$  at  $S$ , both points different from  $P$ ,  $Q$  and  $B$ . Suppose  $CR$  cuts  $\\mathcal{C}_1$  again at  $X$  and  $CS$  cuts  $\\mathcal{C}_2$  again at  $Y$ . Let  $Z$  be a point on line  $XY$ . Prove  $SZ$  is parallel to  $QX$  if and only if  $PZ$  is parallel to  $RX$ .","t":[{"b":2,"e":0.571,"k":"flat","v":0.29687,"x":0.35489,"p":[[0,48,0.0,0.32144,0.11845,0.2857,0.28571,0.32164,0.0,0.71429,1,0,0,1,0,1,0,0,22,0,0,6,0,0,1,0,0,1,0,0,0,0,0],[4,48,0.0833,0.30358,0.19725,0.25,0.28571,0.4286,0.0,0.85714,4,0,0,4,1,3,0,0,15,0,0,5,0,0,1,1,0,1,0,0,1,0,0],[8,48,0.1667,0.29687,0.13376,0.2857,0.28571,0.37489,0.0,0.57143,3,0,0,3,0,2,0,0,18,0,1,6,0,0,2,0,0,0,0,0,0,0,0],[12,48,0.25,0.34819,0.16045,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,3,0,1,14,0,1,8,0,0,1,0,0,3,0,0,0,0,0],[16,48,0.3333,0.32143,0.12877,0.2857,0.2857,0.42857,0.0,0.71429,2,0,1,2,0,1,0,0,18,0,0,10,0,0,0,0,0,1,0,0,0,0,0],[20,48,0.4167,0.3103,0.10809,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,18,0,1,8,0,0,1,0,0,0,0,0,0,0,0],[24,48,0.5,0.32586,0.1347,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,19,0,0,4,0,0,5,0,0,0,0,0,0,0,0],[28,48,0.5833,0.3192,0.11432,0.2857,0.28571,0.42857,0.0,0.57143,2,0,0,2,0,0,0,1,18,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[32,48,0.6667,0.30802,0.11352,0.2857,0.28571,0.42857,0.0,0.571,2,0,0,2,0,1,0,0,20,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[36,48,0.75,0.33036,0.16917,0.2857,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,0,0,0,15,0,0,10,0,0,1,0,0,2,0,0,0,0,0],[40,48,0.8333,0.31026,0.13902,0.2857,0.28571,0.30354,0.0,0.71429,1,0,0,1,0,4,0,0,19,0,1,5,0,0,0,0,0,2,0,0,0,0,0],[44,48,0.9167,0.35257,0.12882,0.2857,0.28571,0.42857,0.14,0.71429,0,0,0,0,0,1,0,0,21,0,0,6,0,0,2,0,0,2,0,0,0,0,0],[48,48,1.0,0.35489,0.10017,0.28571,0.28571,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,19,0,1,10,0,0,1,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.29023,"x":0.35936,"p":[[0,22,0.0,0.33052,0.15741,0.28571,0.28571,0.42857,0.0,0.857,2,0,1,2,0,2,0,0,17,0,0,8,0,0,2,0,0,0,0,0,1,0,0],[4,22,0.1818,0.3192,0.13714,0.28571,0.28571,0.42857,0.0,0.71429,3,0,0,3,0,0,0,0,17,0,1,10,0,0,0,0,0,1,0,0,0,0,0],[8,22,0.3636,0.29023,0.12624,0.2857,0.28571,0.42857,0.0,0.43,3,0,1,3,0,3,0,0,16,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.35491,0.1467,0.2857,0.28571,0.42858,0.0,0.85714,1,0,0,1,0,1,0,0,16,0,1,11,0,0,0,0,0,1,0,0,1,0,0],[16,22,0.7273,0.35513,0.14664,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,17,0,1,9,0,0,0,0,0,3,0,0,0,0,0],[20,22,0.9091,0.33927,0.13714,0.2857,0.28571,0.28571,0.2857,1.0,0,1,0,0,0,0,0,0,25,0,0,5,0,0,1,0,0,0,0,0,0,0,1],[22,22,1.0,0.35936,0.14001,0.2857,0.28571,0.42857,0.2857,0.85714,0,0,0,0,0,0,0,0,22,0,0,7,0,0,1,0,0,0,1,0,1,0,0]]}]},{"i":"6aeed64221237849","q":"Let $A B C$ be a triangle with circumcircle $\\omega$. The internal angle bisectors of $\\angle A B C$ and $\\angle A C B$ intersect $\\omega$ at $X \\neq B$ and $Y \\neq C$, respectively. Let $K$ be a point on $C X$ such that $\\angle K A C=90^{\\circ}$. Similarly, let $L$ be a point on $B Y$ such that $\\angle L A B=90^{\\circ}$. Let $S$ be the midpoint of $\\operatorname{arc} C A B$ of $\\omega$. Prove that $S K=S L$.","t":[{"b":6,"e":0.28571,"k":"flat","v":0.15625,"x":0.26785,"p":[[0,19,0.0,0.24777,0.2067,0.10714,0.28571,0.28571,0.0,0.92857,8,0,4,8,0,4,0,0,15,0,0,1,0,0,3,0,0,0,0,0,0,1,0],[4,19,0.2105,0.23661,0.25155,0.0,0.28571,0.28571,0.0,1.0,11,1,0,11,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0,2,0,1],[8,19,0.4211,0.15625,0.20628,0.0,0.14286,0.28571,0.0,1.0,15,1,1,15,0,6,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[12,19,0.6316,0.24553,0.08917,0.2857,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,3,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.23214,0.09942,0.24999,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,4,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.26785,0.07784,0.2857,0.28571,0.28571,0.0,0.42857,2,0,0,2,0,1,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.09821,"x":0.23659,"p":[[0,13,0.0,0.23659,0.16979,0.14286,0.28571,0.28571,0.0,0.57143,7,0,0,7,0,5,0,0,16,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[4,13,0.3077,0.16071,0.21053,0.0,0.07143,0.2857,0.0,0.85714,16,0,5,16,0,5,0,0,7,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[8,13,0.6154,0.09821,0.13092,0.0,0.0,0.17857,0.0,0.4286,19,0,6,19,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.1875,0.12078,0.10714,0.2857,0.28571,0.0,0.28571,8,0,0,8,0,6,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.22768,0.10012,0.14286,0.28571,0.28571,0.0,0.28571,4,0,0,4,0,5,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c3d946018768e3e1","q":"For a positive integer  $n$ , denote by  $g(n)$  the number of strictly ascending triples chosen from the set  $\\{1, 2, ..., n\\}$ . Find the least positive integer  $n$  such that the following holds:*The number  $g(n)$  can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference  $336$ .*","t":[{"b":3,"e":0.42857,"k":"rising","v":0.01339,"x":0.33247,"p":[[0,8,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,30,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.33028,0.173,0.14289,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,8,0,0,11,0,0,5,0,0,6,0,0,1,0,0,0,0,0],[8,8,1.0,0.33247,0.14876,0.2857,0.28571,0.42857,0.0,0.64,2,0,0,2,0,3,0,0,14,0,0,9,0,0,3,1,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05356,"p":[[0,55,0.0,0.05356,0.17031,0.0,0.0,0.0,0.0,0.71429,29,0,29,29,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,55,0.0727,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,31,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,31,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,30,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,55,0.4364,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,30,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[32,55,0.5818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.03572,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,27,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,55,0.7273,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,31,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[44,55,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"23ba4f67b515be74","q":"Let  $a$ ,  $b$ , and  $c$  be real numbers satisfying  $a+b+c=1$ . If the minimum value of   $$ 21\\sqrt{a^2+2}+4\\sqrt{7b^2+1}+7\\sqrt{9c^2+2} $$   can be expressed as  $m/n$ , where  $m$  and  $n$  are relatively prime positive integers, find  $m+n$ .","t":[{"b":2,"e":1.0,"k":"rising","v":0.71429,"x":1.0,"p":[[0,36,0.0,0.72321,0.43144,0.32143,1.0,1.0,0.0,1.0,8,22,7,8,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,22],[4,36,0.1111,0.71429,0.44032,0.10714,1.0,1.0,0.0,1.0,8,22,5,8,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,22],[8,36,0.2222,0.90625,0.29148,1.0,1.0,1.0,0.0,1.0,3,29,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[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],[16,36,0.4444,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29],[20,36,0.5556,0.90625,0.29148,1.0,1.0,1.0,0.0,1.0,3,29,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[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]]},{"b":5,"e":1.0,"k":"rising","v":0.83036,"x":1.0,"p":[[0,40,0.0,0.83036,0.3597,1.0,1.0,1.0,0.0,1.0,4,26,3,4,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,26],[4,40,0.1,0.9375,0.19865,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,29],[8,40,0.2,0.84821,0.32525,1.0,1.0,1.0,0.0,1.0,2,26,2,2,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,26],[12,40,0.3,0.87946,0.31966,1.0,1.0,1.0,0.0,1.0,3,28,3,3,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[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],[20,40,0.5,0.84375,0.36309,1.0,1.0,1.0,0.0,1.0,5,27,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[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],[28,40,0.7,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[32,40,0.8,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"e26121a4d9383dbc","q":"Let  $\\vartriangle ABC$  have  $AB = 14$ ,  $BC = 30$ ,  $AC = 40$  and  $\\vartriangle AB'C'$  with  $AB' = 7\\sqrt6$ ,  $B'C' = 15\\sqrt6$ ,  $AC' = 20\\sqrt6$  such that  $\\angle BAB' = \\frac{5\\pi}{12}$  . The lines  $BB'$   and  $CC'$   intersect at point  $D$ . Let  $O$  be the circumcenter of  $\\vartriangle BCD$ , and let  $O' $  be the circumcenter of  $\\vartriangle B'C'D$ . Then the length of segment  $OO'$  can be expressed as  $\\frac{a+b \\sqrt{c}}{ d}$  , where  $a$ ,  $b$ ,  $c$ , and  $d$  are positive integers such that  $a$  and  $d$  are relatively prime, and  $c$  is not divisible by the square of any prime. Find  $a+b+c+d$","t":[{"b":0,"e":0.0,"k":"flat","v":0.59821,"x":0.65625,"p":[[0,6,0.0,0.59821,0.45937,0.0,1.0,1.0,0.0,1.0,10,17,9,10,0,1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,17],[4,6,0.6667,0.65625,0.42237,0.14286,1.0,1.0,0.0,1.0,7,17,7,7,0,2,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,17]]},{"b":3,"e":0.71429,"k":"falling","v":0.17411,"x":0.8125,"p":[[0,40,0.0,0.8125,0.35792,0.92857,1.0,1.0,0.0,1.0,4,24,4,4,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,24],[4,40,0.1,0.59821,0.42173,0.10714,0.71429,1.0,0.0,1.0,8,13,8,8,0,2,0,0,1,0,0,1,0,0,1,0,0,4,0,0,2,0,13],[8,40,0.2,0.58036,0.41945,0.14286,0.71429,1.0,0.0,1.0,7,13,6,7,0,4,0,0,1,0,0,0,0,0,3,0,0,3,0,0,1,0,13],[12,40,0.3,0.38839,0.44354,0.0,0.07143,1.0,0.0,1.0,16,9,13,16,0,1,0,0,2,0,0,1,0,0,0,0,0,2,0,0,1,0,9],[16,40,0.4,0.17411,0.2618,0.0,0.0,0.28571,0.0,0.71429,19,0,17,19,0,4,0,0,2,0,0,2,0,0,0,0,0,5,0,0,0,0,0],[20,40,0.5,0.25446,0.30875,0.0,0.0,0.60714,0.0,0.71429,17,0,14,17,0,2,0,0,2,0,0,1,0,0,2,0,0,8,0,0,0,0,0],[24,40,0.6,0.22768,0.2854,0.0,0.14286,0.28571,0.0,1.0,12,1,9,12,0,11,0,0,2,0,0,0,0,0,1,0,0,5,0,0,0,0,1],[28,40,0.7,0.51784,0.24936,0.39286,0.57143,0.71429,0.0,0.71429,4,0,2,4,0,1,0,0,3,0,0,2,0,0,7,0,0,15,0,0,0,0,0],[32,40,0.8,0.4375,0.22851,0.28571,0.42857,0.60714,0.0,0.71429,2,0,1,2,0,5,0,0,5,0,0,5,0,0,7,0,0,8,0,0,0,0,0],[36,40,0.9,0.45536,0.20025,0.28571,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,6,0,0,3,0,0,13,0,0,5,0,0,0,0,0],[40,40,1.0,0.41517,0.20315,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,9,0,0,8,0,0,3,0,0,7,0,0,0,0,0]]}]},{"i":"67ce87e515b48c2b","q":"Let $\\mathcal{C}$ be a circle, $P$ a point outside it, $A$ and $B$ the points of tangency of the two tangents to $\\mathcal{C}$ passing through $P$. Let $K$ be any point on $(AB)$, distinct from $A$ and $B$. We call $T$ the second intersection of $\\mathcal{C}$ and the circumcircle of triangle $PBK$. Furthermore, we call $P'$ the symmetric point of $P$ with respect to $A$.\nShow that $\\widehat{\\mathrm{PBT}}=\\widehat{\\mathrm{P}'\\mathrm{KA}}$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.04018,"x":0.09366,"p":[[0,34,0.0,0.07143,0.08748,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.14286,19,0,1,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,2,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,1,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.09366,0.08454,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.28571,13,0,1,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.04018,0.06424,0.0,0.0,0.14286,0.0,0.143,23,0,8,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.05357,"x":0.10705,"p":[[0,17,0.0,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.42857,22,0,1,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.05795,0.07006,0.0,0.0,0.14286,0.0,0.1429,19,0,1,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fe351236a8848e7d","q":"Let $ABC$ be a triangle and $M$ a point on $[B, C]$. Let $\\omega$ be a circle tangent to $(AB)$ at $T$ and to $(BC)$ at $K$, and tangent to the circumcircle of $AMC$ at $P$. Show that if $(KT) \\parallel (AM)$, then the circumcircles of $KPC$ and $APT$ are tangent at $P$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.00884,"x":0.0625,"p":[[0,18,0.0,0.05348,0.0927,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.04909,0.11627,0.0,0.0,0.0,0.0,0.571,25,0,1,25,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0625,0.11812,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.02679,"x":0.05795,"p":[[0,13,0.0,0.03562,0.07974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,2,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.05795,0.09346,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a7bf4a989b188362","q":"Given a positive integer  $n\\ge 3$ , Ar\u00e1ndano and Banana play a game. Initially, numbers  $1,2,3,\\dots,n$  are written on the blackboard. Alternatingly and starting with Ar\u00e1ndano, the players erase numbers from the board one at a time, until exactly three numbers remain on the board. Banana wins the game if the last three numbers on the board are the sides of a nondegenerate triangle, and Ar\u00e1ndano wins otherwise.\nDetermine, in terms of  $n$ , who has a winning strategy.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,41,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,20,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,21,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"351525f8296496d7","q":"Let  $ABCD$  be a convex quadrilateral whose vertices do not lie on a circle. Let  $A'B'C'D'$  be a quadrangle such that  $A',B', C',D'$  are the centers of the circumcircles of triangles  $BCD,ACD,ABD$ , and  $ABC$ . We write  $T (ABCD) = A'B'C'D'$ . Let us define  $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$ **(a)** Prove that  $ABCD$  and  $A''B''C''D''$  are similar.**(b)**The ratio of similitude depends on the size of the angles of  $ABCD$ . Determine this ratio.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.08928,"x":0.20536,"p":[[0,27,0.0,0.10268,0.12492,0.0,0.0,0.2857,0.0,0.28571,18,0,9,18,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.15625,0.17261,0.0,0.07143,0.28571,0.0,0.42857,16,0,15,16,0,3,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.08928,0.13716,0.0,0.0,0.2857,0.0,0.42857,22,0,17,22,0,1,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.12054,0.14334,0.0,0.0,0.2857,0.0,0.4286,17,0,15,17,0,5,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.20536,0.16728,0.0,0.14288,0.28571,0.0,0.57143,9,0,9,9,0,8,0,0,8,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[20,27,0.7407,0.15625,0.14445,0.0,0.14286,0.2857,0.0,0.4286,13,0,13,13,0,5,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.08482,"x":0.25,"p":[[0,26,0.0,0.09375,0.13175,0.0,0.0,0.1786,0.0,0.42857,20,0,16,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.08482,0.13296,0.0,0.0,0.14286,0.0,0.4286,21,0,7,21,0,5,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.13839,0.15355,0.0,0.07143,0.28571,0.0,0.42857,16,0,13,16,0,4,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.16964,0.15335,0.0,0.14286,0.28571,0.0,0.42857,12,0,9,12,0,6,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.16964,0.14032,0.0,0.14286,0.28571,0.0,0.42857,10,0,9,10,0,9,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.25,0.13363,0.14286,0.28571,0.28571,0.0,0.42857,5,0,4,5,0,4,0,0,17,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.23661,0.12681,0.14286,0.2857,0.28571,0.0,0.42857,4,0,2,4,0,8,0,0,15,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ab213e2d0d9c577b","q":"Let  $X =\\{ X_1 , X_2 , ...\\}$  be a countable set of points in space. Show that there is a positive sequence  $\\{a_k\\}$  such that for any point  $Z\\not\\in X$  the distance between the point Z and the set  $\\{X_1,X_2 , ...,X_k\\}$  is at least  $a_k$  for infinitely many k.","t":[{"b":2,"e":0.0,"k":"flat","v":0.03125,"x":0.1131,"p":[[0,9,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,25,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.05357,0.09943,0.0,0.0,0.14286,0.0,0.42857,23,0,21,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,24,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.1131,0.07314,0.10714,0.14286,0.14286,0.0,0.33333,8,0,0,8,0,23,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.04464,"x":0.04464,"p":[[0,6,0.0,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,24,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"330e259289212b8c","q":"Let  $n$  and  $k$  be positive integers and let   $$ T = \\{ (x,y,z) \\in \\mathbb{N}^3 \\mid 1 \\leq x,y,z \\leq n \\} $$  be the length  $n$  lattice cube. Suppose that  $3n^2 - 3n + 1 + k$  points of  $T$  are colored red such that if  $P$  and  $Q$  are red points and  $PQ$  is parallel to one of the coordinate axes, then the whole line segment  $PQ$  consists of only red points. \n\nProve that there exists at least  $k$  unit cubes of length  $1$ , all of whose vertices are colored red.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.04465,"p":[[0,16,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.02233,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.04465,0.11538,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.03562,"p":[[0,15,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.03562,0.15144,0.0,0.0,0.0,0.0,0.857,29,0,0,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4b450854a5a5bb57","q":"Let  $n$  be an integer and  $n \\geq 2$ ,  $x_1, x_2, \\cdots , x_n$  are arbitrary real number, find the maximum value of   $$ 2\\sum_{1\\leq i<j \\leq n}\\left \\lfloor x_ix_j \\right \\rfloor-\\left (  n-1 \\right )\\sum_{i=1}^{n}\\left \\lfloor x_i^2 \\right \\rfloor   $$","t":[{"b":4,"e":0.42857,"k":"flat","v":0.21427,"x":0.71429,"p":[[0,30,0.0,0.57589,0.33213,0.28571,0.42857,1.0,0.0,1.0,1,10,1,1,0,1,0,0,12,0,0,3,0,0,1,0,0,3,0,0,1,0,10],[4,30,0.1333,0.29464,0.25738,0.14286,0.28571,0.42857,0.0,1.0,7,1,0,7,0,6,0,0,10,0,0,3,0,0,2,0,0,2,0,0,1,0,1],[8,30,0.2667,0.21427,0.17125,0.0,0.28571,0.28571,0.0,0.71429,9,0,0,9,0,4,0,0,16,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,30,0.4,0.49107,0.26471,0.28571,0.57143,0.71429,0.0,1.0,2,2,0,2,0,3,0,0,7,0,0,3,0,0,6,0,0,8,0,0,1,0,2],[16,30,0.5333,0.58036,0.28557,0.42857,0.57143,0.71429,0.14286,1.0,0,7,0,0,0,5,0,0,1,0,0,9,0,0,3,0,0,7,0,0,0,0,7],[20,30,0.6667,0.65625,0.18509,0.42857,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,8,0,0,3,0,0,14,0,0,3,0,3],[24,30,0.8,0.66518,0.17353,0.57142,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,7,0,0,6,0,0,14,0,0,1,0,4],[28,30,0.9333,0.68302,0.15041,0.57143,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,16,0,0,0,0,4],[30,30,1.0,0.71429,0.13832,0.71429,0.71429,0.71429,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,20,0,0,1,0,4]]},{"b":7,"e":0.71429,"k":"flat","v":0.34375,"x":0.79018,"p":[[0,46,0.0,0.59375,0.33714,0.28571,0.71429,0.89286,0.0,1.0,3,8,1,3,0,2,0,0,5,0,0,3,0,0,2,0,0,6,0,0,3,0,8],[4,46,0.087,0.43303,0.3416,0.24999,0.35714,0.71429,0.0,1.0,7,5,1,7,0,1,0,0,8,0,0,5,0,0,2,0,0,2,0,0,2,0,5],[8,46,0.1739,0.34375,0.24707,0.14286,0.28571,0.46429,0.0,1.0,3,1,0,3,0,7,0,0,12,0,0,2,0,0,3,0,0,3,0,0,1,0,1],[12,46,0.2609,0.65622,0.23108,0.42857,0.64286,0.85714,0.14286,1.0,0,6,0,0,0,1,0,0,0,0,0,10,0,0,5,0,0,6,0,0,4,0,6],[16,46,0.3478,0.66964,0.30606,0.42857,0.71429,1.0,0.0,1.0,1,10,1,1,0,3,0,0,2,0,0,3,0,0,4,0,0,6,0,0,3,0,10],[20,46,0.4348,0.63839,0.25996,0.57143,0.71429,0.85704,0.0,1.0,1,4,1,1,0,2,0,0,3,0,0,1,0,0,6,0,0,10,0,0,5,0,4],[24,46,0.5217,0.65179,0.24206,0.42857,0.71429,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,1,0,0,8,0,0,5,0,0,8,0,0,3,0,6],[28,46,0.6087,0.79018,0.18552,0.71429,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,8,0,0,9,0,9],[32,46,0.6957,0.66964,0.21558,0.42857,0.71429,0.75,0.28571,1.0,0,6,0,0,0,0,0,0,2,0,0,7,0,0,4,0,0,11,0,0,2,0,6],[36,46,0.7826,0.62052,0.25407,0.42857,0.64286,0.85714,0.14286,1.0,0,4,0,0,0,1,0,0,5,0,0,7,0,0,3,0,0,5,0,0,7,0,4],[40,46,0.8696,0.52677,0.18012,0.42857,0.49979,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,6,0,0,10,0,0,7,0,0,7,0,0,1,0,1],[44,46,0.9565,0.61603,0.23808,0.42857,0.57143,0.85714,0.28571,1.0,0,4,0,0,0,0,0,0,6,0,0,6,0,0,5,0,0,6,0,0,5,0,4],[46,46,1.0,0.61158,0.26782,0.28571,0.57121,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,9,0,0,5,0,0,3,0,0,2,0,0,9,0,4]]}]},{"i":"a208c73bdd6fa6a9","q":"The hexagon  $ABLCDK$  is inscribed and the line  $LK$  intersects the segments  $AD, BC, AC$  and  $BD$  in points  $M, N, P$  and  $Q$ , respectively. Prove that  $NL \\cdot KP \\cdot MQ = KM \\cdot  PN \\cdot  LQ$ .","t":[{"b":4,"e":0.0,"k":"falling","v":0.03571,"x":0.45534,"p":[[0,16,0.0,0.29463,0.38121,0.0,0.0,0.57111,0.0,1.0,17,5,0,17,0,1,0,0,4,0,0,1,0,0,2,0,0,1,0,0,1,0,5],[4,16,0.25,0.41948,0.35348,0.105,0.35715,0.71429,0.0,1.0,8,5,0,8,0,4,0,0,4,0,0,2,0,0,4,0,0,5,0,0,0,0,5],[8,16,0.5,0.45534,0.37701,0.0,0.42859,0.71429,0.0,1.0,10,6,0,10,0,0,0,0,5,0,0,2,0,0,1,0,0,8,0,0,0,0,6],[12,16,0.75,0.38384,0.33782,0.0,0.28571,0.71429,0.0,1.0,11,1,1,11,0,1,0,0,5,0,0,1,0,0,3,0,0,7,0,0,3,0,1],[16,16,1.0,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.41071,"p":[[0,9,0.0,0.41071,0.39407,0.0,0.28571,0.75,0.0,1.0,12,6,0,12,0,1,0,0,4,0,0,2,0,0,2,0,0,3,0,0,2,0,6],[4,9,0.4444,0.21875,0.335,0.0,0.0,0.28571,0.0,1.0,20,2,0,20,0,1,0,0,4,0,0,0,0,0,0,0,0,4,0,0,1,0,2],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4725452a2add60d5","q":"Let  $n$  be a positive integer . If  $\\sigma$  is a permutation of the first  $n$  positive integers , let  $S(\\sigma)$  be the set of all distinct sums of the form  $\\sum_{i=k}^{l} \\sigma(i)$  where  $1 \\leq k \\leq l \\leq n$  .**(a)** Exhibit a permutation  $\\sigma$  of the first  $n$  positive integers such that  $|S(\\sigma)|\\geq \\left \\lfloor{\\frac{(n+1)^2}{4}}\\right \\rfloor  $ .**(b)** Show that  $|S(\\sigma)|>\\frac{n\\sqrt{n}}{4\\sqrt{2}}$  for all permutations  $\\sigma$  of the first  $n$  positive integers .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.21428,"x":0.30804,"p":[[0,17,0.0,0.27678,0.17105,0.2857,0.28571,0.28571,0.0,0.85714,5,0,0,5,0,2,0,0,19,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[4,17,0.2353,0.27677,0.10673,0.28571,0.28571,0.28571,0.0,0.571,3,0,0,3,0,0,0,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.30804,0.16016,0.28571,0.28571,0.42857,0.0,0.85714,3,0,0,3,0,2,0,0,18,0,0,7,0,0,1,0,0,0,0,0,1,0,0],[12,17,0.7059,0.26339,0.08827,0.28571,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,0,0,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.21428,0.13832,0.0,0.28571,0.28571,0.0,0.42857,9,0,1,9,0,0,0,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.25893,0.10971,0.2857,0.2857,0.28571,0.0,0.4286,4,0,1,4,0,1,0,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.2857,"k":"flat","v":0.24553,"x":0.26786,"p":[[0,11,0.0,0.26339,0.13882,0.28571,0.28571,0.28571,0.0,0.71429,5,0,0,5,0,0,0,0,24,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,11,0.3636,0.25893,0.10972,0.2857,0.28571,0.28571,0.0,0.4286,4,0,1,4,0,1,0,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.24553,0.11971,0.2857,0.28571,0.28571,0.0,0.42857,5,0,0,5,0,2,0,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.26786,0.12753,0.2857,0.28571,0.28571,0.0,0.4286,5,0,0,5,0,0,0,0,21,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ec1fb1b8947ded17","q":"Let $A B C$ be a scalene triangle. Let $K_{a}, L_{a}$, and $M_{a}$ be the respective intersections with $B C$ of the internal angle bisector, external angle bisector, and the median from $A$. The circumcircle of $A K_{a} L_{a}$ intersects $A M_{a}$ a second time at a point $X_{a}$ different from $A$. Define $X_{b}$ and $X_{c}$ analogously. Prove that the circumcenter of $X_{a} X_{b} X_{c}$ lies on the Euler line of $A B C$.","t":[{"b":0,"e":0.28571,"k":"flat","v":0.21428,"x":0.28571,"p":[[0,29,0.0,0.21428,0.11845,0.14289,0.28571,0.28571,0.0,0.28571,7,0,2,7,0,2,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.25893,0.08328,0.28571,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.25446,0.06901,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,5,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.2634,0.0724,0.2857,0.28571,0.28571,0.0,0.286,2,0,1,2,0,1,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.27232,0.05486,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.27232,0.05486,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.28571,1e-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],[28,29,0.9655,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.26339,0.05187,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.21428,"x":0.26785,"p":[[0,17,0.0,0.22312,0.10069,0.14286,0.2857,0.28571,0.0,0.28571,4,0,2,4,0,6,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.24107,0.12595,0.24999,0.2857,0.28571,0.0,0.57143,5,0,0,5,0,3,0,0,22,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,17,0.4706,0.22768,0.11214,0.2857,0.28571,0.28571,0.0,0.28571,6,0,2,6,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.21428,0.09449,0.14286,0.2857,0.28571,0.0,0.28571,3,0,0,3,0,10,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.25893,0.06622,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,4,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b336fc125f119c95","q":"Let  $\\Delta ABC$  be a scalene triangle. Points  $D,E$  lie on side  $\\overline{AC}$  in the order,  $A,E,D,C$ . Let the parallel through  $E$  to  $BC$  intersect  $\\odot (ABD)$  at  $F$ , such that,  $E$  and  $F$  lie on the same side of  $AB$ . Let the parallel through  $E$  to  $AB$  intersect  $\\odot (BDC)$  at  $G$ , such that,  $E$  and  $G$  lie on the same side of  $BC$ . Prove, Points  $D,F,E,G$  are concyclic","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04447,"p":[[0,29,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.04447,0.06596,0.0,0.0,0.14071,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,29,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.00884,0.03424,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.14286,"k":"flat","v":0.02223,"x":0.13831,"p":[[0,35,0.0,0.05349,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.1292,0.04156,0.14286,0.14286,0.14286,0.0,0.143,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.11608,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.13831,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.1292,0.04156,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0dfe69c0eccb0f9d","q":"Let  $O$  be the circumcenter of triangle  $ABC$ . A line through  $O$  intersects the sides  $CA$  and  $CB$  at points  $D$  and  $E$  respectively, and meets the circumcircle of  $ABO$  again at point  $P \\neq O$  inside the triangle. A point  $Q$  on side  $AB$  is such that  $\\frac{AQ}{QB}=\\frac{DP}{PE}$ . Prove that  $\\angle APQ = 2\\angle CAP$ . \n\n*Proposed by Du\u0014san Djuki\u0013c*","t":[{"b":4,"e":0.14286,"k":"flat","v":0.01339,"x":0.06241,"p":[[0,39,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.0267,0.06606,0.0,0.0,0.0,0.0,0.2857,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.04241,0.06404,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,1,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.04018,0.06424,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.03125,"p":[[0,19,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,2,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"644bea839948cd7b","q":"Let  $n \\ge 3$  be an integer. Let  $\\mathcal{P}$  denote the set of vertices of a regular  $n$ -gon on the plane. A polynomial  $f(x, y)$  of two variables with real coefficients is called  $\\textit{regular}$  if   $$ \\mathcal{P} = \\{(u, v) \\in \\mathbb{R}^2 \\, | \\, f(u, v) = 0 \\}. $$   Find the smallest possible value of the degree of a regular polynomial.\n\n*Proposed by Navid Safaei*","t":[{"b":3,"e":0.0,"k":"flat","v":0.16508,"x":0.24106,"p":[[0,13,0.0,0.24106,0.25362,0.0,0.14286,0.42857,0.0,0.85714,12,0,10,12,0,6,0,0,4,0,0,4,0,0,3,0,0,2,0,0,1,0,0],[4,13,0.3077,0.16508,0.2146,0.0,0.14286,0.14286,0.0,0.71429,13,0,7,13,0,13,0,0,1,0,0,0,0,0,3,0,0,2,0,0,0,0,0],[8,13,0.6154,0.23661,0.16982,0.14286,0.21428,0.28571,0.0,0.71429,5,0,5,5,0,11,0,0,9,0,0,5,0,0,1,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.43,"k":"flat","v":0.16071,"x":0.36607,"p":[[0,25,0.0,0.21878,0.24741,0.0,0.14286,0.42858,0.0,0.85714,13,0,9,13,0,7,0,0,3,0,0,3,0,0,4,0,0,1,0,0,1,0,0],[4,25,0.16,0.16071,0.20124,0.0,0.07143,0.2857,0.0,0.57143,16,0,14,16,0,6,0,0,4,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[8,25,0.32,0.16954,0.18706,0.0,0.14286,0.2857,0.0,0.57143,12,0,11,12,0,11,0,0,4,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[12,25,0.48,0.36607,0.14698,0.28571,0.42857,0.42857,0.0,0.71429,1,0,1,1,0,4,0,0,7,0,0,18,0,0,0,0,0,2,0,0,0,0,0],[16,25,0.64,0.24103,0.19054,0.105,0.2857,0.42857,0.0,0.71429,8,0,7,8,0,7,0,0,7,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[20,25,0.8,0.35715,0.19233,0.28571,0.42857,0.42857,0.0,0.85714,3,0,3,3,0,4,0,0,6,0,0,16,0,0,0,0,0,2,0,0,1,0,0],[24,25,0.96,0.36597,0.18546,0.28571,0.42857,0.42858,0.0,0.85714,2,0,2,2,0,4,0,0,8,0,0,14,0,0,1,0,0,2,0,0,1,0,0],[25,25,1.0,0.33036,0.1448,0.2857,0.28571,0.42857,0.0,0.71429,2,0,2,2,0,3,0,0,13,0,0,12,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"23642dd8b4003b4e","q":"Let  $ABCD$  be a square with side length  $2015$ . A disk with unit radius is packed neatly inside corner  $A$  (i.e. tangent to both  $\\overline{AB}$  and  $\\overline{AD}$ ). Alice kicks the disk, which bounces off  $\\overline{CD}$ ,  $\\overline{BC}$ ,  $\\overline{AB}$ ,  $\\overline{DA}$ ,  $\\overline{DC}$  in that order, before landing neatly into corner  $B$ . What is the total distance the center of the disk travelled?\n\n*Proposed by Evan Chen*","t":[{"b":4,"e":1.0,"k":"volatile","v":0.53125,"x":1.0,"p":[[0,3,0.0,0.53125,0.49902,0.0,1.0,1.0,0.0,1.0,15,17,2,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[3,3,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":0.0,"k":"volatile","v":0.0,"x":0.57589,"p":[[0,7,0.0,0.57589,0.48377,0.0,1.0,1.0,0.0,1.0,12,18,2,12,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[4,7,0.5714,0.5625,0.49608,0.0,1.0,1.0,0.0,1.0,14,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6001cf9c267defc3","q":"The infinite sequence \\( a_1, a_2, \\ldots \\) is defined by \\( a_1 = 1 \\) and, for each \\( n \\geq 1 \\), the number \\( a_{n+1} \\) is the smallest positive integer greater than \\( a_n \\) that has the following property: for each \\( k \\in \\{1, 2, \\ldots, n\\} \\), the number \\( a_{n+1} + a_k \\) is not a perfect square. Prove that, for all \\( n \\), it holds that \\( a_n \\leq (n - 1)^2 + 1 \\).","t":[{"b":5,"e":0.4286,"k":"flat","v":0.2365,"x":0.44629,"p":[[0,32,0.0,0.37704,0.32167,0.14286,0.28571,0.57143,0.0,1.0,4,4,0,4,0,10,0,1,4,0,0,3,0,0,3,0,0,2,0,0,1,0,4],[4,32,0.125,0.24107,0.22429,0.10714,0.21428,0.28571,0.0,1.0,8,1,0,8,0,8,0,0,9,0,0,4,0,0,1,0,0,1,0,0,0,0,1],[8,32,0.25,0.2365,0.18425,0.14286,0.2857,0.28571,0.0,0.85714,6,0,0,6,0,9,0,0,11,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[12,32,0.375,0.44629,0.34779,0.14286,0.28571,0.85704,0.0,1.0,2,6,0,2,0,10,0,0,5,0,0,5,0,0,0,0,0,1,0,0,3,0,6],[16,32,0.5,0.30801,0.18245,0.14286,0.28571,0.42857,0.0,0.71429,4,0,0,4,0,6,0,0,8,0,0,10,0,0,3,0,0,1,0,0,0,0,0],[20,32,0.625,0.26776,0.1547,0.14286,0.2857,0.42857,0.0,0.571,3,0,0,3,0,11,0,0,6,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[24,32,0.75,0.27679,0.10677,0.14289,0.28571,0.32143,0.14286,0.4286,0,0,0,0,0,10,0,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.26098,0.14144,0.14286,0.28571,0.42857,0.0,0.4286,3,0,0,3,0,10,0,0,8,0,1,10,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.29005,0.15766,0.14286,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,12,0,0,8,0,0,7,0,0,4,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.25668,"x":0.44418,"p":[[0,38,0.0,0.28571,0.25754,0.14286,0.2857,0.32143,0.0,1.0,7,2,0,7,0,5,0,0,12,0,0,5,0,0,0,0,0,0,0,0,1,0,2],[4,38,0.1053,0.27455,0.23965,0.14286,0.2857,0.28571,0.0,1.0,5,1,0,5,0,9,0,1,10,0,0,4,0,0,0,0,0,0,0,0,2,0,1],[8,38,0.2105,0.32134,0.28576,0.14286,0.28571,0.42857,0.0,1.0,4,3,0,4,0,10,0,0,9,0,0,4,0,0,0,0,0,1,0,0,1,0,3],[12,38,0.3158,0.33919,0.27146,0.14286,0.28571,0.46431,0.0,1.0,3,2,0,3,0,10,0,0,9,0,0,2,0,0,3,0,0,2,0,0,1,0,2],[16,38,0.4211,0.25668,0.23812,0.0,0.2857,0.32143,0.0,0.857,10,0,0,10,0,4,0,0,10,0,0,2,0,1,2,0,0,2,0,0,1,0,0],[20,38,0.5263,0.25895,0.14913,0.14286,0.2143,0.32143,0.0,0.57143,1,0,0,1,0,15,0,0,8,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[24,38,0.6316,0.26337,0.24248,0.0,0.2143,0.42857,0.0,0.85714,9,0,0,9,0,7,0,0,6,0,0,4,0,0,3,0,0,2,0,0,1,0,0],[28,38,0.7368,0.44418,0.27416,0.2857,0.42857,0.71429,0.0,1.0,2,1,0,2,1,4,0,0,8,0,0,4,0,0,3,0,0,6,0,0,3,0,1],[32,38,0.8421,0.38838,0.25059,0.1429,0.28571,0.57143,0.0,1.0,2,1,0,2,0,7,0,0,8,0,0,5,0,0,5,0,0,2,0,0,2,0,1],[36,38,0.9474,0.33704,0.21294,0.14289,0.28571,0.4286,0.0,0.85714,2,0,0,2,1,7,0,0,9,0,0,6,0,0,5,0,0,0,0,0,2,0,0],[38,38,1.0,0.34149,0.2103,0.14286,0.28571,0.4286,0.0,1.0,1,1,0,1,1,8,0,0,9,0,0,6,0,0,5,0,0,1,0,0,0,0,1]]}]},{"i":"d8aa8e7d90ae86a5","q":"Let $A B C$ be a triangle with incenter $I$, incircle $\\gamma$ and circumcircle $\\Gamma$. Let $M, N, P$ be the midpoints of $\\overline{B C}, \\overline{C A}, \\overline{A B}$ and let $E, F$ be the tangency points of $\\gamma$ with $\\overline{C A}$ and $\\overline{A B}$, respectively. Let $U, V$ be the intersections of line $E F$ with line $M N$ and line $M P$, respectively, and let $X$ be the midpoint of $\\operatorname{arc} B A C$ of $\\Gamma$. (a) Prove that $I$ lies on ray $C V$. (b) Prove that line $X I$ bisects $\\overline{U V}$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.12946,"x":0.27678,"p":[[0,35,0.0,0.25669,0.10693,0.2857,0.28571,0.28571,0.0,0.57143,3,0,3,3,1,1,0,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,35,0.1143,0.27678,0.04971,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.27678,0.04971,0.2857,0.28571,0.28571,0.0,0.28571,1,0,1,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.26111,0.06838,0.2857,0.28571,0.28571,0.0,0.28571,1,0,1,1,1,2,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.25892,0.06621,0.2857,0.28571,0.28571,0.0,0.28571,1,0,1,1,0,4,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.21875,0.12364,0.14286,0.28571,0.28571,0.0,0.42857,6,0,4,6,0,5,0,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.12946,0.13533,0.0,0.07143,0.28571,0.0,0.28571,16,0,15,16,0,3,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.19643,"x":0.24098,"p":[[0,5,0.0,0.19643,0.12752,0.0,0.2857,0.28571,0.0,0.28571,9,0,7,9,0,2,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.24098,0.1098,0.2857,0.28571,0.28571,0.0,0.4286,5,0,5,5,0,1,0,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"91422038f8879606","q":"Vishal starts with  $n$  copies of the number  $1$  written on the board. Every minute, he takes two numbers  $a, b$  and replaces them with either  $a+b$  or  $\\min(a^2, b^2)$ . After  $n-1$  there is  $1$  number on the board. Let the maximal possible value of this number be  $f(n)$ . Prove  $2^{n/3}<f(n)\\leq 3^{n/3}$ .","t":[{"b":0,"e":0.4286,"k":"flat","v":0.46874,"x":0.53122,"p":[[0,11,0.0,0.50892,0.15541,0.42857,0.4286,0.71429,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,11,0,0,6,0,0,9,0,0,0,0,0],[4,11,0.3636,0.53122,0.1197,0.42857,0.571,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,14,0,0,10,0,0,7,0,0,0,0,0],[8,11,0.7273,0.5089,0.10675,0.42857,0.42859,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,19,0,0,8,0,0,5,0,0,0,0,0],[11,11,1.0,0.46874,0.13473,0.42857,0.42859,0.4642,0.2857,0.71429,0,0,0,0,0,0,0,0,5,0,0,19,0,0,2,0,0,6,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"rising","v":0.53567,"x":0.70981,"p":[[0,11,0.0,0.53567,0.17856,0.42857,0.571,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,4,0,0,10,0,0,6,0,0,9,0,0,2,0,0],[4,11,0.3636,0.56693,0.17307,0.42857,0.571,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,13,0,0,5,0,0,9,0,0,2,0,1],[8,11,0.7273,0.70981,0.19393,0.57132,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,12,0,0,4,0,6],[11,11,1.0,0.7098,0.17673,0.57143,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,6,0,0,3,0,0,13,0,0,6,0,4]]}]},{"i":"4a7dcdbcc837389b","q":"The diagonals of the convex quadrilateral $ABCD$ are perpendicular and intersect at $O$. The perpendicular to $(AB)$ passing through $O$ intersects $(AB)$ at $M$ and $(CD)$ at $M'$. The perpendicular to $(BC)$ passing through $O$ intersects $(BC)$ at $N$ and $(DA)$ at $N'$. The perpendicular to $(CD)$ passing through $O$ intersects $(CD)$ at $P$ and $(AB)$ at $P'$. The perpendicular to $(DA)$ passing through $O$ intersects $(DA)$ at $Q$ and $(BC)$ at $Q'$.\n\nProve that the points $M, N, P, Q, M', N', P', Q'$ are concyclic.","t":[{"b":1,"e":1.0,"k":"flat","v":0.04909,"x":0.14732,"p":[[0,19,0.0,0.04909,0.16209,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[4,19,0.2105,0.12058,0.27463,0.0,0.0,0.0,0.0,1.0,26,2,2,26,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,2],[8,19,0.4211,0.05802,0.15911,0.0,0.0,0.0,0.0,0.57143,28,0,1,28,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,19,0.6316,0.08473,0.21973,0.0,0.0,0.0,0.0,1.0,26,1,1,26,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[16,19,0.8421,0.09822,0.22428,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[19,19,1.0,0.14732,0.28231,0.0,0.0,0.17857,0.0,1.0,23,2,0,23,0,1,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,2]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.14719,"p":[[0,34,0.0,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.14719,0.2408,0.0,0.0,0.2857,0.0,0.71429,22,0,1,22,0,1,0,0,2,0,0,2,0,0,3,0,0,2,0,0,0,0,0],[8,34,0.2353,0.04463,0.12072,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,34,0.3529,0.04464,0.18707,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,34,0.4706,0.04464,0.14914,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[20,34,0.5882,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.01777,0.0777,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.00884,0.03424,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d69c837382376212","q":"Let  $K$  and  $N > K$  be fixed positive integers. Let  $n$  be a positive integer and let  $a_1, a_2, ..., a_n$  be distinct integers. Suppose that whenever  $m_1, m_2, ..., m_n$  are integers, not all equal to  $0$ , such that  $\\mid{m_i}\\mid \\le K$  for each  $i$ , then the sum  $$ \\sum_{i = 1}^{n} m_ia_i $$  is not divisible by  $N$ . What is the largest possible value of  $n$ ?\n\n*Proposed by Ilija Jovcevski, North Macedonia*","t":[{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.32593,"p":[[0,27,0.0,0.32593,0.30143,0.0,0.42857,0.42857,0.0,1.0,12,3,12,12,0,0,0,0,1,0,0,14,0,0,2,0,0,0,0,0,0,0,3],[4,27,0.1481,0.08929,0.21651,0.0,0.0,0.0,0.0,1.0,26,1,13,26,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[8,27,0.2963,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,5,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,16,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.0,"x":0.25893,"p":[[0,17,0.0,0.25893,0.25614,0.0,0.42857,0.42857,0.0,1.0,14,1,14,14,0,1,0,0,0,0,0,14,0,0,2,0,0,0,0,0,0,0,1],[4,17,0.2353,0.24558,0.25063,0.0,0.42857,0.42857,0.0,1.0,15,1,14,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1],[8,17,0.4706,0.11161,0.23072,0.0,0.0,0.0,0.0,1.0,25,1,21,25,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2c1193a30ff317bc","q":"Two circles  $\\Gamma_1$  and  $\\Gamma_2$ are given with centres  $O_1$  and  $O_2$  and common exterior tangents  $\\ell_1$  and  $\\ell_2$ . The line  $\\ell_1$  intersects  $\\Gamma_1$  in  $A$  and  $\\Gamma_2$  in  $B$ . Let  $X$  be a point on segment  $O_1O_2$ , not lying on  $\\Gamma_1$  or  $\\Gamma_2$ . The segment  $AX$  intersects  $\\Gamma_1$  in  $Y \\ne A$  and the segment  $BX$  intersects  $\\Gamma_2$  in  $Z \\ne B$ . Prove that the line through  $Y$  tangent to  $\\Gamma_1$  and the line through  $Z$  tangent to  $\\Gamma_2$  intersect each other on  $\\ell_2$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,32,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.01786,"x":0.04455,"p":[[0,11,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6f5a8b09f27ea496","q":"Let  $I, O, \\omega, \\Omega$  be the incenter, circumcenter, the incircle, and the circumcircle, respectively, of a scalene triangle  $ABC$ . The incircle  $\\omega$  is tangent to side  $BC$  at point  $D$ . Let  $S$  be the point on the circumcircle  $\\Omega$  such that  $AS, OI, BC$  are concurrent. Let  $H$  be the orthocenter of triangle  $BIC$ . Point  $T$  lies on  $\\Omega$  such that  $\\angle ATI$  is a right angle. Prove that the points  $D, T, H, S$  are concyclic.\n\n*Proposed by ltf0501*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,41,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,2,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e85b335d4bd9fb5e","q":"Let  $p \\ge 5$  be a prime number. Prove that there exists an integer  $a$  with  $1 \\le a \\le p-2$  such that neither  $a^{p-1} -1$  nor  $(a+1)^{p-1} -1$  is divisible by  $p^2$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,6,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,3,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.04911,0.16213,0.0,0.0,0.0,0.0,0.71429,29,0,3,29,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,15,0.0,0.04015,0.1393,0.0,0.0,0.0,0.0,0.571,29,0,2,29,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,15,0.2667,0.07589,0.21124,0.0,0.0,0.0,0.0,0.85714,27,0,6,27,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3ca396f2419a7af3","q":"The graph  $G$  with 2014 vertices doesn\u2019t contain any 3-cliques. If the set of the degrees of the vertices of  $G$  is  $\\{1,2,...,k\\}$ , find the greatest possible value of  $k$ .","t":[{"b":4,"e":0.42857,"k":"flat","v":0.50892,"x":0.59822,"p":[[0,13,0.0,0.5536,0.21051,0.42857,0.5005,0.57143,0.0,1.0,1,4,0,1,0,0,0,0,0,0,0,15,0,0,9,0,0,3,0,0,0,0,4],[4,13,0.3077,0.59822,0.23266,0.42857,0.57143,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,0,0,0,13,0,0,7,0,0,4,0,0,2,0,5],[8,13,0.6154,0.5357,0.16751,0.42857,0.49979,0.57143,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,15,0,0,10,0,0,4,0,0,0,0,2],[12,13,0.9231,0.50892,0.13803,0.42857,0.42857,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,21,0,0,7,0,0,2,0,0,1,0,1],[13,13,1.0,0.50892,0.10676,0.42857,0.4286,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,16,0,0,11,0,0,4,0,0,0,0,0]]},{"b":7,"e":0.57143,"k":"flat","v":0.47768,"x":0.5625,"p":[[0,17,0.0,0.5357,0.13363,0.42857,0.49979,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,10,0,0,5,0,0,0,0,1],[4,17,0.2353,0.53571,0.20203,0.42857,0.57143,0.57143,0.0,1.0,1,3,0,1,0,1,0,0,0,0,0,12,0,0,13,0,0,2,0,0,0,0,3],[8,17,0.4706,0.5625,0.25238,0.42857,0.57143,0.71429,0.0,1.0,2,4,2,2,0,1,0,0,1,0,0,10,0,0,5,0,0,9,0,0,0,0,4],[12,17,0.7059,0.52676,0.14479,0.42857,0.49979,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,14,0,0,10,0,0,5,0,0,0,0,1],[16,17,0.9412,0.51339,0.1551,0.42857,0.42857,0.57143,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,22,0,0,5,0,0,3,0,0,0,0,2],[17,17,1.0,0.47768,0.08459,0.42857,0.42857,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,20,0,0,10,0,0,1,0,0,0,0,0]]}]},{"i":"4bff882bc7e64251","q":"Two circles  $\\omega_1,\\omega_2$  intersect at  $A,B$ . An arbitrary line through  $B$  meets  $\\omega_1,\\omega_2$  at  $C,D$  respectively. The points  $E,F$  are chosen on  $\\omega_1,\\omega_2$  respectively so that  $CE=CB,\\ BD=DF$ . Suppose that  $BF$  meets  $\\omega_1$  at  $P$ , and  $BE$  meets  $\\omega_2$  at  $Q$ . Prove that  $A,P,Q$  are collinear.\n\n*Proposed by Iman Maghsoudi*","t":[{"b":2,"e":0.0,"k":"flat","v":0.01339,"x":0.03563,"p":[[0,15,0.0,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.03563,0.08738,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.02223,"x":0.08929,"p":[[0,7,0.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.08036,0.10677,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,11,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.08929,0.12243,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9f891079bbbbbb46","q":"The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to  $ 100$ . How many cells are there that are on the distance  $ 50$  from each of the three cells?","t":[{"b":2,"e":0.71429,"k":"flat","v":0.54464,"x":0.64731,"p":[[0,45,0.0,0.61606,0.13571,0.57143,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,1,0,0,2,0,0,11,0,0,17,0,0,0,0,0],[4,45,0.0889,0.54464,0.15746,0.42857,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,3,0,0,8,0,0,9,0,0,11,0,0,0,0,0],[8,45,0.1778,0.59821,0.12595,0.53569,0.57143,0.71429,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,7,0,0,9,0,0,15,0,0,0,0,0],[12,45,0.2667,0.64286,0.10102,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,8,0,0,20,0,0,0,0,0],[16,45,0.3556,0.64731,0.0944,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,20,0,0,0,0,0],[20,45,0.4444,0.60713,0.08749,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,18,0,0,11,0,0,0,0,0],[24,45,0.5333,0.62947,0.1063,0.57143,0.64286,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,12,0,0,15,0,0,1,0,0],[28,45,0.6222,0.57588,0.10403,0.53539,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,8,0,0,15,0,0,9,0,0,0,0,0],[32,45,0.7111,0.625,0.11152,0.57143,0.64286,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,11,0,0,15,0,0,1,0,0],[36,45,0.8,0.61607,0.10374,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,12,0,0,15,0,0,0,0,0],[40,45,0.8889,0.62052,0.09852,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,13,0,0,15,0,0,0,0,0],[44,45,0.9778,0.62058,0.10471,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,19,0,0,10,0,0,0,0,1],[45,45,1.0,0.60268,0.09268,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,17,0,0,11,0,0,0,0,0]]},{"b":5,"e":0.71429,"k":"flat","v":0.5625,"x":0.66964,"p":[[0,43,0.0,0.5625,0.11811,0.42857,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,12,0,0,10,0,0,10,0,0,0,0,0],[4,43,0.093,0.62945,0.14223,0.57132,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,6,0,0,18,0,0,0,0,1],[8,43,0.186,0.57588,0.12619,0.57132,0.57143,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,6,0,0,15,0,0,10,0,0,0,0,0],[12,43,0.2791,0.6607,0.12753,0.71429,0.71429,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0,26,0,0,0,0,0],[16,43,0.3721,0.63393,0.10677,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,19,0,0,0,0,0],[20,43,0.4651,0.60714,0.10101,0.57143,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,14,0,0,13,0,0,0,0,0],[24,43,0.5581,0.64729,0.09441,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,0,20,0,0,0,0,0],[28,43,0.6512,0.63838,0.10706,0.57143,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,7,0,0,20,0,0,0,0,0],[32,43,0.7442,0.61606,0.12596,0.57143,0.64286,0.71429,0.14286,0.71429,0,0,0,0,0,1,0,0,0,0,0,3,0,0,12,0,0,16,0,0,0,0,0],[36,43,0.8372,0.66964,0.09062,0.71429,0.71429,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,25,0,0,0,0,0],[40,43,0.9302,0.66517,0.0846,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,20,0,0,1,0,0],[43,43,1.0,0.61158,0.09607,0.57142,0.57143,0.71429,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,4,0,0,15,0,0,13,0,0,0,0,0]]}]},{"i":"2eb1549df2f3773e","q":"Three circles  $k_1, k_2$ , and  $k_3$  intersect in point  $O$ . Let  $A, B$ , and  $C$  be the second intersection points (other than  $O$ ) of  $k_2$  and  $k_3, k_1$  and  $k_3$ , and  $k_1$  and  $k_2$ , respectively. Assume that  $O$  lies inside of the triangle  $ABC$ . Let lines  $AO,BO$ , and  $CO$  intersect circles  $k_1, k_2$ , and  $k_3$  for a second time at points  $A', B'$ , and  $C'$ , respectively. If  $|XY|$  denotes the length of segment  $XY$ , prove that   $\\frac{|AO|}{|AA'|}+\\frac{|BO|}{|BB'|}+\\frac{|CO|}{|CC'|}= 1$","t":[{"b":6,"e":1.0,"k":"flat","v":0.86161,"x":0.99107,"p":[[0,18,0.0,0.86161,0.32827,1.0,1.0,1.0,0.0,1.0,4,25,2,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,25],[4,18,0.2222,0.92857,0.18558,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,5,0,24],[8,18,0.4444,0.91517,0.18509,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,4,0,24],[12,18,0.6667,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[16,18,0.8889,0.95536,0.11538,1.0,1.0,1.0,0.42857,1.0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,26],[18,18,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]},{"b":7,"e":1.0,"k":"flat","v":0.84597,"x":0.96875,"p":[[0,13,0.0,0.90625,0.24383,1.0,1.0,1.0,0.0,1.0,1,26,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,26],[4,13,0.3077,0.84597,0.30521,0.96429,1.0,1.0,0.0,1.0,1,24,0,1,0,2,0,0,2,0,0,0,0,0,1,0,0,0,1,0,1,0,24],[8,13,0.6154,0.92857,0.14726,0.85714,1.0,1.0,0.42857,1.0,0,23,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,6,0,23],[12,13,0.9231,0.94643,0.09279,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,6,0,23],[13,13,1.0,0.96875,0.08552,1.0,1.0,1.0,0.57143,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,27]]}]},{"i":"158bb2c32b54cb94","q":"Let  $m,n\\ge 2$  be given integers. Prove that there exist positive integers  $a_1<a_2<\\ldots<a_m$  so that for any  $1\\le i<j\\le m$  the number  $\\frac{a_j}{a_j-a_i}$  is an integer divisible by  $n$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00223,"x":0.05134,"p":[[0,65,0.0,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,20,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.04013,0.05627,0.0,0.0,0.07143,0.0,0.14286,20,0,13,20,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,23,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,65,0.1846,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,21,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,65,0.2462,0.02009,0.04803,0.0,0.0,0.0,0.0,0.14286,27,0,8,27,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.0357,0.05922,0.0,0.0,0.07111,0.0,0.14286,23,0,8,23,2,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,9,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.01554,0.0425,0.0,0.0,0.0,0.0,0.14286,28,0,5,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,4,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,65,0.5538,0.03795,0.05919,0.0,0.0,0.07143,0.0,0.1429,22,0,2,22,3,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,65,0.6154,0.0379,0.06179,0.0,0.0,0.08821,0.0,0.14286,23,0,5,23,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.02902,0.05315,0.0,0.0,0.01786,0.0,0.1429,24,0,4,24,3,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,65,0.7385,0.05134,0.0897,0.0,0.0,0.08929,0.0,0.42857,21,0,2,21,3,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.04222,0.06375,0.0,0.0,0.14,0.0,0.1429,22,0,10,22,1,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.0357,0.05646,0.0,0.0,0.07143,0.0,0.14286,22,0,2,22,4,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.02009,0.04803,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.01116,0.03149,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,8,0.0,0.01339,0.03762,0.0,0.0,0.0,0.0,0.14286,28,0,21,28,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c6a86d5a78b4dd78","q":"For each positive integer  $n$ , let  $f(n)$  denote the smallest possible value of   $$ |A_1 \\cup A_2 \\cup \\dots \\cup A_n| $$   where  $A_1, A_2, A_3 \\dots A_n$  are sets such that  $A_i \\not\\subseteq A_j$  and  $|A_i| \\neq |A_j|$  whenever  $i \\neq j$ . Determine  $f(n)$  for each positive integer  $n$ .","t":[{"b":0,"e":0.0,"k":"volatile","v":0.13391,"x":0.30802,"p":[[0,6,0.0,0.30802,0.30326,0.0,0.28571,0.4286,0.0,1.0,12,2,8,12,0,2,0,0,3,0,0,8,0,0,3,0,0,1,0,0,1,0,2],[4,6,0.6667,0.13391,0.23399,0.0,0.0,0.28571,0.0,0.85714,21,0,18,21,0,2,0,0,6,0,0,0,0,0,1,0,0,0,0,0,2,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.13839,"x":0.24107,"p":[[0,14,0.0,0.22321,0.33681,0.0,0.0,0.32143,0.0,1.0,19,3,18,19,0,2,0,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,3],[4,14,0.2857,0.17409,0.26178,0.0,0.0,0.28571,0.0,0.71429,20,0,16,20,0,2,0,0,3,0,0,1,0,0,2,0,0,4,0,0,0,0,0],[8,14,0.5714,0.17854,0.26721,0.0,0.0,0.28571,0.0,0.85714,19,0,12,19,0,3,0,0,3,0,0,2,0,0,2,0,0,1,0,0,2,0,0],[12,14,0.8571,0.24107,0.33586,0.0,0.0,0.28571,0.0,1.0,17,3,15,17,0,2,0,0,6,0,0,1,0,0,0,0,0,2,0,0,1,0,3],[14,14,1.0,0.13839,0.14053,0.0,0.07143,0.28571,0.0,0.28571,16,0,15,16,0,1,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"904f6174395690a1","q":"The point  $M$  is the middle of the side  $BC$  of the acute-angled triangle  $ABC$  and the points  $E$  and  $F$  are respectively perpendicular foot of  $M$  to the sides  $AC$  and  $AB$ . The points  $X$  and  $Y$  lie on the plane such that  $\\triangle XEC\\sim\\triangle CEY$  and  $\\triangle BYF\\sim\\triangle XBF$ (The vertices of triangles with this order are corresponded in the similarities) and the points  $E$  and  $F$  <u>don't</u>[neither] lie on the line  $XY$ .  Prove that  $XY\\perp AM$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"d947feb620f9224e","q":"Let $A B C D$ be a cyclic quadrilateral with $|A D|=|B D|$. Let $M$ be the intersection of $A C$ and $B D$. Let $I$ be the incenter of $\\triangle B C M$. Let $N$ be the second intersection of $A C$ with the circumcircle of $\\triangle B M I$. Prove that $|A N| \\cdot|N C| = |C D| \\cdot|B N|$.","t":[{"b":3,"e":0.28571,"k":"flat","v":0.17384,"x":0.45089,"p":[[0,18,0.0,0.29911,0.3524,0.0,0.14286,0.60714,0.0,1.0,14,2,1,14,0,5,0,0,2,0,0,1,0,0,2,0,0,3,0,0,3,0,2],[4,18,0.2222,0.20079,0.29636,0.0,0.07,0.17857,0.0,1.0,16,2,6,16,0,8,0,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,2],[8,18,0.4444,0.45089,0.38484,0.14286,0.28571,0.89286,0.0,1.0,7,8,5,7,0,3,0,0,9,0,0,1,0,0,0,0,0,3,0,0,1,0,8],[12,18,0.6667,0.23659,0.25405,0.10714,0.14286,0.28571,0.0,1.0,8,2,1,8,0,12,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,2],[16,18,0.8889,0.21875,0.2082,0.14286,0.14286,0.28571,0.0,1.0,6,1,0,6,0,13,0,0,10,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[18,18,1.0,0.17384,0.15046,0.105,0.14286,0.2857,0.0,0.71429,8,0,0,8,0,13,0,0,9,0,0,1,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.57143,"k":"rising","v":0.14272,"x":0.45533,"p":[[0,30,0.0,0.27664,0.30689,0.0,0.14286,0.46418,0.0,1.0,10,2,1,10,0,10,0,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,2],[4,30,0.1333,0.24098,0.32034,0.0,0.14143,0.42858,0.0,1.0,15,3,0,15,0,6,0,0,2,0,0,2,0,0,3,0,0,1,0,0,0,0,3],[8,30,0.2667,0.14272,0.26215,0.0,0.0,0.14286,0.0,0.85714,23,0,1,23,0,2,0,0,1,0,0,0,0,0,3,0,0,2,0,0,1,0,0],[12,30,0.4,0.42404,0.21861,0.28571,0.5705,0.57143,0.0,0.71429,5,0,0,5,0,1,0,0,3,0,0,6,0,0,15,0,0,2,0,0,0,0,0],[16,30,0.5333,0.37503,0.25442,0.14286,0.28571,0.57143,0.0,0.85714,5,0,0,5,0,4,0,0,8,0,0,3,0,0,7,0,0,3,0,0,2,0,0],[20,30,0.6667,0.37941,0.25151,0.14286,0.42836,0.57143,0.0,0.85714,6,0,0,6,0,3,0,0,7,0,0,0,0,0,13,0,0,2,0,0,1,0,0],[24,30,0.8,0.45533,0.18362,0.28571,0.57121,0.57143,0.0,0.85714,1,0,0,1,0,2,0,0,8,0,0,3,0,0,16,0,0,1,0,0,1,0,0],[28,30,0.9333,0.36158,0.23139,0.24999,0.42857,0.57143,0.0,0.71429,7,0,0,7,0,1,0,0,7,0,0,3,0,0,13,0,0,1,0,0,0,0,0],[30,30,1.0,0.43292,0.22166,0.28571,0.57141,0.57143,0.0,0.71429,5,0,0,5,0,1,0,0,3,0,0,4,0,0,17,0,0,2,0,0,0,0,0]]}]},{"i":"ba032e1e6c598379","q":"Three schools have  $200$  students each. Every student has at least one friend in each school (if the student  $a$  is a friend of the student  $b$  then  $b$  is a friend of  $a$ ).\nIt is known that there exists a set  $E$  of  $300$  students (among the  $600$ ) such that for any school  $S$  and any two students  $x,y\\in E$  but not in  $S$ , the number of friends in  $S$  of  $x$  and  $y$  are different.\nShow that one can find a student in each school such that they are friends with each other.","t":[{"b":3,"e":0.85714,"k":"flat","v":0.69195,"x":0.85713,"p":[[0,12,0.0,0.74998,0.20517,0.71429,0.85714,0.85714,0.0,1.0,1,1,0,1,0,0,0,0,0,0,0,4,0,0,2,0,0,3,0,0,21,0,1],[4,12,0.3333,0.69195,0.25029,0.53539,0.85714,0.85714,0.0,1.0,2,1,1,2,0,0,0,0,0,0,0,6,0,0,3,0,0,2,0,0,18,0,1],[8,12,0.6667,0.74554,0.27603,0.82143,0.85714,0.85714,0.0,1.0,3,4,3,3,0,0,0,0,0,0,0,2,0,0,2,0,0,1,0,0,20,0,4],[12,12,1.0,0.85713,0.05051,0.85714,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,28,0,2]]},{"b":4,"e":0.85714,"k":"flat","v":0.71874,"x":0.86607,"p":[[0,27,0.0,0.75897,0.24331,0.67857,0.85714,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,1,0,0,5,0,0,1,0,0,1,0,0,17,0,6],[4,27,0.1481,0.78571,0.2369,0.85714,0.85714,0.85714,0.0,1.0,2,4,2,2,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,22,0,4],[8,27,0.2963,0.73214,0.26426,0.67857,0.85714,0.85714,0.0,1.0,2,4,2,2,0,0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,18,0,4],[12,27,0.4444,0.78125,0.22299,0.85714,0.85714,0.85714,0.0,1.0,2,1,2,2,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,26,0,1],[16,27,0.5926,0.73214,0.21943,0.42857,0.85714,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,2,0,0,7,0,0,0,0,0,2,0,0,18,0,3],[20,27,0.7407,0.71874,0.27313,0.57132,0.85714,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,0,0,0,4,0,0,2,0,0,2,0,0,17,0,4],[24,27,0.8889,0.86607,0.03458,0.85714,0.85714,0.85714,0.85714,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,0,2],[27,27,1.0,0.85714,0.08748,0.85714,0.85714,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,26,0,4]]}]},{"i":"bf16df3736f9838a","q":"Let $\\left(a_{n}\\right)_{n \\geqslant 1}$ be a sequence of strictly positive integers such that $a_{1}$ and $a_{2}$ are coprime and, for all $n \\geqslant 1, a_{n+2}=a_{n} a_{n+1}+1$. Show that for any integer $m>1$, there exists $n>m$ such that $a_{m}^{m} \\mid a_{n}^{n}$. Is the result still true when $m=1$?","t":[{"b":3,"e":0.28571,"k":"rising","v":0.30356,"x":0.45978,"p":[[0,7,0.0,0.30356,0.16267,0.14286,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,7,0,0,8,0,0,11,0,0,3,0,0,0,0,0,0,0,0],[4,7,0.5714,0.30803,0.19918,0.14286,0.35714,0.4286,0.0,0.57143,6,0,0,6,0,5,0,0,5,0,0,10,0,0,6,0,0,0,0,0,0,0,0],[7,7,1.0,0.45978,0.14164,0.42857,0.4998,0.57143,0.0,0.57143,1,0,1,1,0,1,0,0,4,0,0,10,0,0,16,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.1158,"x":0.31239,"p":[[0,20,0.0,0.27232,0.19019,0.14286,0.2857,0.42858,0.0,0.71429,4,0,0,4,0,11,0,0,7,0,0,5,0,0,4,0,0,1,0,0,0,0,0],[4,20,0.2,0.26776,0.21357,0.14286,0.14293,0.42857,0.0,0.71429,6,0,0,6,0,11,0,0,4,0,0,4,0,0,6,0,0,1,0,0,0,0,0],[8,20,0.4,0.23652,0.17722,0.14286,0.14286,0.42857,0.0,0.57143,4,0,0,4,0,16,0,0,3,0,0,5,0,0,4,0,0,0,0,0,0,0,0],[12,20,0.6,0.31239,0.21265,0.14286,0.28571,0.4286,0.0,0.71429,6,0,0,6,0,4,0,0,9,0,0,6,0,0,5,0,0,2,0,0,0,0,0],[16,20,0.8,0.14732,0.11564,0.0,0.14286,0.2857,0.0,0.42857,9,0,0,9,0,14,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.1158,0.12073,0.0,0.14,0.14287,0.0,0.4286,14,0,0,14,0,11,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fe20be8465513b95","q":"The incircle of a triangle  $ABC$  touches the sides  $BC,CA,AB$  at points  $D,E,F$ , respectively. Let  $X$  be a point on the incircle, different from the points  $D,E,F$ . The lines  $XD$  and  $EF,XE$  and  $FD,XF$  and  $DE$  meet at points  $J,K,L$ , respectively. Let further  $M,N,P$  be points on the sides  $BC,CA,AB$ , respectively, such that the lines  $AM,BN,CP$  are concurrent. Prove that the lines  $JM,KN$  and  $LP$  are concurrent.\n\n*Dinu Serbanescu*","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.09822,"p":[[0,10,0.0,0.09822,0.11538,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.09588,0.11516,0.0,0.0355,0.14286,0.0,0.42857,16,0,0,16,1,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,45,0.0,0.06679,0.08722,0.0,0.0,0.14286,0.0,0.2857,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,6,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.07367,0.1002,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,1,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.05804,0.10013,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.05804,0.11769,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.01777,0.05904,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,1,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,45,0.9778,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[45,45,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cbfde49da032883b","q":"The prism with the regular octagonal base and with all edges of the length equal to  $1$  is given. The points  $M_{1},M_{2},\\cdots,M_{10}$  are the midpoints of all the faces of the prism. For the point  $P$  from the inside of the prism denote by  $P_{i}$  the intersection point (not equal to  $M_{i}$ ) of the line  $M_{i}P$  with the surface of the prism. Assume that the point  $P$  is so chosen that all associated with  $P$  points  $P_{i}$  do not belong to any edge of the prism and on each face lies exactly one point  $P_{i}$ . Prove that \\[\\sum_{i=1}^{10}\\frac{M_{i}P}{M_{i}P_{i}}=5\\]","t":[{"b":3,"e":0.28571,"k":"flat","v":0.26339,"x":0.33482,"p":[[0,15,0.0,0.28125,0.10403,0.2857,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,3,0,0,26,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,15,0.2667,0.33482,0.16213,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,2,0,0,25,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[8,15,0.5333,0.26339,0.06298,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.28127,0.04351,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.27232,"x":0.40178,"p":[[0,18,0.0,0.27232,0.13054,0.2857,0.28571,0.28571,0.0,0.71429,2,0,0,2,0,5,0,0,22,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,18,0.2222,0.37051,0.17442,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,1,0,0,22,0,0,3,0,0,3,0,0,2,0,0,0,0,1],[8,18,0.4444,0.40178,0.20652,0.28571,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,1,0,0,19,0,0,5,0,0,4,0,0,0,0,0,1,0,2],[12,18,0.6667,0.33481,0.1163,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,27,0,0,0,0,0,4,0,0,1,0,0,0,0,0],[16,18,0.8889,0.37941,0.12674,0.28571,0.28571,0.42858,0.2857,0.71429,0,0,0,0,0,0,0,0,19,0,0,6,0,0,6,0,0,1,0,0,0,0,0],[18,18,1.0,0.39284,0.17127,0.28571,0.28571,0.57111,0.2857,1.0,0,1,0,0,0,0,0,0,21,0,0,2,0,0,7,0,0,1,0,0,0,0,1]]}]},{"i":"96ad3afea079a9c1","q":"The vertices  $ A$  and  $ B$  of an equilateral triangle  $ ABC$  lie on a circle  $k$  of radius  $1$ , and the vertex  $ C$  is in the interior of the circle  $ k$ . A point  $ D$ , different from  $ B$ , lies on  $ k$  so that  $ AD\\equal{}AB$ . The line  $ DC$  intersects  $ k$  for the second time at point  $ E$ . Find the length of the line segment  $ CE$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.05348,"x":0.09375,"p":[[0,39,0.0,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.143,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.0759,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.03571,"x":0.09375,"p":[[0,18,0.0,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9623bd9096544505","q":"Let $\\Gamma$ be the circumcircle of the isosceles triangle $ABC$ with $C$ as the vertex. Inside the side $\\overline{BC}$, there lies a point $M$. There exists a point $N$ on the ray $AM$ such that $M$ is between $A$ and $N$ and $|AN|=|AC|$. The circumcircle of triangle $CMN$ intersects $\\Gamma$ at two distinct points $C$ and $P$. The lines $AB$ and $CP$ intersect at a point $Q$. Prove that $\\measuredangle BMQ = \\measuredangle QMN$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,29,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"c4d7193d695f4570","q":"Let  $S$  be a set of  $n$  points in the plane such that any two points of  $S$  are at least  $1$  unit apart.\r\nProve there is a subset  $T$  of  $S$  with at least  $\\frac{n}{7}$  points such that any two points of  $T$  are at least  $\\sqrt{3}$  units apart.","t":[{"b":2,"e":0.0,"k":"flat","v":0.16517,"x":0.17389,"p":[[0,7,0.0,0.16517,0.09517,0.14286,0.14286,0.14286,0.0,0.571,2,0,2,2,0,25,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.17389,0.16608,0.14286,0.14286,0.14286,0.0,0.71429,6,0,5,6,0,20,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.19186,"x":0.25438,"p":[[0,8,0.0,0.19186,0.17717,0.14214,0.14286,0.17857,0.0,0.57143,7,0,7,7,0,17,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[4,8,0.5,0.23659,0.2219,0.14286,0.14286,0.2857,0.0,1.0,3,1,3,3,0,20,0,0,3,0,0,2,0,0,1,0,0,2,0,0,0,0,1],[8,8,1.0,0.25438,0.1948,0.14286,0.14286,0.28571,0.0,0.85714,2,0,2,2,0,17,0,0,6,0,0,3,0,0,2,0,0,1,0,0,1,0,0]]}]},{"i":"d145573cb9f24799","q":"It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that:\r\n1: if 2 squares are adjacent then one of them is marked.\r\n2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked.\r\n\r\nFind the minimun number of squares we most mark.","t":[{"b":0,"e":0.71429,"k":"volatile","v":0.19195,"x":0.63393,"p":[[0,7,0.0,0.19195,0.3285,0.0,0.0,0.42858,0.0,1.0,23,2,18,23,0,0,0,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,2],[4,7,0.5714,0.59372,0.20551,0.42857,0.57143,0.71429,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,7,0,0,5,0,0,10,0,0,3,0,2],[7,7,1.0,0.63393,0.2111,0.42857,0.71429,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,5,0,0,4,0,0,5,0,0,10,0,0,6,0,2]]},{"b":6,"e":0.0,"k":"volatile","v":0.20535,"x":0.45536,"p":[[0,12,0.0,0.25446,0.3792,0.0,0.0,0.35714,0.0,1.0,20,5,18,20,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,5],[4,12,0.3333,0.45536,0.41563,0.0,0.28571,0.85714,0.0,1.0,12,7,11,12,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,3,0,7],[8,12,0.6667,0.20535,0.33299,0.0,0.0,0.4286,0.0,1.0,22,2,17,22,0,0,0,0,1,0,0,2,0,0,3,0,0,0,0,0,2,0,2]]}]},{"i":"99afaf89ff718a44","q":"Let  $ABCD$  be a parallelogram. A line  $\\ell$  intersects lines  $AB,~ BC,~ CD, ~DA$  at four different points  $E,~ F,~ G,~ H,$  respectively. The circumcircles of triangles  $AEF$  and  $AGH$  intersect again at  $P$ . The circumcircles of triangles  $CEF$  and  $CGH$  intersect again at  $Q$ . Prove that the line  $P Q$  bisects the diagonal  $BD$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.02232,"x":0.16509,"p":[[0,27,0.0,0.14278,0.15972,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,10,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,27,0.1481,0.15624,0.20314,0.0,0.14286,0.2857,0.0,0.85714,15,0,1,15,0,8,0,0,4,0,0,3,0,0,1,0,0,0,0,0,1,0,0],[8,27,0.2963,0.16509,0.2262,0.0,0.14286,0.1429,0.0,0.85714,14,0,1,14,0,11,0,0,2,0,0,2,0,0,0,0,0,2,0,0,1,0,0],[12,27,0.4444,0.12036,0.20549,0.0,0.0,0.1429,0.0,0.85714,19,0,1,19,0,7,0,0,3,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[16,27,0.5926,0.08481,0.13763,0.0,0.0,0.14286,0.0,0.571,20,0,0,20,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,27,0.7407,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.04018,0.13475,0.0,0.0,0.0,0.0,0.71429,28,0,2,28,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.714,"k":"rising","v":0.1517,"x":0.60483,"p":[[0,13,0.0,0.1517,0.24728,0.0,0.0,0.17857,0.0,1.0,18,1,1,18,0,6,0,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,1],[4,13,0.3077,0.19632,0.25442,0.0,0.14286,0.2857,0.0,1.0,13,2,1,13,0,7,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[8,13,0.6154,0.5424,0.36151,0.14286,0.71429,0.85704,0.0,1.0,5,5,0,5,0,5,0,1,1,0,0,0,0,0,1,0,0,10,0,0,4,0,5],[12,13,0.9231,0.58926,0.28959,0.39286,0.71429,0.857,0.0,1.0,3,3,1,3,0,0,0,0,5,0,0,4,0,0,2,0,0,9,0,0,6,0,3],[13,13,1.0,0.60483,0.21943,0.41068,0.64286,0.74996,0.2857,1.0,0,1,0,0,0,0,0,0,7,0,1,2,0,0,6,0,0,8,0,0,7,0,1]]}]},{"i":"b6a7d506400eab8f","q":"The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?","t":[{"b":1,"e":1.0,"k":"rising","v":0.54908,"x":0.85713,"p":[[0,22,0.0,0.54908,0.29257,0.42857,0.571,0.74996,0.0,1.0,3,3,0,3,0,2,0,0,2,0,0,8,0,0,3,0,0,6,0,0,5,0,3],[4,22,0.1818,0.70088,0.27049,0.42857,0.71429,1.0,0.14286,1.0,0,10,0,0,0,1,0,0,3,0,0,6,0,0,3,0,0,4,0,0,5,0,10],[8,22,0.3636,0.74552,0.2135,0.57143,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,0,0,0,3,0,0,6,0,0,7,0,0,7,0,8],[12,22,0.5455,0.69632,0.29196,0.42857,0.78564,1.0,0.14,1.0,0,10,0,0,0,3,0,0,2,0,0,5,0,0,2,0,0,4,0,0,6,0,10],[16,22,0.7273,0.84375,0.17261,0.82132,0.85714,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,0,12,0,12],[20,22,0.9091,0.85713,0.11845,0.71429,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,10,0,11],[22,22,1.0,0.79464,0.08702,0.71429,0.78569,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,14,0,2]]},{"b":4,"e":1.0,"k":"rising","v":0.43302,"x":0.8013,"p":[[0,20,0.0,0.43302,0.37369,0.0,0.42857,0.85714,0.0,1.0,11,2,0,11,0,1,0,0,3,0,0,2,0,0,2,0,0,4,0,0,7,0,2],[4,20,0.2,0.62053,0.37561,0.28571,0.85707,1.0,0.0,1.0,5,9,0,5,0,2,0,0,3,0,0,1,0,0,3,0,0,1,0,0,8,0,9],[8,20,0.4,0.66517,0.37899,0.28571,0.85705,1.0,0.0,1.0,5,13,0,5,0,1,0,0,3,0,0,1,0,0,1,0,0,4,0,0,4,0,13],[12,20,0.6,0.68969,0.32372,0.571,0.71429,1.0,0.0,1.0,2,12,0,2,0,3,0,0,1,0,0,1,0,0,4,1,0,6,0,0,2,0,12],[16,20,0.8,0.79688,0.18133,0.71429,0.82157,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,11,1,0,7,0,9],[20,20,1.0,0.8013,0.26773,0.67836,0.89286,1.0,0.0,1.0,1,15,0,1,0,0,0,0,3,0,0,0,0,0,4,0,0,2,0,0,6,1,15]]}]},{"i":"3798e6d7e941d9ed","q":"Let  $p,q \\in \\mathbb N^{\\ast}$ ,  $p,q \\geq 2$ . We say that a set  $X$  has the property  $\\left( \\mathcal S \\right)$  if no matter how we choose  $p$  subsets  $B_i \\subset X$ ,  $i = \\overline{1,n}$ , not necessarily distinct, each with  $q$  elements, there is a subset  $Y \\subset X$  with  $p$  elements s.t. the intersection of  $Y$  with each of the  $B_i$ 's has an element at most,  $i=\\overline{1,p}$ . Prove that:\r\n\r\n(a) if  $p=4,q=3$  then any set composed of  $9$  elements doesn't have  $\\left( \\mathcal S \\right)$ ;\r\n\r\n(b) any set  $X$  composed of  $pq-q$  elements doesn't have the property  $\\left( \\mathcal S \\right)$ ;\r\n\r\n(c) any set  $X$  composed of  $pq-q+1$  elements has the property  $\\left( \\mathcal S \\right)$ .\r\n\r\n*Dan Schwarz*","t":[{"b":2,"e":0.0,"k":"flat","v":0.01786,"x":0.23214,"p":[[0,56,0.0,0.23214,0.13716,0.14286,0.2857,0.28571,0.0,0.71429,5,0,3,5,0,5,0,0,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,56,0.0714,0.07125,0.10089,0.0,0.0,0.14286,0.0,0.28571,20,0,6,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,56,0.1429,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.2857,22,0,12,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,56,0.2143,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.42857,21,0,8,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,56,0.2857,0.05349,0.09271,0.0,0.0,0.14286,0.0,0.42857,22,0,5,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,56,0.3571,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.42857,21,0,6,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,56,0.4286,0.08482,0.12299,0.0,0.0,0.14286,0.0,0.42857,20,0,4,20,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,56,0.5,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,13,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,56,0.5714,0.09804,0.14908,0.0,0.0,0.14286,0.0,0.71429,18,0,6,18,0,9,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,56,0.6429,0.0892,0.1324,0.0,0.0,0.14286,0.0,0.42857,20,0,3,20,0,6,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,56,0.7143,0.08911,0.06903,0.0,0.14286,0.14286,0.0,0.1429,12,0,5,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,56,0.7857,0.10706,0.10099,0.0,0.14286,0.14286,0.0,0.4286,12,0,3,12,0,17,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,56,0.8571,0.10715,0.11845,0.0,0.14286,0.14286,0.0,0.42857,14,0,1,14,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,56,0.9286,0.11143,0.11697,0.0,0.14286,0.14287,0.0,0.4286,13,0,0,13,0,15,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,56,1.0,0.13393,0.11259,0.0,0.14286,0.14286,0.0,0.4286,9,0,3,9,0,18,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.28571,"k":"flat","v":0.09822,"x":0.29911,"p":[[0,45,0.0,0.20535,0.15542,0.0,0.2857,0.28571,0.0,0.71429,9,0,6,9,0,3,0,0,19,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,45,0.0889,0.14732,0.12103,0.0,0.14286,0.28571,0.0,0.28571,11,0,2,11,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.09822,0.12078,0.0,0.0,0.14286,0.0,0.42857,17,0,8,17,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.13384,0.11258,0.0,0.14286,0.2857,0.0,0.28571,11,0,3,11,0,12,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.11607,0.10374,0.0,0.14286,0.1429,0.0,0.28571,12,0,1,12,0,14,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.24982,0.0877,0.2857,0.2857,0.28571,0.0,0.28571,3,0,2,3,0,2,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.29017,0.06667,0.2857,0.28571,0.28571,0.0,0.42857,1,0,1,1,0,0,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.28126,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,45,0.7111,0.29464,0.07936,0.2857,0.28571,0.28571,0.0,0.4286,1,0,1,1,0,1,0,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[36,45,0.8,0.29911,0.05486,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[40,45,0.8889,0.28571,0.06186,0.2857,0.28571,0.28571,0.0,0.42857,1,0,1,1,0,0,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e7c2a52e6ec25453","q":"Let  $n\\in N$   $n\\geq2$  and the set  $X$  with  $n+1$  elements. The ordered sequences  $(a_{1}, a_{2},\\ldots,a_{n})$  and  $(b_{1},b_{2},\\ldots b_{n})$  of distinct elements of  $X$  are said to be  $\\textit{separated}$  if there exists  $i\\neq j$  such that  $a_{i}=b_{j}$ . Determine the maximal number of ordered sequences of  $n$  elements from  $X$  such that any two of them are  $\\textit{separated}$ .\r\nNote: ordered means that, for example  $(1,2,3)\\neq(2,3,1)$ .","t":[{"b":0,"e":1.0,"k":"rising","v":0.80357,"x":1.0,"p":[[0,19,0.0,0.80357,0.31491,0.57143,1.0,1.0,0.0,1.0,2,21,1,2,0,1,0,0,1,0,0,1,0,0,4,0,0,1,0,0,1,0,21],[4,19,0.2105,0.98661,0.07457,1.0,1.0,1.0,0.57143,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[8,19,0.4211,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[19,19,1.0,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":3,"e":1.0,"k":"rising","v":0.78571,"x":0.94195,"p":[[0,12,0.0,0.78571,0.34442,0.57143,1.0,1.0,0.0,1.0,2,22,2,2,0,2,0,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,22],[4,12,0.3333,0.85268,0.26362,0.82143,1.0,1.0,0.14286,1.0,0,23,0,0,0,1,0,0,3,0,0,0,0,0,3,0,0,1,0,0,1,0,23],[8,12,0.6667,0.88839,0.24415,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,26],[12,12,1.0,0.94195,0.14226,1.0,1.0,1.0,0.571,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,27]]}]},{"i":"8b66e2864e277d1d","q":"Let $A B C$ be a triangle with $\\angle B A C \\neq 90^{\\circ}$. Let $O$ be the circumcenter of the triangle $A B C$ and let $\\Gamma$ be the circumcircle of the triangle $B O C$. Suppose that $\\Gamma$ intersects the line segment $A B$ at $P$ different from $B$, and the line segment $A C$ at $Q$ different from $C$. Let $O N$ be a diameter of the circle $\\Gamma$. Prove that the quadrilateral $A P N Q$ is a parallelogram.","t":[{"b":1,"e":0.571,"k":"flat","v":0.17411,"x":0.49102,"p":[[0,20,0.0,0.28125,0.31234,0.0,0.14286,0.46429,0.0,1.0,12,1,0,12,0,6,0,0,4,0,0,2,0,0,1,0,0,4,0,0,2,0,1],[4,20,0.2,0.30357,0.32093,0.0,0.14286,0.71429,0.0,0.85714,14,0,3,14,0,4,0,0,0,0,0,2,0,0,3,0,0,8,0,0,1,0,0],[8,20,0.4,0.17411,0.26422,0.0,0.0,0.2857,0.0,0.85714,19,0,3,19,0,3,0,0,4,0,0,1,0,0,2,0,0,1,0,0,2,0,0],[12,20,0.6,0.31249,0.30812,0.0,0.2857,0.57111,0.0,1.0,11,1,2,11,0,4,0,0,4,0,0,4,0,0,4,0,0,1,0,0,3,0,1],[16,20,0.8,0.49102,0.22284,0.39286,0.571,0.57143,0.0,0.85714,2,0,0,2,0,2,0,0,4,0,0,6,0,0,11,0,0,4,0,0,3,0,0],[20,20,1.0,0.38837,0.21197,0.25,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,6,0,0,7,0,0,5,0,0,8,0,0,4,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"volatile","v":0.01786,"x":0.33927,"p":[[0,10,0.0,0.16518,0.26027,0.0,0.0,0.2857,0.0,0.85714,20,0,0,20,0,3,0,0,2,0,0,2,0,0,3,0,0,0,0,0,2,0,0],[4,10,0.4,0.33927,0.3229,0.0,0.42857,0.60714,0.0,0.85714,13,0,0,13,0,2,0,0,0,0,0,5,0,0,4,0,0,5,0,0,3,0,0],[8,10,0.8,0.08482,0.21086,0.0,0.0,0.0,0.0,0.85714,27,0,3,27,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,0],[10,10,1.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e6e79592b32ed25f","q":"Let  $k$  be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white. \n\tA turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly  $k$  times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs.\n\tLet  $m$  be the minimal  $k$  value such that Vera can guarantee that she wins no matter what Marco does. For  $k=m$ , let  $t$  be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most  $t$  turns. Compute  $100m + t$ .\n\n*Proposed by Ashwin Sah*","t":[{"b":6,"e":0.14286,"k":"flat","v":0.12946,"x":0.12946,"p":[[0,5,0.0,0.12946,0.13054,0.0,0.14286,0.14286,0.0,0.71429,9,0,8,9,0,20,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.08482,"x":0.11607,"p":[[0,13,0.0,0.09821,0.08328,0.0,0.14286,0.14286,0.0,0.28571,12,0,10,12,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,5,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,8,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,3,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.2857,8,0,4,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2860a394c9e3fa09","q":"Let  $ABC$  be a triangle with incenter  $I$  and let  $AI$  meet  $BC$  at  $D$ . Let  $E$  be a point on the segment  $AC$ , such that  $CD=CE$  and let  $F$  be on the segment  $AB$  such that  $BF=BD$ . Let  $(CEI) \\cap (DFI)=P \\neq I$  and  $(BFI) \\cap (DEI)=Q \\neq I$ . Prove that  $PQ \\perp BC$ .\n\n*Proposed by Leonardo Franchi, Italy*","t":[{"b":1,"e":0.28571,"k":"flat","v":0.0,"x":0.1741,"p":[[0,65,0.0,0.09375,0.09181,0.0,0.14286,0.14286,0.0,0.2857,14,0,2,14,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,2,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,65,0.1846,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,65,0.2462,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,3,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,65,0.5538,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,65,0.6154,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,65,0.7385,0.01777,0.05904,0.0,0.0,0.0,0.0,0.2857,29,0,4,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.1741,0.07771,0.14286,0.14286,0.2857,0.0,0.28571,2,0,1,2,0,21,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.06696,"x":0.13839,"p":[[0,16,0.0,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,1,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.08027,0.08696,0.0,0.07,0.14286,0.0,0.28571,16,0,0,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.12947,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,5,0,3,5,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,1,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"432cfa9c2c20f516","q":"We call a natural number venerable if the sum of all its divisors, including  $1$ , but not including the number itself, is  $1$  less than this number. Find all the venerable numbers, some exact degree of which is also venerable.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.17402,"x":0.23214,"p":[[0,28,0.0,0.21429,0.10102,0.14286,0.2143,0.28571,0.0,0.4286,2,0,0,2,0,14,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.19179,0.07679,0.14286,0.1429,0.2857,0.0,0.28571,1,0,0,1,0,19,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.23214,0.15043,0.14286,0.2857,0.28571,0.0,0.71429,4,0,1,4,0,10,0,0,15,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[12,28,0.4286,0.17402,0.09271,0.14286,0.14286,0.2857,0.0,0.28571,4,0,0,4,0,17,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.2143,0.07143,0.14286,0.2143,0.28571,0.14286,0.286,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.18741,0.09744,0.14286,0.1429,0.28571,0.0,0.28571,4,0,0,4,0,14,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.20509,0.09425,0.14286,0.2857,0.28571,0.0,0.28571,3,0,0,3,0,12,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.18286,0.10255,0.14286,0.14288,0.28571,0.0,0.28571,5,0,0,5,0,13,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.16063,"x":0.2232,"p":[[0,36,0.0,0.20081,0.11219,0.14286,0.1429,0.28571,0.0,0.4286,4,0,0,4,0,13,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.18081,0.09778,0.14286,0.1429,0.2857,0.0,0.28571,4,0,0,4,1,14,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.19196,0.11071,0.14286,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,9,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.2232,0.11254,0.14286,0.2857,0.28571,0.0,0.571,2,0,0,2,0,13,0,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,36,0.4444,0.19643,0.10564,0.14286,0.1429,0.2857,0.0,0.4286,4,0,0,4,0,13,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.19197,0.09181,0.14286,0.1429,0.28571,0.0,0.28571,3,0,0,3,0,15,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.18742,0.09744,0.14286,0.14288,0.28571,0.0,0.42857,3,0,0,3,0,17,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.20536,0.08702,0.14286,0.1429,0.28571,0.0,0.42857,1,0,0,1,0,17,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.20759,0.08992,0.14286,0.2857,0.28571,0.0,0.28571,2,0,0,2,1,12,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.16063,0.10566,0.14286,0.14286,0.2857,0.0,0.42857,6,0,0,6,0,17,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c37e27d047857495","q":"Let  $n$  be a positive integer. Anna and Beatrice play a game with a deck of  $n$  cards labelled with the numbers  $1, 2,...,n$ . Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if  $k$  denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the  $k$  topmost cards. If, after rearranging, the topmost card shows the number k again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.","t":[{"b":4,"e":0.0,"k":"flat","v":0.02679,"x":0.02679,"p":[[0,4,0.0,0.02679,0.09061,0.0,0.0,0.0,0.0,0.42857,29,0,9,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,14,0.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,10,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,9,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c12dddcb4a86673d","q":"Let  $x_1$  and  $x_2$  be positive integers. On a straight line,  $y_1$  white segments and  $y_2$  black segments are given, with  $y_1 \\ge x_1$  and  $y_2 \\ge x_2$ . Suppose that no two segments of the same colour intersect (and do not have common ends). Moreover, suppose that for any choice of  $x_1$  white segments and  $x_2$  black segments, some pair of selected segments will intersect. Prove that  $(y_1-x_1)(y_2-x_2)<x_1x_2$ .\n*Proposed by G. Chelnokov*","t":[{"b":5,"e":0.14286,"k":"flat","v":0.16518,"x":0.28571,"p":[[0,15,0.0,0.22322,0.16342,0.10714,0.2857,0.28571,0.0,0.57143,8,0,1,8,0,6,0,0,11,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.16518,0.17169,0.0,0.14286,0.28571,0.0,0.42857,15,0,2,15,0,3,0,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.23214,0.13716,0.14286,0.28571,0.28571,0.0,0.42857,5,0,1,5,0,8,0,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.25,0.12877,0.14286,0.28571,0.28571,0.0,0.42857,3,0,0,3,0,9,0,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.28571,0.11845,0.14286,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,8,0,0,13,0,0,10,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.22321,"p":[[0,19,0.0,0.09375,0.14555,0.0,0.0,0.14286,0.0,0.57143,20,0,4,20,0,6,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,19,0.2105,0.15179,0.18189,0.0,0.0,0.28571,0.0,0.57143,17,0,5,17,0,3,0,0,6,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[8,19,0.4211,0.15625,0.15714,0.0,0.14286,0.28571,0.0,0.42857,14,0,2,14,0,5,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.22321,0.14258,0.14286,0.2857,0.28571,0.0,0.42857,6,0,1,6,0,8,0,0,12,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3aa06b60ffa040c3","q":"Two circles  $\\omega_1$  and  $\\omega_2$ , centered at  $O_1$  and  $O_2$ , respectively, meet at points  $A$  and  $B$ . A line through  $B$  intersects  $\\omega_1$  again at  $C$  and  $\\omega_2$  again at  $D$ . The tangents to  $\\omega_1$  and  $\\omega_2$  at  $C$  and  $D$ , respectively, meet at  $E$ , and the line  $AE$  intersects the circle  $\\omega$  through  $AO_1O_2$  at  $F$ . Prove that the length of segment  $EF$  is equal to the diameter of  $\\omega$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,12,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03116,"p":[[0,24,0.0,0.03116,0.05889,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"052cebbabc72e815","q":"The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (\u2220A = \u2220A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of \u25b3ABC.","t":[{"b":0,"e":0.14,"k":"flat","v":0.02679,"x":0.1607,"p":[[0,24,0.0,0.06688,0.10085,0.0,0.0,0.14286,0.0,0.28571,21,0,2,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.07143,0.12877,0.0,0.0,0.14286,0.0,0.4286,23,0,0,23,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.09821,0.10971,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.1607,0.15868,0.0,0.14286,0.2857,0.0,0.57143,12,0,0,12,0,8,0,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[24,24,1.0,0.10259,0.1299,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,9,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.08929,"p":[[0,41,0.0,0.08929,0.13716,0.0,0.0,0.14286,0.0,0.57143,20,0,1,20,0,6,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,41,0.0976,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.05357,0.12753,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,41,0.3902,0.0625,0.11811,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.05356,0.12747,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,41,0.5854,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.0267,0.0751,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.08482,0.11769,0.0,0.0,0.14287,0.0,0.28571,20,0,0,20,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.08929,0.11152,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9981476264632d4c","q":"Let $A B C$ be an acute triangle with circumcircle $\\omega$ such that $A B<A C$. Let $M$ be the midpoint of the arc $B C$ of $\\omega$ containing the point $A$, and let $X \\neq M$ be the other point on $\\omega$ such that $A X=A M$. Points $E$ and $F$ are chosen on sides $A C$ and $A B$ of the triangle $A B C$, respectively, such that $E X=E C$ and $F X=F B$. Prove that $A E=A F$.","t":[{"b":2,"e":0.1429,"k":"flat","v":0.14732,"x":0.24553,"p":[[0,29,0.0,0.1607,0.18811,0.0,0.14286,0.1786,0.0,0.57143,13,0,5,13,0,11,0,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[4,29,0.1379,0.15179,0.14698,0.0,0.14286,0.14286,0.0,0.57143,9,0,5,9,0,17,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[8,29,0.2759,0.14732,0.12619,0.0,0.14286,0.1786,0.0,0.57143,9,0,8,9,0,15,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,29,0.4138,0.18732,0.13576,0.14286,0.14286,0.2857,0.0,0.57143,4,0,0,4,0,19,0,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,29,0.5517,0.24553,0.15663,0.14286,0.14288,0.28571,0.14286,0.85714,0,0,0,0,0,17,0,0,12,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[20,29,0.6897,0.18295,0.08922,0.14286,0.14286,0.17857,0.0,0.42857,1,0,0,1,0,23,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.1874,0.12077,0.14286,0.14286,0.2857,0.0,0.571,3,0,0,3,0,20,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[28,29,0.9655,0.21875,0.12363,0.14286,0.14288,0.28571,0.0,0.57143,2,0,0,2,0,16,0,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[29,29,1.0,0.24552,0.14387,0.14286,0.14286,0.28571,0.0,0.57143,1,0,0,1,0,16,0,0,9,0,0,3,0,0,3,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01563,"x":0.11607,"p":[[0,18,0.0,0.10268,0.1439,0.0,0.07143,0.14286,0.0,0.57143,16,0,10,16,0,13,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,18,0.2222,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,19,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.01563,0.04277,0.0,0.0,0.0,0.0,0.1429,28,0,18,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.11607,0.19378,0.0,0.0,0.21429,0.0,0.57143,23,0,17,23,0,1,0,0,0,0,0,7,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"fb3f9c8d21083d02","q":"Let  $m \\ge 2$  be an integer. Find the smallest integer  $n>m$  such that for any partition of the set  $\\{m,m+1,\\cdots,n\\}$  into two subsets, at least one subset contains three numbers  $a, b, c$  such that  $c=a^{b}$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,27,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,45,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,23,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,45,0.0889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,25,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,45,0.1778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,45,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,28,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,45,0.3556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,45,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,45,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,45,0.6222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cfc36ca6f9b27d6b","q":"Let  $I$  be the incenter,  $A_1$  and  $B_1$  midpoints of sides  $BC$  and  $AC$  of a triangle  $\\Delta ABC$ . Denote by  $M$  and  $N$  the midpoints of the arcs  $AC$  and  $BC$  of circumcircle of  $\\Delta ABC$  which do contain the other vertex of the triangle. If points  $M$ ,  $I$  and  $N$  are collinear prove that:\n\\begin{align*}\n\\angle AIB_1=\\angle BIA_1=90^{\\circ}\n\\end{align*}","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,19,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,5,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e0b04ae0ccee203d","q":"Let  $p$  and  $c$  be an prime and a composite, respectively. Prove that there exist two integers  $m,n,$  such that  $$ 0<m-n<\\frac{\\textup{lcm}(n+1,n+2,\\cdots,m)}{\\textup{lcm}(n,n+1,\\cdots,m-1)}=p^c. $$","t":[{"b":3,"e":0.2857,"k":"rising","v":0.02679,"x":0.22321,"p":[[0,29,0.0,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,24,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.07589,0.17491,0.0,0.0,0.0,0.0,0.71429,25,0,24,25,0,3,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[8,29,0.2759,0.12054,0.23176,0.0,0.0,0.03571,0.0,0.71429,24,0,21,24,0,1,0,0,1,0,0,3,0,0,0,0,0,3,0,0,0,0,0],[12,29,0.4138,0.08034,0.20494,0.0,0.0,0.0,0.0,0.71429,27,0,18,27,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[16,29,0.5517,0.13839,0.19393,0.0,0.0,0.28571,0.0,0.71429,20,0,15,20,0,0,0,0,7,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[20,29,0.6897,0.1875,0.16146,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,6,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.11607,0.14032,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,6,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.1383,0.12103,0.0,0.14286,0.2857,0.0,0.42857,11,0,0,11,0,12,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.22321,0.13803,0.14286,0.2857,0.28571,0.0,0.42857,6,0,0,6,0,7,0,0,14,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.07141,"x":0.07141,"p":[[0,3,0.0,0.07141,0.1988,0.0,0.0,0.0,0.0,0.857,27,0,22,27,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0]]}]},{"i":"199adb8f3f86ff2c","q":"You are given  $n \\ge 2$  distinct positive integers. Let's call a pair of these integers *elegant* if their sum is an integer power of  $2$ . For every  $n$  find the largest possible number of elegant pairs.\n\n*Proposed by Oleksiy Masalitin*","t":[{"b":0,"e":1.0,"k":"flat","v":0.56691,"x":0.73208,"p":[[0,30,0.0,0.64728,0.223,0.571,0.71429,0.71429,0.0,1.0,1,5,0,1,0,0,0,0,2,0,0,4,0,0,6,0,0,14,0,0,0,0,5],[4,30,0.1333,0.61156,0.18293,0.571,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,5,0,0,10,0,0,11,0,0,3,0,1],[8,30,0.2667,0.7232,0.17834,0.71429,0.71429,0.85714,0.2857,1.0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,2,0,0,14,0,0,7,0,4],[12,30,0.4,0.64732,0.18893,0.42857,0.71429,0.71429,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,11,0,0,3,0,0,11,0,0,4,0,3],[16,30,0.5333,0.56691,0.1906,0.42857,0.4998,0.71429,0.14286,1.0,0,2,0,0,0,1,0,0,0,0,0,15,0,0,5,0,0,7,0,0,2,0,2],[20,30,0.6667,0.70089,0.18681,0.57142,0.71429,0.85714,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,13,0,0,4,0,5],[24,30,0.8,0.73208,0.18477,0.571,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,4,0,0,6,0,0,11,0,0,4,0,7],[28,30,0.9333,0.6964,0.18125,0.571,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,10,0,0,8,0,3],[30,30,1.0,0.64277,0.19235,0.42857,0.57143,0.74996,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,10,0,0,5,0,0,4,0,4]]},{"b":5,"e":0.71429,"k":"flat","v":0.558,"x":0.6919,"p":[[0,9,0.0,0.59151,0.17966,0.42857,0.64286,0.71429,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,8,0,1,5,0,0,13,0,0,3,0,0],[4,9,0.4444,0.6919,0.17899,0.571,0.71429,0.74996,0.42857,1.0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,8,0,0,11,0,0,3,0,5],[8,9,0.8889,0.558,0.16113,0.42857,0.571,0.71429,0.2857,0.857,0,0,0,0,0,0,0,0,3,0,0,12,0,0,3,0,0,13,0,0,1,0,0],[9,9,1.0,0.57135,0.17127,0.42859,0.57121,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,4,0,0,7,0,0,9,0,0,10,0,0,1,0,1]]}]},{"i":"a10aca95f1a2a40a","q":"The incircle of triangle  $ABC$  for which  $AB\\neq AC$ , is tangent to sides  $BC,CA$  and  $AB$  in points  $D,E$  and  $F$  respectively. Perpendicular from  $D$  to  $EF$  intersects side  $AB$  at  $X$ , and the second intersection point of circumcircles of triangles  $AEF$  and  $ABC$  is  $T$ . Prove that  $TX\\perp TF$ .\n\n*Proposed By Pedram Safaei*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[8,20,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":[[0,46,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,46,0.087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,46,0.2609,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,46,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,46,0.4348,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,46,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,46,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"77dda718d1a1a55b","q":"Let $ABC$ be a triangle, $\\Gamma$ its circumcircle. Let $\\omega_{A}$ be the circle internally tangent to $(AB)$, $(AC)$, and $\\Gamma$. Denote $T_{A}$ as the point of tangency of $\\Gamma$ with $\\omega_{A}$. Similarly, define $T_{B}$ and $T_{C}$.\nShow that $\\left(A T_{A}\\right)$, $\\left(B T_{B}\\right)$, and $\\left(C T_{C}\\right)$ are concurrent.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.10259,"x":0.14268,"p":[[0,26,0.0,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.14286,9,0,1,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,3,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.10714,0.08748,0.0,0.14286,0.14286,0.0,0.42857,10,0,5,10,0,21,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.1292,0.0548,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,27,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.13375,0.04969,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.13384,0.07086,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.09821,"x":0.14714,"p":[[0,28,0.0,0.09821,0.06622,0.0,0.14286,0.14286,0.0,0.14286,10,0,1,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.12482,0.04718,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,1,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.14714,0.04353,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.14706,0.02492,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2e198484bb65a0a7","q":"Triangle  $\\Delta ABC$  is inscribed in circle  $\\Omega$ . The tangency point of  $\\Omega$  and the  $A$ -mixtilinear circle of  $\\Delta ABC$  is  $T$ . Points  $E$ ,  $F$  were chosen on  $AC$ ,  $AB$  respectively so that  $EF\\parallel BC$  and  $(TEF)$  is tangent to  $\\Omega$ . Let  $\\omega$  denote the  $A$ -excircle of  $\\Delta AEF$ , which is tangent to sides  $EF$ ,  $AE$ ,  $AF$  at  $K$ ,  $Y$ ,  $Z$  respectively. Line  $AT$  intersects  $\\omega$  at two points  $P$ ,  $Q$  with  $P$  between  $A$  and  $Q$ . Let  $QK$  and  $YZ$  intersect at  $V$ , and let the tangent to  $\\omega$  at  $P$  and the tangent to  $\\Omega$  at  $T$  intersect at  $U$ . Prove that  $UV\\parallel BC$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"7b927aece8c522b4","q":"There is a deck of cards placed at every points  $A_1, A_2, \\ldots , A_n$  and  $O$ , where  $n \\geq 3$ . We can do one of the following two operations at each step: $1)$  If there are more than 2 cards at some points  $A_i$ , we can withdraw three cards from that deck and place one each at  $A_{i-1}, A_{i+1}$  and  $O$ . (Here  $A_0=A_n$  and  $A_{n+1}=A_1$ ); $2)$  If there are more than or equal to  $n$  cards at point  $O$ , we can withdraw  $n$  cards from that deck and place one each at  $A_1, A_2, \\ldots , A_n$ .\nShow that if the total number of cards is more than or equal to  $n^2+3n+1$ , we can make the number of cards at every points more than or equal to  $n+1$  after finitely many steps.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,9,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,31,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"b43acfd4a740f881","q":"Let $A B C$ be a triangle and let $\\omega$ be its circumcircle. Let $\\ell_{B}$ and $\\ell_{C}$ be two parallel lines passing through $B$ and $C$ respectively. The lines $\\ell_{B}$ and $\\ell_{C}$ intersect with $\\omega$ for the second time at the points $D$ and $E$ respectively, with $D$ belonging on the arc $A B$, and $E$ on the arc $A C$. Suppose that $D A$ intersects $\\ell_{C}$ at $F$, and $E A$ intersects $\\ell_{B}$ at $G$. If $O, O_{1}$ and $O_{2}$ are the circumcenters of the triangles $A B C, A D G$ and $A E F$ respectively, and $P$ is the center of the circumcircle of the triangle $O O_{1} O_{2}$, prove that $O P$ is parallel to $\\ell_{B}$ and $\\ell_{C}$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,17,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,2,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,49,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d0cd57e9f88e9b64","q":"Let  $n$  be a positive integer. Initially a few positive integers are written on the blackboard. On one move Igor chooses two numbers  $a, b$  of the same parity on the blackboard and writes  $\\frac{a+b} {2}$ . After a few moves the numbers on the blackboard were exactly  $1, 2, \\ldots, n$ . Find the smallest possible number of positive integers that were initially written on the blackboard.","t":[{"b":3,"e":0.85714,"k":"rising","v":0.16071,"x":0.77226,"p":[[0,34,0.0,0.16071,0.27372,0.0,0.0,0.17868,0.0,0.85714,21,0,6,21,0,3,0,0,2,0,0,1,0,0,1,0,0,2,0,0,2,0,0],[4,34,0.1176,0.52662,0.24326,0.28571,0.5712,0.71429,0.0,1.0,1,1,1,1,0,3,0,0,5,0,0,4,0,0,7,0,0,8,0,0,3,0,1],[8,34,0.2353,0.52677,0.20334,0.42857,0.5005,0.71407,0.0,0.85714,1,0,0,1,0,0,0,0,5,0,0,10,0,0,6,0,0,6,0,0,4,0,0],[12,34,0.3529,0.58456,0.20638,0.42857,0.64071,0.71429,0.14,0.85714,0,0,0,0,0,2,0,0,3,0,0,6,0,0,5,0,0,11,0,0,5,0,0],[16,34,0.4706,0.70534,0.2141,0.71429,0.71429,0.85714,0.0,1.0,1,2,0,1,0,0,0,0,2,0,0,2,0,0,2,0,0,12,0,0,11,0,2],[20,34,0.5882,0.74997,0.13363,0.71429,0.71429,0.857,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,14,0,0,9,0,3],[24,34,0.7059,0.76783,0.15466,0.71429,0.857,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,5,0,0,8,0,0,13,0,4],[28,34,0.8235,0.77226,0.13302,0.71429,0.85707,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,8,0,0,16,0,2],[32,34,0.9412,0.75444,0.15252,0.71429,0.71429,0.85714,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,10,0,0,12,0,3],[34,34,1.0,0.75891,0.16147,0.71429,0.85707,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,8,0,0,16,0,2]]},{"b":5,"e":0.28571,"k":"flat","v":0.1873,"x":0.29464,"p":[[0,35,0.0,0.18749,0.26349,0.0,0.0,0.28571,0.0,1.0,18,1,4,18,0,2,0,0,5,0,0,3,0,0,1,0,0,2,0,0,0,0,1],[4,35,0.1143,0.23643,0.16992,0.14286,0.14288,0.42857,0.0,0.71429,5,0,3,5,0,12,0,0,6,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[8,35,0.2286,0.20982,0.14279,0.14286,0.14288,0.28571,0.0,0.4286,6,0,3,6,0,11,0,0,9,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.1873,0.14478,0.14214,0.14286,0.2857,0.0,0.571,6,0,1,6,0,15,0,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,35,0.4571,0.26777,0.13725,0.14286,0.2857,0.42857,0.0,0.57143,2,0,0,2,0,10,0,0,11,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[20,35,0.5714,0.20079,0.1551,0.14214,0.14286,0.2857,0.0,0.571,7,0,2,7,0,12,0,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[24,35,0.6857,0.24997,0.15967,0.14286,0.2857,0.42857,0.0,0.571,4,0,0,4,0,11,0,0,8,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[28,35,0.8,0.29464,0.15542,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,11,0,0,12,0,0,7,0,0,1,0,0,0,0,0,1,0,0],[32,35,0.9143,0.24545,0.14833,0.14286,0.2857,0.28571,0.0,0.57143,3,0,1,3,0,12,0,0,10,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[35,35,1.0,0.22316,0.15954,0.14286,0.14286,0.42857,0.0,0.571,5,0,1,5,0,14,0,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"cebd6f08deb52481","q":"In a meeting of  $4042$  people, there are  $2021$  couples, each consisting of two people. Suppose that  $A$  and  $B$ , in the meeting, are friends when they know each other. For a positive integer  $n$ , each people chooses an integer from  $-n$  to  $n$  so that the following conditions hold. (Two or more people may choose the same number). \n\n\n- Two or less people chose  $0$ , and if exactly two people chose  $0$ , they are coupled.\n- Two people are either coupled or don't know each other if they chose the same number.\n- Two people are either coupled or know each other if they chose two numbers that sum to  $0$ .\n\nDetermine the least possible value of  $n$  for which such number selecting is always possible.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,42,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,4,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,42,0.1905,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.02232,0.08827,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,2,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,4,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.01107,"p":[[0,6,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.01107,0.03587,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9ace6ac40f354f6d","q":"The sequence of real numbers $a_{0}, a_{1}, a_{2}, \\ldots$ is defined recursively by  $$ a_{0}=-1, \\quad \\sum_{k=0}^{n} \\frac{a_{n-k}}{k+1}=0 \\quad \\text { for } \\quad n \\geq 1 $$  Show that $a_{n}>0$ for $n \\geq 1$.  (Poland)","t":[{"b":5,"e":0.14286,"k":"flat","v":0.15625,"x":0.51786,"p":[[0,18,0.0,0.29912,0.30797,0.14286,0.14286,0.2857,0.14286,1.0,0,5,0,0,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[4,18,0.2222,0.51786,0.41458,0.14286,0.21429,1.0,0.0,1.0,1,13,0,1,0,15,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,13],[8,18,0.4444,0.45518,0.40803,0.14286,0.14286,1.0,0.0,1.0,1,11,0,1,0,18,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,11],[12,18,0.6667,0.32589,0.30978,0.14286,0.14286,0.28571,0.14286,1.0,0,5,0,0,0,20,0,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,5],[16,18,0.8889,0.15625,0.04164,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.16518,0.08073,0.14286,0.14286,0.14286,0.14286,0.57143,0,0,0,0,0,29,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"volatile","v":0.54902,"x":0.99554,"p":[[0,4,0.0,0.54902,0.4059,0.14286,0.28571,1.0,0.14,1.0,0,14,0,0,0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,14],[4,4,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"a6167905d0a047f7","q":"Let $A, B, C, D, E, F$ be points on a circle with $A E \\| B D$ and $B C \\| D F$. By reflection across the line $C E$, the point $D$ is mapped to $X$. Show that $X$ is as far from the line $E F$ as $B$ is from $A C$.","t":[{"b":0,"e":0.0,"k":"falling","v":0.09375,"x":0.55801,"p":[[0,57,0.0,0.55801,0.34692,0.28571,0.57143,0.85714,0.0,1.0,5,7,1,5,0,0,0,0,7,0,0,1,0,0,4,0,0,5,0,0,3,0,7],[4,57,0.0702,0.33482,0.38731,0.0,0.14288,0.5,0.0,1.0,12,7,0,12,0,5,0,0,5,0,0,2,0,0,0,0,0,1,0,0,0,0,7],[8,57,0.1404,0.22321,0.25738,0.0,0.14286,0.28571,0.0,1.0,13,1,1,13,0,4,0,0,9,0,0,2,0,0,0,0,0,3,0,0,0,0,1],[12,57,0.2105,0.1651,0.16017,0.0,0.14286,0.2857,0.0,0.4286,11,0,0,11,0,12,0,0,2,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[16,57,0.2807,0.12054,0.12931,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,14,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,57,0.3509,0.09375,0.12169,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,8,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,57,0.4211,0.14733,0.13592,0.0,0.14286,0.1429,0.0,0.4286,10,0,0,10,0,15,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,57,0.4912,0.11161,0.12745,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,57,0.5614,0.13839,0.15355,0.0,0.14286,0.2857,0.0,0.4286,15,0,0,15,0,7,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[36,57,0.6316,0.14277,0.15972,0.0,0.14286,0.2857,0.0,0.57143,14,0,0,14,0,9,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[40,57,0.7018,0.15179,0.14699,0.0,0.14286,0.1786,0.0,0.4286,11,0,0,11,0,13,0,0,3,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[44,57,0.7719,0.10706,0.11843,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,14,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,57,0.8421,0.10268,0.12492,0.0,0.07143,0.14287,0.0,0.4286,16,0,0,16,0,11,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.14286,0.13832,0.0,0.14286,0.14287,0.0,0.4286,11,0,0,11,0,14,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.17857,0.13832,0.10714,0.14286,0.2857,0.0,0.5714,8,0,0,8,0,11,0,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[57,57,1.0,0.15625,0.14445,0.0,0.14286,0.2857,0.0,0.4286,11,0,0,11,0,11,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"falling","v":0.09366,"x":0.75441,"p":[[0,37,0.0,0.47321,0.36147,0.14286,0.35714,0.75,0.0,1.0,6,6,0,6,0,3,0,0,7,0,0,2,0,0,1,0,0,5,0,0,2,0,6],[4,37,0.1081,0.61607,0.36672,0.28571,0.71429,1.0,0.0,1.0,4,11,0,4,0,2,0,0,5,0,0,0,0,0,3,0,0,5,0,0,2,0,11],[8,37,0.2162,0.65623,0.329,0.28571,0.85707,1.0,0.0,1.0,1,10,1,1,0,1,0,0,9,0,0,1,0,0,2,0,0,1,0,0,7,0,10],[12,37,0.3243,0.75441,0.28625,0.57132,0.857,1.0,0.0,1.0,1,12,1,1,0,1,0,0,3,0,0,1,0,0,3,0,0,3,0,0,8,0,12],[16,37,0.4324,0.65175,0.31327,0.42857,0.71429,0.89286,0.0,1.0,2,8,1,2,0,2,0,0,3,0,0,2,0,0,5,0,0,4,0,0,6,0,8],[20,37,0.5405,0.21875,0.1889,0.0,0.2143,0.28571,0.0,0.714,9,0,0,9,0,7,0,0,10,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[24,37,0.6486,0.12947,0.16115,0.0,0.0,0.2857,0.0,0.4286,17,0,0,17,0,6,0,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.09366,0.12679,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,9,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.16063,0.17406,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,11,0,0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[36,37,0.973,0.15617,0.15715,0.0,0.14286,0.28571,0.0,0.4286,13,0,0,13,0,8,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.20982,0.17852,0.10714,0.14288,0.2857,0.0,0.71429,8,0,0,8,0,10,0,0,8,0,0,4,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"62f8bc606600faf4","q":"Let  $n\\in\\mathbb{N}$ ,  $n\\geq 2$ . Find all values of  $k\\in\\mathbb{N}$ ,  $k\\geq 1$ , for which the following statement holds:   $$ \\text{\"If }A\\in\\mathcal{M}_n(\\mathbb{C})\\text{ is such that }A^kA^*=A\\text{, then }A=A^*\\text{.\"} $$   (here,  $A^*$  denotes the conjugate transpose of  $A$ ).","t":[{"b":1,"e":0.0,"k":"falling","v":0.68297,"x":0.83481,"p":[[0,8,0.0,0.83481,0.20239,0.71429,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,2,0,0,3,0,0,4,0,0,7,0,15],[4,8,0.5,0.68297,0.30039,0.42859,0.71429,1.0,0.0,1.0,1,10,1,1,0,1,0,0,5,0,0,2,0,0,4,0,0,4,0,0,5,0,10]]},{"b":4,"e":0.71429,"k":"flat","v":0.77231,"x":0.9375,"p":[[0,34,0.0,0.77231,0.24964,0.67857,0.85714,1.0,0.0,1.0,1,10,1,1,0,0,0,0,1,0,0,4,0,0,2,0,0,3,0,0,11,0,10],[4,34,0.1176,0.88393,0.14032,0.85714,0.85714,1.0,0.42857,1.0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,11,0,15],[8,34,0.2353,0.85713,0.12877,0.85711,0.85714,1.0,0.42857,1.0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,17,0,9],[12,34,0.3529,0.86607,0.19865,0.85714,0.85714,1.0,0.0,1.0,1,14,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,13,0,14],[16,34,0.4706,0.7991,0.16311,0.71429,0.85714,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,13,0,7],[20,34,0.5882,0.88392,0.0974,0.85711,0.85714,1.0,0.71429,1.0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,16,0,11],[24,34,0.7059,0.875,0.12753,0.71429,0.85714,1.0,0.57143,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,9,0,14],[28,34,0.8235,0.875,0.13716,0.85714,0.85714,1.0,0.42857,1.0,0,11,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,18,0,11],[32,34,0.9412,0.9375,0.07087,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[34,34,1.0,0.8616,0.13115,0.85711,0.85714,1.0,0.42857,1.0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,4,0,0,16,0,10]]}]},{"i":"7bccefb6851459aa","q":"Let  $T$  be a triangulation of a  $100$ -gon.We construct  $P(T)$  by copying the same  $100$ -gon and drawing a diagonal if it was not drawn in  $T$  an there is a quadrilateral with this diagonal and two other vertices so that all the sides and diagonals(Except the one we are going to draw) are present in  $T$ .Let  $f(T)$  be the number of intersections of diagonals in  $P(T)$ .Find the minimum and maximum of  $f(T)$ .","t":[{"b":2,"e":0.57143,"k":"rising","v":0.09375,"x":0.41071,"p":[[0,39,0.0,0.09375,0.08459,0.0,0.14286,0.14286,0.0,0.28571,13,0,11,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,6,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.16955,0.12597,0.14286,0.14286,0.14286,0.0,0.57143,3,0,2,3,0,25,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,39,0.3077,0.17839,0.13837,0.14286,0.14286,0.14286,0.0,0.57143,3,0,3,3,0,24,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[16,39,0.4103,0.19616,0.13727,0.14286,0.14286,0.14286,0.0,0.57143,1,0,1,1,0,25,0,0,2,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[20,39,0.5128,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,2,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.33033,0.197,0.14286,0.28571,0.42858,0.14286,0.85714,0,0,0,0,0,13,0,0,6,0,0,6,0,0,5,0,0,1,0,0,1,0,0],[28,39,0.7179,0.31687,0.18475,0.14286,0.35714,0.42858,0.0,0.57143,1,0,1,1,0,14,0,0,1,0,0,9,0,0,7,0,0,0,0,0,0,0,0],[32,39,0.8205,0.3258,0.15261,0.14286,0.42857,0.42857,0.14,0.57143,0,0,0,0,0,12,0,0,2,0,0,15,0,0,3,0,0,0,0,0,0,0,0],[36,39,0.9231,0.2991,0.1488,0.14286,0.2857,0.42857,0.14286,0.57143,0,0,0,0,0,12,0,0,9,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[39,39,1.0,0.41071,0.16656,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,7,0,0,2,0,0,12,0,0,10,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.07143,"x":0.13839,"p":[[0,29,0.0,0.07143,0.07986,0.0,0.0,0.14286,0.0,0.2857,17,0,15,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.11607,0.1729,0.0,0.14286,0.14286,0.0,1.0,12,1,10,12,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,29,0.2759,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,4,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.13383,0.15538,0.0,0.14286,0.14286,0.0,0.714,11,0,8,11,0,17,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[16,29,0.5517,0.11607,0.06621,0.14286,0.14286,0.14286,0.0,0.2857,7,0,6,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.14286,4,0,1,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.12938,0.04161,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d81d076f6c7b45d7","q":"Let  $ X$  be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset  $ Y$  of  $ X$  such that  $ a \\minus{} b \\plus{} c \\minus{} d \\plus{} e$  is not divisible by 47 for any  $ a,b,c,d,e \\in Y.$ \r\n\r\n*Author: Gerhard W\u00f6ginger, Netherlands*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.02679,"x":0.15625,"p":[[0,31,0.0,0.14732,0.2382,0.0,0.07143,0.14286,0.0,1.0,16,1,14,16,0,10,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[4,31,0.129,0.15617,0.23788,0.0,0.07,0.14286,0.0,1.0,16,1,16,16,0,10,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[8,31,0.2581,0.15625,0.31003,0.0,0.0,0.14286,0.0,1.0,22,3,20,22,0,4,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[12,31,0.3871,0.125,0.3004,0.0,0.0,0.0,0.0,1.0,25,3,24,25,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[16,31,0.5161,0.04911,0.16602,0.0,0.0,0.0,0.0,0.85714,28,0,27,28,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[20,31,0.6452,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,28,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.125,"x":0.20088,"p":[[0,21,0.0,0.125,0.21651,0.0,0.0,0.14286,0.0,1.0,18,1,16,18,0,9,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,1],[4,21,0.1905,0.18302,0.22651,0.0,0.14286,0.14286,0.0,1.0,11,1,10,11,0,14,0,0,1,0,0,2,0,0,3,0,0,0,0,0,0,0,1],[8,21,0.381,0.20088,0.17442,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,25,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[12,21,0.5714,0.18295,0.10856,0.14286,0.14286,0.14286,0.0,0.57143,1,0,1,1,0,25,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,21,0.7619,0.16054,0.07788,0.14286,0.14286,0.14286,0.0,0.42857,2,0,2,2,0,25,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7031f2971adb9cef","q":"Let  $n\\ge 2$  be a positive integer. For any integer  $a$ , let  $P_a(x)$  denote the polynomial  $x^n+ax$ . Let  $p$  be a prime number and define the set  $S_a$  as the set of residues mod  $p$  that  $P_a(x)$  attains. That is,   $$ S_a=\\{b\\mid 0\\le b\\le p-1,\\text{  and there is }c\\text{ such that }P_a(c)\\equiv b \\pmod{p}\\}. $$  Show that the expression  $\\frac{1}{p-1}\\sum\\limits_{a=1}^{p-1}|S_a|$  is an integer.\n\n*Proposed by fattypiggy123*","t":[{"b":1,"e":0.14286,"k":"flat","v":0.08929,"x":0.14286,"p":[[0,5,0.0,0.08929,0.08564,0.0,0.14286,0.14286,0.0,0.28571,14,0,6,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.12045,0.08071,0.14214,0.14286,0.14286,0.0,0.42857,7,0,3,7,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.08929,"x":0.20089,"p":[[0,25,0.0,0.11152,0.07768,0.0,0.14286,0.14286,0.0,0.28571,9,0,6,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.14286,12,0,7,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.20089,0.26693,0.14286,0.14286,0.14286,0.0,1.0,7,3,6,7,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[12,25,0.48,0.12054,0.17536,0.0,0.14286,0.14286,0.0,1.0,12,1,10,12,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,25,0.64,0.15179,0.17105,0.14286,0.14286,0.14286,0.0,1.0,6,1,4,6,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[20,25,0.8,0.13393,0.16728,0.10714,0.14286,0.14286,0.0,1.0,8,1,7,8,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,25,0.96,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,7,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,7,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"22e7af6b2ba9b998","q":"Let  $S$  be a set of  $n$  points in the coordinate plane.  Say that a pair of points is *aligned* if the two points have the same  $x$ -coordinate or  $y$ -coordinate.  Prove that  $S$  can be partitioned into disjoint subsets such that (a) each of these subsets is a collinear set of points, and (b) at most  $n^{3/2}$  unordered pairs of distinct points in  $S$  are aligned but not in the same subset.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,10,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,3,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,10,0.0,0.04464,0.14032,0.0,0.0,0.0,0.0,0.71429,28,0,5,28,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,10,0.4,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"bfb5bcdf53898aa3","q":"There is given a convex quadrilateral  $ ABCD$ . Prove that there exists a point  $ P$  inside the quadrilateral such that\n\\[\n\\angle PAB \\plus{} \\angle PDC \\equal{} \\angle PBC \\plus{} \\angle PAD \\equal{} \\angle PCD \\plus{} \\angle PBA \\equal{} \\angle PDA \\plus{} \\angle PCB = 90^{\\circ}\n\\]\nif and only if the diagonals  $ AC$  and  $ BD$  are perpendicular.\n\n*Proposed by Dusan Djukic, Serbia*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0267,"p":[[0,36,0.0,0.0267,0.08317,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0267,"p":[[0,25,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"c5bdcfa627d38aaf","q":"The diagonals of a cyclic quadrilateral meet at point  $M$ . A circle  $\\omega$  touches segments  $MA$  and  $MD$  at points  $P,Q$  respectively and touches the circumcircle of  $ABCD$  at point  $X$ . Prove that  $X$  lies on the radical axis of circles  $ACQ$  and  $BDP$ .\n\n*(Proposed by Ivan Frolov)*","t":[{"b":6,"e":0.14286,"k":"flat","v":0.12045,"x":0.14286,"p":[[0,31,0.0,0.12482,0.04718,0.14286,0.14286,0.14286,0.0,0.14286,4,0,1,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.14286,1e-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],[24,31,0.7742,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.12018,"x":0.14286,"p":[[0,33,0.0,0.12018,0.05173,0.14,0.14286,0.14286,0.0,0.14286,5,0,1,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,33,0.6061,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4935006ee8119002","q":"Let  $a,b,c,d$  be odd integers such that  $0<a<b<c<d$  and  $ad=bc$ . Prove that if  $a+d=2^k$  and  $b+c=2^m$  for some integers  $k$  and  $m$ , then  $a=1$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.05348,"p":[[0,6,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.05348,0.09271,0.0,0.0,0.14286,0.0,0.42857,22,0,5,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,42,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,42,0.1905,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,42,0.381,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,2,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,42,0.9524,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[42,42,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6a4c19d23b18ae42","q":"Let $P(x), Q(x), R(x)$ and $S(x)$ be non-constant polynomials with real coefficients such that $P(Q(x))=R(S(x))$. Suppose that the degree of $P(x)$ is divisible by the degree of $R(x)$.\nProve that there is a polynomial $T(x)$ with real coefficients such that $P(x)=R(T(x))$.\n\n## Iran, NaVid Safaei","t":[{"b":1,"e":0.0,"k":"flat","v":0.06696,"x":0.1183,"p":[[0,38,0.0,0.08929,0.06916,0.0,0.14286,0.14286,0.0,0.1429,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.09589,0.06581,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,1,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.08929,0.06682,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,2,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.1183,0.0524,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,1,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.08259,0.06941,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.09143,0.06731,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,1,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.1183,0.0524,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,1,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.09598,0.06588,0.0,0.14286,0.14286,0.0,0.14286,10,0,0,10,1,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.06696,0.07129,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.01339,"x":0.08036,"p":[[0,26,0.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.14286,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.05357,0.06682,0.0,0.0,0.14286,0.0,0.14286,19,0,0,19,2,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2dfc6b1121d0a50a","q":"Let $ABC$ be a non-isosceles triangle at $B$. Let $D$ be the foot of the angle bisector of $\\angle ABC$. Let $M$ be the midpoint of the arc $\\widehat{AC}$ containing $B$ on the circumcircle of triangle $ABC$. The circumcircle of triangle $BDM$ intersects the segment $[AB]$ at a point $K$ distinct from $B$. Let $J$ be the symmetric point of $A$ with respect to point $K$. The line $(DJ)$ intersects the line $(AM)$ at a point $O$. Show that the points $J, B, M$, and $O$ are concyclic.","t":[{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.14284,"p":[[0,18,0.0,0.09375,0.21312,0.0,0.0,0.0,0.0,0.85714,25,0,1,25,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,0],[4,18,0.2222,0.13393,0.24728,0.0,0.0,0.07143,0.0,0.85714,24,0,3,24,0,0,0,0,1,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[8,18,0.4444,0.14284,0.29879,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,1],[12,18,0.6667,0.04009,0.07337,0.0,0.0,0.035,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.13836,"p":[[0,55,0.0,0.10714,0.26487,0.0,0.0,0.0,0.0,1.0,27,1,2,27,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,1],[4,55,0.0727,0.13836,0.26598,0.0,0.0,0.07143,0.0,0.85714,24,0,3,24,0,0,0,0,3,0,0,0,0,0,2,0,0,1,0,0,2,0,0],[8,55,0.1455,0.0625,0.18877,0.0,0.0,0.0,0.0,0.85714,28,0,3,28,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[12,55,0.2182,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,8,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.01784,0.09935,0.0,0.0,0.0,0.0,0.571,31,0,8,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,55,0.3636,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,6,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,7,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,13,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,10,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.02223,0.06281,0.0,0.0,0.0,0.0,0.2857,28,0,11,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"98bcfdd3ee95b54a","q":"We choose 100 points in the coordinate plane. Let  $N$  be the number of triples  $(A,B,C)$  of distinct chosen points such that  $A$  and  $B$  have the same  $y$ -coordinate, and  $B$  and  $C$  have the same  $x$ -coordinate. Find the greatest value that  $N$  can attain considering all possible ways to choose the points.","t":[{"b":3,"e":0.57143,"k":"flat","v":0.44197,"x":0.48214,"p":[[0,8,0.0,0.45536,0.09061,0.42857,0.42857,0.57143,0.28571,0.57143,0,0,0,0,0,0,0,0,4,0,0,18,0,0,10,0,0,0,0,0,0,0,0],[4,8,0.5,0.44197,0.08268,0.42857,0.42857,0.4286,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,21,0,0,7,0,0,0,0,0,0,0,0],[8,8,1.0,0.48214,0.07784,0.42857,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,18,0,0,13,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.42857,"k":"flat","v":0.41518,"x":0.47098,"p":[[0,17,0.0,0.47098,0.11691,0.42857,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,1,18,0,0,8,0,0,1,0,0,1,0,0],[4,17,0.2353,0.41518,0.09689,0.28571,0.42857,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,9,0,0,17,0,0,6,0,0,0,0,0,0,0,0],[8,17,0.4706,0.45982,0.09268,0.42857,0.42857,0.57143,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,17,0,0,11,0,0,0,0,0,0,0,0],[12,17,0.7059,0.43302,0.07561,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,23,0,0,5,0,0,0,0,0,0,0,0],[16,17,0.9412,0.42858,0.07143,0.42857,0.42857,0.4286,0.28571,0.57143,0,0,0,0,0,0,0,0,4,0,0,24,0,0,4,0,0,0,0,0,0,0,0],[17,17,1.0,0.42856,0.0714,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,24,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"8c1afed0c311e195","q":"Two circles are internally tangent at a point  $A$ . Let  $C$  be a point on the smaller circle other than  $A$ . The tangent line to the smaller circle at  $C$  meets the bigger circle at  $D$  and  $E$ ; and the line  $AC$  meets the bigger circle at  $A$  and  $P$ . Show that the line  $PE$  is tangent to the circle through  $A$ ,  $C$ , and  $E$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,16,0.0,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,2,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.01786,"p":[[0,11,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"322c65b866da09a9","q":"The incircle of the acute-angled triangle  $ABC$  is tangent to the sides  $AB, BC, CA$  at points  $C_1, A_1, B_1,$  respectively, and  $I$  is the incenter. Let the midpoint of side  $BC$  be  $M.$  Let  $J$  be the foot of the altitude drawn from  $M$  to  $C_1B_1.$  The tangent drawn from  $B$  to the circumcircle of  $\\triangle BIC$  intersects  $IJ$  at  $X.$  If the circumcircle of  $\\triangle AXI$  intersects  $AB$  at  $Y,$  prove that  $BY = BM.$","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,39,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[4,18,0.2222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"dea85b87538ff220","q":"There are  $5$  boxes arranged in a circle. At first, there is one a box containing one ball, while the other boxes are empty. At each step, we can do one of the following two operations:\ni. select one box that is not empty, remove one ball from the box and add one ball into both boxes next to the box,\nii. select an empty box next to a non-empty box, from the box the non-empty one moves one ball to the empty box.\nIs it possible, that after a few steps, obtained conditions where each box contains exactly  $17^{5^{2016}}$  balls?","t":[{"b":0,"e":1.0,"k":"volatile","v":0.11161,"x":1.0,"p":[[0,28,0.0,0.25,0.40248,0.0,0.0,0.42857,0.0,1.0,22,6,0,22,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,6],[4,28,0.1429,0.11161,0.29609,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[8,28,0.2857,0.25447,0.36375,0.0,0.0,0.42858,0.0,1.0,20,4,0,20,0,0,0,0,0,0,0,5,0,0,2,0,0,0,0,0,1,0,4],[12,28,0.4286,0.34821,0.44883,0.0,0.0,1.0,0.0,1.0,19,9,0,19,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,9],[16,28,0.5714,0.24107,0.39031,0.0,0.0,0.42857,0.0,1.0,22,6,0,22,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,6],[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":1,"e":0.42857,"k":"flat","v":0.0,"x":0.26339,"p":[[0,100,0.0,0.16964,0.34337,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,4],[4,100,0.04,0.23661,0.38731,0.0,0.0,0.42857,0.0,1.0,21,6,0,21,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,6],[8,100,0.08,0.17857,0.34069,0.0,0.0,0.10714,0.0,1.0,24,4,0,24,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[12,100,0.12,0.19196,0.31866,0.0,0.0,0.42857,0.0,1.0,22,3,0,22,0,0,0,0,0,0,0,6,0,0,1,0,0,0,0,0,0,0,3],[16,100,0.16,0.15179,0.31326,0.0,0.0,0.0,0.0,1.0,25,3,1,25,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,3],[20,100,0.2,0.08929,0.25692,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[24,100,0.24,0.07589,0.24996,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[28,100,0.28,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,100,0.32,0.12054,0.30117,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[36,100,0.36,0.08036,0.25738,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[40,100,0.4,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,100,0.44,0.08929,0.25692,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[48,100,0.48,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,100,0.52,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,100,0.56,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,100,0.6,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,100,0.64,0.04018,0.16457,0.0,0.0,0.0,0.0,0.85714,30,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[68,100,0.68,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,100,0.72,0.1875,0.21261,0.0,0.0,0.42857,0.0,0.4286,18,0,0,18,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0],[76,100,0.76,0.26339,0.2055,0.0,0.42857,0.42857,0.0,0.42857,12,0,0,12,0,0,0,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[80,100,0.8,0.26339,0.1992,0.0,0.42857,0.42857,0.0,0.4286,11,0,0,11,0,1,0,0,2,0,0,18,0,0,0,0,0,0,0,0,0,0,0],[84,100,0.84,0.18304,0.20896,0.0,0.0,0.42857,0.0,0.42857,18,0,0,18,0,0,0,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[88,100,0.88,0.22768,0.21387,0.0,0.42857,0.42857,0.0,0.4286,15,0,0,15,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[92,100,0.92,0.22322,0.2111,0.0,0.35714,0.42857,0.0,0.4286,15,0,0,15,0,0,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,0,0],[96,100,0.96,0.20982,0.22864,0.0,0.0,0.42857,0.0,0.71429,17,0,0,17,0,0,0,0,0,0,0,14,0,0,0,0,0,1,0,0,0,0,0],[100,100,1.0,0.1875,0.2126,0.0,0.0,0.42857,0.0,0.42857,18,0,0,18,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6e15752de6acc967","q":"There are  $100$  cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.04455,"p":[[0,29,0.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,4,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.04455,0.09052,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.05348,"p":[[0,11,0.0,0.05348,0.0927,0.0,0.0,0.14071,0.0,0.28571,23,0,3,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,10,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"215aff06ca390e5b","q":"Triangle  $ABC$  is an scalene triangle. Let  $I$  the incenter,  $\\Omega$  the circumcircle,  $E$  the  $A$ -excenter of triangle  $ABC$ . Let  $\\Gamma$  the circle centered at  $E$  and passes  $A$ .  $\\Gamma$  and  $\\Omega$  intersect at point  $D(\\neq A)$ , and the perpendicular line of  $BC$  which passes  $A$  meets  $\\Gamma$  at point  $K(\\neq A)$ .  $L$  is the perpendicular foot from  $I$  to  $AC$ . Now if  $AE$  and  $DK$  intersects at  $F$ , prove that  $BE\\cdot CI=2\\cdot CF\\cdot CL$ .","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"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],[4,10,0.4,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,11,0.0,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,1,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,11,0.3636,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4abe39e8a3251da2","q":"Let  $ p$  and  $ q$  be relatively prime positive integers. A subset  $ S$  of  $ \\{0, 1, 2, \\ldots \\}$  is called **ideal** if  $ 0 \\in S$  and for each element  $ n \\in S,$  the integers  $ n \\plus{} p$  and  $ n \\plus{} q$  belong to  $ S.$  Determine the number of ideal subsets of  $ \\{0, 1, 2, \\ldots \\}.$","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.01786,"p":[[0,8,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,15,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,8,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,15,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,21,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,9,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1da855d08c65d706","q":"We are given  $n$  distinct rectangles in the plane. Prove that between the  $4n$  interior angles formed by these rectangles at least  $4\\sqrt n$  are distinct.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,126,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,126,0.0317,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,126,0.0635,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,126,0.0952,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,126,0.127,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,126,0.1587,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,126,0.1905,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,126,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,126,0.254,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,126,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,126,0.3175,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,126,0.3492,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,126,0.381,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,126,0.4127,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,126,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,126,0.4762,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,126,0.5079,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,126,0.5397,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,126,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,126,0.6032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,126,0.6349,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,126,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,126,0.6984,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,126,0.7302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,126,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,126,0.7937,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,126,0.8254,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,126,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,126,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,126,0.9206,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,126,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,126,0.9841,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[126,126,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,125,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,125,0.032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,125,0.064,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,125,0.096,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,125,0.128,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,125,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,125,0.192,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,125,0.224,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,125,0.256,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,125,0.288,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,125,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,125,0.352,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,125,0.384,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,125,0.416,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,125,0.448,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,125,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,125,0.512,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,125,0.544,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,125,0.576,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,125,0.608,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,125,0.64,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,125,0.672,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,125,0.704,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[92,125,0.736,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,125,0.768,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,125,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[104,125,0.832,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[108,125,0.864,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[112,125,0.896,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[116,125,0.928,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[120,125,0.96,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[124,125,0.992,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[125,125,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d7765581e3e685f8","q":"The incenter of triangle  $ABC$  is  $ I$ . the circumcircle of  $ABC$  is tangent to  $BC$ ,  $CA$ ,  $AB$  at  $T, E, F$ .  $R$  is a point on  $BC$  . Let the  $C$ -excenter of  $\\vartriangle CER$  be  $L$ . Prove that points  $L,T,F$  are collinear if and only if  $B,E,A,R$  are concyclic.\n\n*proposed by kyou46*","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.08472,"p":[[0,13,0.0,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,13,0.3077,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.04009,0.0816,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.08472,0.20153,0.0,0.0,0.14071,0.0,1.0,23,1,0,23,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[13,13,1.0,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"flat","v":0.04911,"x":0.19642,"p":[[0,32,0.0,0.12946,0.29093,0.0,0.0,0.14286,0.0,1.0,23,3,0,23,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,32,0.125,0.07589,0.21124,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,32,0.25,0.0625,0.18536,0.0,0.0,0.0,0.0,1.0,26,1,1,26,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,32,0.375,0.08482,0.1984,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,32,0.5,0.04911,0.17717,0.0,0.0,0.0,0.0,1.0,27,1,1,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,32,0.625,0.11606,0.23536,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,5,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,1],[24,32,0.75,0.19642,0.28737,0.0,0.0,0.42857,0.0,1.0,17,2,0,17,0,6,0,0,0,0,0,5,0,0,1,0,0,1,0,0,0,0,2],[28,32,0.875,0.14732,0.27545,0.0,0.0,0.2857,0.0,1.0,22,2,0,22,0,1,0,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,2],[32,32,1.0,0.14286,0.32927,0.0,0.0,0.0,0.0,1.0,25,4,0,25,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4]]}]},{"i":"e13fab30c7a44d31","q":"Let $A B C$ be an acute triangle satisfying the condition $A B>B C$ and $A C>B C$. Denote by $O$ and $H$ the circumcenter and the orthocenter, respectively, of the triangle $A B C$. Suppose that the circumcircle of the triangle $A H C$ intersects the line $A B$ at $M$ different from $A$, and that the circumcircle of the triangle $A H B$ intersects the line $A C$ at $N$ different from $A$. Prove that the circumcenter of the triangle $M N H$ lies on the line $O H$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.06688,"x":0.1741,"p":[[0,10,0.0,0.06696,0.11837,0.0,0.0,0.14286,0.0,0.42857,23,0,2,23,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.06696,0.11285,0.0,0.0,0.14286,0.0,0.28571,23,0,2,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.06688,0.10085,0.0,0.0,0.14286,0.0,0.28571,21,0,1,21,0,7,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.1741,0.13235,0.0,0.2857,0.28571,0.0,0.28571,11,0,0,11,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.03125,"x":0.08036,"p":[[0,27,0.0,0.0759,0.11837,0.0,0.0,0.1429,0.0,0.28571,22,0,3,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,3,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.06679,0.08722,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.08036,0.1234,0.0,0.0,0.14287,0.0,0.42857,21,0,0,21,0,5,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.07589,0.12869,0.0,0.0,0.14287,0.0,0.42857,23,0,0,23,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.04456,0.07513,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63f393b31a64012f","q":"Let  $\\mathcal{P}$  be a partition of  $\\{1,2,\\dots ,2024\\}$  into sets of two elements, such that for any  $\\{a,b\\}\\in\\mathcal{P}$ , either  $|a-b|=1$  or  $|a-b|=506$ . Suppose that  $\\{1518,1519\\}\\in\\mathcal{P}$ . Determine the pair of  $505$  in the partition.","t":[{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.06696,"p":[[0,7,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,11,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,13,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.14277,"p":[[0,16,0.0,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,10,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,16,0.25,0.10714,0.29233,0.0,0.0,0.0,0.0,1.0,27,3,18,27,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,16,0.5,0.09375,0.29148,0.0,0.0,0.0,0.0,1.0,29,3,12,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[12,16,0.75,0.14277,0.32733,0.0,0.0,0.035,0.0,1.0,24,4,10,24,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4]]}]},{"i":"9b5535350eac3f52","q":"The points  $D, E, F$  lie respectively on the sides  $BC$ ,  $CA$ ,  $AB$  of the triangle ABC such that  $F B = BD$ ,  $DC = CE$ , and the lines  $EF$  and  $BC$  are parallel. Tangent to the circumscribed circle of triangle  $DEF$  at point  $F$  intersects line  $AD$  at point  $P$ . Perpendicular bisector of segment   $EF$  intersects the segment  $AC$  at  $Q$ . Prove that the lines  $P Q$  and  $BC$  are parallel.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.03125,"x":0.14286,"p":[[0,28,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.13831,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.14286,1e-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],[24,28,0.8571,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.04446,"x":0.13839,"p":[[0,21,0.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.04446,0.06595,0.0,0.0,0.14071,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.05804,0.11214,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,1,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c9dc60b500162967","q":"The incircle of a scalene triangle  $ABC$  touches the sides  $BC, CA$ , and  $AB$  at points  $D, E$ , and  $F$ , respectively. Triangles  $APE$  and  $AQF$  are constructed outside the triangle so that \\[AP =PE, AQ=QF, \\angle APE=\\angle ACB,\\text{ and }\\angle AQF =\\angle ABC.\\]Let  $M$  be the midpoint of  $BC$ . Find  $\\angle QMP$  in terms of the angles of the triangle  $ABC$ .\n\n*Iran, Shayan Talaei*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,13,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.0,"k":"flat","v":0.0067,"x":0.03795,"p":[[0,41,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.02455,0.0524,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.03795,0.06182,0.0,0.0,0.08929,0.0,0.1429,23,0,0,23,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.02902,0.05607,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,41,0.9756,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"64dd02d96d0f6fb5","q":"a) Prove that every function of the form  $$ f(x)=\\frac{a_{0}}{2}+\\cos(x)+\\sum_{n=2}^{N}a_{n}\\cos(nx) $$  with  $|a_{0}|<1$  has positive as well as negative values in the period  $[0,2\\pi)$ .\nb) Prove that the function  $$ F(x)=\\sum_{n=1}^{100}\\cos(n^{\\frac{3}{2}}x) $$  has at least  $40$  zeroes in the interval  $(0,1000)$ .","t":[{"b":0,"e":0.42857,"k":"flat","v":0.70088,"x":0.83928,"p":[[0,13,0.0,0.70088,0.25345,0.42857,0.71429,1.0,0.42857,1.0,0,12,0,0,0,0,0,0,0,0,0,13,0,0,1,0,0,6,0,0,0,0,12],[4,13,0.3077,0.78572,0.25754,0.53572,0.92857,1.0,0.14286,1.0,0,16,0,0,0,1,0,0,0,0,0,7,0,0,1,0,0,4,0,0,3,0,16],[8,13,0.6154,0.80357,0.25442,0.67857,1.0,1.0,0.14286,1.0,0,18,0,0,0,1,0,0,0,0,0,6,0,0,1,0,0,5,0,0,1,0,18],[12,13,0.9231,0.83928,0.20748,0.71429,0.92857,1.0,0.42857,1.0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0,3,0,0,7,0,16],[13,13,1.0,0.78125,0.24481,0.67857,0.85714,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,2,0,0,5,0,0,1,0,0,7,0,0,2,0,15]]},{"b":4,"e":0.42857,"k":"flat","v":0.62497,"x":0.76339,"p":[[0,10,0.0,0.76339,0.27341,0.53572,0.9285,1.0,0.0,1.0,1,16,0,1,0,0,0,0,0,0,0,7,0,0,3,0,0,4,0,0,1,0,16],[4,10,0.4,0.66515,0.21012,0.42857,0.71429,0.71429,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,10,0,0,5,0,0,10,0,0,0,0,7],[8,10,0.8,0.62497,0.18472,0.42857,0.71429,0.71429,0.2857,1.0,0,3,0,0,0,0,0,0,1,0,0,10,0,0,4,0,0,13,0,0,1,0,3],[10,10,1.0,0.72768,0.20316,0.57143,0.71429,0.85714,0.42857,1.0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,3,0,0,9,0,0,6,0,7]]}]},{"i":"e7dd48364ed67ec9","q":"Let $A, B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of the line segment $A B$. Let $O_{1}$ be the circle tangent to the line $A B$ at $P$ and tangent to the circle $O$. Let $\\ell$ be the tangent line, different from the line $A B$, to $O_{1}$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $\\ell$ and $O$. Let $Q$ be the midpoint of the line segment $B C$ and $O_{2}$ be the circle tangent to the line $B C$ at $Q$ and tangent to the line segment $A C$. Prove that the circle $O_{2}$ is tangent to the circle $O$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.04464,"x":0.06695,"p":[[0,4,0.0,0.06695,0.13821,0.0,0.0,0.14286,0.0,0.714,22,0,0,22,0,8,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,4,1.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.08035,"p":[[0,21,0.0,0.06697,0.10705,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.08035,0.22569,0.0,0.0,0.0,0.0,1.0,25,1,1,25,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[8,21,0.381,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"468996dca02cd253","q":"Two circles  $\\Omega$  and  $\\Gamma$  are internally tangent at the point  $B$ . The chord  $AC$  of  $\\Gamma$  is tangent to  $\\Omega$  at the point  $L$ , and the segments  $AB$  and  $BC$  intersect  $\\Omega$  at the points  $M$  and  $N$ . Let  $M_1$  and  $N_1$  be the reflections of  $M$  and  $N$  about the line  $BL$ ; and let  $M_2$  and  $N_2$  be the reflections of  $M$  and  $N$  about the line  $AC$ . The lines  $M_1M_2$  and  $N_1N_2$  intersect at the point  $K$ .\nProve that the lines  $BK$  and  $AC$  are perpendicular.\n\n*(M. Karpuk)*","t":[{"b":2,"e":0.0,"k":"flat","v":0.02223,"x":0.13392,"p":[[0,22,0.0,0.13392,0.17101,0.0,0.07143,0.1786,0.0,0.57143,16,0,0,16,0,8,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,22,0.1818,0.11134,0.1323,0.0,0.07,0.14287,0.0,0.4286,16,0,0,16,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.11161,0.15458,0.0,0.0,0.17857,0.0,0.4286,19,0,2,19,0,5,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.02223,0.06281,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.03116,0.06888,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.04893,0.06761,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.4286,"k":"flat","v":0.19643,"x":0.3527,"p":[[0,18,0.0,0.19643,0.16656,0.0,0.14286,0.32143,0.0,0.42857,10,0,1,10,0,8,0,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.29911,0.10926,0.2857,0.28571,0.42857,0.0,0.4286,2,0,0,2,0,2,0,0,19,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.33036,0.07523,0.2857,0.28571,0.42857,0.1429,0.4286,0,0,0,0,0,1,0,0,20,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.33929,0.07784,0.28571,0.28571,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,21,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[16,18,0.8889,0.3527,0.0797,0.28571,0.28586,0.42857,0.2857,0.571,0,0,0,0,0,0,0,0,18,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[18,18,1.0,0.33482,0.07668,0.2857,0.28571,0.42857,0.1429,0.4286,0,0,0,0,0,1,0,0,19,0,0,12,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"228cd16e398054bc","q":"Two circles  $\\omega_1$  and  $\\omega_2$  meeting at point  $A$  and a line  $a$  are given. Let  $BC$  be an arbitrary chord of  $\\omega_2$  parallel to  $a$ , and  $E$ ,  $F$  be the second common points of  $AB$  and  $AC$  respectively with  $\\omega_1$ . Find the locus of common points of lines  $BC$  and  $EF$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,30,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,6,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,4,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.1429,"k":"rising","v":0.00893,"x":0.16956,"p":[[0,11,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.03572,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.16956,0.0558,0.14286,0.14286,0.1429,0.14,0.28571,0,0,0,0,0,26,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2a35212fbe8c6c11","q":"Let  $G$  be a simple, undirected, connected graph with  $100$  vertices and  $2013$  edges. It is given that there exist two vertices  $A$  and  $B$  such that it is not possible to reach  $A$  from  $B$  using one or two edges. We color all edges using  $n$  colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of  $n$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.09375,"x":0.12491,"p":[[0,5,0.0,0.12491,0.16268,0.0,0.14143,0.14286,0.0,0.71429,15,0,15,15,0,11,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,5,0.8,0.09375,0.17354,0.0,0.0,0.14286,0.0,0.71429,22,0,22,22,0,5,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.04902,"x":0.24107,"p":[[0,19,0.0,0.11607,0.14914,0.0,0.14286,0.14286,0.0,0.71429,15,0,14,15,0,11,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,19,0.2105,0.04902,0.11626,0.0,0.0,0.0,0.0,0.42857,26,0,23,26,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.20973,0.12369,0.14286,0.14286,0.17857,0.14,0.57143,0,0,0,0,0,24,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[12,19,0.6316,0.23214,0.1171,0.14286,0.14286,0.28571,0.14286,0.42857,0,0,0,0,0,19,0,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.18732,0.09749,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,26,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.24107,0.12595,0.14286,0.14286,0.42857,0.14286,0.42857,0,0,0,0,0,19,0,0,4,0,0,9,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8d716f501ef5911a","q":"There\u00b4s a ping pong tournament with  $n\\geq 3$  participants that we\u00b4ll call  $1, 2, \\dots n$ . The tournament rules are the following ones: at the start, all the players form a line, ordered from  $1$  to  $n$ . Players  $1$  and  $2$  play the first match. The winner is at the beginning of the line and the loser is placed behind the last person in the line.In the next play, the two who at that moment are the first two in line face each other, the winner is first in line and the loser goes to the end of the line, just behind the last loser. And so on. After  $N$  matches, the tournament ends.Player number  $1$  won  $a_1$  matches, player number  $2$  won  $a_2$ , and so on till player  $n$ , that has won  $a_n$  matches (it is trivial that  $a_1+a_2+\\dots+a_n=N)$ .Determine how many games each player has lost, based on  $a_1, a_2, \\dots , a_n$","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.00893,"p":[[0,7,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,10,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,9,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,70,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,12,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,70,0.0571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,70,0.1143,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,12,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,70,0.1714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,70,0.2286,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,70,0.2857,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,7,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,70,0.3429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,70,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,70,0.4571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,70,0.5143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,70,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,70,0.6286,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,70,0.6857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,70,0.7429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,70,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,70,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,70,0.9143,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,70,0.9714,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[70,70,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4ba2df1b1c1a22ec","q":"What is the largest value for  $m$  for which I can find nonnegative integers  $a_1, a_2, \\ldots, a_m < 2024$  such that for all indices  $i>j,$   $17$  divides  $\\tbinom{a_i}{a_j}$ ?","t":[{"b":4,"e":0.0,"k":"falling","v":0.15179,"x":0.55804,"p":[[0,34,0.0,0.55804,0.47697,0.0,1.0,1.0,0.0,1.0,12,17,0,12,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,17],[4,34,0.1176,0.18303,0.13474,0.0,0.28571,0.28571,0.0,0.28571,11,0,0,11,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.19643,0.13243,0.0,0.28571,0.28571,0.0,0.28571,10,0,0,10,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.20974,0.12369,0.105,0.28571,0.28571,0.0,0.286,8,0,0,8,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.20536,0.12846,0.0,0.28571,0.28571,0.0,0.28571,9,0,0,9,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.20982,0.12364,0.10714,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,1,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.25893,0.08328,0.28571,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.26786,0.06916,0.28571,0.28571,0.28571,0.0,0.28571,2,0,0,2,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.15179,0.13803,0.0,0.21429,0.28571,0.0,0.28571,14,0,0,14,0,2,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.125,"x":0.85268,"p":[[0,38,0.0,0.85268,0.34531,1.0,1.0,1.0,0.0,1.0,4,27,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[4,38,0.1053,0.125,0.13716,0.0,0.0,0.28571,0.0,0.28571,17,0,0,17,0,2,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.19196,0.20079,0.0,0.2857,0.28571,0.0,1.0,13,1,0,13,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,38,0.3158,0.17411,0.13709,0.0,0.28571,0.28571,0.0,0.28571,12,0,0,12,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.17411,0.13236,0.0,0.28571,0.28571,0.0,0.28571,11,0,0,11,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.23215,0.11152,0.2857,0.28571,0.28571,0.0,0.286,6,0,0,6,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.22768,0.11214,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.21428,0.12372,0.21427,0.28571,0.28571,0.0,0.28571,8,0,0,8,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.25893,0.08328,0.28571,0.28571,0.28571,0.0,0.28571,3,0,0,3,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.26786,0.05923,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,2,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.27678,0.04971,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3deaf38d994c5b6a","q":"Two circles  $\\Gamma_1,\\Gamma_2$  intersect at  $A,B$ . Through  $B$  a straight line  $\\ell$  is drawn and  $\\ell\\cap \\Gamma_1=K,\\ell\\cap\\Gamma_2=M\\;(K,M\\neq B)$ . We are given  $\\ell_1\\parallel AM$  is tangent to  $\\Gamma_1$  at  $Q$ .  $QA\\cap \\Gamma_2=R\\;(\\neq A)$  and further  $\\ell_2$  is tangent to  $\\Gamma_2$  at  $R$ .\nProve that: \n\n\n- $\\ell_2\\parallel AK$\n- $\\ell,\\ell_1,\\ell_2$  have a common point.","t":[{"b":3,"e":0.0,"k":"flat","v":0.06241,"x":0.14696,"p":[[0,19,0.0,0.13391,0.17101,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,2,0,0,10,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,19,0.2105,0.08474,0.13764,0.0,0.0,0.14292,0.0,0.57143,21,0,0,21,0,5,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,19,0.4211,0.07571,0.18887,0.0,0.0,0.035,0.0,1.0,24,1,3,24,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,19,0.6316,0.06241,0.09399,0.0,0.0,0.14286,0.0,0.28571,21,0,3,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.14696,0.1262,0.0,0.14286,0.2857,0.0,0.42857,11,0,0,11,0,10,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.12053,0.13882,0.0,0.0,0.2857,0.0,0.4286,17,0,0,17,0,4,0,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.11598,"p":[[0,23,0.0,0.10268,0.1197,0.0,0.0,0.17868,0.0,0.28571,17,0,1,17,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.09357,0.13169,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,9,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,23,0.8696,0.08929,0.12242,0.0,0.0,0.1429,0.0,0.42857,19,0,0,19,0,7,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.11598,0.14031,0.0,0.0,0.2857,0.0,0.4286,17,0,0,17,0,6,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6e78d7063475ba16","q":"let the incircle of a triangle ABC touch BC,AC,AB at A1,B1,C1 respectively. M and N are the midpoints of AB1 and AC1 respectively. MN meets A1C1 at T . draw two tangents TP and TQ through T to incircle. PQ meets MN at L and B1C1 meets PQ at K . assume I is the center of the incircle .\r\n\r\nprove IK is parallel to AL","t":[{"b":3,"e":0.28571,"k":"flat","v":0.21863,"x":0.30344,"p":[[0,13,0.0,0.21863,0.22582,0.0,0.14286,0.35704,0.0,0.57143,12,0,0,12,0,7,0,0,5,0,0,0,0,0,8,0,0,0,0,0,0,0,0],[4,13,0.3077,0.23214,0.21651,0.0,0.14286,0.42857,0.0,0.57143,10,0,0,10,0,8,0,0,5,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[8,13,0.6154,0.30344,0.18474,0.14286,0.28571,0.571,0.0,0.57143,2,0,0,2,0,10,0,0,11,0,0,0,0,0,9,0,0,0,0,0,0,0,0],[12,13,0.9231,0.29017,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.29464,0.03458,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"rising","v":0.18284,"x":0.5714,"p":[[0,28,0.0,0.20089,0.2392,0.0,0.07143,0.46429,0.0,0.57143,16,0,0,16,0,4,0,0,3,0,0,1,0,0,8,0,0,0,0,0,0,0,0],[4,28,0.1429,0.18284,0.20898,0.0,0.14286,0.28571,0.0,0.57143,13,0,0,13,0,9,0,0,4,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[8,28,0.2857,0.44639,0.1812,0.28571,0.57143,0.57143,0.0,0.57143,1,0,0,1,0,4,0,0,6,0,0,0,0,0,21,0,0,0,0,0,0,0,0],[12,28,0.4286,0.43745,0.171,0.28571,0.571,0.57143,0.0,0.57143,2,0,0,2,0,0,0,0,10,0,0,2,0,0,18,0,0,0,0,0,0,0,0],[16,28,0.5714,0.56695,0.02485,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0],[20,28,0.7143,0.56688,0.02484,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0],[24,28,0.8571,0.5625,0.03458,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0],[28,28,1.0,0.5714,0.0001,0.57143,0.57143,0.57143,0.571,0.57143,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0]]}]},{"i":"3e410a27c717d678","q":"Let  $AD$  and  $BE$  be the altitudes of acute triangle  $ABC.$  The circles with diameters  $AD$  and  $BE$  intersect at points  $S$  and  $T$ . Prove that  $\\angle ACS=\\angle BCT.$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,37,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.00884,0.03424,0.0,0.0,0.0,0.0,0.1429,30,0,7,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,4,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,4,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,2,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,3,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,12,0.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,4,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0dad38326659530c","q":"Let $ABC$ be a triangle and let $J$ be the center of its $A$-excircle. Let $K$ be the symmetric point of $J$ with respect to the segment $[BC]$. Let $E$ and $F$ be the points on the lines $(BJ)$ and $(CJ)$ such that $\\widehat{\\mathrm{EAB}}=\\widehat{\\mathrm{CAF}}=90^{\\circ}$. Show that $\\widehat{\\mathrm{FKE}}+\\widehat{\\mathrm{FJE}}=180^{\\circ}$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,28,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,4,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,5,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.01552,"p":[[0,6,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.01552,0.04248,0.0,0.0,0.0,0.0,0.14286,28,0,8,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e1f27c8fbe88dcb5","q":"Let $ABC$ be a triangle, $H$ the foot of the altitude from $A$, $L$ the foot of the bisector from $B$, and $M$ the midpoint of $[AB]$. We assume that in triangle $HLM$, the two properties hold: $(AH)$ is an altitude and $(BL)$ is a bisector. Show that $(CM)$ is a median.","t":[{"b":3,"e":1.0,"k":"flat","v":0.76339,"x":0.90624,"p":[[0,17,0.0,0.7857,0.23147,0.67857,0.78571,1.0,0.2857,1.0,0,15,0,0,0,0,0,0,1,0,0,5,0,0,2,0,0,8,0,0,1,0,15],[4,17,0.2353,0.76339,0.30222,0.57143,0.85714,1.0,0.0,1.0,2,15,0,2,0,0,0,0,2,0,0,3,0,0,2,0,0,3,0,0,5,0,15],[8,17,0.4706,0.90624,0.15408,0.82132,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,2,0,22],[12,17,0.7059,0.88838,0.18119,0.85711,1.0,1.0,0.4286,1.0,0,21,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,2,0,0,4,0,21],[16,17,0.9412,0.85265,0.19065,0.67857,1.0,1.0,0.42857,1.0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,7,0,0,2,0,0,4,0,18],[17,17,1.0,0.87933,0.17182,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,6,0,0,4,0,19]]},{"b":6,"e":1.0,"k":"flat","v":0.7857,"x":0.87945,"p":[[0,4,0.0,0.7857,0.25255,0.57143,1.0,1.0,0.28571,1.0,0,17,0,0,0,0,0,0,2,0,0,4,0,0,5,0,0,3,0,0,1,0,17],[4,4,1.0,0.87945,0.19271,0.82143,1.0,1.0,0.2857,1.0,0,21,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,2,0,0,3,0,21]]}]},{"i":"8c5c01bb9ad7c765","q":"The sequence  $S_0,S_1,S_2,\\ldots$  is defined by\n- $S_n=1$  for  $0\\le n\\le 2011$ , and\n- $S_{n+2012}=S_{n+2011}+S_n$  for  $n\\ge 0$ .\nProve that  $S_{2011a}-S_a$  is a multiple of  $2011$  for all nonnegative integers  $a$ .","t":[{"b":2,"e":1.0,"k":"flat","v":0.77679,"x":0.98661,"p":[[0,21,0.0,0.91964,0.24206,1.0,1.0,1.0,0.0,1.0,2,26,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,26],[4,21,0.1905,0.77679,0.37276,0.85714,1.0,1.0,0.0,1.0,3,20,0,3,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,20],[8,21,0.381,0.94195,0.16701,1.0,1.0,1.0,0.14286,1.0,0,26,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[12,21,0.5714,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[16,21,0.7619,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[20,21,0.9524,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[21,21,1.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":5,"e":1.0,"k":"rising","v":0.51339,"x":1.0,"p":[[0,28,0.0,0.67411,0.44783,0.0,1.0,1.0,0.0,1.0,9,18,0,9,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,18],[4,28,0.1429,0.65179,0.4642,0.0,1.0,1.0,0.0,1.0,10,19,1,10,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,19],[8,28,0.2857,0.51339,0.4763,0.0,0.64286,1.0,0.0,1.0,13,14,0,13,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,14],[12,28,0.4286,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[16,28,0.5714,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[20,28,0.7143,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"d7ccb611face4017","q":"There are  $2019$  points given in the plane. A child wants to draw  $k$  (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal  $k$ , such that for any initial configuration of points it is possible to draw  $k$  discs with the above property?","t":[{"b":0,"e":0.42857,"k":"flat","v":0.2679,"x":0.42857,"p":[[0,50,0.0,0.29911,0.19019,0.0,0.42857,0.42857,0.0,0.4286,9,0,7,9,0,0,0,0,2,0,0,21,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.2679,0.20128,0.0,0.42857,0.42857,0.0,0.43,11,0,7,11,0,1,0,0,1,0,0,19,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.33036,0.17655,0.39285,0.42857,0.42857,0.0,0.4286,7,0,6,7,0,0,0,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.42857,0.03572,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0],[16,50,0.32,0.40179,0.10374,0.42857,0.42857,0.42857,0.0,0.4286,2,0,2,2,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.39286,0.11294,0.42857,0.42857,0.42857,0.0,0.4286,2,0,2,2,0,1,0,0,0,0,0,29,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.37054,0.14223,0.42857,0.42857,0.42857,0.0,0.4286,4,0,4,4,0,0,0,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.42411,0.02486,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.41072,0.08564,0.42857,0.42857,0.42857,0.0,0.57143,1,0,1,1,0,0,0,0,2,0,0,28,0,0,1,0,0,0,0,0,0,0,0],[40,50,0.8,0.42856,0.03566,0.42857,0.42857,0.42857,0.28571,0.571,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0],[44,50,0.88,0.42411,0.04351,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0],[48,50,0.96,0.42857,0.03571,0.42857,0.42857,0.42857,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0],[50,50,1.0,0.42857,0.06186,0.42857,0.42857,0.42857,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.30804,"x":0.44197,"p":[[0,31,0.0,0.30804,0.19269,0.0,0.42857,0.42857,0.0,0.4286,9,0,6,9,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.32143,0.18898,0.21429,0.42857,0.42857,0.0,0.57143,8,0,5,8,0,0,0,0,1,0,0,22,0,0,1,0,0,0,0,0,0,0,0],[8,31,0.2581,0.4375,0.03458,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0],[12,31,0.3871,0.42857,1e-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],[16,31,0.5161,0.4375,0.03458,0.42857,0.42857,0.42857,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0],[20,31,0.6452,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.42858,1e-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],[28,31,0.9032,0.42857,1e-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],[31,31,1.0,0.44197,0.05486,0.42857,0.42857,0.42857,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"6ef3df1cb84dd36e","q":"There are $2 n$ cards. On each card some real number $x, 1 \\leqslant x \\leqslant 2$, is written (there can be different numbers on different cards). Prove that\nthe cards can be divided into two heaps with sums $s_{1}$ and $s_{2}$ so that $\\frac{n}{n+1} \\leqslant \\frac{s_{1}}{s_{2}} \\leqslant 1$.","t":[{"b":5,"e":0.14286,"k":"falling","v":0.14277,"x":0.70089,"p":[[0,24,0.0,0.70089,0.33189,0.53571,0.85714,1.0,0.0,1.0,3,12,0,3,0,1,0,0,2,0,0,2,0,0,3,0,0,4,0,0,5,0,12],[4,24,0.1667,0.38838,0.35577,0.0,0.28571,0.60714,0.0,1.0,9,5,0,9,0,5,0,0,3,0,0,2,0,0,5,0,0,3,0,0,0,0,5],[8,24,0.3333,0.38839,0.36811,0.10714,0.28571,0.60714,0.0,1.0,8,6,0,8,0,7,0,0,3,0,0,3,0,0,3,0,0,1,0,0,1,0,6],[12,24,0.5,0.45973,0.3925,0.105,0.42857,0.85714,0.0,1.0,8,7,0,8,0,5,0,0,2,0,0,3,0,0,2,0,0,2,0,0,3,0,7],[16,24,0.6667,0.37054,0.36746,0.0,0.2857,0.60714,0.0,1.0,10,6,0,10,0,5,0,0,2,0,0,6,0,0,1,0,0,2,0,0,0,0,6],[20,24,0.8333,0.41062,0.33842,0.14286,0.28571,0.71429,0.0,1.0,4,5,0,4,0,8,0,0,6,0,0,5,0,0,0,0,0,2,0,0,2,0,5],[24,24,1.0,0.14277,0.13363,0.0,0.14286,0.1786,0.0,0.42857,11,0,0,11,0,13,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"flat","v":0.41519,"x":0.55801,"p":[[0,11,0.0,0.55801,0.33571,0.28571,0.57121,1.0,0.0,1.0,2,9,0,2,0,5,0,0,3,0,0,4,0,0,6,0,0,3,0,0,0,0,9],[4,11,0.3636,0.48214,0.32683,0.28571,0.42857,0.74996,0.0,1.0,3,6,0,3,0,4,0,0,6,0,0,8,0,0,1,0,0,2,0,0,2,0,6],[8,11,0.7273,0.47321,0.19377,0.28571,0.42857,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,9,0,0,12,0,0,1,0,0,8,0,0,0,0,1],[11,11,1.0,0.41519,0.14444,0.28571,0.42857,0.4286,0.1429,0.71429,0,0,0,0,0,2,0,0,8,0,0,17,0,0,1,0,0,4,0,0,0,0,0]]}]},{"i":"53be22b95a198418","q":"Let $\\boldsymbol{n}$ be a positive integer.\n\n(a) Prove that there exists a set $S$ of $6 n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32 n^{2}$.\n(b) Prove that every set $T$ of $6 n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9 \\boldsymbol{n}^{2}$.","t":[{"b":2,"e":0.28571,"k":"flat","v":0.0625,"x":0.21428,"p":[[0,15,0.0,0.0625,0.12846,0.0,0.0,0.0,0.0,0.42857,25,0,20,25,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.21428,0.16751,0.0,0.2857,0.28571,0.0,0.4286,11,0,6,11,0,1,0,0,13,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.13393,0.15947,0.0,0.0,0.28571,0.0,0.4286,18,0,10,18,0,1,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.15626,0.16507,0.0,0.07143,0.28571,0.0,0.4286,16,0,12,16,0,1,0,0,11,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.08482,"x":0.0893,"p":[[0,10,0.0,0.0893,0.15048,0.0,0.0,0.17857,0.0,0.4286,23,0,13,23,0,1,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.08482,0.12807,0.0,0.0,0.2857,0.0,0.28571,22,0,11,22,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63dbb8e8341009cf","q":"There are  $100$  cards with numbers from  $1$  to  $100$  on the table.Andriy and Nick took the same number of cards in a way such that the following condition holds:if Andriy has a card with a number  $n$  then Nick has a card with a number  $2n+2$ .What is the maximal number of cards that could be taken by the two guys?","t":[{"b":0,"e":0.71429,"k":"flat","v":0.75891,"x":0.78558,"p":[[0,5,0.0,0.75891,0.18014,0.71429,0.71429,0.85714,0.0,1.0,1,6,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,20,0,0,4,0,6],[4,5,0.8,0.76339,0.06785,0.71429,0.71429,0.85714,0.71429,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,0,11,0,0],[5,5,1.0,0.78558,0.11302,0.71429,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,12,0,3]]},{"b":4,"e":0.71429,"k":"flat","v":0.69196,"x":0.83927,"p":[[0,66,0.0,0.83927,0.15049,0.71429,0.85714,1.0,0.571,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,11,0,0,5,0,13],[4,66,0.0606,0.74985,0.09457,0.71429,0.71429,0.75,0.571,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,22,0,0,6,0,2],[8,66,0.1212,0.73631,0.0725,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,7,0,0],[12,66,0.1818,0.73212,0.05925,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,5,0,0],[16,66,0.2424,0.73661,0.06298,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,25,0,0,6,0,0],[20,66,0.303,0.70979,0.07564,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,26,0,0,3,0,0],[24,66,0.3636,0.73214,0.09279,0.71429,0.71429,0.75,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,21,0,0,8,0,0],[28,66,0.4242,0.69641,0.12242,0.71429,0.71429,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,20,0,0,4,0,1],[32,66,0.4848,0.70089,0.06546,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,25,0,0,2,0,0],[36,66,0.5455,0.72307,0.04978,0.71429,0.71429,0.71429,0.571,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,3,0,0],[40,66,0.6061,0.71848,0.06668,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,25,0,0,4,0,0],[44,66,0.6667,0.74105,0.10973,0.71429,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,18,0,0,9,0,1],[48,66,0.7273,0.70088,0.10326,0.71429,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,26,0,0,3,0,0],[52,66,0.7879,0.70088,0.0827,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,24,0,0,3,0,0],[56,66,0.8485,0.69196,0.10779,0.71429,0.71429,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,1,0,0,3,0,0,24,0,0,3,0,0],[60,66,0.9091,0.70535,0.08702,0.71429,0.71429,0.71429,0.4286,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,23,0,0,4,0,0],[64,66,0.9697,0.70516,0.06133,0.71429,0.71429,0.71429,0.57,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,26,0,0,2,0,0],[66,66,1.0,0.73661,0.0724,0.71429,0.71429,0.71429,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,23,0,0,7,0,0]]}]},{"i":"39762c09a51212b2","q":"Let  $n\\geq 3$  and  $A_1,A_2,\\ldots,A_n$  be points on a circle. Find the largest number of acute triangles that can be considered with vertices in these points.\n\n*G. Eckstein*","t":[{"b":4,"e":0.0,"k":"flat","v":0.08482,"x":0.08482,"p":[[0,4,0.0,0.08482,0.20159,0.0,0.0,0.03571,0.0,1.0,24,1,22,24,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,1]]},{"b":7,"e":0.0,"k":"flat","v":0.02232,"x":0.08929,"p":[[0,9,0.0,0.04911,0.12169,0.0,0.0,0.0,0.0,0.57143,26,0,21,26,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,9,0.4444,0.08929,0.15465,0.0,0.0,0.14286,0.0,0.57143,21,0,19,21,0,6,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,9,0.8889,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,21,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"afa8bb490838e604","q":"Let $\\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys \"ab $\\leftarrow \\mathrm{cd} \\leftarrow \\leftarrow \\mathrm{e} \\leftarrow \\leftarrow \\mathrm{f}$ \", the result is \"faecdb\". We say that a string $B$ is reachable from a string $A$ if it is possible to insert some amount of $\\leftarrow$ 's in $A$, such that typing the resulting characters produces $B$. So, our example shows that \"faecdb\" is reachable from \"abcdef\".  Prove that for any two strings $A$ and $B, A$ is reachable from $B$ if and only if $B$ is reachable from $A$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.23665,"x":0.23665,"p":[[0,4,0.0,0.23665,0.2541,0.0,0.14286,0.42857,0.0,0.85714,13,0,7,13,0,4,0,0,4,0,0,8,0,0,0,0,0,1,0,0,2,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.33469,"x":0.33469,"p":[[0,2,0.0,0.33469,0.31057,0.0,0.28571,0.571,0.0,1.0,11,1,6,11,0,2,0,0,4,0,0,6,0,0,3,0,0,2,0,0,3,0,1]]}]},{"i":"e8c3086450ecfbd8","q":"Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges.","t":[{"b":5,"e":0.0,"k":"flat","v":0.00446,"x":0.04902,"p":[[0,33,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"rising","v":0.00446,"x":0.29,"p":[[0,23,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.16071,0.24419,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,12,0,0,3,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[8,23,0.3478,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.26784,0.16652,0.14286,0.2143,0.28571,0.14286,0.857,0,0,0,0,0,16,0,0,9,0,0,4,0,0,2,0,0,0,0,0,1,0,0],[16,23,0.6957,0.29,0.14519,0.14286,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,11,0,0,8,0,0,10,0,0,2,0,0,0,0,0,0,0,0],[20,23,0.8696,0.26785,0.12239,0.14286,0.28571,0.32143,0.14286,0.571,0,0,0,0,0,13,0,0,11,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[23,23,1.0,0.27679,0.15126,0.14286,0.21428,0.42857,0.14286,0.71429,0,0,0,0,0,16,0,0,4,0,0,11,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"45ebdd667cae0d29","q":"Let $\\Gamma$ be the circumcircle of the acute triangle $ABC$. The bisector of angle $ABC$ intersects $AC$ at point $B_{1}$ and the shorter arc $AC$ of $\\Gamma$ at point $P$. The line through $B_{1}$ perpendicular to $BC$ intersects the shorter arc $BC$ of $\\Gamma$ at $K$. The line through $B$ perpendicular to $AK$ intersects $AC$ at $L$. Prove that $K, L$, and $P$ lie on a line.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,12,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14,"k":"flat","v":0.00447,"x":0.13393,"p":[[0,7,0.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1f7e7e2bdb9419b4","q":"The incircle of a triangle  $ABC$  touches the side  $AB$  and  $AC$  at respectively at  $X$  and  $Y$ . Let  $K$  be the midpoint of the arc  $\\widehat{AB}$  on the circumcircle of  $ABC$ . Assume that  $XY$  bisects the segment  $AK$ . What are the possible measures of angle  $BAC$ ?","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.12946,"p":[[0,19,0.0,0.12946,0.17261,0.0,0.0,0.17857,0.0,0.57143,17,0,4,17,0,7,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,11,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.12946,"p":[[0,8,0.0,0.12946,0.16888,0.0,0.0,0.17857,0.0,0.57143,17,0,3,17,0,7,0,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[4,8,0.5,0.09375,0.14555,0.0,0.0,0.14286,0.0,0.57143,18,0,10,18,0,11,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"608346d91589a02e","q":"Two circles with centers  $O_1$  and  $O_2$  meet at points  $A$  and  $B$ . The bisector of angle  $O_1AO_2$  meets the circles for the second time at points  $C $ and  $D$ . Prove that the distances from the circumcenter of triangle  $CBD$  to  $O_1$  and to  $O_2$  are equal.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,22,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[13,13,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"032bc1e99d8afbee","q":"Two circles are said to be *orthogonal* if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles  $\\omega_1$  and  $\\omega_2$  with radii  $10$  and  $13$ , respectively, are externally tangent at point  $P$ . Another circle  $\\omega_3$  with radius  $2\\sqrt2$  passes through  $P$  and is orthogonal to both  $\\omega_1$  and  $\\omega_2$ . A fourth circle  $\\omega_4$ , orthogonal to  $\\omega_3$ , is externally tangent to  $\\omega_1$  and  $\\omega_2$ . Compute the radius of  $\\omega_4$ .","t":[{"b":2,"e":0.57143,"k":"flat","v":0.82142,"x":0.98214,"p":[[0,17,0.0,0.95088,0.11637,1.0,1.0,1.0,0.571,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,3,0,26],[4,17,0.2353,0.98214,0.04725,1.0,1.0,1.0,0.85714,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,28],[8,17,0.4706,0.91518,0.14223,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,22],[12,17,0.7059,0.82142,0.16751,0.71429,0.85707,1.0,0.57143,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,4,0,13],[16,17,0.9412,0.92411,0.11837,0.85714,1.0,1.0,0.57143,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,6,0,21],[17,17,1.0,0.88392,0.15748,0.85714,1.0,1.0,0.571,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,2,0,0,7,0,18]]},{"b":6,"e":1.0,"k":"flat","v":0.91518,"x":0.96875,"p":[[0,33,0.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[4,33,0.1212,0.93304,0.13356,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[8,33,0.2424,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[12,33,0.3636,0.96429,0.10102,1.0,1.0,1.0,0.57143,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,1,0,28],[16,33,0.4848,0.94196,0.12807,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,1,0,26],[20,33,0.6061,0.93304,0.13356,0.96429,1.0,1.0,0.57143,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,4,0,24],[24,33,0.7273,0.91518,0.14223,0.85714,1.0,1.0,0.57143,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,4,0,22],[28,33,0.8485,0.93304,0.15966,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,25],[32,33,0.9697,0.94643,0.1171,1.0,1.0,1.0,0.57143,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,4,0,25],[33,33,1.0,0.95982,0.0735,0.96429,1.0,1.0,0.71429,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,7,0,24]]}]},{"i":"10cc39086a1d04ca","q":"The incircle of triangle  $ABC$  touches sides  $BC$ ,  $AC$ , and  $AB$  at  $D, E$ , and  $F$  respectively. Let  $\\omega_a, \\omega_b$  and  $\\omega_c$  be the circumcircles of triangles  $EAF, DBF$ , and  $DCE$ , respectively. The lines  $DE$  and  $DF$  cut  $\\omega_a$  at  $E_a\\neq{E}$  and  $F_a\\neq{F}$ , respectively. Let  $r_A$  be the line  $E_{a}F_a$ . Let  $r_B$  and  $r_C$  be defined analogously. Show that the lines  $r_A$ ,  $r_B$ , and  $r_C$  determine a triangle with its vertices on the sides of triangle  $ABC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,32,0.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,1,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.14286,19,0,2,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.03571,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,1,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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]]},{"b":1,"e":0.14286,"k":"flat","v":0.00446,"x":0.13393,"p":[[0,23,0.0,0.06241,0.11252,0.0,0.0,0.14286,0.0,0.57143,21,0,2,21,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,23,0.1739,0.04009,0.08907,0.0,0.0,0.0,0.0,0.42857,25,0,2,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.13375,0.03454,0.14286,0.14286,0.14286,0.0,0.143,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e21a65b44a5edcd0","q":"Let  $c$  be a positive real number. Find all functions  $f:\\mathbb{R}^+\\to\\mathbb{R}^+$  that satisfy   $$ x^2f(xf(y))f(x)f(y)=c $$   for all positive reals  $x$  and  $y$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.22768,"x":0.35714,"p":[[0,24,0.0,0.35714,0.13363,0.28571,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,4,0,0,8,0,0,16,0,0,3,0,0,0,0,0,0,0,0],[4,24,0.1667,0.34357,0.18181,0.24999,0.28571,0.42857,0.14,1.0,0,1,0,0,0,8,0,0,10,0,0,11,0,0,1,0,0,1,0,0,0,0,1],[8,24,0.3333,0.3125,0.12595,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,7,0,0,14,0,0,10,0,0,0,0,0,1,0,0,0,0,0],[12,24,0.5,0.22768,0.10012,0.14286,0.2857,0.28571,0.0,0.4286,2,0,0,2,0,11,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.24107,0.10374,0.14286,0.2857,0.28571,0.0,0.42857,3,0,0,3,0,6,0,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.26339,0.06298,0.28571,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.24553,0.07349,0.25,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,7,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.1429,"k":"flat","v":0.24553,"x":0.36152,"p":[[0,16,0.0,0.36152,0.17502,0.2857,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,4,0,0,12,0,0,11,0,0,1,0,0,2,0,0,1,0,0],[4,16,0.25,0.35268,0.11837,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,17,0,0,10,0,0,2,0,0,1,0,0,0,0,0],[8,16,0.5,0.29911,0.13997,0.28571,0.28571,0.42857,0.0,0.57143,3,0,0,3,0,3,0,0,16,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[12,16,0.75,0.24553,0.09606,0.14286,0.2857,0.28571,0.0,0.42857,2,0,0,2,0,7,0,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.25893,0.0974,0.14286,0.28571,0.28571,0.0,0.4286,1,0,0,1,0,8,0,0,19,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"638e0a4e8bcd5629","q":"There is a convex quadrilateral  $ ABCD $  which satisfies  $ \\angle A=\\angle D $ .\nLet the midpoints of  $ AB, AD, CD $  be  $ L,M,N $ .\nLet's say the intersection point of  $ AC, BD $  be  $ E $  .\nLet's say point  $ F $  which lies on  $ \\overrightarrow{ME} $  satisfies  $ \\overline{ME}\\times \\overline{MF}=\\overline{MA}^{2} $ .\n\nProve that  $ \\angle LFM=\\angle MFN $ . :)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,30,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dfa09c43312354ef","q":"Let $N$ be a positive integer. Prove that there exist three permutations $a_{1}, a_{2}, \\ldots, a_{N}$; $b_{1}, b_{2}, \\ldots, b_{N}$; and $c_{1}, c_{2}, \\ldots, c_{N}$ of $1,2, \\ldots, N$ such that  $$ \\left|\\sqrt{a_{k}}+\\sqrt{b_{k}}+\\sqrt{c_{k}}-2 \\sqrt{N}\\right|<2023 $$  for every $k=1,2, \\ldots, N$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,19,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,10,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"cee97573f86cf9dc","q":"Let  $\\omega$  be the circumcircle of acute triangle  $ABC$  where  $\\angle A<\\angle B$  and  $M,N$  be the midpoints of minor arcs  $BC,AC$  of  $\\omega$  respectively. The line  $PC$  is parallel to  $MN$ , intersecting  $\\omega$  at  $P$  (different from  $C$ ). Let  $I$  be the incentre of  $ABC$  and let  $PI$  intersect  $\\omega$  again at the point  $T$ .\n1) Prove that  $MP\\cdot MT=NP\\cdot NT$ ;\n2) Let  $Q$  be an arbitrary point on minor arc  $AB$  and  $I,J$  be the incentres of triangles  $AQC,BCQ$ . Prove that  $Q,I,J,T$  are concyclic.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.04018,"p":[[0,13,0.0,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,27,0,8,27,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01777,0.05904,0.0,0.0,0.0,0.0,0.2857,29,0,9,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.06687,"p":[[0,17,0.0,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,4,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.06687,0.09432,0.0,0.0,0.14286,0.0,0.42857,19,0,5,19,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"ca5c6373ca6ddfa9","q":"Let $ABC$ be a triangle with $AB \\neq AC$, and let $\\omega$ be the $A$-excircle of $ABC$. We denote by $D, E$ and $F$ the points of tangency of $\\omega$ with $[BC]$, $[AC)$, and $[AB)$. The circumcircle of $AEF$ intersects $(BC)$ at $P$ and $Q$. We denote by $M$ the midpoint of $[AD]$. Show that the circumcircle of $MPQ$ is tangent to $\\omega$.\nNote. We recall that the $A$-excircle of $ABC$ is the unique circle tangent to the segment $[BC]$, to the ray $[AB)$ beyond $B$, and to the ray $[AC)$ beyond $C$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.06688,"p":[[0,24,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.05786,0.06995,0.0,0.0,0.14286,0.0,0.14286,19,0,2,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,2,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.06688,0.09431,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,23,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"967345a7d15d1871","q":"There are  $100$  white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.","t":[{"b":0,"e":0.0,"k":"flat","v":0.16964,"x":0.16964,"p":[[0,4,0.0,0.16964,0.18013,0.0,0.14286,0.32143,0.0,0.4286,15,0,7,15,0,4,0,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0625,"x":0.21429,"p":[[0,28,0.0,0.21429,0.23145,0.0,0.14288,0.42857,0.0,0.71429,15,0,4,15,0,2,0,0,4,0,0,8,0,0,1,0,0,2,0,0,0,0,0],[4,28,0.1429,0.12946,0.17627,0.0,0.0,0.2857,0.0,0.57143,19,0,9,19,0,3,0,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0625,0.14258,0.0,0.0,0.03571,0.0,0.71429,24,0,0,24,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,28,0.4286,0.07588,0.1382,0.0,0.0,0.14286,0.0,0.571,23,0,0,23,0,3,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,28,0.5714,0.10715,0.21129,0.0,0.0,0.14287,0.0,1.0,22,1,0,22,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[20,28,0.7143,0.15179,0.22286,0.0,0.0,0.28571,0.0,1.0,17,1,0,17,0,4,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[24,28,0.8571,0.12054,0.26752,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,2],[28,28,1.0,0.09357,0.20075,0.0,0.0,0.14,0.0,1.0,23,1,0,23,0,3,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"2524623228fd6101","q":"Let  $\\mathbb Z_{\\ge 0}$  be the set of non-negative integers, and let  $f:\\mathbb Z_{\\ge 0}\\times \\mathbb Z_{\\ge 0} \\to \\mathbb Z_{\\ge 0}$  be a bijection such that whenever  $f(x_1,y_1) > f(x_2, y_2)$ , we have  $f(x_1+1, y_1) > f(x_2 + 1, y_2)$  and  $f(x_1, y_1+1) > f(x_2, y_2+1)$ .\n\nLet  $N$  be the number of pairs of integers  $(x,y)$  with  $0\\le x,y<100$ , such that  $f(x,y)$  is odd. Find the smallest and largest possible values of  $N$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.00884,"x":0.01786,"p":[[0,21,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,15,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,17,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,8,0.0,0.04018,0.09606,0.0,0.0,0.0,0.0,0.28571,27,0,15,27,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,23,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d5c13dc66bc0cd4d","q":"We call a set of points *free* if there is no equilateral triangle with the vertices among the points of the set. Prove that every set of  $n$  points in the plane contains a *free* subset with at least  $\\sqrt{n}$  elements.","t":[{"b":6,"e":0.0,"k":"flat","v":0.02679,"x":0.0625,"p":[[0,11,0.0,0.03572,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,11,0.3636,0.0625,0.18536,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[8,11,0.7273,0.03125,0.10555,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[11,11,1.0,0.02679,0.10374,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.07589,"p":[[0,13,0.0,0.07589,0.2448,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,13,0.3077,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.03571,0.11294,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.04018,0.12492,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.04464,0.14032,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"48f5d093d682ccae","q":"Let  $\\omega$  be the circumcircle of an acute angled tirangle  $ABC.$  The line tangent to  $\\omega$  at  $A$  intersects the line  $BC$  at the point  $T.$  Let the midpoint of segment  $AT$  be  $N,$  and the centroid of  $\\triangle ABC$  be the point  $G.$   The other tangent line drawn from  $N$  to  $\\omega$  intersects  $\\omega$  at the point  $L.$  The line  $LG$  meets  $\\omega$  at  $S\\neq L.$  \nProve that  $AS\\parallel BC.$","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,28,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0133,0.04137,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03116,"p":[[0,39,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.02233,0.05188,0.0,0.0,0.0,0.0,0.143,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,39,0.4103,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1d430b5908bc2586","q":"Let  $p \\in \\mathbb{N} \\setminus \\{0, 1\\}$  be a fixed positive integer. Prove that for every  $K > 0$ , there exist infinitely many  $n$  and  $N$  such that there are atleast  $\\dfrac{KN}{\\log(N)}$  primes among the following  $N$  numbers given by\n    \\[n + 1, n + 2^p, n + 3^p, \\cdots, n + N^p.\\]\n\n*Proposed by Bimit Mandal*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,59,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,7,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,59,0.0678,0.02232,0.10171,0.0,0.0,0.0,0.0,0.57143,30,0,2,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,59,0.1356,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,59,0.2034,0.05357,0.15464,0.0,0.0,0.0,0.0,0.57143,28,0,3,28,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,59,0.2712,0.02232,0.08827,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,59,0.339,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,59,0.4746,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,59,0.5424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,59,0.6102,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,59,0.678,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,59,0.7458,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,59,0.8136,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,59,0.8814,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,59,0.9492,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[59,59,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,20,0.0,0.05802,0.1466,0.0,0.0,0.0,0.0,0.57143,26,0,5,26,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,20,0.2,0.06696,0.15146,0.0,0.0,0.0,0.0,0.57143,26,0,8,26,0,1,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dbcb34f954902872","q":"We call a number *pal* if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example,  $122$  and  $34$  are pal but  $304$  and  $12$  are not pal. Prove that there exists a pal number with  $n$  digits,  $n > 1$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.06241,"x":0.12054,"p":[[0,16,0.0,0.09375,0.18423,0.0,0.0,0.14286,0.0,1.0,19,1,1,19,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,16,0.25,0.08036,0.14698,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,7,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,16,0.5,0.08697,0.1052,0.0,0.03571,0.14286,0.0,0.42857,16,0,1,16,1,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.12054,0.09523,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01562,"x":0.16523,"p":[[0,11,0.0,0.16523,0.25786,0.0,0.07143,0.1429,0.0,1.0,16,2,0,16,0,9,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,2],[4,11,0.3636,0.10482,0.20513,0.0,0.0,0.14286,0.0,1.0,19,1,1,19,1,9,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[8,11,0.7273,0.08027,0.18874,0.0,0.0,0.14286,0.0,1.0,22,1,0,22,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[11,11,1.0,0.01562,0.04276,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e37cb2a2ced143f3","q":"The tangent at  $C$  to  $\\Omega$ , the circumcircle of scalene triangle  $ABC$  intersects  $AB$  at  $D$ . Through point  $D$ , a line is drawn that intersects segments  $AC$  and  $BC$  at  $K$  and  $L$  respectively. On the segment  $AB$  points  $M$  and  $N$  are marked such that  $AC \\parallel NL$  and  $BC \\parallel KM$ . Lines  $NL$  and  $KM$  intersect at point  $P$  lying inside the triangle  $ABC$ . Let  $\\omega$  be the circumcircle of  $MNP$ . Suppose  $CP$  intersects  $\\omega$  again at  $Q$ . Show that  $DQ$  is tangent to  $\\omega$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.09822,"p":[[0,22,0.0,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.04242,0.07333,0.0,0.0,0.08926,0.0,0.2857,23,0,0,23,1,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.09822,0.07524,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.02232,"x":0.14491,"p":[[0,7,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.14491,0.02817,0.14286,0.14286,0.14286,0.07143,0.28571,0,0,0,0,1,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6367d16b7d33e518","q":"The columns and the rows of a $3 n \\times 3 n$ square board are numbered $1,2, \\ldots, 3 n$. Every square $(x, y)$ with $1 \\leq x, y \\leq 3 n$ is colored asparagus, byzantium or citrine according as the modulo 3 remainder of $x+y$ is 0 , 1 or 2 respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3 n^{2}$ tokens of each color.  Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.02679,"p":[[0,3,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,10,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"bbb56cc2f4c1aa65","q":"Let $a, b, c, d$ be positive real numbers such that $a b c d=1$. Prove the inequality\n\n$$\n\\frac{1}{\\sqrt{a+2 b+3 c+10}}+\\frac{1}{\\sqrt{b+2 c+3 d+10}}+\\frac{1}{\\sqrt{c+2 d+3 a+10}}+\\frac{1}{\\sqrt{d+2 a+3 b+10}} \\leq 1 .\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.03572,"p":[[0,7,0.0,0.03572,0.08748,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.04911,"p":[[0,87,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,87,0.046,0.03571,0.09449,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,87,0.092,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,87,0.1379,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,87,0.1839,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,87,0.2299,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,87,0.2759,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,87,0.3218,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,87,0.3678,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,87,0.4138,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,87,0.4598,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,87,0.5057,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,87,0.5517,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,87,0.5977,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,87,0.6437,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,87,0.6897,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,87,0.7356,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,87,0.7816,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,87,0.8276,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,87,0.8736,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,87,0.9195,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,87,0.9655,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[87,87,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"644340f6237157f0","q":"Let $P, Q \\in \\mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\\lambda$ such that $P+\\lambda Q$ is the square of a polynomial.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.16518,"x":0.20527,"p":[[0,7,0.0,0.16518,0.18935,0.0,0.14286,0.28571,0.0,0.85714,12,0,0,12,0,11,0,0,4,0,0,4,0,0,0,0,0,0,0,0,1,0,0],[4,7,0.5714,0.20527,0.16345,0.0,0.14286,0.32143,0.0,0.42857,9,0,0,9,0,8,0,0,7,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.17839,0.09456,0.14286,0.14286,0.28571,0.0,0.28571,4,0,0,4,0,16,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.09367,"x":0.20983,"p":[[0,27,0.0,0.09367,0.14985,0.0,0.0,0.14286,0.0,0.42857,21,0,2,21,0,5,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.18741,0.16538,0.0,0.14286,0.32142,0.0,0.4286,10,0,0,10,0,10,0,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.16946,0.16539,0.0,0.14286,0.17857,0.0,0.71429,9,0,0,9,0,15,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,0],[12,27,0.4444,0.20983,0.15146,0.10714,0.2857,0.28571,0.0,0.42857,8,0,0,8,0,7,0,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.20973,0.16749,0.105,0.14286,0.42857,0.0,0.4286,8,0,0,8,0,11,0,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.13839,0.16554,0.0,0.14286,0.14286,0.0,0.4286,15,0,1,15,0,10,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.16964,0.13092,0.14286,0.14286,0.14287,0.0,0.42857,6,0,0,6,0,19,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.11598,0.06618,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ce0481f3a2e92c0a","q":"The king assembled 300 wizards and gave them the following challenge. For this challenge, 25 colors can be used, and they are known to the wizards. Each of the wizards receives a hat of one of those 25 colors. If for each color the number of used hats would be written down then all these number would be different, and the wizards know this. Each wizard sees what hat was given to each other wizard but does not see his own hat. Simultaneously each wizard reports the color of his own hat. Is it possible for the wizards to coordinate their actions beforehand so that at least 150 of them would report correctly?","t":[{"b":1,"e":0.0,"k":"flat","v":0.04911,"x":0.16964,"p":[[0,23,0.0,0.04911,0.15815,0.0,0.0,0.0,0.0,0.71429,29,0,29,29,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,23,0.1739,0.10268,0.1996,0.0,0.0,0.0,0.0,0.71429,25,0,25,25,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[8,23,0.3478,0.07589,0.15561,0.0,0.0,0.0,0.0,0.42857,25,0,25,25,0,2,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.11161,0.18466,0.0,0.0,0.21429,0.0,0.4286,23,0,23,23,0,1,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.10268,0.17582,0.0,0.0,0.14286,0.0,0.42857,23,0,23,23,0,2,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.16964,0.22428,0.0,0.0,0.42857,0.0,0.71429,20,0,20,20,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.04464,"x":0.08929,"p":[[0,8,0.0,0.04464,0.12595,0.0,0.0,0.0,0.0,0.42857,28,0,28,28,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.08929,0.1915,0.0,0.0,0.0,0.0,0.71429,26,0,26,26,0,0,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"c1f3ae7ad7d2a8a2","q":"Let $a, b, c$ be three strictly positive real numbers. Show that\n\n$$\n\\frac{a^{4}+1}{b^{3}+b^{2}+b}+\\frac{b^{4}+1}{c^{3}+c^{2}+c}+\\frac{c^{4}+1}{a^{3}+a^{2}+a} \\geqslant 2\n$$","t":[{"b":5,"e":0.4286,"k":"flat","v":0.13839,"x":0.37947,"p":[[0,28,0.0,0.25,0.28794,0.0,0.14286,0.42857,0.0,1.0,16,1,0,16,0,0,0,0,2,0,0,11,0,0,0,0,0,0,0,0,2,0,1],[4,28,0.1429,0.16518,0.2055,0.0,0.0,0.42857,0.0,0.4286,19,0,0,19,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.25,0.28347,0.0,0.21429,0.42857,0.0,1.0,16,2,0,16,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,2],[12,28,0.4286,0.28125,0.27545,0.0,0.42857,0.42857,0.0,1.0,13,2,0,13,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,2],[16,28,0.5714,0.13839,0.24086,0.0,0.0,0.42857,0.0,1.0,23,1,0,23,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[20,28,0.7143,0.20089,0.28316,0.0,0.0,0.42857,0.0,1.0,19,2,0,19,0,1,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,2],[24,28,0.8571,0.24107,0.24856,0.0,0.35714,0.42857,0.0,1.0,15,1,0,15,0,0,0,0,1,0,0,15,0,0,0,0,0,0,0,0,0,0,1],[28,28,1.0,0.37947,0.09182,0.28571,0.42857,0.42857,0.0,0.4286,1,0,0,1,0,0,0,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.00893,"x":0.27232,"p":[[0,76,0.0,0.1875,0.27067,0.0,0.0,0.42857,0.0,1.0,20,1,0,20,0,0,0,0,1,0,0,9,0,0,0,0,0,0,0,0,1,0,1],[4,76,0.0526,0.12054,0.19269,0.0,0.0,0.42857,0.0,0.42857,23,0,0,23,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[8,76,0.1053,0.19643,0.32093,0.0,0.0,0.42857,0.0,1.0,21,2,0,21,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,2],[12,76,0.1579,0.12946,0.21535,0.0,0.0,0.2857,0.0,0.85714,22,0,0,22,0,1,0,0,2,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[16,76,0.2105,0.13839,0.30615,0.0,0.0,0.0,0.0,1.0,26,2,0,26,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,2],[20,76,0.2632,0.21875,0.3009,0.0,0.0,0.42857,0.0,1.0,19,1,0,19,0,0,0,0,2,0,0,7,0,0,0,0,0,1,0,0,2,0,1],[24,76,0.3158,0.27232,0.32411,0.0,0.0,0.42857,0.0,1.0,17,2,0,17,0,0,0,0,0,0,0,9,0,0,1,0,0,2,0,0,1,0,2],[28,76,0.3684,0.17411,0.28288,0.0,0.0,0.42857,0.0,1.0,22,1,0,22,0,0,0,0,0,0,0,7,0,0,0,0,0,1,0,0,1,0,1],[32,76,0.4211,0.08929,0.17035,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,0,0,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[36,76,0.4737,0.04911,0.18766,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,76,0.5263,0.09375,0.24383,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0],[44,76,0.5789,0.08482,0.15916,0.0,0.0,0.03571,0.0,0.4286,24,0,0,24,0,2,0,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.04911,0.12682,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[52,76,0.6842,0.03125,0.10555,0.0,0.0,0.0,0.0,0.4286,29,0,0,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,76,0.7895,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[64,76,0.8421,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.08036,0.20806,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[76,76,1.0,0.22768,0.20473,0.0,0.28571,0.42857,0.0,0.42857,14,0,0,14,0,0,0,0,3,0,0,15,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e0b9b059ebb8b0d6","q":"The sequence  $p_1, p_2, p_3, ...$  is defined as follows.  $p_1$  and  $p_2$  are primes.  $p_n$  is the greatest prime divisor of  $p_{n-1} + p_{n-2} + 2000$ . Show that the sequence is bounded.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.14268,"p":[[0,9,0.0,0.14268,0.10102,0.105,0.14286,0.17868,0.0,0.28571,8,0,0,8,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.13393,0.10677,0.0,0.14286,0.1786,0.0,0.28571,10,0,1,10,0,14,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.11607,"x":0.17857,"p":[[0,19,0.0,0.12929,0.10324,0.0,0.14286,0.14287,0.0,0.28571,10,0,0,10,0,15,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.17393,0.09938,0.14286,0.14286,0.28571,0.0,0.28571,5,0,1,5,0,15,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.17857,0.06186,0.14286,0.14286,0.1786,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.13375,0.06119,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,26,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.11607,0.07523,0.10714,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d305e688ce95025b","q":"The head of the Mint wants to release 12 coins denominations (each - a natural number rubles) so that any amount from 1 to 6543 rubles could be paid without having to pass, using no more than 8 coins. Can he do it? (If the payment amount you can use a few coins of the same denomination.)","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,13,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[13,13,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bd5c28eb57388d10","q":"Two circles $\\Gamma$ and $\\Gamma^{\\prime}$ may intersect at two distinct points $A$ and $B$. A line through $B$ intersects $\\Gamma$ and $\\Gamma^{\\prime}$ at $C$ and $D$ respectively, such that $B$ lies between $C$ and $D$. Another line through $B$ intersects $\\Gamma$ and $\\Gamma^{\\prime}$ at $E$ and $F$ respectively, such that $E$ lies between $B$ and $F$. It is given that $|C D|=|E F|$. The interior of the segment $C F$ intersects $\\Gamma$ and $\\Gamma^{\\prime}$ at $P$ and $Q$ respectively. Furthermore, let $M$ and $N$ be the midpoints of the arcs $P B$ and $B Q$ of $\\Gamma$ and $\\Gamma^{\\prime}$, not containing $C$ and $F$ respectively. Prove that $C N M F$ is a cyclic quadrilateral.","t":[{"b":6,"e":0.4286,"k":"rising","v":0.08482,"x":0.27669,"p":[[0,14,0.0,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.11152,0.10553,0.0,0.14286,0.14286,0.0,0.4286,11,0,1,11,0,19,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.27669,0.16347,0.14286,0.28571,0.42857,0.0,0.571,3,0,0,3,0,12,0,0,2,0,0,14,0,0,1,0,0,0,0,0,0,0,0],[12,14,0.8571,0.2634,0.14773,0.14286,0.14288,0.42857,0.0,0.57143,1,0,0,1,0,16,0,0,3,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[14,14,1.0,0.26782,0.15884,0.14286,0.2857,0.42857,0.0,0.43,4,0,0,4,0,10,0,0,4,0,0,14,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.15607,"p":[[0,32,0.0,0.09812,0.09736,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.15161,0.13334,0.0,0.14286,0.1429,0.0,0.42857,9,0,0,9,0,16,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.15607,0.14447,0.105,0.14286,0.14286,0.0,0.57143,8,0,0,8,0,19,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,32,0.375,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,32,0.75,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0580ab8b361dd8c8","q":"Two nonnegative integers  $a$  and  $b$  are *tuanis* if the decimal expression of  $a+b$  contains only  $0$  and  $1$  as digits. Let  $A$  and  $B$  be two infinite sets of non negative integers such that  $B$  is the set of all the *tuanis* numbers to elements of the set  $A$  and  $A$  the set of all the *tuanis* numbers to elements of the set  $B$ . Show that in at least one of the sets  $A$  and  $B$  there is an infinite number of pairs  $(x,y)$  such that  $x-y=1$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.12938,"x":0.16063,"p":[[0,5,0.0,0.15607,0.10328,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,26,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,5,0.8,0.16063,0.09281,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,29,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[5,5,1.0,0.12938,0.04162,0.14286,0.14286,0.14286,0.0,0.1429,3,0,0,3,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.12054,"x":0.18732,"p":[[0,5,0.0,0.18732,0.1905,0.105,0.14286,0.1786,0.0,0.85714,8,0,1,8,0,16,0,0,2,0,0,4,0,0,1,0,0,0,0,0,1,0,0],[4,5,0.8,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.1429,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.125,0.05923,0.14286,0.14286,0.14286,0.0,0.2857,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ca946b3520f75816","q":"Two circles  $O_1$  and  $O_2$  intersect at  $A,B$ . Diameter  $AC$  of  $\\odot O_1$  intersects  $\\odot O_2$  at  $E$ , Diameter  $AD$  of  $\\odot O_2$  intersects  $\\odot O_1$  at  $F$ .  $CF$  intersects  $O_2$  at  $H$ ,  $DE$  intersects  $O_1$  at  $G,H$ .  $GH\\cap O_1=P$ . Prove that  $PH=PK$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0133,"p":[[0,55,0.0,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,55,0.0727,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,6,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8e65c1074fe89fa6","q":"Let $ABC$ be a triangle. We denote $P$ as the symmetric point of $B$ with respect to $(AC)$ and $Q$ as the symmetric point of $C$ with respect to $(AB)$.\nLet $T$ be the intersection between $(PQ)$ and the tangent at $A$ to the circumcircle of $(APQ)$.\nShow that the symmetric point of $T$ with respect to $A$ lies on $(BC)$.","t":[{"b":0,"e":0.85714,"k":"rising","v":0.61152,"x":0.99107,"p":[[0,10,0.0,0.61152,0.34311,0.25001,0.57143,1.0,0.14,1.0,0,10,0,0,0,8,0,0,1,0,0,4,0,0,4,0,0,1,0,0,4,0,10],[4,10,0.4,0.98214,0.07784,1.0,1.0,1.0,0.57143,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,30],[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],[10,10,1.0,0.95982,0.15663,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,30]]},{"b":3,"e":1.0,"k":"rising","v":0.47759,"x":0.79911,"p":[[0,34,0.0,0.49107,0.34615,0.14286,0.5,0.85714,0.0,1.0,4,3,4,4,0,7,0,0,3,0,0,2,0,0,1,0,0,6,0,0,6,0,3],[4,34,0.1176,0.47759,0.35653,0.14286,0.42857,0.89286,0.0,1.0,2,8,0,2,0,9,0,0,4,0,0,6,0,0,0,0,0,2,0,0,1,0,8],[8,34,0.2353,0.77678,0.14258,0.71429,0.71429,0.85714,0.28571,1.0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,17,0,0,8,0,5],[12,34,0.3529,0.77232,0.12299,0.71429,0.71429,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,0,11,0,3],[16,34,0.4706,0.75892,0.17656,0.71429,0.71429,0.85714,0.1429,1.0,0,6,0,0,0,1,0,0,0,0,0,1,0,0,2,0,0,16,0,0,6,0,6],[20,34,0.5882,0.75893,0.0974,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,20,0,0,8,0,2],[24,34,0.7059,0.78571,0.13832,0.71429,0.71429,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,15,0,0,8,0,6],[28,34,0.8235,0.79911,0.17445,0.71429,0.71429,1.0,0.14286,1.0,0,9,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,0,0,6,0,9],[32,34,0.9412,0.78125,0.13825,0.71429,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,0,8,0,6],[34,34,1.0,0.76339,0.19103,0.71429,0.71429,0.85714,0.14286,1.0,0,7,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,15,0,0,6,0,7]]}]},{"i":"6650cf3a62c82c0a","q":"You are given a set of $n$ blocks, each weighing at least 1 ; their total weight is $2 n$. Prove that for every real number $r$ with $0 \\leqslant r \\leqslant 2 n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r+2$. (Thailand)","t":[{"b":4,"e":0.28571,"k":"rising","v":0.19642,"x":0.38385,"p":[[0,17,0.0,0.19642,0.24417,0.0,0.0,0.42857,0.0,0.71429,17,0,0,17,0,3,0,0,2,0,0,5,0,0,3,0,0,2,0,0,0,0,0],[4,17,0.2353,0.23215,0.25939,0.0,0.14288,0.57143,0.0,0.85714,14,0,0,14,0,5,0,0,3,0,0,1,0,0,8,0,0,0,0,0,1,0,0],[8,17,0.4706,0.25892,0.27065,0.0,0.14288,0.42858,0.0,1.0,13,1,0,13,0,4,0,0,2,0,0,6,0,0,5,0,0,1,0,0,0,0,1],[12,17,0.7059,0.37946,0.28481,0.0,0.42859,0.57143,0.0,1.0,9,1,0,9,0,1,0,0,3,0,0,5,0,0,10,0,0,2,0,0,1,0,1],[16,17,0.9412,0.30346,0.23894,0.0,0.28571,0.571,0.0,0.71429,9,0,0,9,0,3,0,0,6,0,0,5,0,0,7,0,0,2,0,0,0,0,0],[17,17,1.0,0.38385,0.19386,0.24999,0.42857,0.571,0.0,0.71429,2,0,0,2,0,6,0,0,4,0,0,10,0,0,8,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0758,"x":0.22771,"p":[[0,35,0.0,0.22771,0.30902,0.0,0.0,0.4286,0.0,1.0,18,2,0,18,0,1,0,0,3,0,0,4,0,0,3,0,0,0,0,0,1,0,2],[4,35,0.1143,0.0758,0.15142,0.0,0.0,0.14071,0.0,0.57143,23,0,0,23,0,5,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,35,0.2286,0.0758,0.15557,0.0,0.0,0.14071,0.0,0.57143,23,0,1,23,0,6,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,35,0.3429,0.10268,0.12992,0.0,0.0,0.2857,0.0,0.28571,19,0,1,19,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.08482,0.12807,0.0,0.0,0.2857,0.0,0.28571,22,0,0,22,0,1,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.08928,0.12752,0.0,0.0,0.2857,0.0,0.28571,21,0,1,21,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.09375,0.12169,0.0,0.0,0.1786,0.0,0.28571,19,0,0,19,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.07589,0.11836,0.0,0.0,0.14287,0.0,0.28571,22,0,0,22,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.09375,0.13175,0.0,0.0,0.2857,0.0,0.28571,21,0,0,21,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.125,0.13243,0.0,0.07143,0.2857,0.0,0.28571,16,0,0,16,0,4,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1818695faad95f97","q":"The quadrilateral  $ABCD$  is inscribed in the circle  $\\omega$  with the center  $O$ . Suppose that the angles  $B$  and  $C$  are obtuse and lines  $AD$  and  $BC$  are not parallel. Lines  $AB$  and  $CD$  intersect at point  $E$ . Let  $P$  and  $R$  be the feet of the perpendiculars from the point  $E$  on the lines  $BC$  and  $AD$  respectively.  $Q$  is the intersection point of  $EP$  and  $AD, S$  is the intersection point of  $ER$  and  $BC$ . Let K be the midpoint of the segment  $QS$  . Prove that the points  $E, K$ , and  $O$  are collinear.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,19,0.0,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.01786,"x":0.09357,"p":[[0,9,0.0,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.09357,0.06773,0.0,0.14286,0.14286,0.0,0.1429,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"983b894ef23b724c","q":"Let  $\\triangle ABC$  be an acute triangle with circumcenter  $O$  such that  $AB<AC$ , let  $Q$  be the intersection of the external bisector of  $\\angle A$  with  $BC$ , and let  $P$  be a point in the interior of  $\\triangle ABC$  such that  $\\triangle BPA$  is similar to  $\\triangle APC$ . Show that  $\\angle QPA + \\angle OQB = 90^{\\circ}$ .\n\n*Alex Zhu.*","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,31,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"69f1ef3ee496fc92","q":"Let $S \\subseteq \\mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions:  (i) Both functions are strictly increasing, i.e. $f(x)<f(y)$ and $g(x)<g(y)$ for all $x, y \\in S$ with $x<y$;  (ii) The inequality $f(g(g(x)))<g(f(x))$ holds for all $x \\in S$.  Decide whether there exists a Spanish Couple  (a) on the set $S=\\mathbb{N}$ of positive integers;  (b) on the set $S=\\{a-1 / b: a, b \\in \\mathbb{N}\\}$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.04464,"x":0.20536,"p":[[0,46,0.0,0.12054,0.18595,0.0,0.0,0.17857,0.0,0.57143,21,0,2,21,0,3,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[4,46,0.087,0.08929,0.14617,0.0,0.0,0.14286,0.0,0.4286,21,0,2,21,0,6,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,46,0.1739,0.18295,0.25061,0.0,0.0,0.42857,0.0,0.85714,18,0,3,18,0,4,0,0,0,0,0,6,0,0,2,0,0,1,0,0,1,0,0],[12,46,0.2609,0.125,0.17768,0.0,0.0,0.17857,0.0,0.57143,19,0,4,19,0,5,0,0,2,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[16,46,0.3478,0.19632,0.21354,0.0,0.14286,0.32142,0.0,0.57143,12,0,3,12,0,10,0,0,2,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[20,46,0.4348,0.08929,0.14174,0.0,0.0,0.14286,0.0,0.57143,20,0,3,20,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,46,0.5217,0.20536,0.2257,0.0,0.14286,0.42857,0.0,0.57143,15,0,0,15,0,4,0,0,2,0,0,6,0,0,5,0,0,0,0,0,0,0,0],[28,46,0.6087,0.09804,0.15741,0.0,0.0,0.14286,0.0,0.57143,19,0,1,19,0,9,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,46,0.6957,0.0758,0.107,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,10,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.05331,0.07756,0.0,0.0,0.14071,0.0,0.286,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.08474,0.0701,0.0,0.14286,0.14286,0.0,0.143,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.04464,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.08473,"p":[[0,17,0.0,0.08473,0.16308,0.0,0.0,0.14071,0.0,0.57143,23,0,8,23,0,4,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0758,0.13821,0.0,0.0,0.14071,0.0,0.42857,23,0,3,23,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.03572,0.11294,0.0,0.0,0.0,0.0,0.4286,29,0,1,29,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7ba829ce76e0afee","q":"There is a population $P$ of 10000 bacteria, some of which are friends (friendship is mutual), so that each bacterion has at least one friend and if we wish to assign to each bacterion a coloured membrane so that no two friends have the same colour, then there is a way to do it with 2021 colours, but not with 2020 or less.\n\nTwo friends $A$ and $B$ can decide to merge in which case they become a single bacterion whose friends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are not friends.) It turns out that no matter how we perform one merge or two consecutive merges, in the resulting population it would be possible to assign 2020 colours or less so that no two friends have the same colour. Is it true that in any such population $P$ every bacterium has at least 2021 friends?\n\n## Proposed by Bulgaria","t":[{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.04018,"p":[[0,20,0.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,3,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0267,0.0751,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.01786,"x":0.09375,"p":[[0,31,0.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,3,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0625,0.08703,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.04009,0.07337,0.0,0.0,0.035,0.0,0.28571,24,0,1,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,3,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.09375,0.09182,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,15,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.04455,0.07512,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.28571,22,0,1,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"79c2134dedb10258","q":"The numbers $a_{1}, a_{2}, \\ldots, a_{2 \\text { 2016 }}$ are natural numbers such that for all $n, 1 \\leqslant n \\leqslant 2^{2016}$, the following holds:\n\n$$\na_{n} \\leqslant 2016 \\quad \\text { and } \\quad a_{1} a_{2} \\cdots a_{n}+1 \\text { is a perfect square. }\n$$\n\nProve that one of the numbers $a_{1}, a_{2}, \\ldots, a_{22016}$ is equal to 1. (Du\u0161an \u0110uki\u0107)\n\n## SOLUTIONS","t":[{"b":0,"e":0.0,"k":"flat","v":0.01786,"x":0.02679,"p":[[0,10,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,3,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.01339,"x":0.02679,"p":[[0,6,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,7,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e0261761464e014b","q":"Let $a, b$, and $c$ be positive real numbers satisfying $\\min (a+b, b+c, c+a)>\\sqrt{2}$ and $a^{2}+b^{2}+c^{2}=3$. Prove that  $$ \\frac{a}{(b+c-a)^{2}}+\\frac{b}{(c+a-b)^{2}}+\\frac{c}{(a+b-c)^{2}} \\geq \\frac{3}{(a b c)^{2}} $$  Throughout both solutions, we denote the sums of the form $f(a, b, c)+f(b, c, a)+f(c, a, b)$ by $\\sum f(a, b, c)$.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.1383,"x":0.16518,"p":[[0,17,0.0,0.14732,0.07563,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.15616,0.05488,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,27,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.1383,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.2857,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.16518,0.05187,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.13384,"x":0.14732,"p":[[0,14,0.0,0.13821,0.08363,0.14214,0.14286,0.14286,0.0,0.28571,6,0,0,6,0,21,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.13839,0.08364,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,24,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.13384,0.0497,0.14286,0.14286,0.14286,0.0,0.2857,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.14732,0.06666,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c1c7a5598459aa81","q":"There are  $n$  positive real numbers on the board  $a_1,\\ldots, a_n$ . Someone wants to write  $n$  real numbers  $b_1,\\ldots,b_n$ ,such that: $b_i\\geq a_i$ If  $b_i \\geq b_j$  then  $\\frac{b_i}{b_j}$  is integer.\nProve that it is possible to write such numbers with the condition   $$ b_1 \\cdots b_n \\leq 2^{\\frac{n-1}{2}}a_1\\cdots a_n. $$","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00438,"p":[[0,10,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[10,10,1.0,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"35d36f726692f804","q":"There are many opposition societies in the city of N. \nEach society consists of  $10$  members. It is known that for every  $2004$  societies there is a person belonging to at least  $11$  of them. \nProve that the government can arrest  $2003$  people so that at least one member of each society is arrested.\n\n*Proposed by V.Dolnikov, D.Karpov*","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.13393,"p":[[0,44,0.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,3,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,44,0.1818,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.07589,0.24996,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[16,44,0.3636,0.13393,0.29001,0.0,0.0,0.0,0.0,1.0,25,2,0,25,0,1,0,0,1,0,0,0,0,0,2,0,0,1,0,0,0,0,2],[20,44,0.4545,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,1,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[24,44,0.5455,0.07366,0.23109,0.0,0.0,0.0,0.0,1.0,28,1,1,28,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1],[28,44,0.6364,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.0357,0.11287,0.0,0.0,0.0,0.0,0.571,28,0,0,28,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,44,0.8182,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[44,44,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.07143,"p":[[0,20,0.0,0.02677,0.10367,0.0,0.0,0.0,0.0,0.571,29,0,1,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,20,0.2,0.07143,0.22868,0.0,0.0,0.0,0.0,1.0,29,1,0,29,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[8,20,0.4,0.04687,0.12829,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,20,0.6,0.04018,0.11971,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,20,0.8,0.03571,0.12877,0.0,0.0,0.0,0.0,0.71429,28,0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[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]]}]},{"i":"989b98577c158edf","q":"Let $ABC$ be a triangle, and $\\Gamma$ its circumcircle. Let $M$ be the midpoint of the arc $BC$ not containing $A$. A circle $\\mathscr{C}$ is tangent to $[AB), [AC)$ at $D$ and $E$ respectively, and internally tangent to $\\Gamma$ at $F$. Show that $(DE), (BC)$, and $(FM)$ are concurrent.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,21,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.00893,"x":0.02232,"p":[[0,11,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,11,0.7273,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6b18be1b2cf465f9","q":"Let $S$ be a finite set, and let $\\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f \\circ g \\circ f \\neq g \\circ f \\circ g$ for every $g$ in $\\mathcal{A}$ with $g \\neq f$. Show that $f(T)=T$. (India)","t":[{"b":0,"e":0.0,"k":"volatile","v":0.00446,"x":0.57587,"p":[[0,33,0.0,0.0357,0.10708,0.0,0.0,0.0,0.0,0.571,27,0,5,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,33,0.1212,0.08482,0.22263,0.0,0.0,0.0,0.0,0.85714,26,0,11,26,0,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0],[8,33,0.2424,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,10,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.1741,0.28734,0.0,0.0,0.28571,0.0,0.857,22,0,3,22,0,1,0,0,2,0,0,0,0,0,2,0,0,4,0,0,1,0,0],[16,33,0.4848,0.04911,0.15815,0.0,0.0,0.0,0.0,0.71429,28,0,3,28,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,33,0.6061,0.11607,0.25862,0.0,0.0,0.0,0.0,0.71429,26,0,9,26,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0],[24,33,0.7273,0.57587,0.23278,0.57132,0.71429,0.71429,0.0,0.857,3,0,0,3,0,1,0,0,1,0,0,2,0,0,6,0,0,18,0,0,1,0,0],[28,33,0.8485,0.2767,0.31735,0.0,0.07,0.57143,0.0,0.85714,16,0,13,16,0,3,0,0,0,0,0,1,0,0,5,0,0,6,0,0,1,0,0],[32,33,0.9697,0.16518,0.28372,0.0,0.0,0.14286,0.0,0.71429,22,0,14,22,0,3,0,0,0,0,0,0,0,0,1,0,0,6,0,0,0,0,0],[33,33,1.0,0.03125,0.10555,0.0,0.0,0.0,0.0,0.57143,28,0,8,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00893,"x":0.14286,"p":[[0,33,0.0,0.04464,0.15746,0.0,0.0,0.0,0.0,0.71429,29,0,11,29,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[4,33,0.1212,0.06697,0.18205,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[8,33,0.2424,0.14286,0.28794,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,1,0,0],[12,33,0.3636,0.04464,0.14032,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,33,0.4848,0.0625,0.17835,0.0,0.0,0.0,0.0,0.71429,27,0,2,27,0,2,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[20,33,0.6061,0.03125,0.12745,0.0,0.0,0.0,0.0,0.71429,29,0,1,29,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,33,0.7273,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[32,33,0.9697,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e8d8362f9f52bf14","q":"Triangle  $ABC$  has incenter  $I$ . The line through  $I$  perpendicular to  $AI$  meets the circumcircle of  $ABC$  at points  $P$  and  $Q$ , where  $P$  and  $B$  are on the same side of  $AI$ . Let  $X$  be the point such that  $PX$  //  $CI$  and  $QX$  //  $BI$ . Show that  $P B, QC$ , and  $IX$  intersect at a common point.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"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],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,28,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0cd64c56313af462","q":"There exists a unique pair of polynomials  $(P(x),Q(x))$  such that\n\\begin{align*}\nP(Q(x))&= P(x)(x^2-6x+7) \nQ(P(x))&= Q(x)(x^2-3x-2)\n\\end{align*}\nCompute  $P(10)+Q(-10)$ .\n\n*Proposed by Connor Gordon*","t":[{"b":2,"e":0.14286,"k":"falling","v":0.1383,"x":0.75893,"p":[[0,140,0.0,0.62946,0.40226,0.14286,0.85714,1.0,0.0,1.0,3,16,2,3,0,7,0,0,0,0,0,3,0,0,2,0,0,1,0,0,0,0,16],[4,140,0.0286,0.5625,0.39599,0.14286,0.42857,1.0,0.14286,1.0,0,13,0,0,0,13,0,0,1,0,0,3,0,0,0,0,0,1,0,0,1,0,13],[8,140,0.0571,0.50884,0.38629,0.14286,0.42857,1.0,0.14,1.0,0,11,0,0,0,15,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,11],[12,140,0.0857,0.42857,0.35714,0.14286,0.14286,0.78571,0.14286,1.0,0,8,0,0,0,17,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,8],[16,140,0.1143,0.625,0.41764,0.14286,1.0,1.0,0.14286,1.0,0,17,0,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,17],[20,140,0.1429,0.75893,0.38038,0.35714,1.0,1.0,0.0,1.0,1,22,0,1,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,22],[24,140,0.1714,0.25447,0.28735,0.14286,0.14286,0.14286,0.0,1.0,1,4,0,1,0,26,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[28,140,0.2,0.3125,0.33204,0.14286,0.14286,0.32143,0.0,1.0,3,5,0,3,0,19,0,0,2,0,0,2,0,0,0,0,0,0,0,0,1,0,5],[32,140,0.2286,0.19643,0.20748,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[36,140,0.2571,0.25893,0.25614,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,25,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,3],[40,140,0.2857,0.23214,0.20438,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,25,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[44,140,0.3143,0.15625,0.07457,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[48,140,0.3429,0.22322,0.21706,0.14286,0.14286,0.14286,0.14286,1.0,0,2,0,0,0,27,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[52,140,0.3714,0.20982,0.2172,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[56,140,0.4,0.25437,0.28959,0.14286,0.14286,0.14286,0.0,1.0,2,4,0,2,0,24,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[60,140,0.4286,0.17411,0.11143,0.14286,0.14286,0.14286,0.14286,0.71429,0,0,0,0,0,29,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[64,140,0.4571,0.17411,0.15865,0.14286,0.14286,0.14286,0.0,1.0,1,1,0,1,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[68,140,0.4857,0.28125,0.29121,0.14286,0.14286,0.17857,0.0,1.0,1,4,0,1,0,23,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,4],[72,140,0.5143,0.20089,0.21387,0.14286,0.14286,0.14286,0.0,1.0,1,2,0,1,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[76,140,0.5429,0.16518,0.0724,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,29,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[80,140,0.5714,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[84,140,0.6,0.13839,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,140,0.6286,0.14732,0.08364,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[92,140,0.6571,0.1383,0.06666,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[96,140,0.6857,0.23214,0.26904,0.14286,0.14286,0.14286,0.0,1.0,2,3,0,2,0,26,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[100,140,0.7143,0.32589,0.35756,0.14286,0.14286,0.14286,0.0,1.0,1,7,0,1,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7],[104,140,0.7429,0.22759,0.27169,0.14286,0.14286,0.14286,0.0,1.0,3,3,0,3,0,25,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3],[108,140,0.7714,0.24107,0.25364,0.14286,0.14286,0.14286,0.14286,1.0,0,3,0,0,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3],[112,140,0.8,0.22768,0.2547,0.14286,0.14286,0.14286,0.0,1.0,1,3,0,1,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3],[116,140,0.8286,0.21429,0.22304,0.14286,0.14286,0.14286,0.0,1.0,2,2,0,2,0,25,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[120,140,0.8571,0.16518,0.12428,0.14286,0.14286,0.14286,0.14286,0.85714,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[124,140,0.8857,0.14732,0.08364,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,27,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[128,140,0.9143,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[132,140,0.9429,0.16518,0.08828,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[136,140,0.9714,0.15625,0.05486,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[140,140,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":1.0,"k":"volatile","v":0.48661,"x":0.94643,"p":[[0,4,0.0,0.48661,0.39907,0.14286,0.35714,1.0,0.0,1.0,4,11,2,4,0,9,0,0,3,0,0,4,0,0,0,0,0,1,0,0,0,0,11],[4,4,1.0,0.94643,0.15872,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,27]]}]},{"i":"27069b8851eb8786","q":"The sum of four real numbers is  $9$  and the sum of their squares is  $21$ . Prove that these numbers can be denoted by  $a, b, c, d$  so that  $ab-cd \\ge 2$  holds.","t":[{"b":1,"e":0.57143,"k":"rising","v":0.13393,"x":0.34148,"p":[[0,18,0.0,0.1517,0.19542,0.0,0.07,0.2857,0.0,0.71429,16,0,2,16,0,7,0,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[4,18,0.2222,0.13393,0.21998,0.0,0.0,0.2857,0.0,0.71429,22,0,1,22,0,1,0,0,3,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[8,18,0.4444,0.20983,0.2448,0.0,0.14286,0.42858,0.0,0.71429,15,0,0,15,0,5,0,0,2,0,0,4,0,0,4,0,0,2,0,0,0,0,0],[12,18,0.6667,0.18972,0.18525,0.0,0.14286,0.42857,0.0,0.57143,12,0,1,12,1,6,0,0,4,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[16,18,0.8889,0.28572,0.22588,0.10714,0.28571,0.42857,0.0,0.71429,8,0,0,8,0,6,0,0,3,0,0,11,0,0,1,0,0,3,0,0,0,0,0],[18,18,1.0,0.34148,0.2133,0.14296,0.42857,0.4642,0.0,0.71429,5,0,0,5,1,3,0,0,6,0,0,9,0,0,6,0,0,2,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.14284,"p":[[0,15,0.0,0.12491,0.23622,0.0,0.0,0.1429,0.0,1.0,22,1,2,22,0,3,0,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,1],[4,15,0.2667,0.14284,0.21126,0.0,0.0,0.2857,0.0,0.71429,19,0,0,19,0,4,0,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[8,15,0.5333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b2dbcf05ee268419","q":"Two circles  $O_1$  and  $O_2$  intersect at  $A,B$ . Bisector of outer angle  $\\angle O_1AO_2$  intersects  $O_1$  at  $C$ ,  $O_2$  at  $D$ .  $P$  is a point on  $\\odot(BCD)$ ,  $CP\\cap O_1=E,DP\\cap O_2=F$ . Prove that  $PE=PF$ .","t":[{"b":0,"e":0.2857,"k":"rising","v":0.02232,"x":0.24553,"p":[[0,12,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,2,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.11152,0.15038,0.0,0.0,0.2857,0.0,0.57143,19,0,2,19,0,3,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,12,0.6667,0.12053,0.13882,0.0,0.0,0.2857,0.0,0.28571,18,0,1,18,0,1,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.24553,0.09606,0.2857,0.2857,0.28571,0.0,0.28571,4,0,0,4,0,1,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.05804,"x":0.11161,"p":[[0,4,0.0,0.05804,0.12299,0.0,0.0,0.0,0.0,0.42857,26,0,2,26,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.11161,0.13709,0.0,0.0,0.2857,0.0,0.28571,19,0,0,19,0,1,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"71c051f9580f2904","q":"Eight consecutive positive integers  are divided into 2 sets, such that the sum of the squares of the elements in the first set is equal to the sum of the squares of the elements in the second set. Prove that the sum of the lements in the first set is equal to the sum of the elements in the second one.","t":[{"b":6,"e":0.42857,"k":"rising","v":0.22322,"x":0.50893,"p":[[0,12,0.0,0.24545,0.14833,0.14286,0.14286,0.42857,0.14,0.71429,0,0,0,0,0,20,0,0,3,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[4,12,0.3333,0.25893,0.1448,0.14286,0.14288,0.42857,0.14286,0.71429,0,0,0,0,0,17,0,0,6,0,0,8,0,0,0,0,0,1,0,0,0,0,0],[8,12,0.6667,0.22322,0.11258,0.14286,0.14286,0.28571,0.14286,0.4286,0,0,0,0,0,20,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.50893,0.15542,0.42857,0.42857,0.60714,0.28571,1.0,0,1,0,0,0,0,0,0,2,0,0,20,0,0,2,0,0,7,0,0,0,0,1]]},{"b":7,"e":0.14286,"k":"flat","v":0.20089,"x":0.27232,"p":[[0,18,0.0,0.26785,0.14613,0.14286,0.21435,0.42857,0.0,0.571,1,0,1,1,0,15,0,0,4,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[4,18,0.2222,0.22089,0.13292,0.14286,0.14286,0.2857,0.0,0.71429,1,0,1,1,0,18,0,1,8,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[8,18,0.4444,0.27232,0.20628,0.14286,0.14286,0.28571,0.0,1.0,1,1,1,1,0,16,0,0,8,0,0,4,0,0,0,0,0,2,0,0,0,0,1],[12,18,0.6667,0.23214,0.15047,0.14286,0.14286,0.28571,0.14286,0.71429,0,0,0,0,0,22,0,0,3,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[16,18,0.8889,0.20089,0.10012,0.14286,0.14286,0.2857,0.0,0.4286,1,0,1,1,0,20,0,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.2009,0.09354,0.14286,0.14286,0.2857,0.14286,0.4286,0,0,0,0,0,22,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dd866aff97139e31","q":"We say a function $f: \\mathbb{Z}_{\\geq 0} \\times \\mathbb{Z}_{\\geq 0} \\rightarrow \\mathbb{Z}$ is great if for any nonnegative integers $m$ and $n$,  $$ f(m+1, n+1) f(m, n)-f(m+1, n) f(m, n+1)=1 $$  If $A=\\left(a_{0}, a_{1}, \\ldots\\right)$ and $B=\\left(b_{0}, b_{1}, \\ldots\\right)$ are two sequences of integers, we write $A \\sim B$ if there exists a great function $f$ satisfying $f(n, 0)=a_{n}$ and $f(0, n)=b_{n}$ for every nonnegative integer $n$ (in particular, $a_{0}=b_{0}$ ). Prove that if $A, B, C$, and $D$ are four sequences of integers satisfying $A \\sim B$, $B \\sim C$, and $C \\sim D$, then $D \\sim A$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.0625,"x":0.13384,"p":[[0,14,0.0,0.08464,0.11205,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,10,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0625,0.08703,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.07813,0.08631,0.0,0.03571,0.14286,0.0,0.28571,16,0,0,16,1,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.13384,0.10824,0.0,0.14286,0.16082,0.0,0.357,10,0,0,10,0,14,0,1,6,0,1,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.05357,"p":[[0,25,0.0,0.05357,0.12242,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.04009,0.07337,0.0,0.0,0.035,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.04911,0.09182,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.04909,0.11627,0.0,0.0,0.0,0.0,0.571,25,0,0,25,0,5,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,25,0.64,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"23c4f0deb5f45cf9","q":"Let $X$ be a set of 10000 integers, none of them is divisible by 47 . Prove that there exists a 2007-element subset $Y$ of $X$ such that $a-b+c-d+e$ is not divisible by 47 for any $a, b, c, d, e \\in Y$.  (Netherlands)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.07589,"x":0.13839,"p":[[0,9,0.0,0.13839,0.16554,0.0,0.14286,0.14286,0.0,0.71429,12,0,9,12,0,15,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,9,0.4444,0.07589,0.09439,0.0,0.0,0.14286,0.0,0.28571,18,0,17,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.12054,0.13882,0.0,0.14286,0.14286,0.0,0.71429,12,0,10,12,0,16,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.13393,"x":0.14277,"p":[[0,7,0.0,0.13393,0.26229,0.0,0.0,0.14286,0.0,1.0,19,2,18,19,0,9,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[4,7,0.5714,0.14277,0.17128,0.0,0.14286,0.14286,0.0,0.71429,13,0,11,13,0,13,0,0,1,0,0,4,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"b28f2710b1e1bed6","q":"The integers from  $1$  to  $n$  are written, one on each of  $n$  cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.\nWhat is th smallest value of  $n$  for which the second player can ensure that he competes both his moves?","t":[{"b":1,"e":1.0,"k":"flat","v":0.49997,"x":0.54911,"p":[[0,9,0.0,0.54911,0.36441,0.25,0.57143,0.85714,0.0,1.0,7,7,0,7,0,1,0,0,1,0,0,4,0,0,4,0,0,5,0,0,3,0,7],[4,9,0.4444,0.49997,0.35892,0.10714,0.57143,0.71429,0.0,1.0,8,5,3,8,0,2,0,0,1,0,0,1,0,0,7,0,0,6,0,0,2,0,5]]},{"b":6,"e":0.0,"k":"falling","v":0.32133,"x":0.53124,"p":[[0,13,0.0,0.53124,0.32583,0.39285,0.57143,0.71429,0.0,1.0,6,5,1,6,0,1,0,0,1,0,0,5,0,0,4,0,0,10,0,0,0,0,5],[4,13,0.3077,0.49998,0.40247,0.0,0.57143,0.85704,0.0,1.0,11,7,4,11,0,0,0,0,1,0,0,2,0,0,3,0,0,5,0,0,3,0,7],[8,13,0.6154,0.50889,0.29218,0.39286,0.4998,0.71429,0.0,1.0,5,2,2,5,0,0,0,0,3,0,0,8,0,0,4,0,0,6,0,0,4,0,2],[12,13,0.9231,0.32133,0.35718,0.0,0.07,0.71429,0.0,1.0,16,1,8,16,0,1,0,0,1,0,0,2,0,0,2,0,0,6,0,0,3,0,1]]}]},{"i":"e4ee15a84bd4cbb3","q":"Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x)$. Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1$. Prove that $Q(x)$ is a constant polynomial.","t":[{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.04464,"p":[[0,10,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,1,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.03572,0.08749,0.0,0.0,0.0,0.0,0.4286,26,0,2,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03563,"p":[[0,49,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.03563,0.08737,0.0,0.0,0.0,0.0,0.42857,26,0,1,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.0134,0.04165,0.0,0.0,0.0,0.0,0.143,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.03563,0.08738,0.0,0.0,0.0,0.0,0.4286,26,0,0,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c1ae74a3f97f40ec","q":"The incircle of acute triangle  $ABC$  touches  $BC, AC$ , and  $AB$  at points  $D, E$ , and  $F$ , respectively. Let  $P$  be the second intersection of line  $AD$  and the incircle. The line through  $P$  tangent to the incircle intersects  $AB$  and  $AC$  at points  $M$  and  $N$ , respectively. Given that  $\\overline{AB} = 8, \\overline{AC} = 10$ , and  $\\overline{AN} = 4$ , let  $\\overline{AM} = \\tfrac{a}{b}$  where  $a$  and  $b$  are positive coprime integers. What is  $a + b$ ?","t":[{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.20089,"p":[[0,31,0.0,0.16518,0.23176,0.0,0.0,0.42857,0.0,0.71429,20,0,12,20,0,1,0,0,2,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[4,31,0.129,0.20089,0.25966,0.0,0.0,0.42857,0.0,0.71429,19,0,15,19,0,0,0,0,3,0,0,3,0,0,5,0,0,2,0,0,0,0,0],[8,31,0.2581,0.19196,0.25657,0.0,0.0,0.57143,0.0,0.57143,20,0,10,20,0,0,0,0,2,0,0,1,0,0,9,0,0,0,0,0,0,0,0],[12,31,0.3871,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,24,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.10268,0.12492,0.0,0.0,0.2857,0.0,0.28571,18,0,13,18,0,5,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,30,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.04911,"x":0.1875,"p":[[0,17,0.0,0.08929,0.1948,0.0,0.0,0.0,0.0,0.57143,26,0,19,26,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[4,17,0.2353,0.1875,0.22428,0.0,0.07143,0.32143,0.0,0.57143,16,0,7,16,0,4,0,0,4,0,0,2,0,0,6,0,0,0,0,0,0,0,0],[8,17,0.4706,0.14732,0.20973,0.0,0.0,0.28571,0.0,0.57143,20,0,12,20,0,1,0,0,5,0,0,2,0,0,4,0,0,0,0,0,0,0,0],[12,17,0.7059,0.04911,0.14555,0.0,0.0,0.0,0.0,0.57143,28,0,23,28,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"d7781d1e98be3591","q":"Let $A B C$ be an acute, scalene triangle with circumcenter $O$ and symmedian point $K$. Let $X$ be the point on the circumcircle of triangle $B O C$ such that $\\angle A X O=90^{\\circ}$. Assume that $X \\neq K$. The hyperbola passing through $B, C, O, K$, and $X$ intersects the circumcircle of triangle $A B C$ at points $U$ and $V$, distinct from $B$ and $C$. Prove that $U V$ is the perpendicular bisector of $A X$.\nThe symmedian point of triangle $A B C$ is the intersection of the reflections of $B$-median and $C$-median across the angle bisectors of $\\angle A B C$ and $\\angle A C B$, respectively.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,38,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,25,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a86c95a54d18a5c5","q":"There's infinity of the following blocks on the table: $1*1 , 1*2 , 1*3 ,.., 1*n$ . We have a  $n*n$  table and Ali chooses some of these blocks so that the sum of their area is at least  $n^2$ . Then , Amir tries to cover the  $n*n$  table so that none of blocks go out of the table and they don't overlap and he wanna maximize the covered area in the  $n*n$  table with those blocks chosen by Ali. Let  $k$  be the maximum coverable area independent of Ali's choice. Prove that:  $$ n^2 - \\lceil \\frac{n^2}{4} \\rceil \\leq k \\leq n^2 - \\lfloor \\frac{n^2}{8} \\rfloor $$  *Note : the blocks can be placed only vertically or horizontally.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,9,0.0,0.05357,0.15872,0.0,0.0,0.0,0.0,0.85714,26,0,1,26,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,7,0.0,0.07589,0.14279,0.0,0.0,0.14286,0.0,0.57143,22,0,1,22,0,7,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5efabd5581948c8d","q":"We define a $w$-strip as the set of all points in the plane that are between or on two parallel lines on a mutual distance $w$. Let $S$ be a set of $n$ points in the plane such that any three points from $S$ can be covered by a 1 -strip. Show that the entire set $S$ can be covered by a 2 -strip.\n(Romania)","t":[{"b":1,"e":0.0,"k":"flat","v":0.04911,"x":0.07581,"p":[[0,9,0.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,2,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.06241,0.07077,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.07581,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.04455,"x":0.09375,"p":[[0,21,0.0,0.06241,0.07927,0.0,0.0,0.14286,0.0,0.28571,19,0,1,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.1429,14,0,1,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.05786,0.06995,0.0,0.0,0.14286,0.0,0.1429,19,0,0,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f19ce463e5577147","q":"The jury of an olympiad has 30 members in the beginning. Each member of the jury thinks that some of his colleagues are competent, while all the others are not, and these opinions do not change. At the beginning of every session a voting takes place, and those members who are not competent in the opinion of more than one half of the voters are excluded from the jury for the rest of the olympiad. Prove that after at most 15 sessions there will be no more exclusions. (Note that nobody votes about his own competence.)","t":[{"b":0,"e":0.42857,"k":"flat","v":0.03125,"x":0.14721,"p":[[0,14,0.0,0.03125,0.10555,0.0,0.0,0.0,0.0,0.42857,29,0,16,29,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.08929,0.22798,0.0,0.0,0.0,0.0,1.0,27,1,14,27,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[8,14,0.5714,0.04911,0.15815,0.0,0.0,0.0,0.0,0.71429,28,0,13,28,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[12,14,0.8571,0.0625,0.15947,0.0,0.0,0.0,0.0,0.71429,26,0,11,26,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[14,14,1.0,0.14721,0.22437,0.0,0.0,0.21429,0.0,0.71429,20,0,1,20,0,4,0,0,0,0,0,4,0,0,3,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,7,0.0,0.03571,0.10714,0.0,0.0,0.0,0.0,0.42857,28,0,15,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.04464,0.18013,0.0,0.0,0.0,0.0,1.0,29,1,9,29,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3997d9521df9f012","q":"We distribute weights of $1 \\mathrm{~g}, 2 \\mathrm{~g}, \\ldots, 200 \\mathrm{~g}$ on the two pans of a balance so that each pan contains 100 weights.\n\nProve that it is possible to exchange 50 weights from one pan with 50 weights from the other pan so that the balance becomes balanced.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,13,0.0,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,13,0.3077,0.07589,0.2448,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,13,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"261e391689ecb360","q":"Let $\\Gamma_{1}$ and $\\Gamma_{2}$ be two circles intersecting at $M$ and $N$. Let $A$ and $B$ be the points of tangency of these two circles with their nearest common external tangent to $M$. Let $C$ and $D$ be the symmetrics of $A$ and $B$ with respect to $M$, and $E$ and $F$ the points of intersection (other than $M$) of the circumcircle of $MCD$ with $\\Gamma_{1}$ and $\\Gamma_{2}$ respectively. Show that the circumcircles of $MEF$ and $NEF$ have the same radius.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,12,0.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fc86e98497cbb758","q":"The quadrilateral  $ABCD$  is inscribed in a circle. In the segment  $AB$  there is a point  $Q$  such that  $AQ = CD$ , and in the segment  $AD$ , the point  $P$  such that  $AP = BC$ . In what ratio is the segment  $AC$  divided by segment  $PQ$ ?","t":[{"b":4,"e":0.0,"k":"flat","v":0.23213,"x":0.34822,"p":[[0,7,0.0,0.29911,0.29957,0.0,0.28571,0.42857,0.0,1.0,13,2,9,13,0,0,0,0,4,0,0,10,0,0,1,0,0,1,0,0,1,0,2],[4,7,0.5714,0.23213,0.25189,0.0,0.07143,0.42857,0.0,0.71429,16,0,9,16,0,1,0,0,1,0,0,9,0,0,3,0,0,2,0,0,0,0,0],[7,7,1.0,0.34822,0.16342,0.28571,0.42857,0.42857,0.0,0.57143,4,0,0,4,0,3,0,0,2,0,0,21,0,0,2,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.18303,"x":0.37499,"p":[[0,10,0.0,0.29017,0.3633,0.0,0.0,0.4643,0.0,1.0,17,4,7,17,0,1,0,0,1,0,0,5,0,0,2,0,0,1,0,0,1,0,4],[4,10,0.4,0.37499,0.3809,0.0,0.35714,0.60714,0.0,1.0,13,5,7,13,0,2,0,0,1,0,0,4,0,0,4,0,0,1,0,0,2,0,5],[8,10,0.8,0.18303,0.14826,0.0,0.2857,0.28571,0.0,0.42857,12,0,2,12,0,1,0,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.21427,0.14282,0.10714,0.2857,0.28571,0.0,0.571,8,0,2,8,0,3,0,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"b2d0320d224a2b43","q":"Let $ABCD$ be a convex quadrilateral such that the lines $(AD)$ and $(BC)$ are not parallel. Suppose that the circles with diameters $[AB]$ and $[CD]$ intersect at two points $E$ and $F$ located inside $ABCD$. Let $\\Gamma_{E}$ be the circle passing through the orthogonal projections of $E$ onto $(AB)$, $(BC)$, and $(CD)$, and $\\Gamma_{F}$ be the circle passing through the orthogonal projections of $F$ onto $(CD)$, $(DA)$, and $(AB)$.\n\nProve that the midpoint of $[EF]$ lies on the line passing through the two points of intersection of $\\Gamma_{E}$ and $\\Gamma_{F}$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,28,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c5a3249b44633fcb","q":"Let  $\\omega_1$  and  $\\omega_2$  be two non-intersecting circles. Let one of its internal tangents touches  $\\omega_1$  and  $\\omega_2$  at  $A_1$  and  $A_2$ , respectively, and let one of its external tangents touches  $\\omega_1$  and  $\\omega_2$  at  $B_1$  and  $B_2$ , respectively. Prove that if  $A_1B_2 = A_2B_1$ , then    $A_1B_2 \\perp A_2B_1$ .","t":[{"b":1,"e":0.14286,"k":"falling","v":0.13839,"x":0.50892,"p":[[0,12,0.0,0.42857,0.36246,0.0,0.42859,0.71429,0.0,1.0,12,1,4,12,0,0,0,0,1,0,0,4,0,0,0,0,0,9,0,0,5,0,1],[4,12,0.3333,0.49107,0.36585,0.0,0.71429,0.74996,0.0,1.0,9,3,4,9,0,1,0,0,2,0,0,3,0,0,0,0,0,9,0,0,5,0,3],[8,12,0.6667,0.50892,0.35163,0.10714,0.71429,0.85704,0.0,1.0,8,2,5,8,0,1,0,0,1,0,0,5,0,0,0,0,0,8,0,0,7,0,2],[12,12,1.0,0.13839,0.145,0.0,0.14286,0.2857,0.0,0.42857,14,0,0,14,0,8,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.71429,"k":"rising","v":0.30803,"x":0.73214,"p":[[0,31,0.0,0.4866,0.33092,0.14286,0.4286,0.71429,0.0,1.0,7,3,2,7,0,2,0,0,0,0,0,8,0,0,0,0,0,10,0,0,2,0,3],[4,31,0.129,0.30803,0.37475,0.0,0.07143,0.71429,0.0,1.0,16,3,11,16,0,3,0,0,1,0,0,2,0,0,0,0,0,5,0,0,2,0,3],[8,31,0.2581,0.66504,0.21309,0.71321,0.71429,0.74996,0.0,0.85714,2,0,1,2,0,0,0,0,0,0,0,4,0,0,1,0,0,17,0,0,8,0,0],[12,31,0.3871,0.69629,0.1275,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,21,0,0,6,0,0],[16,31,0.5161,0.64731,0.17851,0.64264,0.71429,0.71429,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,7,0,0,0,0,0,20,0,0,4,0,0],[20,31,0.6452,0.68749,0.1357,0.71429,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,20,0,0,6,0,0],[24,31,0.7742,0.71429,0.12372,0.71429,0.71429,0.75,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,20,0,0,8,0,0],[28,31,0.9032,0.73214,0.11709,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,19,0,0,10,0,0],[31,31,1.0,0.73213,0.10566,0.71429,0.71429,0.85714,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,20,0,0,9,0,0]]}]},{"i":"c2f2775260104aa0","q":"The inscribed circle of triangle ABC touches its sides at points  $A_1$ ,  $B_1,$    $C_1$ , and the exscribed circles touch its sides at points  $A_2$ ,  $B_2$ ,  $C_2$  respectively. Circle  $a_1$  passes through points  $A$ ,  $B_1$  and  $C_1$ , and circle  $a_2$  passes through  $A$ ,  $B_2$  and  $C_2$ . These circles intersect for the second time at point  $A_3$ . Circles  $b_1$ ,  $b_2$  and  $c_1$ ,  $c_2$ , as well as points  $B_3$  and  $C_3$ , are constructed similarly. Prove that lines  $AA_3$ ,  $BB_3$ ,  $CC_3$  are parallel.","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.02232,"p":[[0,4,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,13,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.02232,"x":0.02232,"p":[[0,9,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,16,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"63e129d0a8f87279","q":"a)Prove that  $\\frac{1}{2}+\\frac{1}{3}+...+\\frac{1}{{{2}^{m}}}<m$ , for any  $m\\in {{\\mathbb{N}}^{*}}$ .\nb)Let  ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$  be the prime numbers less than  ${{2}^{100}}$ . Prove that $\\frac{1}{{{p}_{1}}}+\\frac{1}{{{p}_{2}}}+...+\\frac{1}{{{p}_{n}}}<10$","t":[{"b":0,"e":0.2857,"k":"flat","v":0.33482,"x":0.46429,"p":[[0,5,0.0,0.46428,0.18211,0.28571,0.42857,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,15,0,0,2,0,0,7,0,0,8,0,0,0,0,0],[4,5,0.8,0.46429,0.17127,0.28571,0.42857,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,7,0,0,7,0,0,5,0,0,1,0,0],[5,5,1.0,0.33482,0.11633,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,26,0,0,3,0,0,1,0,0,2,0,0,0,0,0]]},{"b":1,"e":0.57143,"k":"flat","v":0.45535,"x":0.58033,"p":[[0,5,0.0,0.45535,0.20341,0.2857,0.28571,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,17,0,0,3,0,0,3,0,0,7,0,0,2,0,0],[4,5,0.8,0.49105,0.1954,0.28571,0.4286,0.60714,0.2857,0.85714,0,0,0,0,0,0,0,0,12,0,0,5,0,0,7,0,0,5,0,0,3,0,0],[5,5,1.0,0.58033,0.17473,0.57143,0.57143,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,3,0,0,1,0,0,17,0,0,7,0,0,3,0,0]]}]},{"i":"79519adb14e70825","q":"The vertices of a convex  $2550$ -gon are colored black and white as follows: black, white, two black, two white, three black, three white, ..., 50 black, 50 white. Dania divides the polygon into quadrilaterals with diagonals that have no common points. Prove that there exists a quadrilateral among these, in which two adjacent vertices are black and the other two are white. \n\n*D. Rudenko*","t":[{"b":1,"e":0.57143,"k":"flat","v":0.23213,"x":0.41071,"p":[[0,10,0.0,0.40625,0.38484,0.0,0.28571,0.78571,0.0,1.0,10,8,6,10,0,0,0,0,10,0,0,2,0,0,1,0,0,1,0,0,0,0,8],[4,10,0.4,0.23213,0.28958,0.0,0.2857,0.28571,0.0,1.0,14,3,3,14,0,1,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,3],[8,10,0.8,0.41071,0.24419,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,0,0,0,21,0,0,3,0,0,1,0,0,1,0,0,3,0,2],[10,10,1.0,0.36383,0.12024,0.28571,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,19,0,0,5,0,1,6,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"falling","v":0.21428,"x":0.40625,"p":[[0,15,0.0,0.40625,0.4448,0.0,0.28571,1.0,0.0,1.0,14,11,3,14,0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,11],[4,15,0.2667,0.27232,0.37005,0.0,0.0,0.32144,0.0,1.0,17,5,7,17,0,1,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,5],[8,15,0.5333,0.23213,0.32683,0.0,0.0,0.28571,0.0,1.0,17,4,5,17,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[12,15,0.8,0.24553,0.33736,0.0,0.07143,0.28571,0.0,1.0,16,4,2,16,0,1,0,0,10,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[15,15,1.0,0.21428,0.11845,0.14286,0.28571,0.28571,0.0,0.28571,7,0,0,7,0,2,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"01e7145a2ab86d10","q":"Let $ABC$ be an isosceles triangle at $A$, $D$ the foot of the altitude from $A$, and $M$ a point inside $ADC$ such that $\\widehat{AMB}$ is obtuse and $\\widehat{DBM} + \\widehat{DAM} = \\widehat{MCB}$. The lines $(CM)$ and $(AD)$ intersect at $P$, and the lines $(BM)$ and $(AD)$ intersect at $Q$. Let $S$ be a point on $[AB]$ and $R$ a point on $[AM)$ that is not on $[AM]$ such that $\\widehat{SQB} = \\widehat{DPC}$ and $\\widehat{MRQ} = 2 \\widehat{QAM}$. Show that $QRS$ is isosceles.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,23,0.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.04018,0.1197,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,12,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"20c24fc45b8d0827","q":"Two fixed circles are given on the plane, one of them lies inside the other one. From a point  $C$  moving arbitrarily on the external circle, draw two chords  $CA, CB$  of the larger circle such that they tangent to the smalaler one. Find the locus of the incenter of triangle  $ABC$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.05804,"x":0.10268,"p":[[0,49,0.0,0.07572,0.07113,0.0,0.14,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,6,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.0758,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,1,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.09358,0.06773,0.0,0.14286,0.14286,0.0,0.1429,11,0,3,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.1429,9,0,1,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.08902,0.06896,0.0,0.14286,0.14286,0.0,0.14286,12,0,2,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.08027,0.0708,0.0,0.14286,0.14286,0.0,0.143,14,0,2,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,1,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.08474,0.0701,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,0,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.07581,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.07581,0.07121,0.0,0.14143,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.08483,0.07017,0.0,0.14286,0.14286,0.0,0.1429,13,0,1,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[49,49,1.0,0.09813,0.06616,0.0,0.14286,0.14286,0.0,0.1429,10,0,1,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.04018,"x":0.08929,"p":[[0,10,0.0,0.08929,0.09279,0.0,0.14286,0.14286,0.0,0.42857,14,0,1,14,0,17,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,2,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,2,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2e06b8dcce9144df","q":"The points  $M$  and  $N$  are on the side  $BC$  of  $\\Delta ABC$ , so that  $BM=CN$  and  $M$  is between  $B$  and  $N$ . Points  $P\\in AN$  and  $Q\\in AM$  are such that  $\\angle PMC=\\angle MAB$  and  $\\angle QNB=\\angle NAC$ . Prove that  $\\angle QBC=\\angle PCB$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0267,"p":[[0,14,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0267,0.05558,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.02223,"p":[[0,22,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.01313,0.04081,0.0,0.0,0.0,0.0,0.14,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"6121abce8a447fd7","q":"Let $ABC$ be a triangle with all angles acute. Let $F$ be the foot of the altitude from $A$ in $\\triangle ABC$, and let $P$ be a point on the segment $[AF]$. We denote $D$ as the point on $(BC)$ such that $(PD)$ is parallel to $(AC)$, and then $X$ as the point where the circumcircle of $\\triangle ABD$ and the circle centered at $D$ passing through $A$ intersect. Similarly, we denote $E$ as the point on $(BC)$ such that $(PE)$ is parallel to $(AB)$, and then $Y$ as the point where the circumcircle of $\\triangle ACE$ and the circle centered at $E$ passing through $A$ intersect.\nProve that the points $B, C, X$, and $Y$ are concyclic.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.08482,"p":[[0,12,0.0,0.08482,0.21086,0.0,0.0,0.0,0.0,0.71429,26,0,1,26,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[4,12,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.03125,"x":0.11607,"p":[[0,5,0.0,0.11607,0.22142,0.0,0.0,0.14286,0.0,0.71429,23,0,2,23,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,0,0,0,0],[4,5,0.8,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"10b320ce61849adf","q":"The incircle of a non-isosceles triangle  $ABC$  has centre  $I$  and is tangent to   $BC$  and  $CA$  in  $D$  and  $E$ , respectively. Let  $H$  be the orthocentre of  $ABI$ , let  $K$  be the intersection of  $AI$  and  $BH$  and let  $L$  be the intersection of  $BI$  and  $AH$ . Show that the circumcircles of  $DKH$  and  $ELH$  intersect on the incircle of  $ABC$ .","t":[{"b":1,"e":0.14286,"k":"flat","v":0.01786,"x":0.17411,"p":[[0,17,0.0,0.05804,0.11769,0.0,0.0,0.14286,0.0,0.57143,23,0,4,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,13,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.17411,0.13236,0.14286,0.14286,0.1786,0.0,0.57143,5,0,5,5,0,19,0,0,6,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,17,0.7059,0.14723,0.04352,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.03116,"x":0.125,"p":[[0,26,0.0,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.2857,19,0,1,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.08482,0.14223,0.0,0.0,0.14286,0.0,0.71429,19,0,4,19,0,10,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,26,0.3077,0.03116,0.05889,0.0,0.0,0.0,0.0,0.14286,25,0,6,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.09375,0.08459,0.0,0.14286,0.14286,0.0,0.28571,13,0,10,13,0,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.10714,0.07986,0.0,0.14286,0.14286,0.0,0.28571,10,0,7,10,0,20,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.09821,0.07523,0.0,0.14286,0.14286,0.0,0.28571,11,0,9,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f3f35c30d4c63d62","q":"The non-isosceles triangle  $ABC$  is inscribed in the circle \u03c9. The tangent to this circle at the point  $C$  intersects the line  $AB$  at the point  $D$ . Let the bisector of the angle  $CDB$  intersect the segments  $AC$  and  $BC$  at the points  $K$  and  $L$ , respectively. On the side  $AB$ , the point  $M$  is taken such that  $AK / BL = AM / BM$ . Let the perpendiculars from the point  $M$  to the lines  $KL$  and  $DC$  intersect the lines  $AC$  and  $DC$  at the points  $P$  and  $Q$ , respectively. Prove that the angle  $CQP$  is half of the angle  $ACB$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.03125,"x":0.14732,"p":[[0,26,0.0,0.06241,0.10055,0.0,0.0,0.14286,0.0,0.28571,22,0,1,22,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.04911,0.09181,0.0,0.0,0.03571,0.0,0.28571,24,0,9,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0758,0.1128,0.0,0.0,0.14286,0.0,0.28571,21,0,8,21,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,3,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.05804,0.11214,0.0,0.0,0.0,0.0,0.28571,25,0,9,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,26,0.7692,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,5,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,26,0.9231,0.14732,0.13115,0.0,0.14286,0.28571,0.0,0.28571,13,0,0,13,0,5,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[26,26,1.0,0.13839,0.12619,0.0,0.14286,0.2857,0.0,0.28571,13,0,0,13,0,7,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,14,0.0,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,3,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"5b4e807da4ef5333","q":"Let $\\omega_{1}, \\omega_{2}$ be two circles tangent to each other at a point $T$, such that $\\omega_{1}$ is inside $\\omega_{2}$. Let $M$ and $N$ be two distinct points on $\\omega_{1}$, different from $T$. Let [AB] and [CD] be two chords of the circle $\\omega_{2}$ passing through $M$ and $N$ respectively. Suppose that the segments [BD], $[A C]$, and $[M N]$ intersect at a point $K$.\nShow that (TK) is the bisector of the angle $\\widehat{M T N}$.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,11,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"0efbd8a2bffb0cf0","q":"There are  $10$  cups, each having  $10$  pebbles in them. Two players  $A$  and  $B$  play a game, repeating the following in order each move: $\\bullet$   $B$  takes one pebble from each cup and redistributes them as  $A$  wishes.  $\\bullet$  After  $B$  distributes the pebbles, he tells how many pebbles are in each cup to  $A$ . Then  $B$  destroys all the cups having no pebbles. $\\bullet$   $B$  switches the places of two cups without telling  $A$ .\n\nAfter finitely many moves,  $A$  can guarantee that  $n$  cups are destroyed. Find the maximum possible value of  $n$ .\n(Note that  $A$  doesn't see the cups while playing.)\n\n*Proposed by Emre Osman*","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,29,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,9,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,21,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,24,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02455,"p":[[0,5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.02455,0.09211,0.0,0.0,0.0,0.0,0.5,29,0,1,29,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5585eb49bc4ed3de","q":"Let  $q$  be a positive integer which is not a perfect cube. Prove that there exists a positive constant  $C$  such that for all natural numbers  $n$ , one has  $$ \\{ nq^{\\frac{1}{3}} \\} + \\{ nq^{\\frac{2}{3}} \\} \\geq Cn^{-\\frac{1}{2}} $$  where  $\\{ x \\}$  denotes the fractional part of  $x$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.08482,"x":0.10714,"p":[[0,7,0.0,0.08482,0.11769,0.0,0.0,0.14286,0.0,0.28571,20,0,17,20,0,5,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.10714,0.11845,0.0,0.07143,0.17857,0.0,0.28571,16,0,15,16,0,8,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.09374,"x":0.10268,"p":[[0,6,0.0,0.10268,0.12992,0.0,0.0,0.2857,0.0,0.28571,19,0,15,19,0,3,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.09374,0.13646,0.0,0.0,0.14286,0.0,0.571,19,0,17,19,0,7,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"61b6c5eddb8a2cb8","q":"Let  $a_{1}, a_{2}, a_{3}, \\cdots$  be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form  $a_{i}+2a_{j}+4a_{k}$ , where  $i, j, $  and  $k$  are not necessarily distinct. Determine  $a_{1998}$ .","t":[{"b":1,"e":1.0,"k":"rising","v":0.34821,"x":1.0,"p":[[0,37,0.0,0.42857,0.41955,0.0,0.28571,1.0,0.0,1.0,11,10,5,11,0,2,0,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,10],[4,37,0.1081,0.50446,0.45314,0.0,0.5,1.0,0.0,1.0,11,13,5,11,0,3,0,0,1,0,0,1,0,0,2,0,0,0,0,0,1,0,13],[8,37,0.2162,0.34821,0.42848,0.0,0.07143,1.0,0.0,1.0,16,9,9,16,0,1,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,9],[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":3,"e":1.0,"k":"rising","v":0.47321,"x":1.0,"p":[[0,41,0.0,0.47321,0.47305,0.0,0.21429,1.0,0.0,1.0,14,12,7,14,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,12],[4,41,0.0976,0.47768,0.46649,0.0,0.28571,1.0,0.0,1.0,13,13,6,13,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,13],[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]]}]},{"i":"88d69ef3ec4b92e7","q":"We call the positive integer  $n$  a  $3-$ *stratum* number if we can divide the set of its positive divisors into  $3$  subsets such that the sum of each subset is equal to the others. $a)$  Find a  $3-$ stratum number. $b)$  Prove that there are infinitely many  $3-$ stratum numbers.","t":[{"b":2,"e":1.0,"k":"rising","v":0.77232,"x":1.0,"p":[[0,30,0.0,0.77232,0.28982,0.60721,0.85714,1.0,0.2857,1.0,0,14,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,9,0,14],[4,30,0.1333,0.82142,0.28794,0.85711,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,5,0,20],[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],[24,30,0.8,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":5,"e":1.0,"k":"rising","v":0.6875,"x":1.0,"p":[[0,28,0.0,0.6875,0.30812,0.28571,0.85714,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,10,0,0,2,0,0,1,0,0,1,0,0,7,0,11],[4,28,0.1429,0.76786,0.31894,0.28571,1.0,1.0,0.2857,1.0,0,19,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,0,3,0,19],[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]]}]},{"i":"f7c96f4b76db1ce3","q":"Let  $n$  be a positive integer. What is the largest  $k$  for which there exist  $n\\times n$  matrices  $M_1,\\dots,M_k$  and  $N_1,\\dots,N_k$  with real entries such that for all  $i$  and  $j,$  the matrix product  $M_iN_j$  has a zero entry somewhere on its diagonal if and only if  $i\\ne j?$","t":[{"b":3,"e":0.0,"k":"flat","v":0.09375,"x":0.26339,"p":[[0,26,0.0,0.09375,0.12169,0.0,0.0,0.17857,0.0,0.28571,19,0,5,19,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.26339,0.14773,0.14286,0.28571,0.28571,0.0,0.57143,4,0,3,4,0,5,0,0,18,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[8,26,0.3077,0.20536,0.13333,0.10714,0.28571,0.28571,0.0,0.42857,8,0,2,8,0,4,0,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.25446,0.12745,0.24999,0.28571,0.28571,0.0,0.71429,3,0,1,3,0,5,0,0,22,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,26,0.6154,0.25,0.12372,0.25,0.28571,0.28571,0.0,0.57143,4,0,3,4,0,4,0,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,26,0.7692,0.20982,0.18552,0.0,0.2857,0.28571,0.0,0.57143,11,0,4,11,0,3,0,0,14,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[24,26,0.9231,0.14286,0.15152,0.0,0.14286,0.2857,0.0,0.4286,14,0,11,14,0,8,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"rising","v":0.11607,"x":0.32589,"p":[[0,44,0.0,0.11607,0.15746,0.0,0.0,0.2857,0.0,0.57143,19,0,6,19,0,3,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,44,0.0909,0.15178,0.12846,0.0,0.14286,0.28571,0.0,0.28571,12,0,6,12,0,6,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.15625,0.13533,0.0,0.14286,0.28571,0.0,0.4286,12,0,6,12,0,6,0,0,13,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.22319,0.17099,0.0,0.28571,0.28571,0.0,0.57143,9,0,6,9,0,3,0,0,16,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[16,44,0.3636,0.21428,0.13363,0.14286,0.2857,0.28571,0.0,0.57143,6,0,2,6,0,7,0,0,17,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,44,0.4545,0.30357,0.1171,0.2857,0.28571,0.28571,0.0,0.71429,1,0,1,1,0,2,0,0,24,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[24,44,0.5455,0.28571,0.06186,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,44,0.6364,0.31697,0.08552,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[32,44,0.7273,0.30804,0.06297,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,28,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,44,0.8182,0.32588,0.08914,0.2857,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,1,0,0,23,0,0,6,0,0,2,0,0,0,0,0,0,0,0],[40,44,0.9091,0.32589,0.11971,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,2,0,0,24,0,0,2,0,0,3,0,0,1,0,0,0,0,0],[44,44,1.0,0.29464,0.06121,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"99789ab8fe40805f","q":"Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.","t":[{"b":5,"e":0.2857,"k":"flat","v":0.07143,"x":0.21875,"p":[[0,11,0.0,0.17409,0.17396,0.0,0.21428,0.28571,0.0,0.57143,14,0,2,14,0,2,0,0,13,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,11,0.3636,0.13839,0.13592,0.0,0.14286,0.2857,0.0,0.28571,15,0,0,15,0,3,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.21875,0.11285,0.14289,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,3,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.28571,"k":"flat","v":0.12491,"x":0.22768,"p":[[0,8,0.0,0.12491,0.16655,0.0,0.0,0.2857,0.0,0.57143,18,0,0,18,0,4,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,8,0.5,0.17847,0.20824,0.0,0.14143,0.28571,0.0,0.71429,15,0,0,15,0,3,0,0,10,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[8,8,1.0,0.22768,0.11214,0.2857,0.28571,0.28571,0.0,0.28571,6,0,0,6,0,1,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"06d943c78bd54e88","q":"Let $a, b, c$ be the sides of a triangle. Prove that  $$ \\frac{\\sqrt{b+c-a}}{\\sqrt{b}+\\sqrt{c}-\\sqrt{a}}+\\frac{\\sqrt{c+a-b}}{\\sqrt{c}+\\sqrt{a}-\\sqrt{b}}+\\frac{\\sqrt{a+b-c}}{\\sqrt{a}+\\sqrt{b}-\\sqrt{c}} \\leq 3 . $$  (Korea)","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.2232,"p":[[0,28,0.0,0.10268,0.25313,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,1],[4,28,0.1429,0.16964,0.2911,0.0,0.0,0.2857,0.0,1.0,22,1,0,22,0,1,0,0,3,0,0,0,0,0,1,0,0,4,0,0,0,0,1],[8,28,0.2857,0.2232,0.34243,0.0,0.0,0.46418,0.0,1.0,21,3,0,21,0,0,0,0,2,0,0,1,0,0,3,0,0,2,0,0,0,0,3],[12,28,0.4286,0.13839,0.25123,0.0,0.0,0.07143,0.0,0.71429,24,0,0,24,0,0,0,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,0],[16,28,0.5714,0.174,0.31689,0.0,0.0,0.2857,0.0,1.0,22,3,0,22,0,1,0,0,4,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[20,28,0.7143,0.20982,0.303,0.0,0.0,0.32144,0.0,0.85714,20,0,0,20,0,0,0,0,4,0,0,1,0,0,1,0,0,4,0,0,2,0,0],[24,28,0.8571,0.13393,0.2111,0.0,0.0,0.28571,0.0,0.71429,21,0,0,21,0,2,0,0,3,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[28,28,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.09821,"p":[[0,27,0.0,0.09374,0.22189,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[4,27,0.1481,0.09821,0.24074,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[8,27,0.2963,0.05802,0.15911,0.0,0.0,0.0,0.0,0.57143,28,0,0,28,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[12,27,0.4444,0.06696,0.2082,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0],[16,27,0.5926,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c913b9753dd39d99","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1 \\leq k \\leq n$ is good if there exists some $\\ell$ with $1 \\leq \\ell \\leq m$ such that  $$ a_{k}+a_{k+1}+\\cdots+a_{k+\\ell-1} \\geq 0 $$  where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that  $$ \\sum_{k \\in T} a_{k} \\geq 0 $$","t":[{"b":3,"e":0.14286,"k":"flat","v":0.03572,"x":0.09822,"p":[[0,5,0.0,0.03572,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.09822,0.10972,0.0,0.14286,0.14286,0.0,0.42857,15,0,0,15,0,13,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.02009,"x":0.08482,"p":[[0,90,0.0,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,90,0.0444,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.42857,20,0,0,20,0,9,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,90,0.0889,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.42857,22,0,1,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,90,0.1333,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.57143,20,0,1,20,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,90,0.1778,0.06473,0.11758,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,1,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,90,0.2222,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,90,0.2667,0.04464,0.13092,0.0,0.0,0.0,0.0,0.71429,26,0,1,26,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,90,0.3111,0.06697,0.11285,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,90,0.3556,0.0759,0.13356,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,8,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,90,0.4,0.06242,0.11253,0.0,0.0,0.14286,0.0,0.42857,22,0,1,22,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[40,90,0.4444,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,22,0,1,22,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,90,0.4889,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,90,0.5333,0.05804,0.10013,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[52,90,0.5778,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,90,0.6222,0.05802,0.11763,0.0,0.0,0.14286,0.0,0.571,23,0,1,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,90,0.6667,0.04464,0.15335,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[64,90,0.7111,0.02009,0.04803,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,90,0.7556,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,90,0.8,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,90,0.8444,0.02902,0.09081,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[80,90,0.8889,0.07143,0.0945,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[84,90,0.9333,0.0759,0.08737,0.0,0.0,0.14286,0.0,0.28571,17,0,0,17,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,90,0.9778,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[90,90,1.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"53c05b0b4ae6051c","q":"Let $n \\geq 2$ be an integer. Let $P\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ be a nonconstant $n$-variable polynomial with real coefficients. Assuming that $P$ vanishes whenever two of its arguments are equal, prove that $P$ has at least $n$ ! terms.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"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],[4,10,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,15,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a4314c8d49b3ad34","q":"Let $b_{1}, \\ldots, b_{n}$ be non-negative real numbers such that $b_{1}+b_{2}+\\ldots+b_{n}=2$. Let $a_{0}, a_{1}, \\ldots, a_{n}$ be real numbers such that $a_{0}=a_{n}=0$. Suppose that for all $i$ between 1 and $n$ inclusive,\n\n$$\n\\left|a_{i}-a_{i-1}\\right| \\leqslant b_{i}\n$$\n\nShow that we have\n\n$$\n\\left(a_{0}+a_{1}\\right) b_{1}+\\left(a_{1}+a_{2}\\right) b_{2}+\\ldots+\\left(a_{n-1}+a_{n}\\right) b_{n} \\leqslant 2\n$$","t":[{"b":0,"e":0.14286,"k":"flat","v":0.125,"x":0.24553,"p":[[0,34,0.0,0.24553,0.26782,0.0,0.14286,0.28571,0.0,1.0,10,2,0,10,0,8,0,0,7,0,0,1,0,0,4,0,0,0,0,0,0,0,2],[4,34,0.1176,0.22321,0.25738,0.0,0.14286,0.42857,0.0,1.0,13,1,0,13,0,7,0,0,3,0,0,2,0,0,6,0,0,0,0,0,0,0,1],[8,34,0.2353,0.17402,0.22229,0.0,0.14286,0.17857,0.0,0.85714,13,0,0,13,0,11,0,0,3,0,0,1,0,0,2,0,0,1,0,0,1,0,0],[12,34,0.3529,0.23659,0.23852,0.0,0.14286,0.32143,0.0,0.85714,11,0,0,11,0,6,0,0,7,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[16,34,0.4706,0.125,0.13243,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,12,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,34,0.5882,0.18304,0.28175,0.0,0.0,0.28571,0.0,1.0,17,2,0,17,0,5,0,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,2],[24,34,0.7059,0.16509,0.22619,0.0,0.14143,0.28571,0.0,1.0,15,1,0,15,0,8,0,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[28,34,0.8235,0.1607,0.2105,0.0,0.14286,0.17857,0.0,0.85714,14,0,0,14,0,10,0,0,4,0,0,0,0,0,3,0,0,0,0,0,1,0,0],[32,34,0.9412,0.13821,0.06666,0.14286,0.14286,0.14286,0.0,0.28571,4,0,0,4,0,25,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.125,0.08564,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.57143,"k":"rising","v":0.21875,"x":0.60266,"p":[[0,12,0.0,0.25893,0.2299,0.0,0.28571,0.32143,0.0,0.85714,9,0,0,9,0,5,0,0,10,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[4,12,0.3333,0.21875,0.22864,0.0,0.14286,0.28571,0.0,1.0,11,1,0,11,0,6,0,0,9,0,0,2,0,0,3,0,0,0,0,0,0,0,1],[8,12,0.6667,0.32143,0.32143,0.0,0.21429,0.57143,0.0,1.0,11,2,0,11,0,5,0,0,3,0,0,1,0,0,8,0,0,0,0,0,2,0,2],[12,12,1.0,0.60266,0.1461,0.57143,0.57143,0.57143,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,22,0,0,4,0,0,1,0,2]]}]},{"i":"948d693fc972c92d","q":"There are  $100$  points on the circumference of a circle, arbitrarily labelled by  $1,2,\\ldots,100$ . For each three points, call their triangle **clockwise** if the increasing order of them is in clockwise order. Prove that it is impossible to have exactly  $2017$  **clockwise** triangles.","t":[{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.21875,"p":[[0,40,0.0,0.21875,0.40404,0.0,0.0,0.03571,0.0,1.0,24,6,9,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,6],[4,40,0.1,0.08928,0.25441,0.0,0.0,0.0,0.0,1.0,28,1,16,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[8,40,0.2,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,28,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,30,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,40,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,30,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.27679,"p":[[0,30,0.0,0.23214,0.36553,0.0,0.0,0.42857,0.0,1.0,21,4,9,21,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0,4],[4,30,0.1333,0.27679,0.39599,0.0,0.0,0.60714,0.0,1.0,20,5,7,20,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,5],[8,30,0.2667,0.04464,0.1448,0.0,0.0,0.0,0.0,0.57143,29,0,15,29,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,30,0.4,0.08929,0.24679,0.0,0.0,0.0,0.0,1.0,28,1,5,28,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,1],[16,30,0.5333,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,22,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,30,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,15,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,6,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,6,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7f6f1c6055baf555","q":"The prime number  $p$  and a positive integer  $k$  are given. Assume that  $P(x)\\in \\mathbb Z[X]$  is a polynomial with coefficients in the set  $\\{0,1,\\cdots,p-1\\}$  with least degree which satisfies the following property: There exists a permutaion of numbers  $1,2,\\cdots,p-1$  around a circle such that for any  $k$  consecutive numbers  $a_1,a_2,\\cdots,a_k$  one has  $$ p | P(a_1)+P(a_2)+\\cdots+ P(a_k). $$  Prove that  $P(x)$  is of the form  $ax^d+b$ .\n\n\nProposed by *Yahya Motevassel*","t":[{"b":3,"e":0.2857,"k":"flat","v":0.14732,"x":0.24098,"p":[[0,3,0.0,0.24098,0.15341,0.14286,0.2857,0.28571,0.14,1.0,0,1,0,0,0,15,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[3,3,1.0,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.19196,"x":0.22759,"p":[[0,10,0.0,0.22759,0.07027,0.14286,0.28571,0.28571,0.14,0.28571,0,0,0,0,0,13,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.19196,0.07668,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,22,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.21875,0.10092,0.14286,0.14286,0.28571,0.14286,0.4286,0,0,0,0,0,19,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.22312,0.07945,0.14286,0.2857,0.28571,0.14,0.42857,0,0,0,0,0,15,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"12bbda1712b905f9","q":"We have  $n$  points in the plane, no three on a line.\r\nWe call  $k$  of them good if they form a convex polygon and there is no other point in the convex polygon.\r\nSuppose that for a fixed  $k$  the number of  $k$  good points is  $c_k$ .\r\nShow that the following sum is independent of the structure of points and only depends on  $n$  :\r\n\\[ \\sum_{i=3}^n (-1)^i c_i \\]","t":[{"b":3,"e":0.14286,"k":"flat","v":0.01339,"x":0.08473,"p":[[0,16,0.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.42857,24,0,2,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.08473,0.07866,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.14286,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.09375,"p":[[0,28,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.06679,0.08722,0.0,0.0,0.14286,0.0,0.28571,19,0,1,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.09375,0.09852,0.0,0.14286,0.14286,0.0,0.42857,14,0,0,14,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.09348,0.06766,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.07589,0.07129,0.0,0.14286,0.14286,0.0,0.1429,15,0,0,15,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8889fecf6ea4c209","q":"Let $d$ be a positive integer. Show that for every integer $S$, there exists an integer $n>0$ and a sequence $\\epsilon_{1}, \\epsilon_{2}, \\ldots, \\epsilon_{n}$, where for any $k, \\epsilon_{k}=1$ or $\\epsilon_{k}=-1$, such that\n\n$$\nS=\\epsilon_{1}(1+d)^{2}+\\epsilon_{2}(1+2 d)^{2}+\\epsilon_{3}(1+3 d)^{2}+\\cdots+\\epsilon_{n}(1+n d)^{2}\n$$","t":[{"b":4,"e":0.0,"k":"flat","v":0.08036,"x":0.19643,"p":[[0,19,0.0,0.19196,0.16213,0.14286,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,11,0,0,13,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,19,0.2105,0.17848,0.16753,0.0,0.21428,0.28571,0.0,0.71429,12,0,0,12,0,4,0,0,14,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,19,0.4211,0.18294,0.12495,0.14214,0.14286,0.2857,0.0,0.42857,7,0,0,7,0,11,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.19196,0.10479,0.14286,0.2143,0.28571,0.0,0.28571,5,0,0,5,0,11,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.19643,0.1171,0.14286,0.28571,0.28571,0.0,0.42857,6,0,0,6,0,9,0,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.18303,"x":0.27678,"p":[[0,10,0.0,0.18303,0.11425,0.14286,0.2143,0.28571,0.0,0.28571,7,0,0,7,0,9,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.19643,0.12242,0.14286,0.2857,0.28571,0.0,0.42857,7,0,0,7,0,7,0,0,17,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.27678,0.03458,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fdc3266a3e2d4945","q":"Let $S$ be a set containing $n^{2}+n-1$ elements. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,28,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"aec4518f3808cdd2","q":"Let $a_{1}, \\ldots, a_{r}$ be positive real numbers. For $n>r$, we inductively define  $$ a_{n}=\\max _{1 \\leq k \\leq n-1}\\left(a_{k}+a_{n-k}\\right) $$  Prove that there exist positive integers $\\ell \\leq r$ and $N$ such that $a_{n}=a_{n-\\ell}+a_{\\ell}$ for all $n \\geq N$. (Iran)","t":[{"b":4,"e":0.57143,"k":"rising","v":0.27223,"x":0.54462,"p":[[0,27,0.0,0.29909,0.13051,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,7,0,0,20,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[4,27,0.1481,0.2991,0.14878,0.2857,0.28571,0.28571,0.0,0.71429,1,0,1,1,0,6,0,0,20,0,0,0,0,0,4,0,0,1,0,0,0,0,0],[8,27,0.2963,0.27223,0.09701,0.2857,0.28571,0.28571,0.14,0.57143,0,0,0,0,0,7,0,0,23,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,27,0.4444,0.50444,0.20197,0.28571,0.57121,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,7,0,0,5,0,0,12,0,0,3,0,0,2,0,1],[16,27,0.5926,0.53124,0.1439,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,2,0,0,0,0,0,7,0,0,21,0,0,1,0,0,0,0,1],[20,27,0.7407,0.51784,0.10564,0.42859,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,5,0,0,22,0,0,1,0,0,0,0,0],[24,27,0.8889,0.52231,0.1365,0.42857,0.57143,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,2,0,0,7,0,0,21,0,0,0,0,0,0,0,1],[27,27,1.0,0.54462,0.07522,0.57132,0.57143,0.57143,0.28571,0.71429,0,0,0,0,0,0,0,0,1,0,0,5,0,0,25,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.26339,"x":0.36605,"p":[[0,12,0.0,0.3482,0.22284,0.28571,0.28571,0.28571,0.0,1.0,1,2,1,1,0,5,0,0,19,0,0,0,0,0,4,0,0,1,0,0,0,0,2],[4,12,0.3333,0.36605,0.13801,0.2857,0.28571,0.46418,0.14286,0.71429,0,0,0,0,0,1,0,0,21,0,0,2,0,0,7,0,0,1,0,0,0,0,0],[8,12,0.6667,0.27668,0.12346,0.24999,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,7,0,0,20,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,12,1.0,0.26339,0.05187,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,5,0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7db15e7fbf49c49f","q":"Let  $f$  and  $g$  be  $2 \\pi$ -periodic integrable functions such that in some neighborhood of  $0$ ,  $g(x) = f(ax)$  with some  $a \\neq 0$ . Prove that the Fourier series of  $f$  and  $g$  are simultaneously convergent or divergent at  $0$ .","t":[{"b":2,"e":0.57143,"k":"flat","v":0.46424,"x":0.49554,"p":[[0,11,0.0,0.49554,0.13825,0.42857,0.4286,0.57143,0.14286,0.85714,0,0,0,0,0,2,0,0,0,0,0,15,0,0,12,0,0,2,0,0,1,0,0],[4,11,0.3636,0.46424,0.07135,0.42857,0.42857,0.571,0.2857,0.57143,0,0,0,0,0,0,0,0,1,0,0,22,0,0,9,0,0,0,0,0,0,0,0],[8,11,0.7273,0.48659,0.07014,0.42857,0.42857,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0],[11,11,1.0,0.48661,0.07016,0.42857,0.42857,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,19,0,0,13,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.57143,"k":"flat","v":0.45982,"x":0.54015,"p":[[0,19,0.0,0.45982,0.06901,0.42857,0.42857,0.4643,0.28571,0.57143,0,0,0,0,0,0,0,0,1,0,0,23,0,0,8,0,0,0,0,0,0,0,0],[4,19,0.2105,0.54015,0.13708,0.42857,0.57143,0.57143,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,15,0,0,2,0,0,1,0,1],[8,19,0.4211,0.48657,0.12296,0.42857,0.42857,0.571,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,21,0,0,8,0,0,1,0,0,0,0,1],[12,19,0.6316,0.47322,0.08328,0.42857,0.42857,0.46431,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,24,0,0,6,0,0,2,0,0,0,0,0],[16,19,0.8421,0.4732,0.06619,0.42857,0.42857,0.57141,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0],[19,19,1.0,0.46427,0.06183,0.42857,0.42857,0.4642,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0]]}]},{"i":"9f795da06ff08fab","q":"The point $P$ lies in the interior of triangle $ABC$ and satisfies\n\n$$\n\\varangle B P C-\\varangle B A C=\\varangle C P A-\\varangle C B A=\\varangle A P B-\\varangle A C B .\n$$\n\nProve that then the following holds:\n\n$$\n\\overline{P A} \\cdot \\overline{B C}=\\overline{P B} \\cdot \\overline{A C}=\\overline{P C} \\cdot \\overline{A B}\n$$\n\nFirst, one observes that $\\varangle B P C=60^{\\circ}+\\alpha, \\varangle C P A=60^{\\circ}+\\beta, \\varangle A P B=60^{\\circ}+\\gamma$. By symmetry, it suffices to show one of the two claimed equations. (Hint: For an angle $\\varangle X Y Z$ with $0^{\\circ}<\\varangle X Y Z<180^{\\circ}$ in the mathematically positive sense, set $\\varangle Z Y X:=$ $180^{\\circ}-\\varangle X Y Z$. With this convention, for example, for four pairwise distinct points $W, X, Y, Z$ on a circle, it always holds that $\\varangle X Y Z=\\varangle X W Z$ ).","t":[{"b":3,"e":0.14286,"k":"flat","v":0.14268,"x":0.14286,"p":[[0,10,0.0,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.1383,"x":0.14277,"p":[[0,9,0.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.1383,0.02485,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.1429,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2a982b7e5ef755c9","q":"Let  $n>1$  be an integer, and let  $k$  be the number of distinct prime divisors of  $n$ . Prove that there exists an integer  $a$ ,  $1<a<\\frac{n}{k}+1$ , such that  $n \\mid a^2-a$ .","t":[{"b":2,"e":0.28571,"k":"flat","v":0.15625,"x":0.24553,"p":[[0,6,0.0,0.15625,0.09006,0.14286,0.14286,0.1786,0.0,0.28571,5,0,4,5,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.21875,0.07973,0.14286,0.2857,0.28571,0.0,0.28571,1,0,1,1,0,13,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.24553,0.06423,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,9,0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.14286,"x":0.23205,"p":[[0,30,0.0,0.1875,0.09061,0.14286,0.14286,0.2857,0.0,0.28571,3,0,1,3,0,16,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.17411,0.09932,0.14286,0.14286,0.28571,0.0,0.28571,5,0,0,5,0,15,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.18304,0.08917,0.14286,0.14286,0.2857,0.0,0.28571,3,0,0,3,0,17,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.23205,0.0995,0.14286,0.2857,0.28571,0.0,0.57143,1,0,0,1,0,12,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,30,0.5333,0.21429,0.07986,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,14,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.22312,0.09414,0.14286,0.2857,0.28571,0.0,0.42857,2,0,1,2,0,11,0,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.17839,0.06196,0.14286,0.14286,0.17857,0.14,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c27e12b1cc489f82","q":"Let $A B C$ be a triangle. Let $\\theta$ be a fixed angle for which  $$ \\theta<\\frac{1}{2} \\min (\\angle A, \\angle B, \\angle C) $$  Points $S_{A}$ and $T_{A}$ lie on segment $B C$ such that $\\angle B A S_{A}=\\angle T_{A} A C=\\theta$. Let $P_{A}$ and $Q_{A}$ be the feet from $B$ and $C$ to $\\overline{A S_{A}}$ and $\\overline{A T_{A}}$ respectively. Then $\\ell_{A}$ is defined as the perpendicular bisector of $\\overline{P_{A} Q_{A}}$. Define $\\ell_{B}$ and $\\ell_{C}$ analogously by repeating this construction two more times (using the same value of $\\theta$ ). Prove that $\\ell_{A}, \\ell_{B}$, and $\\ell_{C}$ are concurrent or all parallel.","t":[{"b":5,"e":0.14286,"k":"flat","v":0.00446,"x":0.09375,"p":[[0,71,0.0,0.0625,0.13333,0.0,0.0,0.03571,0.0,0.57143,24,0,5,24,0,5,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,71,0.0563,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,5,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,71,0.1127,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,71,0.169,0.03562,0.09439,0.0,0.0,0.0,0.0,0.42857,27,0,2,27,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,71,0.2254,0.0625,0.12339,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,4,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,71,0.2817,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,1,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,71,0.338,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,71,0.3944,0.09375,0.15815,0.0,0.0,0.14286,0.0,0.57143,20,0,1,20,0,8,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,71,0.4507,0.05348,0.07774,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,71,0.507,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,71,0.5634,0.03125,0.08553,0.0,0.0,0.0,0.0,0.4286,27,0,1,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,71,0.6197,0.08481,0.13762,0.0,0.0,0.14286,0.0,0.571,20,0,1,20,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[48,71,0.6761,0.03562,0.07129,0.0,0.0,0.0,0.0,0.2857,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,71,0.7324,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,71,0.7887,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.1429,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,71,0.8451,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,71,0.9014,0.05348,0.07774,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,71,0.9577,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[71,71,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01777,"x":0.08481,"p":[[0,24,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,5,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,2,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.05804,0.1063,0.0,0.0,0.14286,0.0,0.42857,23,0,5,23,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.08481,0.16307,0.0,0.0,0.14286,0.0,0.57143,23,0,1,23,0,4,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,24,0.6667,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,5,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,23,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f96c6207c9db110d","q":"Let $ABC$ be a triangle, and $D, E, F$ the feet of the altitudes from $A, B, C$ respectively. We also define $H$ as the orthocenter of $ABC$, $O$ as the center of its circumcircle, and $X$ as the point on the line $(EF)$ that satisfies $XA = XD$. Show that the lines $(AX)$ and $(OH)$ are perpendicular.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,12,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,24,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"45178f084247e552","q":"Two circles in the plane do not intersect and do not lie inside each other. We choose diameters  $A_1B_1$  and  $A_2B_2$  of these circles such that the segments  $A_1A_2$  and  $B_1B_2'$  intersect. Let  $A$  and  $B$  be the midpoints of the segments  $A_1A_2$  and  $B_1B_2$ , and  $C$  be the intersection point of these segments. Prove that the orthocenter of the triangle  $ABC$  belongs to a fixed line that does not depend on the choice of diameters.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.02679,"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],[4,64,0.0625,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,64,0.125,0.0267,0.06606,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,64,0.1875,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,64,0.25,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,5,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,64,0.3125,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,64,0.375,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,64,0.4375,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,2,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,64,0.5,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,64,0.5625,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,64,0.625,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,64,0.6875,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,64,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,64,0.8125,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,64,0.875,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,64,0.9375,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,64,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.04465,"p":[[0,19,0.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a2053cb3c382a6ff","q":"Let $m, n \\geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_{1}, X_{2}, \\ldots, X_{m}$ be pairwise distinct non-empty, not necessary disjoint subsets of $X$. A function $f: X \\rightarrow$ $\\{1,2, \\ldots, n+1\\}$ is called nice if there exists an index $k$ such that  $$ \\sum_{x \\in X_{k}} f(x)>\\sum_{x \\in X_{i}} f(x) \\text { for all } i \\neq k $$  Prove that the number of nice functions is at least $n^{n}$. (Germany)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.0133,"x":0.05357,"p":[[0,33,0.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,9,0.0,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1420edf9bfd26ed3","q":"Let $p$ be a prime number and let $s$ be an integer with $0<s<p$. Prove that there exist integers $m$ and $n$ with $0<m<n<p$ and  $$ \\left\\{\\frac{s m}{p}\\right\\}<\\left\\{\\frac{s n}{p}\\right\\}<\\frac{s}{p} $$  if and only if $s$ is not a divisor of $p-1$.","t":[{"b":0,"e":1.0,"k":"flat","v":0.86591,"x":0.95088,"p":[[0,23,0.0,0.87499,0.17406,0.71429,1.0,1.0,0.42857,1.0,0,19,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,7,0,0,3,0,19],[4,23,0.1739,0.86591,0.14715,0.82132,0.85714,1.0,0.571,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,10,0,14],[8,23,0.3478,0.9375,0.11259,0.85714,1.0,1.0,0.57143,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,5,0,23],[12,23,0.5217,0.93302,0.1129,0.85714,1.0,1.0,0.571,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,6,0,22],[16,23,0.6957,0.9375,0.13333,1.0,1.0,1.0,0.4286,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,2,0,25],[20,23,0.8696,0.95088,0.09853,1.0,1.0,1.0,0.71429,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,3,0,25],[23,23,1.0,0.9241,0.11286,0.85711,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,5,0,21]]},{"b":3,"e":1.0,"k":"flat","v":0.86607,"x":1.0,"p":[[0,40,0.0,0.86607,0.20183,0.71429,1.0,1.0,0.28571,1.0,0,20,0,0,0,0,0,0,1,0,0,2,0,0,1,0,0,6,0,0,2,0,20],[4,40,0.1,0.96428,0.07987,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,0,26],[8,40,0.2,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[12,40,0.3,0.98661,0.05486,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[16,40,0.4,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,40,0.5,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[24,40,0.6,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[32,40,0.8,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[36,40,0.9,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"e36601f6a1555be1","q":"Triangle  $ABC$  is inscribed in circle  $\\omega$ .  A circle with chord  $BC$  intersects segments  $AB$  and  $AC$  again at  $S$  and  $R$ , respectively.  Segments  $BR$  and  $CS$  meet at  $L$ , and rays  $LR$  and  $LS$  intersect  $\\omega$  at  $D$  and  $E$ , respectively.  The internal angle bisector of  $\\angle BDE$  meets line  $ER$  at  $K$ .  Prove that if  $BE = BR$ , then  $\\angle ELK = \\tfrac{1}{2} \\angle BCD$ .\n\n*Proposed by Evan Chen*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,18,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,3,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"eafaeae2d561ab1c","q":"The real numbers $x, y, z, m, n$ are positive, such that $m+n \\geq 2$. Prove that\n\n$$\n\\begin{gathered}\nx \\sqrt{y z(x+m y)(x+n z)}+y \\sqrt{x z(y+m x)(y+n z)}+z \\sqrt{x y(z+m x)(x+n y)} \\leq \\\\\n\\frac{3(m+n)}{8}(x+y)(y+z)(z+x) .\n\\end{gathered}\n$$","t":[{"b":1,"e":0.42857,"k":"flat","v":0.30347,"x":0.46423,"p":[[0,29,0.0,0.30347,0.22803,0.14286,0.28571,0.46418,0.0,0.71429,5,0,1,5,0,10,0,0,4,0,0,5,0,0,5,0,0,3,0,0,0,0,0],[4,29,0.1379,0.3348,0.24116,0.14286,0.35714,0.57111,0.0,0.857,5,0,0,5,0,8,0,0,3,0,0,7,0,0,7,0,0,0,0,0,2,0,0],[8,29,0.2759,0.36606,0.23672,0.1429,0.28571,0.42857,0.0,1.0,2,2,0,2,0,7,0,0,8,0,0,8,0,0,4,0,0,1,0,0,0,0,2],[12,29,0.4138,0.3302,0.26809,0.14286,0.2857,0.57111,0.0,0.85714,6,0,0,6,0,8,0,0,5,0,0,3,0,0,6,0,0,1,0,0,3,0,0],[16,29,0.5517,0.36603,0.21702,0.14289,0.42857,0.571,0.0,0.71429,5,0,2,5,0,4,0,0,3,0,0,10,0,0,8,0,0,2,0,0,0,0,0],[20,29,0.6897,0.39729,0.18806,0.2857,0.42857,0.571,0.0,0.71429,2,0,0,2,0,2,0,0,10,0,0,9,0,0,5,0,0,4,0,0,0,0,0],[24,29,0.8276,0.37944,0.19757,0.2857,0.35714,0.42858,0.0,0.85714,1,0,0,1,0,5,0,0,10,0,0,9,0,0,4,0,0,1,0,0,2,0,0],[28,29,0.9655,0.46423,0.1923,0.39286,0.42857,0.57143,0.0,0.85714,1,0,0,1,0,3,0,0,4,0,0,9,0,0,10,0,0,4,0,0,1,0,0],[29,29,1.0,0.43301,0.21571,0.28571,0.42859,0.57143,0.0,0.85714,2,0,0,2,0,3,0,0,6,0,0,9,0,0,7,0,0,3,0,0,2,0,0]]},{"b":2,"e":0.0,"k":"volatile","v":0.0134,"x":0.33477,"p":[[0,5,0.0,0.27231,0.19676,0.14286,0.21428,0.42857,0.0,0.71429,4,0,0,4,0,12,0,0,5,0,0,7,0,0,2,0,0,2,0,0,0,0,0],[4,5,0.8,0.33477,0.19753,0.14286,0.42857,0.4286,0.0,0.71429,4,0,0,4,0,6,0,0,5,0,0,10,0,0,6,0,0,1,0,0,0,0,0],[5,5,1.0,0.0134,0.04165,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8a294ad27925dfa6","q":"Let $n \\geq 0$ be an integer, and $x_{1}, x_{2}, \\ldots, x_{n+1}$ be strictly positive real numbers such that $\\prod_{k=1}^{n+1} x_{i}=1$. Prove that\n\n$$\n\\sqrt[x_{1}]{n}+\\sqrt[x_{2}]{n}+\\cdots+\\sqrt[x_{n+1}]{n} \\geq n \\sqrt[n]{x_{1}}+n \\sqrt[n]{x_{2}}+\\cdots+n \\sqrt[n]{x_{n+1}}\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,7,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,62,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,62,0.0645,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,62,0.129,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,62,0.1935,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,62,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,62,0.3226,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,62,0.3871,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,62,0.4516,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,62,0.5161,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,62,0.5806,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,62,0.6452,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,62,0.7097,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,62,0.7742,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,62,0.8387,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,62,0.9032,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[62,62,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e36cd44848830541","q":"There are  $m \\ge 2$  blue points and  $n \\ge 2$  red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?","t":[{"b":0,"e":0.0,"k":"flat","v":0.08902,"x":0.10715,"p":[[0,11,0.0,0.10715,0.25,0.0,0.0,0.14286,0.0,1.0,23,1,21,23,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1],[4,11,0.3636,0.08902,0.06896,0.0,0.14286,0.14286,0.0,0.1429,12,0,8,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.09348,0.06766,0.0,0.14286,0.14286,0.0,0.14286,11,0,8,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.07142,"x":0.16061,"p":[[0,12,0.0,0.16061,0.31287,0.0,0.0,0.14286,0.0,1.0,21,3,18,21,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[4,12,0.3333,0.07142,0.17853,0.0,0.0,0.03571,0.0,0.85714,24,0,19,24,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0]]}]},{"i":"37c93b76a4b443fd","q":"Let $c$ and $n>c$ be positive integers. Mary's teacher writes $n$ positive integers on a blackboard. Is it true that for all $n$ and $c$ Mary can always label the numbers written by the teacher by $a_{1}, \\ldots, a_{n}$ in such an order that the cyclic product $\\left(a_{1}-a_{2}\\right) \\cdot\\left(a_{2}-a_{3}\\right) \\cdot \\ldots \\cdot\\left(a_{n-1}-a_{n}\\right) \\cdot\\left(a_{n}-a_{1}\\right)$ would be congruent to either 0 or $c$ modulo $n$ ?","t":[{"b":1,"e":0.71429,"k":"rising","v":0.29893,"x":0.66517,"p":[[0,14,0.0,0.30357,0.24679,0.14286,0.14286,0.28571,0.0,0.85714,1,0,0,1,0,16,0,0,8,0,0,1,0,0,0,0,0,3,0,0,3,0,0],[4,14,0.2857,0.35714,0.26726,0.14286,0.28571,0.5,0.0,1.0,2,1,0,2,0,9,0,0,12,0,0,1,0,0,0,0,0,5,0,0,2,0,1],[8,14,0.5714,0.39732,0.25688,0.14286,0.28571,0.71429,0.14286,1.0,0,1,0,0,0,9,0,0,11,0,0,3,0,0,0,0,0,6,0,0,2,0,1],[12,14,0.8571,0.29893,0.22701,0.14286,0.14286,0.28571,0.14,0.71429,0,0,0,0,0,18,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0],[14,14,1.0,0.66517,0.14111,0.71429,0.71429,0.71429,0.0,0.85714,1,0,0,1,0,0,0,0,0,0,0,1,0,0,5,0,0,24,0,0,1,0,0]]},{"b":5,"e":0.14,"k":"falling","v":0.15179,"x":0.33027,"p":[[0,31,0.0,0.3125,0.25862,0.14286,0.14286,0.39286,0.14286,0.85714,0,0,0,0,0,20,0,0,4,0,0,0,0,0,0,0,0,6,0,0,2,0,0],[4,31,0.129,0.33027,0.24344,0.14286,0.21428,0.5,0.14,0.85714,0,0,0,0,0,16,0,0,7,0,0,1,0,0,0,0,0,7,0,0,1,0,0],[8,31,0.2581,0.2008,0.11219,0.14286,0.14286,0.2857,0.14,0.71429,0,0,0,0,0,22,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,31,0.3871,0.1875,0.06622,0.14286,0.14286,0.28571,0.14286,0.28571,0,0,0,0,0,22,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.17848,0.06191,0.14286,0.14286,0.17857,0.14,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.17857,0.07143,0.14286,0.14286,0.14286,0.14286,0.42857,0,0,0,0,0,25,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.17411,0.06901,0.14286,0.14286,0.1786,0.0,0.28571,1,0,0,1,0,23,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fb8707411fea7c13","q":"Two circles meet at points  $A$  and  $B$ . A line through  $B$  intersects the first circle again at  $K$  and the second circle at  $M$ . A line parallel to  $AM$  is tangent to the first circle at  $Q$ . The line  $AQ$  intersects the second circle again at  $R$ . $(a)$  Prove that the tangent to the second circle at  $R$  is parallel to  $AK$ . $(b)$  Prove that these two tangents meet on  $KM$ .","t":[{"b":2,"e":0.14286,"k":"flat","v":0.13384,"x":0.24554,"p":[[0,35,0.0,0.24554,0.16458,0.14286,0.1429,0.42857,0.0,0.4286,5,0,1,5,0,12,0,0,2,0,0,13,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.21428,0.14722,0.14286,0.14286,0.32143,0.0,0.571,3,0,1,3,0,19,0,0,2,0,0,7,0,0,1,0,0,0,0,0,0,0,0],[8,35,0.2286,0.18741,0.14035,0.14286,0.14286,0.1786,0.0,0.42857,5,0,0,5,0,19,0,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.17384,0.10564,0.14286,0.14286,0.14286,0.0,0.4286,2,0,1,2,0,25,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.15161,0.13335,0.105,0.14286,0.14287,0.0,0.4286,8,0,4,8,0,19,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.14724,0.09771,0.14286,0.14286,0.14286,0.0,0.4286,5,0,3,5,0,23,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.14723,0.10999,0.14286,0.14286,0.14286,0.0,0.42857,6,0,1,6,0,22,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.13384,0.10061,0.14214,0.14286,0.14286,0.0,0.42857,7,0,3,7,0,22,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.15143,0.08707,0.14286,0.14286,0.14286,0.0,0.4286,3,0,2,3,0,26,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.14286,0.05051,0.14286,0.14286,0.14286,0.0,0.2857,2,0,1,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.42857,"k":"flat","v":0.20518,"x":0.4151,"p":[[0,9,0.0,0.28572,0.24744,0.14286,0.14288,0.42857,0.0,1.0,4,2,1,4,0,13,0,0,4,0,0,8,0,0,0,0,0,1,0,0,0,0,2],[4,9,0.4444,0.20518,0.1555,0.14214,0.14286,0.42857,0.0,0.4286,6,0,1,6,0,15,0,0,2,0,0,9,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.4151,0.13551,0.42857,0.42857,0.42858,0.14,0.71429,0,0,0,0,0,4,0,0,1,0,0,24,0,0,0,0,0,3,0,0,0,0,0],[9,9,1.0,0.37053,0.11767,0.39286,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,6,0,0,2,0,0,23,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"e29cf36f0dc9c677","q":"You have two blackboards  $A$  and  $B$ . You have to write on them some of the integers greater than or equal to  $2$  and less than or equal to  $20$  in such a way that each number on blackboard  $A$  is co-prime with each number on blackboard  $B.$  Determine the maximum possible value of multiplying the number of numbers written in  $A$  by the number of numbers written in  $B$ .","t":[{"b":2,"e":0.28571,"k":"flat","v":0.29018,"x":0.375,"p":[[0,94,0.0,0.33483,0.16602,0.28571,0.28571,0.28571,0.0,0.85714,1,0,1,1,0,0,0,0,26,0,0,2,0,0,0,0,0,1,0,0,2,0,0],[4,94,0.0426,0.29018,0.08364,0.2857,0.28571,0.28571,0.0,0.57143,1,0,1,1,0,1,0,0,27,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,94,0.0851,0.29464,0.07936,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,94,0.1277,0.32589,0.20897,0.2857,0.28571,0.28571,0.0,1.0,2,2,2,2,0,2,0,0,23,0,0,2,0,0,0,0,0,1,0,0,0,0,2],[16,94,0.1702,0.35268,0.18552,0.28571,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,26,0,0,1,0,0,0,0,0,2,0,0,1,0,1],[20,94,0.2128,0.375,0.21651,0.28571,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,2,0,0,21,0,0,5,0,0,1,0,0,0,0,0,0,0,3],[24,94,0.2553,0.33928,0.13243,0.2857,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,26,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[28,94,0.2979,0.30804,0.08073,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,27,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[32,94,0.3404,0.29464,0.09407,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,3,0,0,26,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[36,94,0.383,0.30357,0.1171,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,3,0,0,25,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[40,94,0.4255,0.32143,0.12877,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,3,0,0,23,0,0,3,0,0,1,0,0,2,0,0,0,0,0],[44,94,0.4681,0.29911,0.06546,0.28571,0.28571,0.28571,0.1429,0.57143,0,0,0,0,0,1,0,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[48,94,0.5106,0.2991,0.05486,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,94,0.5532,0.2991,0.04164,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[56,94,0.5957,0.29912,0.07457,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,26,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[60,94,0.6383,0.29465,0.08703,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,2,0,0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[64,94,0.6809,0.2991,0.10326,0.2857,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[68,94,0.7234,0.29018,0.0563,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,2,0,0,27,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[72,94,0.766,0.29465,0.04972,0.2857,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[76,94,0.8085,0.29911,0.11495,0.28571,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,5,0,0,22,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[80,94,0.8511,0.32946,0.10903,0.28571,0.28571,0.28571,0.2857,0.85714,0,0,0,0,0,0,0,0,25,0,0,6,0,0,0,0,0,0,0,0,1,0,0],[84,94,0.8936,0.29018,0.02486,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[88,94,0.9362,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[92,94,0.9787,0.32053,0.09361,0.2857,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,1,0,0,24,0,0,6,0,0,0,0,0,1,0,0,0,0,0],[94,94,1.0,0.3259,0.09606,0.28571,0.28571,0.32164,0.14286,0.71429,0,0,0,0,0,1,0,0,23,0,0,7,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"flat","v":0.28125,"x":0.33928,"p":[[0,77,0.0,0.33928,0.12242,0.28571,0.28571,0.32143,0.14286,0.71429,0,0,0,0,0,1,0,0,23,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[4,77,0.0519,0.32589,0.11426,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,28,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[8,77,0.1039,0.32588,0.10245,0.2857,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,27,0,0,2,0,0,2,0,0,1,0,0,0,0,0],[12,77,0.1558,0.30804,0.05188,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[16,77,0.2078,0.30357,0.04725,0.2857,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[20,77,0.2597,0.28571,0.03571,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,77,0.3117,0.30357,0.04725,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[28,77,0.3636,0.29018,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,77,0.4156,0.29553,0.06531,0.2857,0.28571,0.28571,0.14286,0.6,0,0,0,0,0,1,0,0,29,0,0,1,0,0,0,1,0,0,0,0,0,0,0],[36,77,0.4675,0.29464,0.04971,0.28571,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[40,77,0.5195,0.2991,0.07457,0.28571,0.28571,0.28571,0.2857,0.71429,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[44,77,0.5714,0.29018,0.02486,0.2857,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,77,0.6234,0.28127,0.02486,0.28571,0.28571,0.28571,0.14286,0.286,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,77,0.6753,0.29018,0.04352,0.28571,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,1,0,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[56,77,0.7273,0.28571,1e-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],[60,77,0.7792,0.30803,0.05187,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,27,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[64,77,0.8312,0.30625,0.10504,0.28571,0.28571,0.28571,0.14286,0.8,0,0,0,0,0,1,0,0,29,0,0,0,0,0,1,0,0,0,0,1,0,0,0],[68,77,0.8831,0.29464,0.03458,0.2857,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[72,77,0.9351,0.29464,0.03458,0.28571,0.28571,0.28571,0.2857,0.4286,0,0,0,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[76,77,0.987,0.28572,0.03571,0.28571,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,1,0,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[77,77,1.0,0.28125,0.02486,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,1,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"596e92a833cbfafe","q":"Let $x, y$ and $z$ be positive real numbers such that $x y z=1$. Prove that\n\n$$\n(1+x)(1+y)(1+z) \\geq 2\\left(1+\\sqrt[3]{\\frac{y}{x}}+\\sqrt[3]{\\frac{z}{y}}+\\sqrt[3]{\\frac{x}{z}}\\right)\n$$","t":[{"b":3,"e":0.14286,"k":"falling","v":0.24991,"x":0.4107,"p":[[0,9,0.0,0.40177,0.21557,0.2857,0.42857,0.4642,0.0,1.0,1,1,1,1,0,6,0,0,6,0,0,11,0,0,3,0,0,4,0,0,0,0,1],[4,9,0.4444,0.4107,0.2519,0.24999,0.42857,0.57141,0.0,1.0,2,2,0,2,0,6,0,0,6,0,0,8,0,0,4,0,0,4,0,0,0,0,2],[8,9,0.8889,0.39732,0.3108,0.14286,0.28571,0.46429,0.0,1.0,3,4,1,3,0,8,0,0,7,0,0,6,0,0,1,0,0,1,0,0,2,0,4],[9,9,1.0,0.24991,0.1238,0.14286,0.2857,0.28571,0.0,0.4286,2,0,0,2,0,11,0,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.34375,"x":0.44641,"p":[[0,8,0.0,0.44641,0.23076,0.28571,0.42857,0.60714,0.0,0.85714,1,0,0,1,0,4,0,0,8,0,0,7,0,0,4,0,0,5,0,0,3,0,0],[4,8,0.5,0.34375,0.26692,0.14286,0.2857,0.42857,0.0,1.0,1,3,1,1,0,13,0,0,7,0,0,4,0,0,3,0,0,1,0,0,0,0,3],[8,8,1.0,0.38393,0.10374,0.28571,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,1,0,0,12,0,0,15,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"d84d26e1297b18e5","q":"Let  $\\omega$  be the circumcircle of an actue triangle  $ABC$  and let  $H$  be the feet of aliitude from  $A$  to  $BC$ . Let  $M$  and  $N$  be the midpoints of the sides  $AC$  and  $AB$ . The lines  $BM$  and  $CN$  intersect each other at  $G$  and intersect  $\\omega$  at  $P$  and  $Q$  respectively. The circles  $(HMG)$  and  $(HNG)$  intersect the segments  $HP$  and  $HQ$  again at  $R$  and  $S$  respectively. Prove that  $PQ\\parallel RS$ .","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00884,"p":[[0,13,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7a0ba46af014cd81","q":"Let  $H$  be the orthocenter of an acute angled triangle  $ABC$ . Circumcircle of the triangle  $ABC$  and the circle of diameter  $[AH]$  intersect at point  $E$ , different from  $A$ . Let  $M$  be the midpoint  of the small arc  $BC$  of the circumcircle of the triangle  $ABC$  and let  $N$  the midpoint of the large arc  $BC$  of the circumcircle of the triangle  $BHC$   Prove that  points  $E, H, M, N$  are concyclic.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.03116,"x":0.08482,"p":[[0,9,0.0,0.03116,0.0589,0.0,0.0,0.0,0.0,0.1429,25,0,2,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,2,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.06697,0.07129,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.03116,"x":0.08027,"p":[[0,20,0.0,0.05786,0.06995,0.0,0.0,0.14286,0.0,0.143,19,0,2,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.03116,0.0589,0.0,0.0,0.0,0.0,0.1429,25,0,3,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.07144,0.07987,0.0,0.0,0.14286,0.0,0.28571,17,0,2,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.1429,16,0,6,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.03572,0.06187,0.0,0.0,0.03571,0.0,0.1429,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e03b890154fe385e","q":"There are  $30$  students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends. \nPS. Here majority means larger than half.","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.04464,"p":[[0,15,0.0,0.04464,0.10374,0.0,0.0,0.0,0.0,0.42857,26,0,16,26,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.04009,0.07337,0.0,0.0,0.035,0.0,0.2857,24,0,9,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,8,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03571,"x":0.04902,"p":[[0,6,0.0,0.03571,0.12877,0.0,0.0,0.0,0.0,0.71429,28,0,19,28,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,6,0.6667,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.28571,22,0,15,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.04464,0.09061,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5297f35e56923545","q":"The incircle of  $\\omega$  of  $\\triangle ABC$  is tangent to  $\\overline{BC}$  at  $X$ . Let  $Y \\neq X$  be the other intersection of  $\\overline{AX}$  with  $\\omega$ . Points  $P$  and  $Q$  lie on   $\\overline{AB}$  and  $\\overline{AC}$ , respectively, so that  $\\overline{PQ}$  is tangent to  $\\omega$  at  $Y$ . Assume that  $AP=3, PB = 4, AC=8$ , and  $AQ = \\tfrac{m}{n}$ , where   $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,9,0.0,0.05804,0.12807,0.0,0.0,0.0,0.0,0.57143,25,0,10,25,0,3,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,9,0.4444,0.10714,0.16366,0.0,0.0,0.2857,0.0,0.57143,21,0,10,21,0,2,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,43,0.0,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,28,0,13,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,43,0.093,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,10,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.10714,0.18898,0.0,0.0,0.17857,0.0,0.57143,23,0,10,23,0,1,0,0,4,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,43,0.2791,0.04463,0.1259,0.0,0.0,0.0,0.0,0.571,28,0,12,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,43,0.3721,0.04018,0.11971,0.0,0.0,0.0,0.0,0.57143,28,0,11,28,0,1,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,43,0.4651,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,13,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.01786,0.07784,0.0,0.0,0.0,0.0,0.4286,30,0,12,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,29,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"987075a2921715e2","q":"Let $n \\geq 2$ be an integer, and let $\\left\\{a_{1}, \\ldots, a_{m}\\right\\}$ denote the $m=\\varphi(n)$ integers less than $n$ and relatively prime to $n$. Assume that every prime divisor of $m$ also divides $n$. Prove that $m$ divides $a_{1}^{k}+\\cdots+a_{m}^{k}$ for every positive integer $k$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.1517,"x":0.21875,"p":[[0,13,0.0,0.1874,0.19703,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,8,0,0,6,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[4,13,0.3077,0.1517,0.23674,0.0,0.0,0.14287,0.0,1.0,17,1,0,17,0,8,0,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,1],[8,13,0.6154,0.16964,0.20341,0.0,0.14286,0.28571,0.0,0.71429,15,0,0,15,0,6,0,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[12,13,0.9231,0.21875,0.24738,0.0,0.14286,0.42857,0.0,1.0,10,1,0,10,0,12,0,0,1,0,0,6,0,0,1,0,0,0,0,0,1,0,1],[13,13,1.0,0.18751,0.22711,0.0,0.14286,0.2857,0.0,1.0,12,1,0,12,0,11,0,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,1]]},{"b":2,"e":0.0,"k":"flat","v":0.02232,"x":0.21875,"p":[[0,18,0.0,0.14731,0.22722,0.0,0.0,0.14286,0.0,1.0,17,1,0,17,0,8,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[4,18,0.2222,0.16063,0.18815,0.0,0.14286,0.17857,0.0,0.71429,13,0,0,13,0,11,0,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[8,18,0.4444,0.15178,0.20803,0.0,0.0,0.28571,0.0,0.57143,18,0,0,18,0,5,0,0,2,0,0,3,0,0,4,0,0,0,0,0,0,0,0],[12,18,0.6667,0.21875,0.26481,0.0,0.14286,0.42857,0.0,0.85714,14,0,0,14,0,7,0,0,1,0,0,5,0,0,2,0,0,1,0,0,2,0,0],[16,18,0.8889,0.21429,0.22016,0.0,0.14286,0.42857,0.0,0.71429,12,0,0,12,0,7,0,0,3,0,0,7,0,0,1,0,0,2,0,0,0,0,0],[18,18,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8aacac15c2fdac71","q":"We call a positive integer  $n$  *amazing* if there exist positive integers  $a, b, c$  such that the equality\n\\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\\]\nholds. Prove that there exist  $2011$  consecutive positive integers which are *amazing*.**Note.** By  $(m, n)$  we denote the greatest common divisor of positive integers  $m$  and  $n$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.11607,"p":[[0,60,0.0,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.28571,22,0,6,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,60,0.0667,0.06697,0.09439,0.0,0.0,0.14286,0.0,0.28571,20,0,2,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,60,0.1333,0.0625,0.08703,0.0,0.0,0.14286,0.0,0.28571,20,0,3,20,0,10,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,60,0.2,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,4,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,60,0.2667,0.11607,0.13092,0.0,0.0,0.28571,0.0,0.28571,17,0,1,17,0,4,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,60,0.3333,0.08036,0.2141,0.0,0.0,0.0,0.0,1.0,25,1,5,25,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[24,60,0.4,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,6,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,60,0.4667,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.28571,20,0,1,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,60,0.5333,0.06696,0.09438,0.0,0.0,0.14286,0.0,0.2857,20,0,1,20,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,60,0.6,0.07589,0.10092,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,60,0.6667,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,60,0.7333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,60,0.8,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,60,0.8667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,60,0.9333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,60,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.03125,"x":0.14723,"p":[[0,29,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,6,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,3,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.0892,0.1056,0.0,0.0,0.14286,0.0,0.28571,17,0,2,17,0,10,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.14723,0.10403,0.105,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,15,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.28571,18,0,1,18,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.1025,0.10242,0.0,0.14143,0.14286,0.0,0.28571,14,0,0,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.11161,0.12234,0.0,0.07143,0.28571,0.0,0.28571,16,0,2,16,0,7,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.125,0.10564,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,14,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.10268,0.10853,0.0,0.14286,0.14286,0.0,0.28571,15,0,0,15,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"27cf681ff6508801","q":"Let $a, b, c, d$ be strictly positive real numbers such that: $a+b+c+d=1$. Show that:\n\n$$\n\\frac{a^{4}}{a^{3}+a^{2} b+a b^{2}+b^{3}}+\\frac{b^{4}}{b^{3}+b^{2} c+b c^{2}+c^{3}}+\\frac{c^{4}}{c^{3}+c^{2} d+c d^{2}+d^{3}}+\\frac{d^{4}}{d^{3}+d^{2} a+d a^{2}+a^{3}} \\geqslant \\frac{1}{4}\n$$\n\nand determine the cases of equality.","t":[{"b":0,"e":0.42857,"k":"flat","v":0.30357,"x":0.52232,"p":[[0,37,0.0,0.47768,0.24643,0.28571,0.42857,0.42858,0.14286,1.0,0,5,0,0,0,1,0,0,10,0,0,14,0,0,1,0,0,1,0,0,0,0,5],[4,37,0.1081,0.52232,0.29582,0.28571,0.42857,0.75,0.14286,1.0,0,7,0,0,0,3,0,0,10,0,0,6,0,0,4,0,0,1,0,0,1,0,7],[8,37,0.2162,0.3616,0.19228,0.2857,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,3,0,0,20,0,0,5,0,0,0,0,0,2,0,0,1,0,1],[12,37,0.3243,0.36607,0.18877,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,21,0,0,5,0,0,0,0,0,2,0,0,1,0,1],[16,37,0.4324,0.37053,0.22263,0.2857,0.28571,0.42857,0.14286,1.0,0,3,0,0,0,4,0,0,18,0,0,6,0,0,1,0,0,0,0,0,0,0,3],[20,37,0.5405,0.33035,0.10374,0.28571,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,20,0,0,9,0,0,0,0,0,1,0,0,0,0,0],[24,37,0.6486,0.36607,0.19865,0.2857,0.28571,0.42857,0.14286,1.0,0,2,0,0,0,3,0,0,19,0,0,6,0,0,1,0,0,1,0,0,0,0,2],[28,37,0.7568,0.375,0.06916,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,12,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.30357,0.06916,0.2857,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[36,37,0.973,0.39731,0.07768,0.42857,0.42857,0.42857,0.14286,0.571,0,0,0,0,0,1,0,0,6,0,0,24,0,0,1,0,0,0,0,0,0,0,0],[37,37,1.0,0.40625,0.06298,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,6,0,0,25,0,0,1,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.26786,"x":0.37946,"p":[[0,6,0.0,0.37946,0.22192,0.28571,0.28571,0.42857,0.0,1.0,1,2,0,1,0,5,0,0,11,0,0,10,0,0,1,0,0,2,0,0,0,0,2],[4,6,0.6667,0.27232,0.04164,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,3,0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.26786,0.04724,0.28571,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"490f1115f092a504","q":"Let $n$ be a positive integer, and let $A$ and $B$ be $n \\times n$ matrices with complex entries such that $A^{2}=B^{2}$. Show that there exists an $n \\times n$ invertible matrix $S$ with complex entries that satisfies $S(A B-B A)=(B A-A B) S$.","t":[{"b":4,"e":0.0,"k":"falling","v":0.00446,"x":0.39732,"p":[[0,14,0.0,0.24999,0.32339,0.0,0.07143,0.42858,0.0,1.0,16,1,0,16,0,4,0,0,2,0,0,3,0,0,1,0,0,2,0,0,3,0,1],[4,14,0.2857,0.39732,0.38088,0.10714,0.28571,0.85714,0.0,1.0,8,5,0,8,0,7,0,0,5,0,0,1,0,0,0,0,0,2,0,0,4,0,5],[8,14,0.5714,0.08929,0.21053,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,4,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[12,14,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.06697,0.08737,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.85714,"k":"flat","v":0.23661,"x":0.37499,"p":[[0,8,0.0,0.23661,0.34183,0.0,0.14286,0.28571,0.0,1.0,15,4,1,15,0,8,0,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,4],[4,8,0.5,0.37499,0.21942,0.28571,0.28571,0.42857,0.0,0.85714,1,0,0,1,0,5,0,0,14,0,0,5,0,0,2,0,0,2,0,0,3,0,0],[8,8,1.0,0.36607,0.17105,0.28571,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,2,0,0,18,0,0,9,0,0,1,0,0,0,0,0,1,0,1]]}]},{"i":"ca0055708ef125b4","q":"We say that two non-negative integers are related if their sum uses only the digits 0 and 1 . For example 22 and 79 are related. Let A and B be two infinite sets of non-negative integers such that: (1) if a $\\square \\mathrm{A}$ and $\\mathrm{b} \\square \\mathrm{B}$, then a and $\\mathrm{b}$ are related, (2) if $\\mathrm{c}$ is related to every member of A, then it belongs to $\\mathrm{B},(3)$ if $\\mathrm{c}$ is related to every member of $\\mathrm{B}$, then it belongs to $\\mathrm{A}$. Show that in one of the sets $\\mathrm{A}, \\mathrm{B}$ we can find an infinite number of pairs of consecutive numbers.","t":[{"b":5,"e":0.0,"k":"flat","v":0.08009,"x":0.12054,"p":[[0,23,0.0,0.09375,0.11633,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.12054,0.11355,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,14,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.09821,0.12595,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,10,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.08018,0.09396,0.0,0.07,0.14286,0.0,0.4286,16,0,2,16,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.08009,0.08684,0.0,0.07,0.14286,0.0,0.28571,16,0,1,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0892,0.07778,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,18,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.11143,0.07764,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.10259,"p":[[0,21,0.0,0.10259,0.10245,0.0,0.14286,0.14286,0.0,0.42857,13,0,0,13,0,16,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.09366,0.09847,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,13,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.06687,0.11279,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.08929,0.15872,0.0,0.0,0.14286,0.0,0.71429,21,0,0,21,0,6,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,21,0.7619,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"408ca329e918e20b","q":"Two circles are intersecting in points  $P$  and  $Q$ . Construct two points  $A$  and  $B$  on these circles so that  $P\\in AB$  and the product  $AP.PB$  is maximal.","t":[{"b":2,"e":0.57143,"k":"rising","v":0.36161,"x":0.79462,"p":[[0,21,0.0,0.49998,0.33502,0.21427,0.64286,0.71429,0.0,1.0,8,1,6,8,0,0,0,0,3,0,0,2,0,0,3,0,0,9,0,0,6,0,1],[4,21,0.1905,0.36161,0.335,0.0,0.28571,0.71429,0.0,1.0,11,1,7,11,0,2,0,0,6,0,0,1,0,0,0,0,0,9,0,0,2,0,1],[8,21,0.381,0.70089,0.13534,0.67857,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,6,0,0,17,0,0,6,0,1],[12,21,0.5714,0.70982,0.13592,0.67857,0.71429,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,5,0,0,15,0,0,8,0,1],[16,21,0.7619,0.70089,0.15303,0.57143,0.71429,0.85714,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,0,0,0,8,0,0,11,0,0,11,0,0],[20,21,0.9524,0.68301,0.17401,0.57143,0.71429,0.85714,0.0,0.85714,1,0,1,1,0,0,0,0,0,0,0,2,0,0,7,0,0,13,0,0,9,0,0],[21,21,1.0,0.79462,0.12342,0.71429,0.85714,0.85714,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,10,0,0,14,0,4]]},{"b":4,"e":0.0,"k":"volatile","v":0.09375,"x":0.57586,"p":[[0,18,0.0,0.57586,0.31437,0.39286,0.71429,0.85714,0.0,1.0,5,2,4,5,0,0,0,0,3,0,0,3,0,0,4,0,0,6,0,0,9,0,2],[4,18,0.2222,0.55804,0.32411,0.39286,0.71429,0.71429,0.0,1.0,5,5,4,5,0,2,0,0,1,0,0,3,0,0,4,0,0,11,0,0,1,0,5],[8,18,0.4444,0.09375,0.18074,0.0,0.0,0.14286,0.0,0.57143,23,0,9,23,0,4,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[12,18,0.6667,0.13839,0.17672,0.0,0.0,0.28571,0.0,0.42857,19,0,4,19,0,1,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"495140017163023d","q":"There are some counters in some cells of  $100\\times 100$  board. Call a cell *nice* if there are an even number of counters in adjacent cells. Can exactly one cell be *nice*?\n\n*K. Knop*","t":[{"b":2,"e":0.0,"k":"flat","v":0.01786,"x":0.06695,"p":[[0,35,0.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.02231,0.10163,0.0,0.0,0.0,0.0,0.571,30,0,1,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,35,0.2286,0.06695,0.14714,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,5,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,35,0.3429,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00893,"x":0.04464,"p":[[0,50,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,1,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[20,50,0.4,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,50,0.48,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,2,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,50,0.56,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,3,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,50,0.72,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"84d37f43410eae74","q":"We have  $4n + 5$  points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form  $n$  empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the  $n$  triangles have the same color.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.04018,"p":[[0,25,0.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,3,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,1,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.02455,0.0524,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.02902,0.06648,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,1,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,1,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.01339,"x":0.06473,"p":[[0,39,0.0,0.05117,0.07605,0.0,0.0,0.14071,0.0,0.28571,21,0,0,21,1,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,39,0.1026,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,39,0.2051,0.04009,0.07337,0.0,0.0,0.035,0.0,0.28571,24,0,1,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,39,0.3077,0.06473,0.07857,0.0,0.0,0.14286,0.0,0.2857,18,0,1,18,1,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,39,0.4103,0.04446,0.06595,0.0,0.0,0.14071,0.0,0.14286,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,39,0.5128,0.05134,0.07625,0.0,0.0,0.14286,0.0,0.2857,21,0,1,21,1,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,39,0.6154,0.03562,0.06171,0.0,0.0,0.035,0.0,0.14286,24,0,3,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,39,0.7179,0.05579,0.06855,0.0,0.0,0.14286,0.0,0.143,19,0,0,19,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.0625,0.06858,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,2,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,39,0.9231,0.0625,0.06858,0.0,0.0,0.14286,0.0,0.14286,17,0,0,17,2,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[39,39,1.0,0.04679,0.06575,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,1,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e11018f052b0685a","q":"We place a certain number of open segments in the plane, none of which are parallel to the $x$ and $y$ axes. These segments are disjoint. Thanima starts moving from $(0,0)$ parallel to the $x$-axis. Each time she encounters a wall, she turns 90 degrees and continues moving without crossing the wall.\nProve that it is impossible for Thanima to visit both sides of all the walls.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.01339,"p":[[0,9,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,6,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8a1c5e4618f261ec","q":"The sphere inscribed in a tetrahedron  $ABCD$  touches face  $ABC$  at point  $H$ . Another sphere touches face  $ABC$  at  $O$  and the planes containing the other three faces at points exterior to the faces. Prove that if  $O$  is the circumcenter of triangle  $ABC$ , then  $H$  is the orthocenter of that triangle.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.09821,"p":[[0,17,0.0,0.09821,0.27067,0.0,0.0,0.0,0.0,1.0,28,2,5,28,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[4,17,0.2353,0.05357,0.14617,0.0,0.0,0.0,0.0,0.57143,27,0,2,27,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.08482,"p":[[0,16,0.0,0.08482,0.18851,0.0,0.0,0.0,0.0,0.57143,25,0,2,25,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[4,16,0.25,0.07142,0.17854,0.0,0.0,0.0,0.0,0.57143,27,0,1,27,0,1,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[8,16,0.5,0.05357,0.16656,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[12,16,0.75,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6cce30773b0fdc80","q":"Let $n$ be a positive integer. Show that the numbers  $$ \\left(\\begin{array}{c} 2^{n}-1 \\\\ 0 \\end{array}\\right), \\quad\\left(\\begin{array}{c} 2^{n}-1 \\\\ 1 \\end{array}\\right), \\quad\\left(\\begin{array}{c} 2^{n}-1 \\\\ 2 \\end{array}\\right), \\quad \\ldots, \\quad\\left(\\begin{array}{c} 2^{n}-1 \\\\ 2^{n-1}-1 \\end{array}\\right) $$  are congruent modulo $2^{n}$ to $1,3,5, \\ldots, 2^{n}-1$ in some order.","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,31,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.04911,0.10479,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":7,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,19,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"67d3d7ed59b31491","q":"Let $O$ denote the circumcentre of an acute-angled triangle $A B C$. A circle $\\Gamma$ passing through vertex $A$ intersects segments $A B$ and $A C$ at points $P$ and $Q$ such that $\\angle B O P=\\angle A B C$ and $\\angle C O Q=\\angle A C B$. Prove that the reflection of $B C$ in the line $P Q$ is tangent to $\\Gamma$.","t":[{"b":0,"e":0.2857,"k":"flat","v":0.06687,"x":0.28571,"p":[[0,23,0.0,0.1875,0.1357,0.0,0.2857,0.28571,0.0,0.28571,11,0,0,11,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.12946,0.13997,0.0,0.0,0.28571,0.0,0.28571,17,0,4,17,0,1,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.1741,0.13235,0.0,0.2857,0.28571,0.0,0.28571,11,0,0,11,0,3,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.06687,0.11831,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.28571,1e-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],[20,23,0.8696,0.28571,1e-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],[23,23,1.0,0.28571,1e-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]]},{"b":6,"e":0.2857,"k":"rising","v":0.10714,"x":0.28571,"p":[[0,38,0.0,0.10714,0.13363,0.0,0.0,0.2857,0.0,0.28571,19,0,1,19,0,2,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,38,0.1053,0.11607,0.14032,0.0,0.0,0.28571,0.0,0.28571,19,0,2,19,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,38,0.2105,0.1384,0.13593,0.0,0.14286,0.28571,0.0,0.286,15,0,6,15,0,3,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.22321,0.11811,0.2857,0.2857,0.28571,0.0,0.28571,7,0,7,7,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,38,0.4211,0.19643,0.13243,0.0,0.28571,0.28571,0.0,0.28571,10,0,9,10,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.21875,0.11837,0.24999,0.28571,0.28571,0.0,0.28571,7,0,7,7,0,1,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,38,0.6316,0.1875,0.1357,0.0,0.28571,0.28571,0.0,0.28571,11,0,11,11,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.1875,0.1357,0.0,0.28571,0.28571,0.0,0.28571,11,0,11,11,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,38,0.8421,0.19642,0.13243,0.0,0.2857,0.28571,0.0,0.28571,10,0,10,10,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,38,0.9474,0.28571,1e-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],[38,38,1.0,0.27678,0.03458,0.2857,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"03ed3fa28059a7de","q":"Let $n$ be a positive integer, and let $x$ and $y$ be positive real numbers such that $x^{n}+y^{n}=1$. Prove that  $$ \\left(\\sum_{k=1}^{n} \\frac{1+x^{2 k}}{1+x^{4 k}}\\right)\\left(\\sum_{k=1}^{n} \\frac{1+y^{2 k}}{1+y^{4 k}}\\right)<\\frac{1}{(1-x)(1-y)} $$  (Estonia)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.19643,"x":0.35268,"p":[[0,15,0.0,0.29009,0.33217,0.0,0.14143,0.71429,0.0,1.0,15,1,0,15,0,4,0,0,1,0,0,0,0,0,3,0,0,8,0,0,0,0,1],[4,15,0.2667,0.19643,0.32488,0.0,0.0,0.17857,0.0,1.0,20,2,0,20,0,4,0,0,1,0,0,1,0,0,0,0,0,3,0,0,1,0,2],[8,15,0.5333,0.33929,0.3549,0.0,0.28571,0.71429,0.0,1.0,14,3,0,14,0,1,0,0,3,0,0,3,0,0,1,0,0,7,0,0,0,0,3],[12,15,0.8,0.35268,0.22299,0.14289,0.28571,0.46429,0.0,0.71429,3,0,0,3,0,6,0,0,10,0,0,5,0,0,2,0,0,6,0,0,0,0,0],[15,15,1.0,0.22322,0.24727,0.0,0.14288,0.32143,0.0,0.71429,14,0,0,14,0,3,0,0,7,0,0,3,0,0,1,0,0,4,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"falling","v":0.0,"x":0.3482,"p":[[0,9,0.0,0.3482,0.34798,0.0,0.21429,0.71429,0.0,1.0,13,2,0,13,0,3,0,0,1,0,0,2,0,0,2,0,0,9,0,0,0,0,2],[4,9,0.4444,0.25,0.30305,0.0,0.0,0.57143,0.0,0.71429,17,0,0,17,0,2,0,0,2,0,0,1,0,0,3,0,0,7,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"997497a2faec5c95","q":"There are  $1 000 000$  soldiers in a line. The sergeant splits the line into  $100$  segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most  $100$  of these numbers are different.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0625,"p":[[0,40,0.0,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,9,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,40,0.1,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,40,0.3,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[20,40,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"815ee919a98de6ee","q":"There are  $n$  players,  $n\\ge 2$ , which are playing a card game with  $np$  cards in  $p$  rounds. The cards are coloured in  $n$  colours and each colour is labelled with the numbers  $1,2,\\ldots ,p$ . The game submits to the following rules:\n[list]each player receives  $p$  cards.\nthe player who begins the first round throws a card and each player has to discard a card of the same colour, if he has one; otherwise they can give an arbitrary card.\nthe winner of the round is the player who has put the greatest card of the same colour as the first one. \nthe winner of the round starts the next round with a card that he selects and the play continues with the same rules.\nthe played cards are out of the game.[/list]\nShow that if all cards labelled with number  $1$  are winners, then  $p\\ge 2n$ .\n\n*Barbu Berceanu*","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,20,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,22,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,20,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,16,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,12,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02232,"x":0.02232,"p":[[0,2,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,24,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d6c725088e6bb347","q":"There are 2017 lines in a plane such that no 3 of them go through the same point. Turbo the snail can slide along the lines in the following fashion: she initially moves on one of the lines and continues moving on a given line until she reaches an intersection of 2 lines. At the intersection, she follows her journey on the other line turning left or right, alternating the direction she chooses at each intersection point she passes. Can it happen that she slides through a line segment for a second time in her journey but in the opposite direction as she did for the first time?\n\nM\u00e1rk Di Giovanni, Hungary","t":[{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.05357,"p":[[0,12,0.0,0.05357,0.12753,0.0,0.0,0.0,0.0,0.57143,26,0,6,26,0,2,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,12,0.3333,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,10,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,6,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,14,0.0,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,28,0,4,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,14,0.2857,0.04911,0.12682,0.0,0.0,0.0,0.0,0.57143,27,0,8,27,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"8c02b5a068308dba","q":"Line  $\\ell$  intersects the right branch of the hyperbola  $y = 1/x$ ,  $x > 0$ , at points  $A$  and  $B$ . Lines  $\\ell_1$   and  $\\ell_2$  , parallel to line  $\\ell$ , intersect the left branch of this hyperbola (  $x < 0$  ) at points  $E, F$  and  $C, D$ , respectively. Segment  $AD$  intersects line  $\\ell_1$  at point  $G$ , and segment  $BC$  intersects line  $\\ell_1$   at point  $H$ . Prove that the lengths of segments  $GE$  and  $HF$  are equal.","t":[{"b":2,"e":0.71429,"k":"flat","v":0.56688,"x":0.74999,"p":[[0,19,0.0,0.74999,0.25506,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,1,0,0,0,0,0,3,0,0,2,0,0,12,0,0,1,0,12],[4,19,0.2105,0.56688,0.26856,0.42857,0.71429,0.71429,0.0,1.0,3,2,0,3,0,2,0,0,2,0,0,3,0,0,2,0,0,18,0,0,0,0,2],[8,19,0.4211,0.63837,0.22012,0.42859,0.71429,0.71429,0.14286,1.0,0,5,0,0,0,1,0,0,2,0,0,6,0,0,6,0,0,11,0,0,1,0,5],[12,19,0.6316,0.72321,0.23402,0.71429,0.71429,0.89286,0.0,1.0,1,8,0,1,0,1,0,0,0,0,0,2,0,0,3,0,0,15,0,0,2,0,8],[16,19,0.8421,0.72321,0.11811,0.71429,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,24,0,0,5,0,1],[19,19,1.0,0.74554,0.09268,0.71429,0.71429,0.71429,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,26,0,0,2,0,3]]},{"b":7,"e":1.0,"k":"volatile","v":0.67408,"x":0.92857,"p":[[0,7,0.0,0.74107,0.21558,0.71429,0.71429,0.89286,0.1429,1.0,0,8,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,14,0,0,4,0,8],[4,7,0.5714,0.67408,0.24546,0.4286,0.71429,0.89286,0.14286,1.0,0,8,0,0,0,1,0,0,2,0,0,6,0,0,5,0,0,8,0,0,2,0,8],[7,7,1.0,0.92857,0.18558,1.0,1.0,1.0,0.2857,1.0,0,27,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,0,1,0,27]]}]},{"i":"45dbcd0ee2eeb5f6","q":"Xenia and Sergey play the following game. Xenia thinks of a positive integer  $N$  not exceeding  $5000$ . Then she fixes  $20$  distinct positive integers  $a_1, a_2, \\cdots, a_{20}$  such that, for each  $k = 1,2,\\cdots,20$ , the numbers  $N$  and  $a_k$  are congruent modulo  $k$ . By a move, Sergey tells Xenia a set  $S$  of positive integers not exceeding  $20$ , and she tells him back the set  $\\{a_k : k \\in S\\}$  without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of?\n\n*Sergey Kudrya, Russia*","t":[{"b":2,"e":0.28571,"k":"flat","v":0.16518,"x":0.33482,"p":[[0,10,0.0,0.19196,0.26871,0.0,0.07143,0.2857,0.0,1.0,16,1,8,16,0,6,0,0,4,0,0,1,0,0,1,0,0,3,0,0,0,0,1],[4,10,0.4,0.16518,0.26752,0.0,0.0,0.2857,0.0,0.85714,20,0,13,20,0,3,0,0,3,0,0,2,0,0,0,0,0,2,0,0,2,0,0],[8,10,0.8,0.33033,0.13087,0.2857,0.28571,0.42857,0.14286,0.71429,0,0,0,0,0,4,0,0,19,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[10,10,1.0,0.33482,0.14111,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,5,0,0,11,0,0,13,0,0,1,0,0,1,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.10259,"x":0.18732,"p":[[0,48,0.0,0.18732,0.20962,0.0,0.14286,0.28571,0.0,0.85714,13,0,9,13,0,6,0,0,8,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[4,48,0.0833,0.13384,0.13803,0.0,0.14286,0.14286,0.0,0.71429,10,0,1,10,0,17,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[8,48,0.1667,0.10268,0.10853,0.0,0.14286,0.14286,0.0,0.28571,15,0,1,15,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,48,0.25,0.1158,0.09733,0.0,0.14286,0.14286,0.0,0.28571,11,0,1,11,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.15624,0.12551,0.10714,0.14286,0.2857,0.0,0.571,8,0,0,8,0,15,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,48,0.4167,0.11161,0.13236,0.0,0.07143,0.14289,0.0,0.4286,16,0,0,16,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.10268,0.09606,0.0,0.14286,0.14287,0.0,0.28571,13,0,1,13,0,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,48,0.5833,0.11152,0.10552,0.0,0.14286,0.14286,0.0,0.28571,13,0,0,13,0,13,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.14277,0.11845,0.0,0.14286,0.2857,0.0,0.42857,10,0,0,10,0,13,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.15178,0.10677,0.10714,0.14286,0.2857,0.0,0.28571,8,0,0,8,0,14,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.16062,0.0928,0.14286,0.14286,0.2857,0.0,0.28571,5,0,0,5,0,18,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.15615,0.16112,0.0,0.14286,0.17857,0.0,0.71429,10,0,1,10,0,14,0,0,6,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[48,48,1.0,0.10259,0.08167,0.0,0.14286,0.14286,0.0,0.28571,11,0,1,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a9105534b679e3f7","q":"Let $x, y, z$ be positive real numbers. Prove that:\n\n$$\n\\left(x^{2}+y+1\\right)\\left(x^{2}+z+1\\right)\\left(y^{2}+z+1\\right)\\left(y^{2}+x+1\\right)\\left(z^{2}+x+1\\right)\\left(z^{2}+y+1\\right) \\geq(x+y+z)^{6}\n$$","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.07143,"p":[[0,11,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,11,0.7273,0.07143,0.24223,0.0,0.0,0.0,0.0,1.0,28,2,0,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[11,11,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.02233,"x":0.06696,"p":[[0,5,0.0,0.04464,0.17655,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,5,0.8,0.06696,0.24218,0.0,0.0,0.0,0.0,1.0,29,2,0,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[5,5,1.0,0.02233,0.05188,0.0,0.0,0.0,0.0,0.143,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"415378969ec6b3ea","q":"There are  $2024$  cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest  $k$  for which the tourist will always be able to visit at least  $k$  cities? During the route, he can return to the cities he has already visited.\n\n*Proposed by Bogdan Rublov*","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,14,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,7,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,65,0.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,5,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,65,0.0615,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,65,0.1231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,65,0.1846,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,65,0.2462,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,65,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,65,0.3692,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,65,0.4308,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,65,0.4923,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,65,0.5538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,65,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,65,0.7385,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,65,0.9846,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3ff2670f2bd663b3","q":"Two points  $A$  and  $B$  and line  $\\ell$  are fixed in the plane so that  $\\ell$  is not perpendicular to  $AB$  and does not intersect the segment  $AB$ . We consider all circles with a centre  $O$  not lying on  $\\ell$ , passing through  $A$  and  $B$  and meeting  $\\ell$  at some points  $C$  and  $D$ . Prove that all the circumcircles of triangles  $OCD$  touch a fixed circle.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,8,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,9,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,9,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,20,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,13,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,16,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,16,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,13,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,8,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c9f199489b586d3b","q":"Let $a_{1}, b_{1}, a_{2}, b_{2}, \\ldots, a_{n}, b_{n}$ be nonnegative real numbers. Prove that  $$ \\sum_{i, j=1}^{n} \\min \\left\\{a_{i} a_{j}, b_{i} b_{j}\\right\\} \\leq \\sum_{i, j=1}^{n} \\min \\left\\{a_{i} b_{j}, a_{j} b_{i}\\right\\} $$","t":[{"b":5,"e":0.28571,"k":"flat","v":0.15179,"x":0.25893,"p":[[0,38,0.0,0.17411,0.14167,0.14286,0.14286,0.14286,0.0,0.85714,2,0,0,2,0,26,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,38,0.1053,0.20088,0.17442,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,26,0,0,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[8,38,0.2105,0.17857,0.09449,0.14286,0.14286,0.14286,0.0,0.4286,1,0,0,1,0,25,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,38,0.3158,0.19643,0.16269,0.14286,0.14286,0.14286,0.14286,1.0,0,1,0,0,0,27,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[16,38,0.4211,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,38,0.5263,0.16518,0.12931,0.14286,0.14286,0.14286,0.0,0.85714,1,0,0,1,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,38,0.6316,0.18295,0.07354,0.14286,0.14286,0.17857,0.14,0.42857,0,0,0,0,0,24,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,38,0.7368,0.22768,0.19516,0.14286,0.14286,0.2857,0.14286,1.0,0,1,0,0,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,1],[32,38,0.8421,0.20527,0.10067,0.14286,0.14286,0.2857,0.0,0.57143,1,0,0,1,0,18,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,38,0.9474,0.25893,0.05576,0.2857,0.28571,0.28571,0.14286,0.286,0,0,0,0,0,6,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[38,38,1.0,0.24107,0.06621,0.14286,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.16964,"x":0.24553,"p":[[0,13,0.0,0.16965,0.08328,0.14286,0.14286,0.1429,0.0,0.42857,2,0,0,2,0,23,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.17857,0.07143,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,22,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.20534,0.12842,0.14286,0.14286,0.2857,0.14286,0.71429,0,0,0,0,0,23,0,0,7,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[12,13,0.9231,0.24553,0.21496,0.14286,0.14286,0.28571,0.0,1.0,1,1,0,1,0,21,0,0,5,0,0,1,0,0,2,0,0,0,0,0,1,0,1],[13,13,1.0,0.16964,0.06621,0.14286,0.14286,0.14287,0.0,0.28571,1,0,0,1,0,24,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6ebf77cc21cef837","q":"Let $n$ be an integer with $n \\geqslant 2$. Does there exist a sequence $\\left(a_{1}, \\ldots, a_{n}\\right)$ of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia)","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0357,"p":[[0,55,0.0,0.0357,0.10708,0.0,0.0,0.0,0.0,0.571,27,0,0,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,55,0.0727,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,55,0.1455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,55,0.2182,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,55,0.2909,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,55,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,55,0.4364,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,55,0.5091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,55,0.5818,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,55,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,55,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,55,0.8727,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,55,0.9455,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,6,0.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d0d894b4e0db98e4","q":"Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board.  (i) If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$.  (ii) If no such pair exists, we write down two times the number 0 .  Prove that, no matter how we make the choices in $(i)$, operation (ii) will be performed only finitely many times.  (Serbia)","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.00893,"p":[[0,7,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,7,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,7,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,7,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,10,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,10,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f83f024f6eaae98d","q":"Let $n \\geq m \\geq 1$ be integers. Prove that  $$ \\sum_{k=m}^{n}\\left(\\frac{1}{k^{2}}+\\frac{1}{k^{3}}\\right) \\geq m \\cdot\\left(\\sum_{k=m}^{n} \\frac{1}{k^{2}}\\right)^{2} $$","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,9,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,17,0.0,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,17,0.2353,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"128ee4ef449ce839","q":"Prove that for positive real numbers $a, b$, and $c$ such that $a+b+c=1$, the inequality\n\n$$\n\\frac{1}{b c+a+\\frac{1}{a}}+\\frac{1}{c a+b+\\frac{1}{b}}+\\frac{1}{a b+c+\\frac{1}{c}} \\leqslant \\frac{27}{31}\n$$\n\nholds.\n\n(Marko Radovanovi\u0107 with modifications)\n\nTime for work 270 minutes.\n\nEach task is worth 7 points.\n\n## SERBIAN MATHEMATICAL OLYMPIAD\n\ncompetition of high school students in mathematics\n\nBelgrade, 13.04.2008.\n\n## Second day","t":[{"b":0,"e":0.14286,"k":"flat","v":0.125,"x":0.14732,"p":[[0,28,0.0,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.14286,1e-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],[8,28,0.2857,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.14732,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.14732,0.02485,0.14286,0.14286,0.14286,0.14286,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.14286,1e-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],[24,28,0.8571,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.13839,"x":0.14286,"p":[[0,5,0.0,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.14286,1e-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":"41127475cfa29fe3","q":"Two types of pieces, bishops and rooks, are to be placed on a  $10\\times 10$  chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop  $B$  is said to *attack* a piece  $P$  if  $B$  and  $P$  are on the same diagonal and there are no pieces between  $B$  and  $P$  on that diagonal; a rook  $R$  is said to attack a piece  $P$  if  $R$  and  $P$  are on the same row or column and there are no pieces between  $R$  and  $P$  on that row or column.\nA piece  $P$  is *chocolate* if no other piece  $Q$  attacks  $P$ .\nWhat is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?\n\n*Proposed by Jos\u00e9 Alejandro Reyes Gonz\u00e1lez*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.09375,"p":[[0,21,0.0,0.09375,0.17717,0.0,0.0,0.0,0.0,0.42857,25,0,14,25,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.09375,0.17717,0.0,0.0,0.0,0.0,0.42857,25,0,12,25,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.06696,0.18205,0.0,0.0,0.0,0.0,0.71429,28,0,24,28,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.06697,"x":0.14732,"p":[[0,9,0.0,0.14732,0.20355,0.0,0.0,0.42857,0.0,0.42857,21,0,12,21,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.10714,0.18898,0.0,0.0,0.07143,0.0,0.57143,24,0,14,24,0,0,0,0,1,0,0,6,0,0,1,0,0,0,0,0,0,0,0],[8,9,0.8889,0.06697,0.15561,0.0,0.0,0.0,0.0,0.4286,27,0,13,27,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1c9f83b592f1ece5","q":"Prove that the equation\n\n$$\n6\\left(6 a^{2}+3 b^{2}+c^{2}\\right)=5 n^{2}\n$$\n\nhas no solutions in integers except $a=b=c=n=0$.","t":[{"b":6,"e":0.57,"k":"flat","v":0.82738,"x":0.92562,"p":[[0,28,0.0,0.91963,0.18539,0.96429,1.0,1.0,0.28571,1.0,0,24,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,5,0,24],[4,28,0.1429,0.82738,0.29565,0.82132,1.0,1.0,0.0,1.0,1,21,1,1,0,2,0,0,0,1,0,1,0,0,2,0,0,1,0,0,3,0,21],[8,28,0.2857,0.92116,0.17556,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,2,1,0,2,0,0,0,0,26],[12,28,0.4286,0.8988,0.19652,1.0,1.0,1.0,0.405,1.0,0,25,0,0,0,0,0,0,0,0,0,2,0,1,3,0,0,1,0,0,0,0,25],[16,28,0.5714,0.90848,0.1715,0.96429,1.0,1.0,0.5,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,2,0,24],[20,28,0.7143,0.92562,0.1565,1.0,1.0,1.0,0.42857,1.0,0,25,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,1,0,0,2,0,25],[24,28,0.8571,0.90625,0.15815,0.85714,1.0,1.0,0.42857,1.0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,3,0,22],[28,28,1.0,0.89584,0.15913,0.82132,1.0,1.0,0.40429,1.0,0,20,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,5,0,0,3,1,20]]},{"b":7,"e":1.0,"k":"flat","v":0.84982,"x":1.0,"p":[[0,19,0.0,0.89062,0.25316,0.96429,1.0,1.0,0.0,1.0,2,24,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,2,0,24],[4,19,0.2105,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[8,19,0.4211,0.84982,0.2695,0.85711,1.0,1.0,0.0,1.0,1,21,1,1,0,0,0,0,3,0,0,0,1,0,0,0,0,2,0,0,4,0,21],[12,19,0.6316,0.86754,0.23685,0.85714,1.0,1.0,0.0,1.0,1,21,0,1,0,0,0,0,0,0,0,2,1,0,1,0,0,2,0,0,4,0,21],[16,19,0.8421,0.94879,0.18219,1.0,1.0,1.0,0.0,1.0,1,28,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,28],[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":"14f570594da675f0","q":"Let $L B C$ be a fixed triangle with $L B=L C$, and let $A$ be a variable point on arc $L B$ of its circumcircle. Let $I$ be the incenter of $\\triangle A B C$ and $\\overline{A K}$ the altitude from $A$. The circumcircle of $\\triangle I K L$ intersects lines $K A$ and $B C$ again at $U \\neq K$ and $V \\neq K$. Finally, let $T$ be the projection of $I$ onto line $U V$. Prove that the line through $T$ and the midpoint of $\\overline{I K}$ passes through a fixed point as $A$ varies.","t":[{"b":5,"e":0.42857,"k":"rising","v":0.0,"x":0.33482,"p":[[0,20,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.01339,0.07457,0.0,0.0,0.0,0.0,0.4286,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.27679,0.20184,0.0,0.42857,0.42857,0.0,0.4286,11,0,1,11,0,0,0,0,1,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.33482,0.17717,0.42857,0.42857,0.42857,0.0,0.4286,7,0,0,7,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,29,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"30344d972484cbc3","q":"Let $t$ be a non-zero natural number. Show that there exists an integer $n>1$ coprime with $t$ such that for any integer $k \\geq 1$, the integer $n^{k}+t$ is not a power (i.e., it is not of the form $m^{r}$ with $m \\geq 1$ and $r \\geq 2$).","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.03348,"p":[[0,27,0.0,0.03125,0.09932,0.0,0.0,0.0,0.0,0.42857,29,0,3,29,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.03348,0.09277,0.0,0.0,0.0,0.0,0.4286,27,0,4,27,1,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,1,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,1,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,27,0.7407,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"rising","v":0.04464,"x":0.29463,"p":[[0,11,0.0,0.05357,0.14617,0.0,0.0,0.0,0.0,0.57143,27,0,4,27,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[4,11,0.3636,0.04464,0.12595,0.0,0.0,0.0,0.0,0.57143,28,0,1,28,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,11,0.7273,0.07143,0.15152,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,2,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[11,11,1.0,0.29463,0.15539,0.14286,0.28571,0.32143,0.14286,0.57143,0,0,0,0,0,12,0,0,12,0,0,2,0,0,6,0,0,0,0,0,0,0,0]]}]},{"i":"5c43a5ce02bf24ec","q":"The sides  $ [AB]$  and  $ [AC]$  of the triangle  $ ABC$  are tangent to the incircle with center  $ I$  of the  $ \\triangle ABC$  at the points  $ M$  and  $ N$ , respectively. The internal bisectors of the  $ \\triangle ABC$  drawn form  $ B$  and  $ C$  intersect the line  $ MN$  at the points  $ P$  and  $ Q$ , respectively. Suppose that  $ F$  is the intersection point of the lines  $ CP$  and  $ BQ$ . Prove that  $ FI\\perp BC$ .","t":[{"b":4,"e":0.14286,"k":"flat","v":0.04464,"x":0.08464,"p":[[0,21,0.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,3,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,4,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.07143,0.07143,0.0,0.07143,0.14286,0.0,0.14286,16,0,1,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.08027,0.07079,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.05358,0.06917,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.08464,0.07002,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.04455,"x":0.08036,"p":[[0,23,0.0,0.04893,0.06761,0.0,0.0,0.14286,0.0,0.1429,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.05804,0.07016,0.0,0.0,0.14286,0.0,0.1429,19,0,2,19,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.08036,0.07087,0.0,0.14286,0.14286,0.0,0.1429,14,0,0,14,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0caf3eb0c6b02558","q":"Prove that\n\n$$\n\\left(a^{2}+2\\right)\\left(b^{2}+2\\right)\\left(c^{2}+2\\right) \\geq 9(a b+b c+c a)\n$$\n\nfor all real numbers $a, b, c>0$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,14,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,0,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[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":0.03571,"p":[[0,21,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.03571,0.11845,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,21,0.5714,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"859069b131b44505","q":"Let $a_{1}, \\ldots, a_{2019}$ be positive integers. Show that the following are equivalent:\n(i) there exists a real number $x$ such that for all $i \\in\\{1, \\ldots, 2019\\}$, we have: $a_{i}=\\lfloor i x\\rfloor$\n(ii) for all $\\boldsymbol{i}, \\mathfrak{j} \\in\\{1, \\ldots, 2019\\}$ satisfying $i+j \\leqslant 2019$, we have: $a: a_{i}+a_{j} \\leqslant a_{i+j} \\leqslant a_{i}+a_{j}+1$.","t":[{"b":2,"e":0.28571,"k":"falling","v":0.29462,"x":0.49552,"p":[[0,13,0.0,0.49552,0.32729,0.2857,0.28571,0.85704,0.0,1.0,1,7,0,1,0,3,0,0,15,0,0,1,0,0,1,0,0,2,0,0,2,0,7],[4,13,0.3077,0.35705,0.26252,0.14289,0.28571,0.32143,0.14,1.0,0,4,0,0,0,9,0,0,15,0,0,3,0,0,1,0,0,0,0,0,0,0,4],[8,13,0.6154,0.41065,0.25199,0.2857,0.28571,0.57143,0.14,1.0,0,2,0,0,0,6,0,0,14,0,0,2,0,0,4,0,0,2,0,0,2,0,2],[12,13,0.9231,0.30355,0.06911,0.2857,0.28571,0.28571,0.2857,0.57143,0,0,0,0,0,0,0,0,30,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[13,13,1.0,0.29462,0.04964,0.2857,0.28571,0.28571,0.2857,0.571,0,0,0,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"flat","v":0.27223,"x":0.51335,"p":[[0,32,0.0,0.41963,0.28557,0.2857,0.28571,0.46418,0.14286,1.0,0,5,0,0,0,5,0,0,17,0,0,2,0,0,2,0,0,0,0,0,1,0,5],[4,32,0.125,0.44187,0.31012,0.2857,0.28571,0.60714,0.0,1.0,1,6,0,1,0,4,0,0,16,0,0,2,0,0,1,0,0,1,0,0,1,0,6],[8,32,0.25,0.34821,0.22286,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,6,0,0,20,0,0,0,0,0,3,0,0,0,0,0,1,0,2],[12,32,0.375,0.39732,0.22794,0.2857,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,1,0,0,24,0,0,0,0,0,0,0,0,3,0,0,3,0,1],[16,32,0.5,0.28122,0.13586,0.25,0.28571,0.28571,0.0,0.71429,1,0,0,1,0,7,0,0,20,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[20,32,0.625,0.27669,0.08716,0.2857,0.28571,0.28571,0.14,0.5714,0,0,0,0,0,6,0,0,23,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,32,0.75,0.51335,0.25718,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,1,0,0,13,0,0,3,0,0,6,0,0,2,0,0,4,0,3],[28,32,0.875,0.49103,0.1921,0.28571,0.42857,0.57143,0.14286,1.0,0,1,0,0,0,1,0,0,8,0,0,8,0,0,10,0,0,2,0,0,2,0,1],[32,32,1.0,0.27223,0.12046,0.14296,0.2857,0.28571,0.14,0.71429,0,0,0,0,0,9,0,0,20,0,0,1,0,0,1,0,0,1,0,0,0,0,0]]}]},{"i":"933da5a0092703a4","q":"Let $\\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\\varphi(n)=m$ has at least 2015 solutions in $n$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,11,0.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.05357,0.1171,0.0,0.0,0.0,0.0,0.42857,26,0,5,26,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.42857,"k":"rising","v":0.02232,"x":0.41964,"p":[[0,36,0.0,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.03125,0.07771,0.0,0.0,0.0,0.0,0.2857,27,0,1,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.03571,0.11845,0.0,0.0,0.0,0.0,0.57143,29,0,0,29,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,36,0.4444,0.05362,0.13256,0.0,0.0,0.0,0.0,0.43,27,0,2,27,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.27679,0.14258,0.14286,0.2857,0.42857,0.0,0.4286,3,0,0,3,0,8,0,0,9,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.39286,0.09449,0.42857,0.42857,0.42857,0.14286,0.57143,0,0,0,0,0,3,0,0,3,0,0,25,0,0,1,0,0,0,0,0,0,0,0],[28,36,0.7778,0.41964,0.06121,0.42857,0.42857,0.42857,0.2857,0.57143,0,0,0,0,0,0,0,0,4,0,0,26,0,0,2,0,0,0,0,0,0,0,0],[32,36,0.8889,0.41069,0.11148,0.42857,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,3,0,0,2,0,0,24,0,0,2,0,0,1,0,0,0,0,0],[36,36,1.0,0.37947,0.11633,0.42857,0.42857,0.42857,0.0,0.57143,1,0,0,1,0,3,0,0,3,0,0,24,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"d25665964d77c96e","q":"Prove that the number of odd numbers in row $n$ is at most twice the number of switch pairs in row $n-1$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.04464,"x":0.25443,"p":[[0,58,0.0,0.12947,0.22689,0.0,0.0,0.1786,0.0,0.71429,23,0,0,23,0,1,0,0,1,0,0,4,0,0,1,0,0,2,0,0,0,0,0],[4,58,0.069,0.17857,0.26487,0.0,0.0,0.42858,0.0,0.71429,21,0,0,21,0,1,0,0,0,0,0,4,0,0,3,0,0,3,0,0,0,0,0],[8,58,0.1379,0.08036,0.15542,0.0,0.0,0.0,0.0,0.4286,25,0,0,25,0,0,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.07143,0.17496,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,2,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[16,58,0.2759,0.12498,0.20121,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,2,0,0,5,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[20,58,0.3448,0.16963,0.23536,0.0,0.0,0.42857,0.0,0.71429,19,0,0,19,0,3,0,0,1,0,0,4,0,0,4,0,0,1,0,0,0,0,0],[24,58,0.4138,0.14283,0.22583,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,0,0,0,3,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[28,58,0.4828,0.04464,0.14032,0.0,0.0,0.0,0.0,0.71429,28,0,0,28,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[32,58,0.5517,0.09375,0.17353,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,1,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[36,58,0.6207,0.08927,0.18119,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,1,0,0,1,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[40,58,0.6897,0.11159,0.18114,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,4,0,0,2,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[44,58,0.7586,0.20981,0.24216,0.0,0.14286,0.32143,0.0,1.0,14,1,0,14,0,4,0,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,1],[48,58,0.8276,0.20978,0.22293,0.0,0.14285,0.42857,0.0,0.57143,16,0,0,16,0,0,0,0,5,0,0,7,0,0,4,0,0,0,0,0,0,0,0],[52,58,0.8966,0.25443,0.23067,0.0,0.28571,0.42857,0.0,0.57143,13,0,0,13,0,1,0,0,4,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[56,58,0.9655,0.19195,0.23311,0.0,0.0,0.42857,0.0,0.57143,18,0,0,18,0,1,0,0,2,0,0,6,0,0,5,0,0,0,0,0,0,0,0],[58,58,1.0,0.11604,0.20953,0.0,0.0,0.07143,0.0,0.57143,24,0,0,24,0,0,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"volatile","v":0.12499,"x":0.29464,"p":[[0,4,0.0,0.12499,0.26663,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,2,0,0,2,0,0,0,0,0,1,0,0,1,0,1],[4,4,1.0,0.29464,0.40867,0.0,0.0,0.71429,0.0,1.0,20,6,0,20,0,0,0,0,1,0,0,1,0,0,1,0,0,3,0,0,0,0,6]]}]},{"i":"f6d2a3801053a620","q":"Let k,m and n be three different positive integers. Prove that \\[ \n\t\\left( k-\\frac{1}{k} \\right)\\left( m-\\frac{1}{m} \\right)\\left( n-\\frac{1}{n} \\right) \\le kmn-(k+m+n). \\]","t":[{"b":2,"e":0.14286,"k":"flat","v":0.09375,"x":0.12054,"p":[[0,18,0.0,0.0982,0.16142,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,7,0,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,18,0.2222,0.09375,0.15815,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,3,0,0,4,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,18,0.4444,0.11607,0.1448,0.0,0.07143,0.14287,0.0,0.42857,16,0,0,16,0,10,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.12054,0.17169,0.0,0.0,0.1429,0.0,0.57143,18,0,0,18,0,8,0,0,0,0,0,5,0,0,1,0,0,0,0,0,0,0,0],[16,18,0.8889,0.12054,0.16793,0.0,0.0,0.2857,0.0,0.4286,20,0,0,20,0,2,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.11607,0.15746,0.0,0.0,0.14286,0.0,0.4286,18,0,0,18,0,7,0,0,2,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"rising","v":0.05357,"x":0.38392,"p":[[0,19,0.0,0.12947,0.1488,0.0,0.14286,0.17868,0.0,0.4286,15,0,0,15,0,9,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.09822,0.18013,0.0,0.0,0.14286,0.0,0.57143,22,0,0,22,0,5,0,0,1,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[8,19,0.4211,0.05357,0.14617,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,2,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,19,0.6316,0.08034,0.13798,0.0,0.0,0.14286,0.0,0.571,21,0,0,21,0,7,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,19,0.8421,0.29463,0.19863,0.14286,0.28571,0.4286,0.0,0.57143,7,0,0,7,0,4,0,0,6,0,0,10,0,0,5,0,0,0,0,0,0,0,0],[19,19,1.0,0.38392,0.16535,0.28571,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,5,0,0,7,0,0,9,0,0,10,0,0,0,0,0,0,0,0]]}]},{"i":"06b3195fad73e057","q":"Let $n, m, k$ and $l$ be positive integers with $n \\neq 1$ such that $n^{k}+m n^{l}+1$ divides $n^{k+l}-1$. Prove that  - $m=1$ and $l=2 k$; or - $l \\mid k$ and $m=\\frac{n^{k-l}-1}{n^{l}-1}$.","t":[{"b":1,"e":0.85714,"k":"flat","v":0.12054,"x":0.26338,"p":[[0,17,0.0,0.2412,0.20959,0.14286,0.14286,0.32464,0.0,0.85714,5,0,0,5,0,15,0,0,4,0,0,4,0,0,2,0,0,1,0,0,1,0,0],[4,17,0.2353,0.18302,0.1829,0.0,0.14286,0.2857,0.0,0.71429,11,0,0,11,0,9,0,0,7,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[8,17,0.4706,0.12054,0.11356,0.0,0.14286,0.14286,0.0,0.4286,12,0,0,12,0,14,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.14268,0.11294,0.0,0.14286,0.1786,0.0,0.42857,9,0,0,9,0,15,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.25893,0.16146,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,14,0,0,12,0,0,2,0,0,1,0,0,2,0,0,0,0,0],[17,17,1.0,0.26338,0.19267,0.14286,0.2857,0.32143,0.0,0.85714,4,0,0,4,0,11,0,0,9,0,0,4,0,0,3,0,0,0,0,0,1,0,0]]},{"b":2,"e":0.14,"k":"flat","v":0.10268,"x":0.19642,"p":[[0,11,0.0,0.19642,0.17402,0.14286,0.14286,0.28571,0.0,0.71429,7,0,0,7,0,14,0,0,7,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[4,11,0.3636,0.10268,0.10248,0.0,0.14286,0.14286,0.0,0.28571,14,0,0,14,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.17402,0.18468,0.0,0.14286,0.2857,0.0,0.71429,12,0,0,12,0,9,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[11,11,1.0,0.13394,0.10062,0.10714,0.14286,0.1429,0.0,0.4286,8,0,0,8,0,19,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7dae9f7f42d16a15","q":"Let $P$ be a polynomial with rational coefficients of degree greater than or equal to 2, and $\\left(q_{n}\\right)_{n \\in \\mathbb{N}}$ a sequence of rationals such that for all $n \\geqslant 0, q_{n}=P\\left(q_{n+1}\\right)$. Show that the sequence $q_{n}$ is periodic from a certain rank.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.29902,"x":0.47753,"p":[[0,28,0.0,0.47753,0.25394,0.28571,0.57143,0.57143,0.0,1.0,4,1,4,4,0,2,0,0,3,0,0,3,0,0,13,0,0,5,0,0,1,0,1],[4,28,0.1429,0.37937,0.30439,0.0,0.42857,0.71429,0.0,0.85714,9,0,9,9,0,4,0,0,2,0,0,2,0,0,5,0,0,9,0,0,1,0,0],[8,28,0.2857,0.29902,0.28657,0.0,0.35714,0.57143,0.0,0.85714,14,0,14,14,0,1,0,0,1,0,0,3,0,0,11,0,0,1,0,0,1,0,0],[12,28,0.4286,0.42407,0.20663,0.28571,0.571,0.57143,0.0,0.57143,5,0,5,5,0,0,0,0,4,0,0,5,0,0,18,0,0,0,0,0,0,0,0],[16,28,0.5714,0.35262,0.20192,0.2857,0.35714,0.571,0.0,0.57143,5,0,5,5,0,2,0,0,9,0,0,5,0,0,11,0,0,0,0,0,0,0,0],[20,28,0.7143,0.38835,0.16061,0.28571,0.42857,0.571,0.0,0.57143,2,0,2,2,0,1,0,0,11,0,0,8,0,0,10,0,0,0,0,0,0,0,0],[24,28,0.8571,0.37499,0.1847,0.28571,0.42857,0.57143,0.0,0.57143,4,0,4,4,0,0,0,0,11,0,0,6,0,0,11,0,0,0,0,0,0,0,0],[28,28,1.0,0.44195,0.16505,0.39286,0.4998,0.57143,0.0,0.57143,2,0,2,2,0,1,0,0,5,0,0,8,0,0,16,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"flat","v":0.42407,"x":0.49552,"p":[[0,9,0.0,0.42407,0.28002,0.14286,0.4998,0.57143,0.0,1.0,7,1,7,7,0,2,0,0,1,0,0,6,0,0,10,0,0,4,0,0,1,0,1],[4,9,0.4444,0.42849,0.23952,0.2857,0.42857,0.57143,0.0,0.85714,4,0,4,4,0,2,0,0,6,0,0,5,0,0,10,0,0,3,0,0,2,0,0],[8,9,0.8889,0.49552,0.27195,0.39286,0.57143,0.71429,0.0,1.0,5,1,5,5,0,1,0,0,2,0,0,4,0,0,10,0,0,7,0,0,2,0,1],[9,9,1.0,0.49104,0.24466,0.28571,0.57143,0.57143,0.0,1.0,2,1,2,2,0,4,0,0,3,0,0,3,0,0,13,0,0,4,0,0,2,0,1]]}]},{"i":"8be25eca3f6e9170","q":"There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme.\n\n*A. Magazinov*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.08036,"p":[[0,43,0.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,5,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,43,0.093,0.08036,0.09407,0.0,0.07143,0.14286,0.0,0.42857,16,0,7,16,0,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,43,0.186,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.42857,24,0,5,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.05357,0.13716,0.0,0.0,0.0,0.0,0.71429,25,0,5,25,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[16,43,0.3721,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,2,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,43,0.5581,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,43,0.6512,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":0.14286,"k":"flat","v":0.03125,"x":0.12491,"p":[[0,7,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,7,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.05802,0.11763,0.0,0.0,0.14286,0.0,0.571,23,0,2,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[7,7,1.0,0.12491,0.04721,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1830e02ec3d84768","q":"Two permutations $a_{1}, a_{2}, \\ldots, a_{2010}$ and $b_{1}, b_{2}, \\ldots, b_{2010}$ of the numbers $1,2, \\ldots, 2010$ are said to intersect if $a_{k}=b_{k}$ for some value of $k$ in the range $1 \\leq k \\leq 2010$. Show that there exist 1006 permutations of the numbers $1,2, \\ldots, 2010$ such that any other such permutation is guaranteed to intersect at least one of these 1006 permutations.","t":[{"b":4,"e":0.0,"k":"volatile","v":0.0625,"x":0.77679,"p":[[0,14,0.0,0.49107,0.47774,0.0,0.57143,1.0,0.0,1.0,15,12,13,15,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,12],[4,14,0.2857,0.55134,0.46881,0.0,0.85714,1.0,0.0,1.0,12,14,10,12,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,1,14],[8,14,0.5714,0.5625,0.45588,0.0,0.85714,1.0,0.0,1.0,12,12,9,12,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,5,0,12],[12,14,0.8571,0.77679,0.35703,0.71429,1.0,1.0,0.0,1.0,5,18,1,5,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,18],[14,14,1.0,0.0625,0.14698,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":1.0,"k":"volatile","v":0.34598,"x":0.93526,"p":[[0,13,0.0,0.48661,0.47764,0.0,0.42856,1.0,0.0,1.0,15,13,11,15,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,13],[4,13,0.3077,0.40179,0.46489,0.0,0.0,1.0,0.0,1.0,17,10,12,17,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,10],[8,13,0.6154,0.34598,0.41769,0.0,0.0,0.75,0.0,1.0,17,6,15,17,0,1,0,0,1,0,0,2,0,0,1,0,0,2,0,0,1,1,6],[12,13,0.9231,0.93526,0.07858,0.85714,1.0,1.0,0.71429,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,1,18],[13,13,1.0,0.9308,0.07027,0.85714,0.96429,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,1,16]]}]},{"i":"dadd6f38b2a97500","q":"Let $I$ be the incenter of a triangle $A B C$ and $\\Gamma$ be its circumcircle. Let the line $A I$ intersect $\\Gamma$ at a point $D \\neq A$. Let $F$ and $E$ be points on side $B C$ and $\\operatorname{arc} B D C$ respectively such that $\\angle B A F=\\angle C A E<\\frac{1}{2} \\angle B A C$. Finally, let $G$ be the midpoint of the segment $I F$. Prove that the lines $D G$ and $E I$ intersect on $\\Gamma$. (Hong Kong)","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,18,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,37,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,37,0.7568,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d9ce279eb0c4d667","q":"Two equal-sized regular  $n$ -gons intersect to form a  $2n$ -gon  $C$ . Prove that the sum of the sides of  $C$  which form part of one  $n$ -gon equals half the perimeter of  $C$ .\n\n*Alternative formulation:*\n\n Let two equal regular  $n$ -gons  $S$  and  $T$  be located in the plane such that their intersection  $S\\cap T$  is a  $2n$ -gon (with  $n\\ge 3$ ). The sides of the polygon  $S$  are coloured in red and the sides of  $T$  in blue. \n\nProve that the sum of the lengths of the blue sides of the polygon  $S\\cap T$  is equal to the sum of the lengths of its red sides.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,18,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,2,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.04911,0.14555,0.0,0.0,0.0,0.0,0.57143,28,0,2,28,0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[16,18,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,32,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"c8d9ae94afbb3335","q":"Let $\\mathrm{C}_{1}, \\mathrm{C}_{2}, \\ldots \\mathrm{C}_{\\mathrm{n}}$ be circles of the same radius arranged in the plane such that they are never tangent to each other and there always exists a path passing through the circles to go from a point on one of them to another (in other words, the circles are connected). Denoting $S$ as the set of intersection points of the circles, show that $|S| \\geqslant n$.","t":[{"b":4,"e":1.0,"k":"flat","v":0.19197,"x":0.45089,"p":[[0,12,0.0,0.3884,0.37667,0.0,0.35714,0.71429,0.0,1.0,10,6,1,10,0,5,0,0,1,0,0,6,0,0,1,0,0,2,0,0,1,0,6],[4,12,0.3333,0.19197,0.21312,0.0,0.14286,0.28571,0.0,0.85714,10,0,3,10,0,13,0,0,3,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[8,12,0.6667,0.35268,0.34253,0.10714,0.1429,0.71429,0.0,1.0,8,3,0,8,0,9,0,0,2,0,0,3,0,0,1,0,0,4,0,0,2,0,3],[12,12,1.0,0.45089,0.36441,0.14286,0.28571,0.85714,0.0,1.0,3,7,0,3,0,9,0,0,6,0,0,3,0,0,1,0,0,0,0,0,3,0,7]]},{"b":5,"e":0.71429,"k":"rising","v":0.27678,"x":0.62713,"p":[[0,14,0.0,0.30796,0.37818,0.0,0.14143,0.60682,0.0,1.0,15,4,2,15,0,5,0,0,0,0,0,2,0,0,2,0,0,2,0,0,2,0,4],[4,14,0.2857,0.27678,0.3443,0.0,0.14286,0.42857,0.0,1.0,14,4,0,14,0,5,0,0,3,0,0,3,0,0,1,0,0,2,0,0,0,0,4],[8,14,0.5714,0.35487,0.38533,0.0,0.14286,0.80357,0.0,1.0,9,6,0,9,0,9,0,0,4,0,0,1,0,0,0,0,0,0,1,0,2,0,6],[12,14,0.8571,0.62713,0.369,0.24999,0.71429,1.0,0.0,1.0,2,12,0,2,0,6,0,0,3,0,0,0,0,0,2,1,0,4,0,0,2,0,12],[14,14,1.0,0.61606,0.3719,0.24999,0.78564,1.0,0.0,1.0,4,9,0,4,0,4,0,0,2,0,0,1,0,0,3,0,0,2,0,0,7,0,9]]}]},{"i":"c860732313978b92","q":"Let  $\\omega$  be the circumcircle of acute triangle  $ABC$ . Two tangents of  $\\omega$  from  $B$  and  $C$  intersect at  $P$ ,  $AP$  and  $BC$  intersect at  $D$ . Point  $E$ ,  $F$  are on  $AC$  and  $AB$  such that  $DE \\parallel BA$  and  $DF \\parallel CA$ .\r\n(1) Prove that  $F,B,C,E$  are concyclic.\r\n\r\n(2) Denote  $A_{1}$  the centre of the circle passing through  $F,B,C,E$ .  $B_{1}$ ,  $C_{1}$  are difined similarly. Prove that  $AA_{1}$ ,  $BB_{1}$ ,  $CC_{1}$  are concurrent.","t":[{"b":1,"e":0.28571,"k":"flat","v":0.15615,"x":0.27901,"p":[[0,6,0.0,0.15615,0.14877,0.0,0.14286,0.2857,0.0,0.571,13,0,13,13,0,5,0,0,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,6,0.6667,0.27901,0.11893,0.2857,0.28571,0.28571,0.0,0.57143,3,0,3,3,0,1,0,0,24,0,1,1,0,0,2,0,0,0,0,0,0,0,0],[6,6,1.0,0.25892,0.07523,0.2857,0.2857,0.28571,0.0,0.42857,1,0,1,1,0,5,0,0,25,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"flat","v":0.19196,"x":0.28571,"p":[[0,19,0.0,0.19196,0.12168,0.10714,0.2857,0.28571,0.0,0.28571,8,0,8,8,0,5,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.20982,0.11836,0.14286,0.2857,0.28571,0.0,0.42857,6,0,6,6,0,6,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.24769,0.08384,0.14286,0.2857,0.28571,0.14,0.42857,0,0,0,0,0,11,0,0,18,0,1,2,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.25893,0.07524,0.24999,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,8,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.27232,0.06546,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,5,0,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.28571,0.06186,0.2857,0.2857,0.28571,0.14286,0.4286,0,0,0,0,0,3,0,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a95f31fb349ec42f","q":"Let $a_{1}, a_{2}, \\ldots, a_{2023}$ be positive real numbers with\n\n$$\na_{1}+a_{2}^{2}+a_{3}^{3}+\\cdots+a_{2023}^{2023}=2023\n$$\n\nShow that\n\n$$\na_{1}^{2023}+a_{2}^{2022}+\\cdots+a_{2022}^{2}+a_{2023}>1+\\frac{1}{2023} .\n$$","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,9,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,24,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2fc66292ef3a740e","q":"There are  $ n \\geq 5$  pairwise different points in the plane. For every point, there are just four points whose distance from which is  $ 1$ . Find the maximum value of  $ n$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.07143,"p":[[0,13,0.0,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,18,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,23,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,22,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.07143,0.10101,0.0,0.0,0.14286,0.0,0.28571,20,0,12,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.02232,"x":0.04911,"p":[[0,15,0.0,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,16,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.04911,0.09181,0.0,0.0,0.03571,0.0,0.28571,24,0,15,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,17,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,9,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,4,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5c094bb0442632f9","q":"We choose random a unitary polynomial of degree  $n$  and coefficients in the set  $1,2,...,n!$ . Prove that the probability for this polynomial to be special is between  $0.71$  and  $0.75$ , where a polynomial  $g$  is called special if for every  $k>1$  in the sequence  $f(1), f(2), f(3),...$  there are infinitely many numbers relatively prime with  $k$ .","t":[{"b":0,"e":0.85714,"k":"flat","v":0.8125,"x":0.91964,"p":[[0,9,0.0,0.8125,0.3163,0.85711,1.0,1.0,0.0,1.0,3,18,1,3,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,7,0,18],[4,9,0.4444,0.89731,0.11426,0.857,0.92857,1.0,0.71429,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,9,0,16],[8,9,0.8889,0.91964,0.10677,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,8,0,19],[9,9,1.0,0.85266,0.10406,0.85711,0.85714,0.85714,0.571,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,6,0,0,18,0,7]]},{"b":1,"e":1.0,"k":"rising","v":0.76339,"x":0.98214,"p":[[0,22,0.0,0.76339,0.33996,0.67857,1.0,1.0,0.0,1.0,3,17,1,3,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,5,0,17],[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],[8,22,0.3636,0.95089,0.08459,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,7,0,23],[12,22,0.5455,0.93749,0.07088,0.85714,1.0,1.0,0.857,1.0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,18],[16,22,0.7273,0.96874,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[20,22,0.9091,0.95535,0.06623,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[22,22,1.0,0.95088,0.07669,0.85714,1.0,1.0,0.71429,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,9,0,22]]}]},{"i":"b39e7e41a354a23e","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}$ be distinct positive integers, $n \\geq 3$. Prove that there exist distinct indices $i$ and $j$ such that $a_{i}+a_{j}$ does not divide any of the numbers $3 a_{1}, 3 a_{2}, \\ldots, 3 a_{n}$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.03571,"x":0.10713,"p":[[0,7,0.0,0.0758,0.1128,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.10713,0.15148,0.0,0.0,0.17857,0.0,0.571,19,0,1,19,0,5,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[7,7,1.0,0.03571,0.07986,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"flat","v":0.02222,"x":0.09375,"p":[[0,11,0.0,0.09375,0.13175,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,7,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.02222,0.10153,0.0,0.0,0.0,0.0,0.571,30,0,0,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,11,0.7273,0.08036,0.1234,0.0,0.0,0.14286,0.0,0.4286,20,0,0,20,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.08036,0.13333,0.0,0.0,0.14286,0.0,0.4286,21,0,0,21,0,7,0,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b144d5d1d7cd4185","q":"Let $\\mathrm{P}, \\mathrm{Q}$ be two non-constant polynomials with real coefficients and coprime. Show that there are at most three real numbers $\\lambda$ such that:\n\n$$\nP+\\lambda Q=R^{2}\n$$\n\nwhere $R \\in \\mathbb{R}[X]$.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.04911,"x":0.13839,"p":[[0,18,0.0,0.08482,0.1551,0.0,0.0,0.07143,0.0,0.57143,24,0,1,24,0,0,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,18,0.2222,0.09375,0.13175,0.0,0.0,0.17868,0.0,0.42857,20,0,1,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.13839,0.15355,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,10,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,18,0.6667,0.12946,0.12556,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,7,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.07143,0.10714,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.04911,0.08458,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0133,"x":0.14286,"p":[[0,11,0.0,0.12946,0.17627,0.0,0.0,0.2857,0.0,0.71429,18,0,0,18,0,4,0,0,7,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[4,11,0.3636,0.13393,0.12846,0.0,0.14286,0.2857,0.0,0.42857,13,0,1,13,0,9,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.14286,0.13363,0.0,0.14286,0.28571,0.0,0.42857,12,0,2,12,0,10,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5e07c1257ca6edc5","q":"Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define\n\n$$\nf_{j}(r)=\\min (j r, n)+\\min \\left(\\frac{j}{r}, n\\right), \\quad \\text { and } \\quad g_{j}(r)=\\min (\\lceil j r\\rceil, n)+\\min \\left(\\left\\lceil\\frac{j}{r}\\right\\rceil, n\\right) \\text {, }\n$$\n\nwhere $\\lceil x\\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that\n\n$$\n\\sum_{j=1}^{n} f_{j}(r) \\leq n^{2}+n \\leq \\sum_{j=1}^{n} g_{j}(r)\n$$","t":[{"b":3,"e":0.42857,"k":"flat","v":0.12947,"x":0.27231,"p":[[0,9,0.0,0.27231,0.23243,0.0,0.42857,0.42857,0.0,0.71429,12,0,1,12,0,1,0,0,2,0,0,14,0,0,1,0,0,2,0,0,0,0,0],[4,9,0.4444,0.2634,0.25533,0.0,0.28571,0.42857,0.0,1.0,13,1,0,13,0,1,0,0,3,0,0,12,0,0,1,0,0,1,0,0,0,0,1],[8,9,0.8889,0.17187,0.17115,0.0,0.14286,0.2857,0.0,0.42857,13,0,0,13,1,5,0,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.12947,0.1488,0.0,0.14286,0.1786,0.0,0.42857,15,0,0,15,0,9,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.24534,"x":0.375,"p":[[0,19,0.0,0.25,0.31542,0.0,0.14286,0.42857,0.0,1.0,15,3,0,15,0,4,0,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,3],[4,19,0.2105,0.24534,0.2266,0.0,0.14286,0.42857,0.0,0.71429,11,0,0,11,0,6,0,0,1,0,0,11,0,0,1,0,0,2,0,0,0,0,0],[8,19,0.4211,0.31247,0.17651,0.14286,0.42857,0.42857,0.0,0.571,5,0,0,5,0,5,0,0,3,0,0,17,0,0,2,0,0,0,0,0,0,0,0],[12,19,0.6316,0.375,0.22232,0.28571,0.42857,0.42857,0.0,1.0,5,1,0,5,0,1,0,0,4,0,0,19,0,0,0,0,0,1,0,0,1,0,1],[16,19,0.8421,0.3259,0.21793,0.14286,0.42857,0.42858,0.0,0.71429,6,0,0,6,0,6,0,0,0,0,0,16,0,0,1,0,0,3,0,0,0,0,0],[19,19,1.0,0.24544,0.1898,0.105,0.14288,0.42857,0.0,0.571,8,0,0,8,0,9,0,0,0,0,0,14,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"bd6179b32a60ec31","q":"On the board we write a series of  $n$  numbers, where  $n \\geq 40$ , and each one of them is equal to either  $1$  or  $-1$ , such that the following conditions both hold:\n\n(i) The sum of every  $40$  consecutive numbers is equal to  $0$ .\n(ii) The sum of every  $42$  consecutive numbers is not equal to  $0$ .\n\nWe denote by  $S_n$  the sum of the  $n$  numbers of the board. Find the maximum possible value of  $S_n$  for all possible values of  $n$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.89285,"x":0.96875,"p":[[0,40,0.0,0.96875,0.06902,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,26],[4,40,0.1,0.95982,0.06424,0.85714,1.0,1.0,0.857,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[8,40,0.2,0.95088,0.06786,0.85714,1.0,1.0,0.857,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,21],[12,40,0.3,0.91518,0.07016,0.85714,0.85714,1.0,0.85714,1.0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19,0,13],[16,40,0.4,0.93303,0.07129,0.85714,1.0,1.0,0.857,1.0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,17],[20,40,0.5,0.89285,0.07143,0.85714,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,9],[24,40,0.6,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[28,40,0.7,0.91071,0.06916,0.85714,0.85714,1.0,0.857,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,12],[32,40,0.8,0.90625,0.07668,0.85714,0.85714,1.0,0.71429,1.0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,19,0,12],[36,40,0.9,0.92856,0.07144,0.85714,0.92857,1.0,0.857,1.0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,16],[40,40,1.0,0.95535,0.07524,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23]]},{"b":6,"e":1.0,"k":"flat","v":0.95982,"x":0.99554,"p":[[0,9,0.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26],[4,9,0.4444,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[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],[9,9,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]}]},{"i":"5ca2810305bd4ded","q":"Let $n$ be a positive integer relatively prime to 6 . We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,33,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,2,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,33,0.2424,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":6,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,40,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.04464,0.15746,0.0,0.0,0.0,0.0,0.71429,29,0,0,29,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,40,0.2,0.06696,0.19556,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,40,0.3,0.05357,0.13716,0.0,0.0,0.0,0.0,0.57143,27,0,3,27,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,40,0.4,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,26,0,2,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,1,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,40,0.7,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"243a7b4f8243f403","q":"Let $n$ be a positive integer. For a real number $x \\geq 1$, it holds that $\\left\\lfloor x^{n+1}\\right\\rfloor$, $\\left\\lfloor x^{n+2}\\right\\rfloor, \\ldots,\\left\\lfloor x^{4 n}\\right\\rfloor$ are all squares of positive integers. Prove that $\\lfloor x\\rfloor$ is also the square of a positive integer.\nWith $\\lfloor z\\rfloor$ we mean the greatest integer less than or equal to $z$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,31,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00894,0.04976,0.0,0.0,0.0,0.0,0.286,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,31,0.5161,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"83242bbfa648b2d5","q":"Let $a_{1}, a_{2}, a_{3}$ be strictly positive integers. For any integer $n \\geqslant 3$, we set\n\n$$\na_{n+1}=\\operatorname{lcm}\\left(a_{n}, a_{n-1}\\right)-\\operatorname{lcm}\\left(a_{n-1}, a_{n-2}\\right)\n$$\n\nwith the understanding that $\\operatorname{lcm}(0, x)=0$ for any integer $x$.\nProve that there exists a natural number $k$ such that $k \\leqslant a_{3}+4$ and $a_{k} \\leqslant 0$.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.13393,"x":0.16518,"p":[[0,24,0.0,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.2857,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.14286,0.08748,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,24,0.3333,0.1517,0.07937,0.14286,0.14286,0.14286,0.0,0.5714,1,0,1,1,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,24,0.5,0.16518,0.12931,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,24,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,24,0.6667,0.16063,0.07786,0.14286,0.14286,0.14286,0.14,0.57143,0,0,0,0,0,30,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,24,0.8333,0.14259,0.00083,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.14277,0.03572,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,30,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.2857,"k":"flat","v":0.05357,"x":0.15625,"p":[[0,43,0.0,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.57143,4,0,0,4,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,43,0.093,0.1517,0.11259,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,23,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,43,0.186,0.14268,0.05051,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,28,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,43,0.2791,0.14705,0.06669,0.14286,0.14286,0.14286,0.0,0.42857,2,0,0,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,43,0.3721,0.1517,0.06123,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,43,0.4651,0.1515,0.08699,0.14286,0.14286,0.14286,0.0,0.571,2,0,0,2,0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,43,0.5581,0.15179,0.10062,0.14286,0.14286,0.14286,0.0,0.57143,3,0,0,3,0,27,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,43,0.6512,0.15616,0.05488,0.14286,0.14286,0.14286,0.14,0.42857,0,0,0,0,0,30,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.15625,0.11495,0.14286,0.14286,0.14286,0.0,0.42857,6,0,0,6,0,20,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.08911,0.15459,0.0,0.0,0.14286,0.0,0.71429,20,0,0,20,0,8,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[40,43,0.9302,0.08929,0.12242,0.0,0.0,0.14286,0.0,0.42857,18,0,0,18,0,10,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ced96046ec7a2a9f","q":"Two-way flights are operated between  $80$  cities in such a way that each city is connected to at least  $7$  other cities by a direct flight and any two cities are connected by a finite sequence of flights. Find the smallest  $k$  such that for any such arrangement of flights it is possible to travel from any city to any other city by a sequence of at most  $k$  flights.","t":[{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.03125,"p":[[0,11,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,25,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,21,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,11,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02232,"x":0.09821,"p":[[0,12,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,20,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,24,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.09821,0.22142,0.0,0.0,0.03571,0.0,1.0,24,1,18,24,0,3,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,1]]}]},{"i":"a1abee8bfd1ddde9","q":"Let  $a_1<a_2<a_3<\\dots$  be positive integers such that  $a_{k+1}$  divides  $2(a_1+a_2+\\dots+a_k)$  for every  $k\\geqslant 1$ . Suppose that for infinitely many primes  $p$ , there exists  $k$  such that  $p$  divides  $a_k$ . Prove that for every positive integer  $n$ , there exists  $k$  such that  $n$  divides  $a_k$ .","t":[{"b":0,"e":0.28571,"k":"flat","v":0.20974,"x":0.41071,"p":[[0,11,0.0,0.27232,0.31209,0.0,0.21428,0.46431,0.0,1.0,15,1,2,15,0,1,0,0,6,0,0,2,0,0,1,0,0,5,0,0,1,0,1],[4,11,0.3636,0.41071,0.26184,0.2857,0.28571,0.60714,0.0,1.0,4,1,1,4,0,2,0,0,11,0,0,3,0,0,4,0,0,6,0,0,1,0,1],[8,11,0.7273,0.35714,0.18211,0.2857,0.28571,0.32144,0.14286,0.85714,0,0,0,0,0,4,0,0,20,0,0,1,0,0,3,0,0,3,0,0,1,0,0],[11,11,1.0,0.20974,0.08744,0.14286,0.14288,0.28571,0.0,0.42857,1,0,0,1,0,16,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.28571,"k":"flat","v":0.11165,"x":0.625,"p":[[0,41,0.0,0.31473,0.32037,0.0,0.21429,0.60714,0.0,0.85714,13,0,1,13,1,2,0,0,1,0,0,5,0,0,2,0,0,5,0,0,3,0,0],[4,41,0.0976,0.20982,0.33879,0.0,0.0,0.60714,0.0,0.85714,23,0,2,23,0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,3,0,0],[8,41,0.1951,0.21874,0.29661,0.0,0.0,0.28571,0.0,0.85714,17,0,1,17,0,3,0,0,5,0,0,0,0,0,2,0,0,2,0,0,3,0,0],[12,41,0.2927,0.11165,0.21653,0.0,0.0,0.14287,0.0,1.0,22,1,3,22,0,3,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[16,41,0.3902,0.625,0.21354,0.53571,0.71429,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,5,0,0,2,0,0,5,0,0,12,0,0,6,0,1],[20,41,0.4878,0.30804,0.29474,0.10714,0.28571,0.28571,0.0,1.0,8,4,0,8,0,1,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,0,4],[24,41,0.5854,0.27232,0.09006,0.2857,0.28571,0.28571,0.0,0.57143,2,0,0,2,0,1,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,41,0.6829,0.26339,0.0724,0.2857,0.28571,0.28571,0.0,0.42857,1,0,0,1,0,4,0,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.27233,0.05486,0.2857,0.28571,0.28571,0.0,0.286,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.27678,0.08702,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,3,0,0,26,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,41,0.9756,0.27232,0.05486,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,1,0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[41,41,1.0,0.26339,0.06297,0.2857,0.2857,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ebbf5bd6eb7c6c41","q":"Let ABC be a triangle and  $M$  be an interior point. Prove that \n\n\\[ \\min\\{MA,MB,MC\\}+MA+MB+MC<AB+AC+BC.\\]","t":[{"b":4,"e":0.14286,"k":"flat","v":0.30793,"x":0.64049,"p":[[0,55,0.0,0.45969,0.32097,0.14286,0.571,0.71429,0.0,1.0,4,1,0,4,0,8,0,0,2,0,0,1,0,0,4,0,0,7,0,0,5,0,1],[4,55,0.0727,0.46872,0.28843,0.14289,0.571,0.71429,0.0,1.0,3,1,0,3,0,6,0,0,3,0,0,3,0,0,7,0,0,5,0,0,4,0,1],[8,55,0.1455,0.30793,0.29478,0.0,0.1429,0.60682,0.0,0.85714,9,0,0,9,0,8,0,0,3,0,0,3,0,0,1,0,0,6,0,0,2,0,0],[12,55,0.2182,0.37944,0.2958,0.10714,0.42857,0.57143,0.0,0.85714,8,0,0,8,0,3,0,0,3,0,0,7,0,0,5,0,0,1,0,0,5,0,0],[16,55,0.2909,0.3839,0.273,0.14286,0.2857,0.60714,0.0,0.85714,2,0,0,2,0,12,0,0,3,0,0,3,0,0,4,0,0,5,0,0,3,0,0],[20,55,0.3636,0.57589,0.28456,0.39285,0.64286,0.85714,0.0,1.0,2,2,0,2,0,3,0,0,3,0,0,3,0,0,5,0,0,7,0,0,7,0,2],[24,55,0.4364,0.43747,0.25984,0.25,0.571,0.60714,0.0,0.85714,5,0,0,5,0,3,0,0,3,0,0,4,0,0,9,0,0,7,0,0,1,0,0],[28,55,0.5091,0.45082,0.27455,0.2857,0.42857,0.57143,0.0,1.0,2,3,0,2,0,5,0,0,6,0,0,6,0,0,6,0,0,3,0,0,1,0,3],[32,55,0.5818,0.491,0.28104,0.1429,0.571,0.71429,0.0,1.0,1,1,0,1,0,8,0,0,2,0,0,4,0,0,6,0,0,5,0,0,5,0,1],[36,55,0.6545,0.41959,0.29864,0.14286,0.571,0.71429,0.0,1.0,6,1,0,6,0,6,0,0,1,0,0,2,0,0,8,0,0,7,0,0,1,0,1],[40,55,0.7273,0.64049,0.19868,0.571,0.71429,0.71429,0.14,1.0,0,1,0,0,0,2,0,0,1,0,0,3,0,0,7,1,0,11,0,0,6,0,1],[44,55,0.8,0.54014,0.14165,0.42857,0.571,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,2,0,0,13,0,0,8,0,0,8,0,0,1,0,0],[48,55,0.8727,0.49102,0.11808,0.42857,0.4286,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,4,0,0,13,0,0,12,0,0,3,0,0,0,0,0],[52,55,0.9455,0.54462,0.11538,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,2,0,0,8,0,0,16,0,0,6,0,0,0,0,0],[55,55,1.0,0.55131,0.1201,0.42857,0.57143,0.60714,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,1,9,0,0,13,0,0,8,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.03572,"x":0.44639,"p":[[0,33,0.0,0.44639,0.29177,0.14286,0.42857,0.60714,0.0,1.0,3,3,0,3,0,6,0,0,4,0,0,5,0,0,6,0,0,4,0,0,1,0,3],[4,33,0.1212,0.38387,0.27063,0.14286,0.42857,0.57143,0.0,1.0,4,1,0,4,0,9,0,0,1,0,0,5,0,0,8,0,0,3,0,0,1,0,1],[8,33,0.2424,0.3124,0.32035,0.105,0.14286,0.57111,0.0,1.0,8,2,0,8,0,11,0,0,2,0,0,2,0,0,2,0,0,3,0,0,2,0,2],[12,33,0.3636,0.29911,0.32214,0.0,0.14286,0.60714,0.0,0.85714,12,0,0,12,0,6,0,0,3,0,0,1,0,0,2,0,0,4,0,0,4,0,0],[16,33,0.4848,0.2812,0.26836,0.10714,0.14286,0.571,0.0,0.85714,8,0,0,8,0,11,0,0,2,0,0,2,0,0,4,0,0,4,0,0,1,0,0],[20,33,0.6061,0.21874,0.23684,0.0,0.14286,0.32143,0.0,0.71429,11,0,0,11,0,10,0,0,3,0,0,2,0,0,3,0,0,3,0,0,0,0,0],[24,33,0.7273,0.09822,0.18708,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,4,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[28,33,0.8485,0.07134,0.09443,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[33,33,1.0,0.09357,0.11626,0.0,0.14,0.14286,0.0,0.57143,15,0,0,15,0,15,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"5fbd4f4ee056e083","q":"Let $n$ be a positive integer and let $S \\subseteq\\{0,1\\}^{n}$ be a set of binary strings of length $n$. Given an odd number $x_{1}, \\ldots, x_{2 k+1} \\in S$ of binary strings (not necessarily distinct), their majority is defined as the binary string $y \\in\\{0,1\\}^{n}$ for which the $i^{\\text {th }}$ bit of $y$ is the most common bit among the $i^{\\text {th }}$ bits of $x_{1}, \\ldots, x_{2 k+1}$. (For example, if $n=4$ the majority of $0000,0000,1101,1100,0101$ is 0100 .) Suppose that for some positive integer $k, S$ has the property $P_{k}$ that the majority of any $2 k+1$ binary strings in $S$ (possibly with repetition) is also in $S$. Prove that $S$ has the same property $P_{k}$ for all positive integers $k$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,19,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"434dcc6253d4d492","q":"Let $x, y$ and $z$ be positive real numbers such that $x y z=1$. Prove the inequality\n\n$$\n\\frac{1}{x(a y+b)}+\\frac{1}{y(a z+b)}+\\frac{1}{z(a x+b)} \\geq 3 \\text {, if: }\n$$\n\na) $a=0$ and $b=1$;\nb) $a=1$ and $b=0$;\nc) $a+b=1$ for $a, b>0$\n\nWhen does the equality hold true?","t":[{"b":0,"e":1.0,"k":"rising","v":0.69639,"x":1.0,"p":[[0,29,0.0,0.70976,0.2409,0.571,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,1,0,0,5,0,0,13,0,0,0,0,0,1,0,12],[4,29,0.1379,0.69639,0.26428,0.53539,0.57143,1.0,0.28571,1.0,0,12,0,0,0,0,0,0,4,0,0,4,0,0,9,0,0,2,0,0,1,0,12],[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],[29,29,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32]]},{"b":7,"e":1.0,"k":"rising","v":0.67408,"x":1.0,"p":[[0,11,0.0,0.70979,0.2612,0.571,0.64286,1.0,0.2857,1.0,0,12,0,0,0,0,0,0,4,0,0,3,0,0,9,0,0,2,0,0,2,0,12],[4,11,0.3636,0.67408,0.29718,0.53539,0.57143,1.0,0.0,1.0,1,13,1,1,0,0,0,0,5,0,0,2,0,0,11,0,0,0,0,0,0,0,13],[8,11,0.7273,0.68302,0.26423,0.4286,0.57143,1.0,0.2857,1.0,0,11,0,0,0,0,0,0,4,0,0,5,0,0,9,0,0,1,0,0,2,0,11],[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":"9d6079a4bc69f659","q":"There is three-dimensional space. For every integer  $n$  we build planes  $ x \\pm y\\pm z = n$ . All space is divided on octahedrons and tetrahedrons.\nPoint  $(x_0,y_0,z_0)$  has rational coordinates but not lies on any plane. Prove, that there is such natural    $k$  , that point  $(kx_0,ky_0,kz_0)$  lies strictly inside the octahedron of partition.","t":[{"b":4,"e":0.14286,"k":"flat","v":0.07589,"x":0.07589,"p":[[0,3,0.0,0.07589,0.13356,0.0,0.0,0.14286,0.0,0.42857,23,0,13,23,0,3,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.08928,"x":0.09822,"p":[[0,8,0.0,0.08928,0.17034,0.0,0.0,0.14286,0.0,0.85714,21,0,15,21,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,8,0.5,0.09822,0.13092,0.0,0.0,0.14287,0.0,0.4286,18,0,14,18,0,8,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c394747e6a2b3461","q":"What is the largest amount of elements that can be taken from the set  $\\{1, 2, ... , 2012, 2013\\}$ , such that within them there are no distinct three, say  $a$ ,  $b$ ,and  $c$ , such that  $a$  is a divisor or multiple of  $b-c$ ?","t":[{"b":0,"e":0.28571,"k":"flat","v":0.17858,"x":0.29017,"p":[[0,13,0.0,0.22768,0.12807,0.25,0.28571,0.28571,0.0,0.42857,7,0,4,7,0,1,0,0,22,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.18303,0.1439,0.0,0.28571,0.28571,0.0,0.42857,11,0,3,11,0,3,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.17858,0.14286,0.0,0.28571,0.28571,0.0,0.42857,11,0,5,11,0,4,0,0,15,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.28571,1e-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],[13,13,1.0,0.29017,0.02486,0.28571,0.28571,0.28571,0.2857,0.42857,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.01786,"x":0.22768,"p":[[0,29,0.0,0.16062,0.13717,0.0,0.2857,0.28571,0.0,0.28571,13,0,2,13,0,2,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.15625,0.13533,0.0,0.21428,0.28571,0.0,0.28571,13,0,4,13,0,3,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.14286,0.13363,0.0,0.14286,0.28571,0.0,0.28571,14,0,5,14,0,4,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.22768,0.12807,0.14286,0.28571,0.28571,0.0,0.57143,6,0,4,6,0,3,0,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,29,0.5517,0.14732,0.17307,0.0,0.0,0.28571,0.0,0.71429,17,0,12,17,0,0,0,0,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,29,0.6897,0.1875,0.13092,0.0,0.28571,0.28571,0.0,0.28571,10,0,6,10,0,2,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.10268,0.13474,0.0,0.0,0.28571,0.0,0.28571,20,0,11,20,0,1,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,11,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d4c9d3ec26bb6cad","q":"There are 20 students in a high school class, and each student has exactly three close friends in the class. Five of the students have bought tickets to an upcoming concert. If any student sees that at least two of their close friends have bought tickets, then they will buy a ticket too.\n\nIs it possible that the entire class buys tickets to the concert?\n\n(Assume that friendship is mutual; if student  $A$  is close friends with student  $B$ , then  $B$  is close friends with  $A$ .)","t":[{"b":1,"e":0.2857,"k":"falling","v":0.05804,"x":0.41518,"p":[[0,23,0.0,0.29464,0.37954,0.0,0.07143,0.71429,0.0,1.0,16,4,2,16,0,3,0,0,4,0,0,0,0,0,0,0,0,3,0,0,2,0,4],[4,23,0.1739,0.22768,0.3551,0.0,0.0,0.28571,0.0,1.0,20,4,3,20,0,1,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,4],[8,23,0.3478,0.3125,0.38538,0.0,0.14286,0.57143,0.0,1.0,14,6,1,14,0,4,0,0,5,0,0,0,0,0,2,0,0,0,0,0,1,0,6],[12,23,0.5217,0.41518,0.41398,0.0,0.28571,0.89286,0.0,1.0,12,8,1,12,0,2,0,0,4,0,0,2,0,0,0,0,0,3,0,0,1,0,8],[16,23,0.6957,0.40178,0.42474,0.0,0.28571,1.0,0.0,1.0,13,9,3,13,0,1,0,0,6,0,0,1,0,0,0,0,0,1,0,0,1,0,9],[20,23,0.8696,0.34821,0.42248,0.0,0.0,0.78571,0.0,1.0,17,8,3,17,0,0,0,0,3,0,0,1,0,0,2,0,0,1,0,0,0,0,8],[23,23,1.0,0.05804,0.17445,0.0,0.0,0.0,0.0,0.85714,28,0,0,28,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0]]},{"b":3,"e":0.0,"k":"falling","v":0.04018,"x":0.37946,"p":[[0,18,0.0,0.2632,0.36094,0.0,0.0,0.46418,0.0,1.0,17,4,5,17,0,3,0,0,3,0,0,1,0,0,2,0,0,1,0,0,1,0,4],[4,18,0.2222,0.37946,0.40502,0.0,0.28571,0.78571,0.0,1.0,13,8,1,13,0,0,0,0,8,0,0,1,0,0,0,0,0,2,0,0,0,0,8],[8,18,0.4444,0.16071,0.29827,0.0,0.0,0.2857,0.0,1.0,23,2,4,23,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0,0,0,2],[12,18,0.6667,0.30356,0.37071,0.0,0.07145,0.60714,0.0,1.0,16,4,2,16,0,1,0,0,5,0,0,0,0,0,2,0,0,3,0,0,1,0,4],[16,18,0.8889,0.13839,0.28456,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,2,0,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,1],[18,18,1.0,0.04018,0.17582,0.0,0.0,0.0,0.0,1.0,29,1,2,29,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]]}]},{"i":"426b7f0a3d8c3a8b","q":"The circumcircle of triangle  $ABC$  has centre  $O$ .  $P$  is the midpoint of  $\\widehat{BAC}$  and  $QP$  is the diameter. Let  $I$  be the incentre of  $\\triangle ABC$  and let  $D$  be the intersection of  $PI$  and  $BC$ . The circumcircle of  $\\triangle AID$  and the extension of  $PA$  meet at  $F$ . The point  $E$  lies on the line segment  $PD$  such that  $DE=DQ$ . Let  $R,r$  be the radius of the inscribed circle and circumcircle of  $\\triangle ABC$ , respectively. \nShow that if  $\\angle AEF=\\angle APE$ , then  $\\sin^2\\angle BAC=\\dfrac{2r}R$","t":[{"b":1,"e":0.28571,"k":"rising","v":0.01339,"x":0.20089,"p":[[0,16,0.0,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.20089,0.11214,0.14286,0.2857,0.28571,0.0,0.28571,6,0,0,6,0,7,0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.18303,0.1197,0.10714,0.2857,0.28571,0.0,0.28571,8,0,0,8,0,7,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,2,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"595e7c8fd16af2ec","q":"Let $H=\\left\\{\\lfloor i \\sqrt{2}\\rfloor: i \\in \\mathbb{Z}_{>0}\\right\\}=\\{1,2,4,5,7, \\ldots\\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \\subset\\{1,2, \\ldots, n\\}$ satisfies $|A| \\geqslant C \\sqrt{n}$, then there exist $a, b \\in A$ such that $a-b \\in H$.  (Brazil)","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,11,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,3,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.01339,"x":0.14286,"p":[[0,27,0.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,2,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.07134,0.07134,0.0,0.07,0.14286,0.0,0.14286,16,0,1,16,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.14286,2e-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":"397394ba3a69ad60","q":"We consider a prism which has the upper and inferior basis the pentagons:  $A_{1}A_{2}A_{3}A_{4}A_{5}$  and  $B_{1}B_{2}B_{3}B_{4}B_{5}$ . Each of the sides of the two pentagons and the segments  $A_{i}B_{j}$  with  $i,j=1,\\ldots$ ,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.","t":[{"b":0,"e":0.0,"k":"falling","v":0.04464,"x":0.45978,"p":[[0,20,0.0,0.31247,0.26348,0.0,0.35716,0.57143,0.0,0.57143,12,0,4,12,0,1,0,0,3,0,0,1,0,0,15,0,0,0,0,0,0,0,0],[4,20,0.2,0.36155,0.27425,0.0,0.571,0.57143,0.0,1.0,9,1,5,9,0,2,0,0,4,0,0,0,0,0,16,0,0,0,0,0,0,0,1],[8,20,0.4,0.45978,0.18806,0.39286,0.57143,0.57143,0.0,0.57143,3,0,1,3,0,1,0,0,4,0,0,2,0,0,22,0,0,0,0,0,0,0,0],[12,20,0.6,0.37054,0.23652,0.14286,0.57143,0.57143,0.0,0.57143,7,0,3,7,0,2,0,0,5,0,0,1,0,0,17,0,0,0,0,0,0,0,0],[16,20,0.8,0.45081,0.21457,0.42857,0.57141,0.57143,0.0,0.57143,5,0,0,5,0,1,0,0,1,0,0,2,0,0,23,0,0,0,0,0,0,0,0],[20,20,1.0,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.41067,"x":0.46427,"p":[[0,8,0.0,0.41067,0.24934,0.24999,0.57143,0.57143,0.0,0.85714,7,0,5,7,0,1,0,0,3,0,0,1,0,0,19,0,0,0,0,0,1,0,0],[4,8,0.5,0.46427,0.22014,0.4286,0.57143,0.57143,0.0,0.857,5,0,3,5,0,0,0,0,2,0,0,2,0,0,22,0,0,0,0,0,1,0,0]]}]},{"i":"0e4cfc06bfafbda2","q":"Let  $\\omega_1$  be the circumcircle of triangle  $ABC$  and  $O$  be its circumcenter. A circle  $\\omega_2$  touches the sides  $AB, AC$ , and touches the arc  $BC$  of  $\\omega_1$  at point  $K$ . Let  $I$  be the incenter of  $ABC$ .\nProve that the line  $OI$  contains the symmedian of triangle  $AIK$ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,36,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,3,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3ac64a67eafb3b24","q":"The internal angle bisectors of  $A$ ,  $B$ , and  $C$  in  $\\triangle ABC$  concur at  $I$  and intersect the circumcircle of  $\\triangle ABC$  at  $L$ ,  $M$ , and  $N$ , respectively. The circle with diameter  $IL$  intersects  $BC$  at  $D$  and  $E$ ; the circle with diameter  $IM$  intersects  $CA$  at  $F$  and  $G$ ; the circle with diameter  $IN$  intersects  $AB$  at  $H$  and  $J$ . Show that  $D$ ,  $E$ ,  $F$ ,  $G$ ,  $H$ , and  $J$  are concyclic.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.07589,"p":[[0,19,0.0,0.04911,0.11071,0.0,0.0,0.0,0.0,0.4286,25,0,9,25,0,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.07589,0.15146,0.0,0.0,0.03571,0.0,0.57143,24,0,1,24,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[8,19,0.4211,0.05804,0.12299,0.0,0.0,0.0,0.0,0.4286,25,0,5,25,0,3,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.03125,0.09268,0.0,0.0,0.0,0.0,0.42857,28,0,9,28,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0133,"x":0.09375,"p":[[0,19,0.0,0.09375,0.15407,0.0,0.0,0.14286,0.0,0.57143,21,0,6,21,0,5,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,19,0.2105,0.08482,0.17076,0.0,0.0,0.03571,0.0,0.71429,24,0,8,24,0,2,0,0,3,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[8,19,0.4211,0.0133,0.05465,0.0,0.0,0.0,0.0,0.2857,30,0,6,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.08929,0.14617,0.0,0.0,0.14286,0.0,0.4286,22,0,0,22,0,3,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"65ebea50110886b3","q":"Let $k$ be the circumcircle of $\\triangle ABC$, and $k_{a}$ the excircle opposite to vertex $A$. The two common tangents of circles $k$ and $k_{a}$ intersect the line $BC$ at points $P$ and $Q$. Prove that $\\varangle P A B = \\varangle Q A C$.\n\n(Du\u0161an \u0110uki\u0107)\n\n## SOLUTIONS","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00884,"p":[[0,32,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[20,32,0.625,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01786,"p":[[0,31,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"4ac81ece85599fbc","q":"Some of the towns in a country are connected with bidirectional paths, where each town can be reached by any other by going through these paths. From each town there are at least  $n \\geq 3$  paths. In the country there is no such route that includes all towns exactly once. Find the least possible number of towns in this country (Answer depends from  $n$ ).","t":[{"b":4,"e":0.2857,"k":"rising","v":0.36607,"x":0.99553,"p":[[0,32,0.0,0.36607,0.45308,0.0,0.0,1.0,0.0,1.0,18,10,0,18,0,0,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,10],[4,32,0.125,0.86607,0.2788,1.0,1.0,1.0,0.2857,1.0,0,26,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,26],[8,32,0.25,0.8616,0.27776,1.0,1.0,1.0,0.28571,1.0,0,25,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,25],[12,32,0.375,0.96428,0.14286,1.0,1.0,1.0,0.28571,1.0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,30],[16,32,0.5,0.98214,0.06916,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,30],[20,32,0.625,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,32,0.75,0.9866,0.05487,1.0,1.0,1.0,0.71429,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,30],[28,32,0.875,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[32,32,1.0,0.95089,0.1411,1.0,1.0,1.0,0.28571,1.0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,2,0,27]]},{"b":6,"e":1.0,"k":"rising","v":0.5,"x":1.0,"p":[[0,28,0.0,0.5,0.46566,0.0,0.35716,1.0,0.0,1.0,13,14,0,13,0,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,14],[4,28,0.1429,0.79018,0.36594,0.71429,1.0,1.0,0.0,1.0,4,23,2,4,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,23],[8,28,0.2857,0.97768,0.12428,1.0,1.0,1.0,0.28571,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[12,28,0.4286,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,28,0.5714,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,28,0.7143,0.97769,0.12423,1.0,1.0,1.0,0.286,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[24,28,0.8571,0.95536,0.1729,1.0,1.0,1.0,0.2857,1.0,0,30,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,30],[28,28,1.0,0.91071,0.23623,1.0,1.0,1.0,0.2857,1.0,0,28,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,28]]}]},{"i":"ddae3cbbc6be9bfe","q":"Prove that there exists a positive integer $n<10^{6}$ such that $5^{n}$ has six consecutive zeros in its decimal representation.","t":[{"b":0,"e":1.0,"k":"rising","v":0.49552,"x":0.97321,"p":[[0,10,0.0,0.49552,0.46151,0.0,0.49979,1.0,0.0,1.0,14,12,0,14,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,0,2,0,12],[4,10,0.4,0.57143,0.4475,0.0,0.85714,1.0,0.0,1.0,11,13,2,11,0,0,0,0,1,0,0,1,0,0,2,0,0,0,0,0,4,0,13],[8,10,0.8,0.95535,0.06622,0.85714,1.0,1.0,0.857,1.0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,22],[10,10,1.0,0.97321,0.05576,1.0,1.0,1.0,0.85714,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,26]]},{"b":7,"e":0.0,"k":"falling","v":0.02679,"x":0.54909,"p":[[0,31,0.0,0.54909,0.45332,0.0,0.71429,1.0,0.0,1.0,12,13,1,12,0,0,0,0,1,0,0,0,0,0,2,0,0,2,0,0,2,0,13],[4,31,0.129,0.44196,0.45926,0.0,0.21429,1.0,0.0,1.0,16,10,3,16,0,0,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,10],[8,31,0.2581,0.24107,0.41869,0.0,0.0,0.21429,0.0,1.0,24,6,4,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,6],[12,31,0.3871,0.36152,0.45459,0.0,0.0,1.0,0.0,1.0,18,9,1,18,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,9],[16,31,0.5161,0.35714,0.43006,0.0,0.0,0.85714,0.0,1.0,18,5,3,18,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,5],[20,31,0.6452,0.26786,0.39569,0.0,0.0,0.75,0.0,1.0,20,3,2,20,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,0,5,0,3],[24,31,0.7742,0.24106,0.38037,0.0,0.0,0.57111,0.0,1.0,21,4,0,21,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,2,0,4],[28,31,0.9032,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.13831,0.09094,0.14214,0.14286,0.14287,0.0,0.28571,7,0,1,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"72102abd876a432c","q":"We are given a collection of  $2^{2^k}$  coins, where  $k$  is a non-negative integer. Exactly one coin is fake.\nWe have an unlimited number of service dogs. One dog is sick but we do not know which one. \nA test consists of three steps: select some coins from the collection of all coins; choose a service dog; the dog smells all of the selected coins at once.  \nA healthy dog will bark if and only if the fake coin is amongst them. Whether the sick dog will bark or not is random. \nDevise a strategy to find the fake coin, using at most  $2^k+k+2$  tests, and prove that it works.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.09821,"p":[[0,8,0.0,0.09821,0.20025,0.0,0.0,0.17857,0.0,1.0,23,1,10,23,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.04912,"x":0.08929,"p":[[0,17,0.0,0.08036,0.19541,0.0,0.0,0.0,0.0,1.0,25,1,13,25,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,17,0.2353,0.08929,0.24936,0.0,0.0,0.0,0.0,1.0,27,2,16,27,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[8,17,0.4706,0.0625,0.18877,0.0,0.0,0.0,0.0,1.0,27,1,11,27,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,17,0.7059,0.07589,0.19556,0.0,0.0,0.0,0.0,1.0,26,1,17,26,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,17,0.9412,0.04912,0.10481,0.0,0.0,0.0,0.0,0.286,26,0,17,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"da8a6bc3e30c8d01","q":"Let  $n{}$  be a positive integer, and let  $\\mathcal{C}$  be a collection of subsets of  $\\{1,2,\\ldots,2^n\\}$  satisfying both of the following conditions:[list=1]\n[*]Every  $(2^n-1)$ -element subset of  $\\{1,2,\\ldots,2^n\\}$  is a member of  $\\mathcal{C}$ , and\n[*]Every non-empty member  $C$  of  $\\mathcal{C}$  contains an element  $c$  such that  $C\\setminus\\{c\\}$  is again a member of  $\\mathcal{C}$ .\n[/list]Determine the smallest size  $\\mathcal{C}$  may have.\n\n*Serbia, Pavle Martinovic \u0301*","t":[{"b":0,"e":0.0,"k":"flat","v":0.05357,"x":0.125,"p":[[0,11,0.0,0.05357,0.09943,0.0,0.0,0.03571,0.0,0.28571,24,0,11,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.09822,0.12078,0.0,0.0,0.1786,0.0,0.28571,18,0,11,18,0,6,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.125,0.15047,0.0,0.0,0.2857,0.0,0.4286,17,0,15,17,0,5,0,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01339,"x":0.08482,"p":[[0,15,0.0,0.08473,0.1509,0.0,0.0,0.14286,0.0,0.57143,22,0,12,22,0,5,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.08482,0.14223,0.0,0.0,0.14286,0.0,0.57143,22,0,12,22,0,3,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0625,0.14698,0.0,0.0,0.0,0.0,0.57143,26,0,24,26,0,2,0,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,15,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,25,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c93fee5a478483fb","q":"Let $a, b$ be integers such that $\\mathrm{pgcd}(\\mathrm{a}, \\mathrm{b})$ has at least two distinct prime factors. Let $S=\\{x \\in \\mathbb{N} \\mid x \\equiv a[b]\\}$. An element of $S$ is said to be irreducible if it cannot be written as a product of at least two elements of $S$ (not necessarily distinct).\nShow that there exists $\\mathrm{N}>0$ such that every element of $S$ can be written as a product of at most N irreducible elements of $S$ (not necessarily distinct).","t":[{"b":2,"e":0.14286,"k":"flat","v":0.07589,"x":0.11143,"p":[[0,11,0.0,0.07589,0.11837,0.0,0.0,0.14286,0.0,0.57143,19,0,7,19,0,11,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,11,0.3636,0.09813,0.09057,0.0,0.14286,0.14286,0.0,0.42857,12,0,3,12,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.11143,0.06894,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.1025,0.08162,0.0,0.14286,0.14286,0.0,0.2857,11,0,1,11,0,19,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.05348,"x":0.09366,"p":[[0,10,0.0,0.06232,0.07067,0.0,0.0,0.14286,0.0,0.1429,18,0,10,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,8,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.09366,0.07662,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.05348,0.06905,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"10fd2d8fa833e877","q":"Let $n$ be an integer greater than 1 , and let $a_{0}, a_{1}, \\ldots, a_{n}$ be real numbers with $a_{1}=a_{n-1}=0$. Prove that for any real number $k$,\n\n$$\n\\left|a_{0}\\right|-\\left|a_{n}\\right| \\leq \\sum_{i=0}^{n-2}\\left|a_{i}-k a_{i+1}-a_{i+2}\\right|\n$$","t":[{"b":6,"e":0.28571,"k":"rising","v":0.08472,"x":0.37946,"p":[[0,9,0.0,0.08472,0.178,0.0,0.0,0.035,0.0,0.71429,24,0,1,24,0,3,0,0,2,0,0,1,0,0,1,0,0,1,0,0,0,0,0],[4,9,0.4444,0.15615,0.23515,0.0,0.0,0.2857,0.0,0.71429,19,0,1,19,0,4,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0,0,0],[8,9,0.8889,0.37946,0.20705,0.28571,0.28571,0.57143,0.0,0.71429,2,0,0,2,0,3,0,0,14,0,0,4,0,0,3,0,0,6,0,0,0,0,0],[9,9,1.0,0.37498,0.19477,0.2857,0.28571,0.57111,0.0,0.71429,1,0,0,1,0,5,0,0,13,0,0,3,0,0,6,0,0,4,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.00446,"x":0.08929,"p":[[0,17,0.0,0.07143,0.19562,0.0,0.0,0.0,0.0,0.71429,27,0,0,27,0,2,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[4,17,0.2353,0.08929,0.19805,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[8,17,0.4706,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.08928,0.20746,0.0,0.0,0.0,0.0,0.857,25,0,2,25,0,2,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[16,17,0.9412,0.05357,0.14174,0.0,0.0,0.0,0.0,0.71429,26,0,1,26,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[17,17,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"dadb9593e241c2e6","q":"Let $n \\geq 3$ be an integer and let $1<a_{1} \\leq a_{2} \\leq a_{3} \\leq \\cdots \\leq a_{n}$ be $n$ real numbers such that $a_{1}+a_{2}+a_{3}+\\cdots+a_{n}=2 n$. Prove that\n\n$$\na_{1} a_{2} \\cdots a_{n-1}+a_{1} a_{2} \\cdots a_{n-2}+\\cdots+a_{1} a_{2}+a_{1}+2 \\leq a_{1} a_{2} \\cdots a_{n}\n$$","t":[{"b":1,"e":0.1429,"k":"flat","v":0.04911,"x":0.17856,"p":[[0,25,0.0,0.08928,0.22515,0.0,0.0,0.14286,0.0,1.0,23,1,0,23,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[4,25,0.16,0.17856,0.30928,0.0,0.0,0.14287,0.0,1.0,20,2,0,20,0,5,0,0,1,0,0,0,0,0,2,0,0,1,0,0,1,0,2],[8,25,0.32,0.09375,0.19759,0.0,0.0,0.14286,0.0,0.85714,22,0,0,22,0,7,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0],[12,25,0.48,0.09375,0.17717,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,3,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[16,25,0.64,0.09821,0.11538,0.0,0.07143,0.14286,0.0,0.42857,16,0,0,16,0,11,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.12946,0.16506,0.0,0.14286,0.14286,0.0,0.85714,13,0,0,13,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[24,25,0.96,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.09375,0.11071,0.0,0.07143,0.14286,0.0,0.4286,16,0,0,16,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,15,0.0,0.05804,0.1984,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,15,0.2667,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"cf4f97a2228d5b52","q":"Two circles in the plane,  $\\gamma_1$  and  $\\gamma_2$ , meet at points  $M$  and  $N$ . Let  $A$  be a point on  $\\gamma_1$ , and let  $D$  be a point on  $\\gamma_2$ . The lines  $AM$  and  $AN$  meet again  $\\gamma_2$  at points  $B$  and  $C$ , respectively, and the lines  $DM$  and  $DN$  meet again  $\\gamma_1$  at points  $E$  and  $F$ , respectively. Assume the order  $M$ ,  $N$ ,  $F$ ,  $A$ ,  $E$  is circular around  $\\gamma_1$ , and the segments  $AB$  and  $DE$  are congruent. Prove that the points  $A$ ,  $F$ ,  $C$  and  $D$  lie on a circle whose centre does not depend on the position of the points  $A$  and  $D$  on the respective circles, subject to the assumptions above. \n\n*****","t":[{"b":1,"e":0.0,"k":"flat","v":0.00661,"x":0.02679,"p":[[0,18,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.02232,0.04871,0.0,0.0,0.0,0.0,0.1429,26,0,5,26,2,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.00661,0.02699,0.0,0.0,0.0,0.0,0.14,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.42857,"k":"flat","v":0.01107,"x":0.07134,"p":[[0,12,0.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.01107,0.03588,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.07134,0.09443,0.0,0.0,0.14286,0.0,0.28571,19,0,1,19,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.03348,0.06913,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,1,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6801381dc5d0f5b9","q":"Let  $\\sigma (n)$  shows the number of positive divisors of  $n$ . Let  $s(n)$  be the number of positive divisors of  $n+1$  such that for every divisor  $a$ ,  $a-1$  is also a divisor of  $n$ . Find the maximum value of  $2s(n)- \\sigma (n) $ .","t":[{"b":1,"e":0.2857,"k":"flat","v":0.25,"x":0.30803,"p":[[0,28,0.0,0.30357,0.28065,0.14286,0.28571,0.28571,0.0,1.0,4,4,0,4,0,8,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,28,0.1429,0.25446,0.13709,0.14286,0.28571,0.28571,0.0,0.85714,1,0,0,1,0,10,0,0,19,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[8,28,0.2857,0.28554,0.24233,0.14286,0.14286,0.28571,0.14,1.0,0,2,0,0,0,17,0,0,11,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[12,28,0.4286,0.29016,0.18721,0.14286,0.28571,0.28571,0.14286,1.0,0,1,0,0,0,10,0,0,19,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[16,28,0.5714,0.29902,0.22412,0.14286,0.2857,0.28571,0.14,1.0,0,2,0,0,0,12,0,0,16,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[20,28,0.7143,0.30803,0.18935,0.2857,0.28571,0.28571,0.14286,1.0,0,2,0,0,0,6,0,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[24,28,0.8571,0.25,0.10714,0.14286,0.28571,0.28571,0.14286,0.71429,0,0,0,0,0,11,0,0,20,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,28,1.0,0.28125,0.04351,0.2857,0.28571,0.28571,0.14286,0.42857,0,0,0,0,0,2,0,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.24107,"x":0.26785,"p":[[0,8,0.0,0.24107,0.22711,0.14286,0.2857,0.28571,0.0,1.0,7,2,0,7,0,6,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,8,0.5,0.24982,0.06217,0.25,0.2857,0.28571,0.14,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"48b52dec5bea15e1","q":"Let's expand a little bit three circles, touching each other externally, so that three pairs of intersection points appear. Denote by $A_{1}, B_{1}, C_{1}$ the three so obtained \"external\" points and by $A_{2}, B_{2}, C_{2}$ the corresponding \"internal\" points. Prove the equality\n\n$$\n\\left|A_{1} B_{2}\\right| \\cdot\\left|B_{1} C_{2}\\right| \\cdot\\left|C_{1} A_{2}\\right|=\\left|A_{1} C_{2}\\right| \\cdot\\left|C_{1} B_{2}\\right| \\cdot\\left|B_{1} A_{2}\\right| .\n$$","t":[{"b":5,"e":1.0,"k":"volatile","v":0.55804,"x":0.9866,"p":[[0,6,0.0,0.55804,0.43354,0.0,0.71429,1.0,0.0,1.0,9,13,0,9,0,2,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,13],[4,6,0.6667,0.95089,0.13651,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,4,0,26],[6,6,1.0,0.9866,0.04165,1.0,1.0,1.0,0.857,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29]]},{"b":6,"e":0.14286,"k":"falling","v":0.25,"x":0.63839,"p":[[0,13,0.0,0.63839,0.40404,0.28571,0.85714,1.0,0.0,1.0,5,15,0,5,0,2,0,0,5,0,0,0,0,0,1,0,0,2,0,0,2,0,15],[4,13,0.3077,0.59375,0.40107,0.24999,0.71429,1.0,0.0,1.0,7,12,0,7,0,1,0,0,3,0,0,1,0,0,3,0,0,3,0,0,2,0,12],[8,13,0.6154,0.25893,0.11538,0.14286,0.2857,0.28571,0.0,0.71429,1,0,0,1,0,8,0,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,13,0.9231,0.25,0.06185,0.25,0.2857,0.28571,0.14286,0.28571,0,0,0,0,0,8,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.30804,0.08828,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,2,0,0,25,0,0,3,0,0,2,0,0,0,0,0,0,0,0]]}]},{"i":"e4ea02ada1346224","q":"Let $n$ be a positive integer that is coprime to 6. We color the vertices of a regular $n$-gon with three colors such that for each color, the number of vertices colored with it is odd.\nProve that there always exists an isosceles triangle whose vertices belong to the vertices of the $n$-gon and are all colored differently.","t":[{"b":1,"e":0.0,"k":"flat","v":0.00893,"x":0.06692,"p":[[0,13,0.0,0.04455,0.08318,0.0,0.0,0.035,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.06692,0.11294,0.0,0.0,0.14286,0.0,0.43,21,0,0,21,0,9,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.0134,0.04165,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.05802,"p":[[0,5,0.0,0.05802,0.11763,0.0,0.0,0.14286,0.0,0.571,23,0,0,23,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"352c51ac22237a4b","q":"Two points  $A$  and  $B$  are given in the plane. A point  $X$  is called their  *preposterous midpoint* if there is a Cartesian coordinate system in the plane such that the coordinates of  $A$  and  $B$  in this system are non-negative, the abscissa of  $X$  is the geometric mean of the abscissae of  $A$  and  $B$ , and the ordinate of  $X$  is the geometric mean of the ordinates  of  $A$  and  $B$ . Find the locus of all the *preposterous midpoints* of  $A$  and  $B$ . \n\n(K. Tyschu)","t":[{"b":6,"e":0.0,"k":"flat","v":0.19187,"x":0.24107,"p":[[0,12,0.0,0.22321,0.16728,0.14286,0.14286,0.28571,0.0,0.71429,4,0,1,4,0,14,0,0,10,0,0,2,0,0,0,0,0,2,0,0,0,0,0],[4,12,0.3333,0.19187,0.11075,0.14286,0.14288,0.28571,0.0,0.42857,5,0,3,5,0,12,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.24107,0.19377,0.14286,0.2857,0.28571,0.0,0.71429,7,0,4,7,0,7,0,0,13,0,0,1,0,0,2,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.17857,"x":0.20517,"p":[[0,18,0.0,0.19186,0.14109,0.105,0.2143,0.2857,0.0,0.571,8,0,4,8,0,8,0,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,18,0.2222,0.20517,0.17108,0.14,0.1429,0.2857,0.0,0.71429,7,0,4,7,0,12,0,0,8,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[8,18,0.4444,0.19187,0.1499,0.14214,0.14286,0.28571,0.0,0.57143,7,0,6,7,0,13,0,0,7,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,18,0.6667,0.20088,0.16695,0.14286,0.14286,0.2857,0.0,0.71429,6,0,3,6,0,15,0,0,6,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[16,18,0.8889,0.17857,0.10714,0.14286,0.14288,0.2857,0.0,0.28571,6,0,3,6,0,12,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.18304,0.08917,0.14286,0.14288,0.2857,0.0,0.28571,3,0,2,3,0,17,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"916ef0250694ecc8","q":"Let $x, y$ and $z$ be positive real numbers such that $\\sqrt{x}+\\sqrt{y}+\\sqrt{z}=1$. Prove that\n\n$$\n\\frac{x^{2}+y z}{\\sqrt{2 x^{2}(y+z)}}+\\frac{y^{2}+z x}{\\sqrt{2 y^{2}(z+x)}}+\\frac{z^{2}+x y}{\\sqrt{2 z^{2}(x+y)}} \\geq 1\n$$","t":[{"b":3,"e":0.0,"k":"flat","v":0.01339,"x":0.16964,"p":[[0,10,0.0,0.09374,0.20702,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0],[4,10,0.4,0.1339,0.26467,0.0,0.0,0.0,0.0,1.0,25,1,0,25,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,1],[8,10,0.8,0.16964,0.30397,0.0,0.0,0.14286,0.0,1.0,22,2,0,22,0,3,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,2],[10,10,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.125,"p":[[0,13,0.0,0.08929,0.23077,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1],[4,13,0.3077,0.10714,0.25,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,1,0,0,0,0,0,1,0,0,2,0,0,1,0,0,0,0,1],[8,13,0.6154,0.125,0.3004,0.0,0.0,0.0,0.0,1.0,27,2,0,27,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,2],[12,13,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"227351a7f589eaec","q":"Prove that for any four positive real numbers $a, b, c, d$ the inequality  $$ \\frac{(a-b)(a-c)}{a+b+c}+\\frac{(b-c)(b-d)}{b+c+d}+\\frac{(c-d)(c-a)}{c+d+a}+\\frac{(d-a)(d-b)}{d+a+b} \\geq 0 $$  holds. Determine all cases of equality.","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.03572,"p":[[0,9,0.0,0.03116,0.06888,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.03572,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.00893,"x":0.03125,"p":[[0,15,0.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,1,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c55d3f8bdbff996b","q":"Let $a$ be a positive integer which is not a square number. Denote by $A$ the set of all positive integers $k$ such that  $$ k=\\frac{x^{2}-a}{x^{2}-y^{2}} $$  for some integers $x$ and $y$ with $x>\\sqrt{a}$. Denote by $B$ the set of all positive integers $k$ such that (1) is satisfied for some integers $x$ and $y$ with $0 \\leqslant x<\\sqrt{a}$. Prove that $A=B$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.00223,"x":0.0133,"p":[[0,9,0.0,0.00223,0.01243,0.0,0.0,0.0,0.0,0.07143,31,0,1,31,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,33,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,33,0.4848,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"1b5d90c128fa1919","q":"Let $m_{1}, m_{2}, \\ldots, m_{n}$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A=\\left(a_{1}, \\ldots, a_{n}\\right)$ and any permutation $w=w_{1}, \\ldots, w_{n}$ of $m_{1}, \\ldots, m_{n}$, define an $A$-inversion of $w$ to be a pair of entries $w_{i}, w_{j}$ with $i<j$ for which one of the following conditions holds:  - $a_{i} \\geq w_{i}>w_{j}$, - $w_{j}>a_{i} \\geq w_{i}$, or - $w_{i}>w_{j}>a_{i}$.  Show that, for any two sequences of integers $A=\\left(a_{1}, \\ldots, a_{n}\\right)$ and $B=\\left(b_{1}, \\ldots, b_{n}\\right)$, and for any positive integer $k$, the number of permutations of $m_{1}, \\ldots, m_{n}$ having exactly $k A$-inversions is equal to the number of permutations of $m_{1}, \\ldots, m_{n}$ having exactly $k B$-inversions.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.09374,"x":0.125,"p":[[0,2,0.0,0.09374,0.12164,0.0,0.07143,0.14286,0.0,0.571,16,0,0,16,0,13,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[2,2,1.0,0.125,0.09279,0.10714,0.14286,0.14286,0.0,0.42857,8,0,0,8,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.125,"p":[[0,48,0.0,0.08482,0.1063,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,48,0.0833,0.08036,0.10062,0.0,0.0,0.14286,0.0,0.42857,17,0,0,17,0,13,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,48,0.1667,0.125,0.1915,0.0,0.14286,0.14286,0.0,1.0,15,1,0,15,0,12,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,48,0.25,0.08482,0.11214,0.0,0.0,0.14286,0.0,0.28571,19,0,1,19,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,48,0.3333,0.08036,0.09407,0.0,0.0,0.14286,0.0,0.2857,17,0,1,17,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.0625,0.10062,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,48,0.5,0.06696,0.13356,0.0,0.0,0.14286,0.0,0.57143,23,0,0,23,0,6,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,48,0.5833,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,48,0.6667,0.04911,0.09852,0.0,0.0,0.03571,0.0,0.42857,24,0,0,24,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,48,0.8333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.02223,0.05167,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f6df242dbafa24bf","q":"Show that for any positive real numbers  $a, b, c$  such that  $a + b + c = ab + bc + ca$ ,\nthe following inequality holds $3 + \\sqrt[3]{\\frac{a^3+1}{2}}+\\sqrt[3]{\\frac{b^3+1}{2}}+\\sqrt[3]{\\frac{c^3+1}{2}}\\leq 2(a+b+c)$ *Proposed by Dorlir Ahmeti, Albania*","t":[{"b":0,"e":0.14286,"k":"flat","v":0.0625,"x":0.13821,"p":[[0,6,0.0,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.143,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.10714,"p":[[0,8,0.0,0.10714,0.17496,0.0,0.14286,0.14286,0.0,1.0,14,1,0,14,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,8,0.5,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"09f82ed9de063a3d","q":"Let $a_{0}$ and $a_{n}$ be distinct divisors of a natural number $m>1$, and the sequence of natural numbers $a_{0}, a_{1}, a_{2}, \\ldots, a_{n}$ satisfies\n\n$$\na_{i+1}=\\left|a_{i} \\pm a_{i-1}\\right| \\quad \\text { for } 0<i<n\n$$\n\nIf the GCD $\\left(a_{0}, \\ldots, a_{n}\\right)=1$, prove that there is a term in the sequence that is less than $\\sqrt{m}$.\n\n(Dushan Dukic)\n\n## SOLUTIONS","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.05358,"p":[[0,36,0.0,0.03572,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,1,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.03572,0.07144,0.0,0.0,0.0,0.0,0.28571,25,0,4,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.05358,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,3,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,1,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,2,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,36,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.04009,"x":0.05357,"p":[[0,5,0.0,0.04009,0.07337,0.0,0.0,0.035,0.0,0.2857,24,0,2,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"492b589e979551cf","q":"On sides  $BC$  and  $AC$  of  $\\triangle ABC$  given are  $D$  and  $E$ , respectively. Let  $F$  ( $F \\neq C$ ) be a point of intersection of circumcircle of  $\\triangle CED$  and line that is parallel to  $AB$  and passing through C. Let  $G$  be a point of intersection of line  $FD$  and side  $AB$ , and let  $H$  be on line  $AB$  such that  $\\angle HDA = \\angle GEB$  and  $H-A-B$ . If  $DG=EH$ , prove that point of intersection of  $AD$  and  $BE$  lie on angle bisector of  $\\angle ACB$ .\n\n*Proposed by Milos Milosavljevic*","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.06473,"p":[[0,32,0.0,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,32,0.125,0.03571,0.06186,0.0,0.0,0.03571,0.0,0.14286,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.06473,0.10618,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,1,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,2,0.0,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,2,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"705fa29ea57db81f","q":"Let $a, b$ and $c$ be positive integers satisfying the equation $(a, b)+[a, b]=2021^{c}$. If $|a-b|$ is a prime number, prove that the number $(a+b)^{2}+4$ is composite.\n\n## Proposed by Serbia","t":[{"b":3,"e":0.57143,"k":"flat","v":0.1383,"x":0.42415,"p":[[0,55,0.0,0.3614,0.2601,0.14286,0.28571,0.4642,0.0,1.0,4,2,2,4,0,6,0,0,7,0,0,7,0,0,5,0,0,0,0,0,1,0,2],[4,55,0.0727,0.42408,0.2157,0.28571,0.42857,0.57111,0.0,1.0,1,2,0,1,0,5,0,0,4,0,0,12,0,0,8,0,0,0,0,0,0,0,2],[8,55,0.1455,0.33925,0.27137,0.10714,0.28571,0.57111,0.0,1.0,8,1,6,8,0,4,0,0,5,0,0,4,0,0,7,0,0,3,0,0,0,0,1],[12,55,0.2182,0.24106,0.23536,0.0,0.14286,0.4642,0.0,0.71429,11,0,9,11,0,7,0,0,4,0,0,2,0,0,7,0,0,1,0,0,0,0,0],[16,55,0.2909,0.25881,0.26352,0.0,0.14286,0.4286,0.0,1.0,10,1,9,10,0,8,0,0,4,0,0,3,0,0,4,0,0,2,0,0,0,0,1],[20,55,0.3636,0.14277,0.18211,0.0,0.14286,0.1786,0.0,0.85714,14,0,12,14,0,10,0,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[24,55,0.4364,0.24991,0.18216,0.0,0.28571,0.42857,0.0,0.5714,9,0,8,9,0,3,0,0,8,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[28,55,0.5091,0.14732,0.19719,0.0,0.0,0.28571,0.0,0.57143,18,0,15,18,0,4,0,0,4,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[32,55,0.5818,0.1383,0.16554,0.0,0.0,0.2857,0.0,0.42857,17,0,15,17,0,4,0,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[36,55,0.6545,0.2232,0.20495,0.0,0.21429,0.42857,0.0,0.571,13,0,12,13,0,3,0,0,2,0,0,13,0,0,1,0,0,0,0,0,0,0,0],[40,55,0.7273,0.19648,0.18819,0.0,0.14286,0.32143,0.0,0.57143,10,0,8,10,0,11,0,0,3,0,0,5,0,0,3,0,0,0,0,0,0,0,0],[44,55,0.8,0.33036,0.13092,0.28571,0.35714,0.42857,0.0,0.57143,2,0,1,2,0,3,0,0,11,0,0,15,0,0,1,0,0,0,0,0,0,0,0],[48,55,0.8727,0.42415,0.16554,0.42857,0.42859,0.57143,0.0,0.57143,3,0,3,3,0,0,0,0,4,0,0,13,0,0,12,0,0,0,0,0,0,0,0],[52,55,0.9455,0.23661,0.23038,0.0,0.28571,0.42857,0.0,0.71429,14,0,14,14,0,1,0,0,4,0,0,9,0,0,3,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"rising","v":0.31241,"x":0.59369,"p":[[0,26,0.0,0.31241,0.16926,0.14286,0.28571,0.42857,0.0,0.57143,3,0,2,3,0,7,0,0,7,0,0,11,0,0,4,0,0,0,0,0,0,0,0],[4,26,0.1538,0.32142,0.17855,0.25,0.28571,0.42857,0.0,0.57143,4,0,2,4,0,4,0,0,10,0,0,8,0,0,6,0,0,0,0,0,0,0,0],[8,26,0.3077,0.40166,0.20033,0.28571,0.42857,0.57111,0.0,0.71429,4,0,1,4,0,2,0,0,3,0,0,12,0,0,9,0,0,2,0,0,0,0,0],[12,26,0.4615,0.5714,0.06186,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,2,0,0,29,0,0,0,0,0,1,0,0],[16,26,0.6154,0.5758,0.02487,0.57142,0.57143,0.57143,0.571,0.71429,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31,0,0,1,0,0,0,0,0],[20,26,0.7692,0.58481,0.08268,0.57143,0.57143,0.57143,0.4286,1.0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,29,0,0,1,0,0,0,0,1],[24,26,0.9231,0.56695,0.14054,0.57143,0.57143,0.57143,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,2,0,0,27,0,0,0,0,0,1,0,1],[26,26,1.0,0.59369,0.08829,0.57143,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,3,0,0]]}]},{"i":"ba14151ab3414f78","q":"Prove that there exists a real constant  $c$  such that for any pair  $(x,y)$  of real numbers, there exist relatively prime integers  $m$  and  $n$  satisfying the relation\n\\[\n\\sqrt{(x-m)^2 + (y-n)^2} < c\\log (x^2 + y^2 + 2).\n\\]","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,25,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[25,25,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,8,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"46c275edc4f8c429","q":"Prove The Following identity: $ \\sum_{j \\equal{} 0}^n \\left ({3n \\plus{} 2 \\minus{} j \\choose j}2^j \\minus{} {3n \\plus{} 1 \\minus{} j \\choose j \\minus{} 1}2^{j \\minus{} 1}\\right ) \\equal{} 2^{3n}$ .\nThe Second term on left hand side is to be regarded zero for j=0.","t":[{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"243f015162c5b15e","q":"Prove that the edges of a simple planar graph can always be oriented such that the outdegree of all vertices is at most three.\n\n*UK Competition Problem*","t":[{"b":3,"e":1.0,"k":"rising","v":0.84821,"x":1.0,"p":[[0,39,0.0,0.84821,0.31326,1.0,1.0,1.0,0.0,1.0,3,25,0,3,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,25],[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],[8,39,0.2051,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[12,39,0.3077,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[16,39,0.4103,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[20,39,0.5128,0.99107,0.0346,1.0,1.0,1.0,0.857,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[24,39,0.6154,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[28,39,0.7179,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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]]},{"b":7,"e":0.0,"k":"falling","v":0.16962,"x":0.99107,"p":[[0,40,0.0,0.85713,0.30515,0.96429,1.0,1.0,0.0,1.0,3,24,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,24],[4,40,0.1,0.8125,0.35614,0.92857,1.0,1.0,0.0,1.0,4,24,0,4,0,1,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,24],[8,40,0.2,0.99107,0.04971,1.0,1.0,1.0,0.71429,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,31],[12,40,0.3,0.91071,0.27837,1.0,1.0,1.0,0.0,1.0,2,29,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29],[16,40,0.4,0.92411,0.2448,1.0,1.0,1.0,0.0,1.0,2,28,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[20,40,0.5,0.69643,0.43558,0.10714,1.0,1.0,0.0,1.0,8,20,0,8,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,20],[24,40,0.6,0.69643,0.42969,0.24999,1.0,1.0,0.0,1.0,7,20,1,7,0,1,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,20],[28,40,0.7,0.37045,0.47231,0.0,0.0,1.0,0.0,1.0,19,11,0,19,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,11],[32,40,0.8,0.16962,0.30393,0.0,0.0,0.10714,0.0,1.0,24,1,0,24,0,0,0,0,0,0,0,1,0,0,2,0,0,4,0,0,0,0,1],[36,40,0.9,0.17857,0.26486,0.0,0.0,0.32143,0.0,0.71429,20,0,0,20,0,2,0,0,2,0,0,2,0,0,2,0,0,4,0,0,0,0,0],[40,40,1.0,0.29464,0.36058,0.0,0.0,0.71429,0.0,1.0,18,1,0,18,0,0,0,0,2,0,0,1,0,0,1,0,0,6,0,0,3,0,1]]}]},{"i":"e2946e9198e48cd6","q":"We are given a positive integer  $ r$  and a rectangular board  $ ABCD$  with dimensions  $ AB \\equal{} 20, BC \\equal{} 12$ . The rectangle is divided into a grid of  $ 20 \\times 12$  unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is  $ \\sqrt {r}$ . The task is to find a sequence of moves leading from the square with  $ A$  as a vertex to the square with  $ B$  as a vertex.\r\n\r\n(a) Show that the task cannot be done if  $ r$  is divisible by 2 or 3.\r\n\r\n(b) Prove that the task is possible when  $ r \\equal{} 73$ . \r\n\r\n(c) Can the task be done when  $ r \\equal{} 97$ ?","t":[{"b":0,"e":0.0,"k":"falling","v":0.25445,"x":0.48659,"p":[[0,14,0.0,0.48659,0.24964,0.42857,0.57143,0.57143,0.0,0.85714,6,0,6,6,0,0,0,0,0,0,0,3,0,0,16,0,0,6,0,0,1,0,0],[4,14,0.2857,0.4107,0.24678,0.32143,0.57143,0.57143,0.0,0.71429,8,0,8,8,0,0,0,0,0,0,0,6,0,0,16,0,0,2,0,0,0,0,0],[8,14,0.5714,0.47768,0.16602,0.42857,0.57143,0.57143,0.0,0.57143,3,0,3,3,0,0,0,0,0,0,0,9,0,0,20,0,0,0,0,0,0,0,0],[12,14,0.8571,0.25445,0.28286,0.0,0.0,0.57143,0.0,0.85714,17,0,17,17,0,0,0,0,1,0,0,3,0,0,10,0,0,0,0,0,1,0,0]]},{"b":6,"e":0.57143,"k":"flat","v":0.46432,"x":0.52679,"p":[[0,14,0.0,0.46432,0.26963,0.42857,0.57143,0.57143,0.0,1.0,7,1,7,7,0,0,0,0,0,0,0,5,0,0,13,0,0,6,0,0,0,0,1],[4,14,0.2857,0.52679,0.06621,0.42857,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,10,0,0,22,0,0,0,0,0,0,0,0],[8,14,0.5714,0.48661,0.14223,0.42857,0.57143,0.57143,0.0,0.57143,2,0,2,2,0,0,0,0,0,0,0,11,0,0,19,0,0,0,0,0,0,0,0]]}]},{"i":"2bea53290b2733c0","q":"MNG   Let $a$ and $b$ be distinct integers greater than 1 . Prove that there exists a positive integer $n$ such that $\\left(a^{n}-1\\right)\\left(b^{n}-1\\right)$ is not a perfect square.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"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],[4,32,0.125,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[12,32,0.375,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,32,1.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8890a737a9386aca","q":"Prove that for any positive integer  $n$ , the least common multiple of the numbers  $1,2,\\ldots,n$  and the least common multiple of the numbers: \\[\\binom{n}{1},\\binom{n}{2},\\ldots,\\binom{n}{n}\\] are equal if and only if  $n+1$  is a prime number. \r\n\r\n*Laurentiu Panaitopol*","t":[{"b":1,"e":1.0,"k":"rising","v":0.07144,"x":0.45086,"p":[[0,18,0.0,0.14731,0.20353,0.0,0.07143,0.17868,0.0,0.71429,16,0,0,16,0,8,0,0,4,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[4,18,0.2222,0.24326,0.30135,0.0,0.14286,0.28571,0.0,1.0,11,2,0,11,0,10,0,1,3,0,0,1,0,0,2,0,0,0,0,0,2,0,2],[8,18,0.4444,0.15624,0.15299,0.0,0.14286,0.2857,0.0,0.571,11,0,0,11,0,12,0,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[12,18,0.6667,0.07144,0.17858,0.0,0.0,0.14286,0.0,1.0,22,1,1,22,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[16,18,0.8889,0.42855,0.29013,0.2857,0.28571,0.571,0.14286,1.0,0,5,0,0,0,5,0,0,17,0,0,1,0,0,2,0,0,1,0,0,1,0,5],[18,18,1.0,0.45086,0.29255,0.2857,0.28571,0.57111,0.14286,1.0,0,5,0,0,0,5,0,0,14,0,0,3,0,0,3,0,0,0,0,0,2,0,5]]},{"b":6,"e":0.71429,"k":"rising","v":0.15625,"x":0.46856,"p":[[0,27,0.0,0.16069,0.16262,0.0,0.14286,0.2857,0.0,0.571,12,0,0,12,0,9,0,0,8,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[4,27,0.1481,0.35266,0.38295,0.0,0.14288,0.71429,0.0,1.0,10,5,0,10,0,9,0,0,2,0,0,0,0,0,1,0,0,3,0,0,2,0,5],[8,27,0.2963,0.18286,0.24023,0.0,0.14286,0.2857,0.0,1.0,11,2,0,11,0,12,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[12,27,0.4444,0.18973,0.25481,0.0,0.14286,0.2857,0.0,1.0,11,2,0,11,1,11,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[16,27,0.5926,0.15625,0.20628,0.0,0.14286,0.1429,0.0,0.85714,12,0,0,12,0,13,0,0,5,0,0,0,0,0,0,0,0,0,0,0,2,0,0],[20,27,0.7407,0.39953,0.24602,0.25,0.28571,0.58929,0.14286,0.85714,0,0,0,0,0,8,0,0,12,0,0,2,0,0,2,1,0,3,0,0,4,0,0],[24,27,0.8889,0.46856,0.30569,0.14286,0.571,0.71407,0.0,1.0,2,3,0,2,0,8,0,0,4,0,0,1,0,0,8,0,0,3,0,0,3,0,3],[27,27,1.0,0.38835,0.25309,0.14286,0.35714,0.4642,0.14286,1.0,0,2,0,0,0,11,0,0,5,0,0,8,0,0,4,0,0,0,0,0,2,0,2]]}]},{"i":"c540d46e09dc6562","q":"On the hypotenuse  $AB$  of the right-angled triangle  $ABC$ , a point  $K$  is chosen such that  $BK = BC$ . Let  $P$  be a point on the perpendicular line from point  $K$  to the line  $CK$ , equidistant from the points  $K$  and  $B$ . Also let  $L$  denote the midpoint of the segment  $CK$ . Prove that line  $AP$  is tangent to the circumcircle of the triangle  $BLP$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,23,0.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,27,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[12,27,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.02223,0.07225,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bf411ac33728008b","q":"Show that the sum of the digits of $2^{2^{2 \\cdot 2023}}$ is greater than 2023 .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,21,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"257c982b8cf31232","q":"Let  $f$  be a function defined on the positive integers with  $f(n) \\ge 0$  and  $f(n) \\le f(n+1)$  for all  $n$ . Prove that if\n\\[\\sum_{n = 1}^{\\infty} \\frac{f(n)}{n^2}\\]\ndiverges, there exists a sequence  $a_1, a_2, \\dots$  such that the sequence  $\\tfrac{a_n}{n}$  hits every natural number, while\n\\[a_{n+m} \\le a_n + a_m + f(n+m)\\]\nholds for every pair  $n$ ,  $m$ .","t":[{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.12054,"p":[[0,6,0.0,0.12054,0.2055,0.0,0.0,0.14286,0.0,0.71429,21,0,10,21,0,4,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,0],[4,6,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.02679,0.09061,0.0,0.0,0.0,0.0,0.42857,29,0,1,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.02232,"x":0.05124,"p":[[0,7,0.0,0.03124,0.10548,0.0,0.0,0.0,0.0,0.571,28,0,10,28,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.05124,0.12645,0.0,0.0,0.0,0.0,0.571,25,0,17,25,1,4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[7,7,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,8,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7e0a84f78b1d6492","q":"We consider the sequences strictely increasing  $(a_0,a_1,...)$  of naturals which have the following property : \r\nFor every natural  $n$ , there is exactly one representation of  $n$  as  $a_i+2a_j+4a_k$ , where  $i,j,k$  can be equal.\r\n\r\nProve that there is exactly a such sequence and find  $a_{2002}$","t":[{"b":5,"e":0.14286,"k":"flat","v":0.12045,"x":0.34821,"p":[[0,13,0.0,0.29455,0.35707,0.0,0.14286,0.57143,0.0,1.0,13,3,12,13,0,7,0,0,2,0,0,1,0,0,2,0,0,1,0,0,3,0,3],[4,13,0.3077,0.34821,0.35344,0.14286,0.14286,0.71429,0.0,1.0,5,4,4,5,0,15,0,0,2,0,0,1,0,0,0,0,0,2,0,0,3,0,4],[8,13,0.6154,0.25447,0.33261,0.0,0.14286,0.32143,0.0,1.0,14,3,14,14,0,6,0,0,4,0,0,2,0,0,1,0,0,0,0,0,2,0,3],[12,13,0.9231,0.12045,0.07237,0.14214,0.14286,0.14286,0.0,0.28571,7,0,2,7,0,23,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.14723,0.06667,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,25,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.17857,"x":0.17857,"p":[[0,6,0.0,0.17857,0.29014,0.0,0.07143,0.14286,0.0,1.0,16,2,14,16,0,10,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,2],[4,6,0.6667,0.17857,0.30723,0.0,0.07143,0.14286,0.0,1.0,16,3,14,16,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,3]]}]},{"i":"6cf89885ff6b8f61","q":"Parallel lines  $\\ell_1$ ,  $\\ell_2$ ,  $\\ell_3$ ,  $\\ell_4$  are evenly spaced in the plane, in that order. Square  $ABCD$  has the property that  $A$  lies on  $\\ell_1$  and  $C$  lies on  $\\ell_4$ . Let  $P$  be a uniformly random point in the interior of  $ABCD$  and let  $Q$  be a uniformly random point on the perimeter of  $ABCD$ . Given that the probability that  $P$  lies between  $\\ell_2$  and  $\\ell_3$  is  $\\frac{53}{100}$  , the probability that  $Q$  lies between  $\\ell_2$  and  $\\ell_3$  can be expressed as  $\\frac{a}{b}$ , where  $a$  and  $b$  are relatively prime positive integers. Compute  $100a + b$ .","t":[{"b":1,"e":1.0,"k":"flat","v":0.62946,"x":0.69196,"p":[[0,23,0.0,0.63393,0.31731,0.57143,0.71429,0.85714,0.0,1.0,4,5,2,4,0,2,0,0,0,0,0,1,0,0,5,0,0,8,0,0,7,0,5],[4,23,0.1739,0.69196,0.16016,0.57143,0.57143,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,6,0,0,5,0,4],[8,23,0.3478,0.68304,0.15865,0.57143,0.71429,0.71429,0.28571,1.0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,12,0,0,11,0,0,4,0,3],[12,23,0.5217,0.65177,0.1234,0.57143,0.57143,0.71429,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,15,0,0,11,0,0,3,0,1],[16,23,0.6957,0.6875,0.14914,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,11,0,0,10,0,0,8,0,1],[20,23,0.8696,0.62946,0.16311,0.57143,0.57143,0.71429,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,1,0,0,19,0,0,4,0,0,4,0,2],[23,23,1.0,0.67411,0.12492,0.57143,0.57143,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,8,0,0,6,0,1]]},{"b":7,"e":0.57143,"k":"rising","v":0.58036,"x":0.75446,"p":[[0,28,0.0,0.60268,0.27603,0.53571,0.71429,0.85714,0.0,1.0,3,2,3,3,0,1,0,0,2,0,0,2,0,0,7,0,0,8,0,0,7,0,2],[4,28,0.1429,0.58036,0.31931,0.53571,0.57143,0.71429,0.0,1.0,5,6,3,5,0,1,0,0,1,0,0,1,0,0,9,0,0,8,0,0,1,0,6],[8,28,0.2857,0.71429,0.20825,0.57143,0.71429,0.85714,0.0,1.0,1,7,1,1,0,0,0,0,0,0,0,0,0,0,12,0,0,9,0,0,3,0,7],[12,28,0.4286,0.74554,0.17029,0.57143,0.71429,1.0,0.57143,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,12,0,0,0,0,9],[16,28,0.5714,0.66518,0.14555,0.57143,0.57143,0.71429,0.4286,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,7,0,0,3,0,3],[20,28,0.7143,0.74107,0.15335,0.57143,0.71429,0.85714,0.57143,1.0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,12,0,0,4,0,6],[24,28,0.8571,0.69643,0.1915,0.57143,0.57143,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,1,0,0,15,0,0,6,0,0,2,0,7],[28,28,1.0,0.75446,0.17215,0.57143,0.71429,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,9,0,0,10,0,0,4,0,8]]}]},{"i":"75b534702f7daa21","q":"Let convex quadrilateral  $ ABCD$  be inscribed in a circle centers at  $ O.$  The opposite sides  $ BA,CD$  meet at  $ H$ , the diagonals  $ AC,BD$  meet at  $ G.$  Let  $ O_{1},O_{2}$  be the circumcenters of triangles  $ AGD,BGC.$   $ O_{1}O_{2}$  intersects  $ OG$  at  $ N.$  The line  $ HG$  cuts the circumcircles of triangles  $ AGD,BGC$  at  $ P,Q$ , respectively. Denote by  $ M$  the midpoint of  $ PQ.$  Prove that  $ NO \\equal{} NM.$","t":[{"b":5,"e":0.28571,"k":"flat","v":0.15624,"x":0.26786,"p":[[0,4,0.0,0.15624,0.22117,0.0,0.0,0.28571,0.0,0.85714,19,0,0,19,0,2,0,0,4,0,0,5,0,0,1,0,0,0,0,0,1,0,0],[4,4,1.0,0.26786,0.18123,0.14286,0.28571,0.42857,0.0,0.57143,7,0,0,7,0,6,0,0,4,0,0,14,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.19642,"p":[[0,9,0.0,0.19642,0.24677,0.0,0.14286,0.28571,0.0,0.85714,15,0,0,15,0,5,0,0,5,0,0,3,0,0,2,0,0,0,0,0,2,0,0],[4,9,0.4444,0.15178,0.24468,0.0,0.0,0.2857,0.0,1.0,19,1,0,19,0,3,0,0,6,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[8,9,0.8889,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"568ec87c7573dacc","q":"Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \\[a_{99}x^{99} + \\cdots + a_1x + a_0\\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?","t":[{"b":0,"e":0.0,"k":"falling","v":0.0,"x":0.17409,"p":[[0,58,0.0,0.17409,0.29391,0.0,0.0,0.21429,0.0,1.0,21,2,1,21,0,3,0,0,0,0,0,2,0,0,4,0,0,0,0,0,0,0,2],[4,58,0.069,0.15625,0.27976,0.0,0.0,0.42857,0.0,1.0,23,2,7,23,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,2],[8,58,0.1379,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,58,0.2069,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,58,0.2759,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,58,0.3448,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,58,0.4138,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,58,0.4828,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,58,0.5517,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,58,0.6207,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,58,0.6897,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,58,0.7586,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,58,0.8276,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,58,0.8966,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[58,58,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,24,0.0,0.03125,0.11143,0.0,0.0,0.0,0.0,0.57143,29,0,8,29,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,24,0.1667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5db2bf9b891dfbd7","q":"Let  $k$  be a positive integer and  $P$  a point in the plane. We wish to draw lines, none passing through  $P$ , in such a way that any ray starting from  $P$  intersects at least  $k$  of these lines. Determine the smallest number of lines needed.","t":[{"b":0,"e":0.57143,"k":"flat","v":0.46872,"x":0.53571,"p":[[0,23,0.0,0.46872,0.28399,0.39285,0.571,0.60714,0.0,1.0,6,2,6,6,0,1,0,0,1,0,0,7,0,0,9,0,0,5,0,0,1,0,2],[4,23,0.1739,0.51337,0.23652,0.42857,0.42859,0.57143,0.0,1.0,3,2,3,3,0,0,0,0,0,0,0,14,0,0,9,0,0,1,0,0,3,0,2],[8,23,0.3478,0.49999,0.08747,0.42857,0.42857,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,18,0,0,12,0,0,2,0,0,0,0,0],[12,23,0.5217,0.50445,0.10089,0.42857,0.42859,0.57143,0.42857,0.857,0,0,0,0,0,0,0,0,0,0,0,18,0,0,12,0,0,1,0,0,1,0,0],[16,23,0.6957,0.52676,0.12077,0.42857,0.571,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,15,0,0,14,0,0,2,0,0,0,0,1],[20,23,0.8696,0.53571,0.10102,0.42857,0.57143,0.57143,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,12,0,0,17,0,0,2,0,0,1,0,0],[23,23,1.0,0.50892,0.08701,0.42857,0.4998,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,16,0,0,14,0,0,2,0,0,0,0,0]]},{"b":3,"e":1.0,"k":"rising","v":0.37499,"x":1.0,"p":[[0,23,0.0,0.54015,0.23887,0.42857,0.57143,0.57143,0.0,1.0,3,3,3,3,0,0,0,0,0,0,0,9,0,0,13,0,0,3,0,0,1,0,3],[4,23,0.1739,0.37499,0.26904,0.0,0.42857,0.57143,0.0,0.85714,10,0,9,10,0,0,0,0,0,0,0,8,0,0,11,0,0,2,0,0,1,0,0],[8,23,0.3478,0.52235,0.1621,0.42857,0.5005,0.57143,0.0,1.0,1,1,1,1,0,0,0,0,0,0,0,15,0,0,10,0,0,5,0,0,0,0,1],[12,23,0.5217,0.96875,0.05906,1.0,1.0,1.0,0.85714,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25],[16,23,0.6957,0.97768,0.06298,1.0,1.0,1.0,0.71429,1.0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,28],[20,23,0.8696,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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":"c29d273268e6c072","q":"Let $\\mathbb{N}$ be the set of positive integers. Let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \\leq f(n) \\leq n+2012$ for all $n$.  Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,19,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,16,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,15,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,17,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.05804,0.11769,0.0,0.0,0.0,0.0,0.42857,25,0,1,25,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02678,"p":[[0,27,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,15,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,27,0.1481,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,15,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,27,0.2963,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,27,0.4444,0.02678,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,15,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,18,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,27,0.7407,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,17,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,27,0.8889,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[27,27,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"00220c3ede2184ff","q":"Point  $S$  is the midpoint of arc  $ACB$  of the circumscribed circle  $k$  around triangle  $ABC$  with  $AC>BC$ . Let  $I$  be the incenter of triangle  $ABC$ . Line  $SI$  intersects  $k$  again at point  $T$ . Let  $D$  be the reflection of  $I$  across  $T$  and  $M$  be the midpoint of side  $AB$ . Line  $IM$  intersects the line through  $D$ , parallel to  $AB$ , at point  $E$ . Prove that  $AE=BD$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.02679,"p":[[0,16,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,2,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,3,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,17,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,1,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.00875,0.03389,0.0,0.0,0.0,0.0,0.14,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"e3c20b85b53c53ec","q":"a) Prove that there are infinitely many pairs  $(m, n)$  of positive integers satisfying the following equality  $[(4 + 2\\sqrt3)m] = [(4 -2\\sqrt3)n]$ b) Prove that if  $(m, n)$  satisfies the equality, then the number  $(n + m)$  is odd.\n\n(I. Voronovich)","t":[{"b":5,"e":1.0,"k":"volatile","v":0.36161,"x":0.97321,"p":[[0,6,0.0,0.36161,0.26483,0.14286,0.42857,0.57143,0.0,1.0,6,1,3,6,0,7,0,0,1,0,0,5,0,0,10,0,0,2,0,0,0,0,1],[4,6,0.6667,0.96652,0.08561,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,1,27],[6,6,1.0,0.97321,0.08328,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,29]]},{"b":7,"e":0.4286,"k":"volatile","v":0.32579,"x":0.65177,"p":[[0,8,0.0,0.32579,0.24026,0.14286,0.28571,0.57111,0.0,0.85714,5,0,0,5,0,9,0,0,3,0,0,6,0,0,6,0,0,2,0,0,1,0,0],[4,8,0.5,0.39276,0.26971,0.14286,0.4286,0.57143,0.0,1.0,5,1,2,5,0,6,0,0,3,0,0,4,0,0,8,0,0,5,0,0,0,0,1],[8,8,1.0,0.65177,0.11259,0.57143,0.71429,0.71429,0.2857,0.857,0,0,0,0,0,0,0,0,1,0,0,2,0,0,8,0,0,20,0,0,1,0,0]]}]},{"i":"21ab4a3d81474883","q":"Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let  $ A$  be the beginning vertex and  $ B$  be the end vertex. Let there be  $ p \\times q \\times r$  cubes on the string  $ (p, q, r \\geq 1).$ \r\n\r\n*(a)* Determine for which values of  $ p, q,$  and  $ r$  it is possible to build a block with dimensions  $ p, q,$  and  $ r.$  Give reasons for your answers.\r\n*(b)* The same question as (a) with the extra condition that  $ A \\equal{} B.$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0625,"x":0.14286,"p":[[0,16,0.0,0.14286,0.13363,0.0,0.14286,0.28571,0.0,0.28571,14,0,7,14,0,4,0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.13393,0.15126,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,2,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0625,0.11258,0.0,0.0,0.03571,0.0,0.28571,24,0,1,24,0,2,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.08919,0.12238,0.0,0.0,0.17857,0.0,0.28571,20,0,0,20,0,4,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.09375,0.13175,0.0,0.0,0.1786,0.0,0.42857,20,0,1,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0892,"x":0.09375,"p":[[0,3,0.0,0.09375,0.12168,0.0,0.0,0.17857,0.0,0.28571,19,0,7,19,0,5,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.0892,0.11706,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,6,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"aa6c8f00405d92c0","q":"Let $f$ be a function from the set of integers to the set of positive integers. Suppose that for any two integers $m$ and $n$, the difference $f(m)-f(n)$ is divisible by $f(m-n)$. Prove that for all integers $m, n$ with $f(m) \\leq f(n)$ the number $f(n)$ is divisible by $f(m)$.","t":[{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.375,"p":[[0,40,0.0,0.34375,0.45577,0.0,0.0,1.0,0.0,1.0,19,10,0,19,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,10],[4,40,0.1,0.17857,0.3607,0.0,0.0,0.03571,0.0,1.0,24,5,0,24,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[8,40,0.2,0.12054,0.2699,0.0,0.0,0.03571,0.0,1.0,24,2,0,24,0,2,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[12,40,0.3,0.21875,0.3813,0.0,0.0,0.28571,0.0,1.0,23,5,0,23,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,5],[16,40,0.4,0.375,0.46941,0.0,0.0,1.0,0.0,1.0,19,10,1,19,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,10],[20,40,0.5,0.10714,0.29233,0.0,0.0,0.0,0.0,1.0,27,3,0,27,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[24,40,0.6,0.1875,0.36323,0.0,0.0,0.03571,0.0,1.0,24,4,2,24,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,4],[28,40,0.7,0.3125,0.43512,0.0,0.0,0.85714,0.0,1.0,20,7,1,20,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,2,0,7],[32,40,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"falling","v":0.00446,"x":0.34821,"p":[[0,34,0.0,0.25,0.40877,0.0,0.0,0.32143,0.0,1.0,22,7,1,22,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,7],[4,34,0.1176,0.26784,0.4161,0.0,0.0,0.57111,0.0,1.0,22,7,1,22,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,7],[8,34,0.2353,0.34821,0.43439,0.0,0.07143,1.0,0.0,1.0,16,9,1,16,0,3,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,9],[12,34,0.3529,0.12947,0.30589,0.0,0.0,0.0,0.0,1.0,26,3,1,26,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[16,34,0.4706,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,34,0.5882,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,0,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"9639b646e3a1ba75","q":"Let  $\\triangle ABC$  have incenter  $I$  and centroid  $G$ . Suppose that  $P_A$  is the foot of the perpendicular from  $C$  to the exterior angle bisector of  $B$ , and  $Q_A$  is the foot of the perpendicular from  $B$  to the exterior angle bisector of  $C$ . Define  $P_B$ ,  $P_C$ ,  $Q_B$ , and  $Q_C$  similarly. Show that  $P_A, P_B, P_C, Q_A, Q_B,$  and  $Q_C$  lie on a circle whose center is on line  $IG$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.00446,"x":0.12945,"p":[[0,32,0.0,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,2,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,32,0.125,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,32,0.25,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,32,0.375,0.02232,0.0724,0.0,0.0,0.0,0.0,0.2857,29,0,3,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,32,0.5,0.04909,0.19756,0.0,0.0,0.0,0.0,1.0,30,1,4,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[20,32,0.625,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.04911,0.13651,0.0,0.0,0.0,0.0,0.71429,26,0,6,26,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,32,0.875,0.12945,0.21533,0.0,0.0,0.2857,0.0,0.71429,22,0,0,22,0,1,0,0,4,0,0,1,0,0,3,0,0,1,0,0,0,0,0],[32,32,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03571,"p":[[0,37,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,37,0.1081,0.03571,0.10102,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,37,0.2162,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,37,0.3243,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,37,0.4324,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,37,0.5405,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,37,0.6486,0.02679,0.10374,0.0,0.0,0.0,0.0,0.5714,29,0,0,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,37,0.7568,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,37,0.8649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,37,0.973,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[37,37,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"77e80c2e58b22e6a","q":"Let the Nagel point of triangle  $ABC$  be  $N$ . We draw lines from  $B$  and  $C$  to  $N$  so that these lines intersect sides  $AC$  and  $AB$  in  $D$  and  $E$  respectively.  $M$  and  $T$  are midpoints of segments  $BE$  and  $CD$  respectively.  $P$  is the second intersection point of circumcircles of triangles  $BEN$  and  $CDN$ .  $l_1$  and  $l_2$  are perpendicular lines to  $PM$  and  $PT$  in points  $M$  and  $T$  respectively. Prove that lines  $l_1$  and  $l_2$  intersect on the circumcircle of triangle  $ABC$ .\n\n*Proposed by Nima Hamidi*","t":[{"b":1,"e":0.0,"k":"flat","v":0.02437,"x":0.125,"p":[[0,8,0.0,0.0892,0.08559,0.0,0.14286,0.14286,0.0,0.28571,14,0,1,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.125,0.09279,0.10714,0.14286,0.14286,0.0,0.42857,8,0,1,8,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.02437,0.052,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.10491,"x":0.25,"p":[[0,21,0.0,0.10491,0.09446,0.0,0.14286,0.14286,0.0,0.28571,12,0,2,12,1,15,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.16063,0.09944,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,22,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.15617,0.10327,0.14286,0.14286,0.14287,0.0,0.4286,5,0,1,5,0,21,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.16509,0.08076,0.14286,0.14286,0.14286,0.0,0.4286,2,0,1,2,0,24,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.25,0.08748,0.14286,0.2857,0.28571,0.0,0.4286,1,0,1,1,0,8,0,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c5c8d706c92f361f","q":"The largest one of numbers  $ p_1^{\\alpha_1}, p_2^{\\alpha_2}, \\cdots, p_t^{\\alpha_t}$  is called a   $ \\textbf{Good Number}$  of positive integer  $ n$ , if  $ \\displaystyle n\\equal{} p_1^{\\alpha_1} \\cdot p_2^{\\alpha_2}  \\cdots p_t^{\\alpha_t}$ , where  $ p_1$ ,  $ p_2$ ,  $ \\cdots$ ,  $ p_t$  are pairwisely different primes and  $ \\alpha_1, \\alpha_2, \\cdots, \\alpha_t$  are positive integers. Let  $ n_1, n_2, \\cdots, n_{10000}$  be  $ 10000$  distinct positive integers such that the  $ \\textbf{Good Numbers}$  of  $ n_1, n_2, \\cdots, n_{10000}$  are all equal. \r\n\r\nProve that there exist integers  $ a_1, a_2, \\cdots, a_{10000}$  such that any two of the following  $ 10000$  arithmetical progressions  $ \\{ a_i, a_i \\plus{} n_i, a_i \\plus{} 2n_i, a_i \\plus{} 3n_i, \\cdots \\}$ ( $ i\\equal{}1,2, \\cdots 10000$ ) have no common terms.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.12054,"x":0.22322,"p":[[0,15,0.0,0.22322,0.2111,0.0,0.21428,0.32143,0.0,0.71429,11,0,11,11,0,5,0,0,8,0,0,5,0,0,1,0,0,2,0,0,0,0,0],[4,15,0.2667,0.20982,0.11837,0.14286,0.14286,0.28571,0.0,0.4286,3,0,3,3,0,15,0,0,10,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,15,0.5333,0.12054,0.12931,0.0,0.14286,0.14286,0.0,0.57143,12,0,12,12,0,16,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[12,15,0.8,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.42857,1,0,1,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.17411,"x":0.18741,"p":[[0,7,0.0,0.17411,0.15458,0.0,0.14286,0.28571,0.0,0.71429,9,0,9,9,0,11,0,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,7,0.5714,0.18741,0.13573,0.105,0.14286,0.28571,0.0,0.42857,8,0,8,8,0,9,0,0,12,0,0,3,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c1621973b4252e83","q":"Let $ABCD$ be a trapezoid with $(AB)$ parallel to $(CD)$. Suppose there are two circles $\\omega_{1}$ and $\\omega_{2}$ inside the trapezoid such that $\\omega_{1}$ is tangent to the sides $[DA]$, $[AB]$, and $[BC]$, and $\\omega_{2}$ is tangent to the sides $[BC]$, $[CD]$, and $[DA]$. Let $\\left(d_{1}\\right)$ be the second tangent (after $(AD)$) to $\\omega_{2}$ passing through $A$, and let $\\left(d_{2}\\right)$ be the second tangent (after $(BC)$) to $\\omega_{1}$ passing through $C$.\nShow that $\\left(d_{1}\\right)$ and $\\left(d_{2}\\right)$ are parallel.","t":[{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.05357,"p":[[0,16,0.0,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,2,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,5,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.02455,0.0524,0.0,0.0,0.0,0.0,0.14286,26,0,8,26,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.04688,0.07494,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,1,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.05357,0.09943,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01116,"x":0.02446,"p":[[0,24,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,6,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.02446,0.0522,0.0,0.0,0.0,0.0,0.14286,26,0,2,26,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.02232,0.06039,0.0,0.0,0.0,0.0,0.28571,27,0,3,27,2,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,24,0.5,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,4,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,4,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,3,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.01339,0.03762,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"812588fa69aaf8b9","q":"Show that \n\n\\[A_n=\\prod_{j=0}^{n-1}\\cfrac{(3j+1)!}{(n+j)!}\\]\n\nis an integer, for any positive integer \\(n\\).","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.13393,"p":[[0,16,0.0,0.13393,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,28,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.10259,0.06418,0.0,0.14286,0.14286,0.0,0.1429,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.11598,"x":0.14286,"p":[[0,11,0.0,0.13839,0.09094,0.14286,0.14286,0.14286,0.0,0.42857,5,0,0,5,0,25,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.11598,0.05572,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"49c5a330f12107f6","q":"Two mathematicians, lost in Berlin, arrived on the corner of Barbarossa street with Martin Luther street and need to arrive on the corner of Meininger street with Martin Luther street. Unfortunately they don't know which direction to go along Martin Luther Street to reach Meininger Street nor how far it is, so they must go fowards and backwards along Martin Luther street until they arrive on the desired corner. What is the smallest value for a positive integer  $k$  so that they can be sure that if there are  $N$  blocks between Barbarossa street and Meininger street then they can arrive at their destination by walking no more than  $kN$  blocks (no matter what  $N$  turns out to be)?","t":[{"b":2,"e":0.0,"k":"flat","v":0.14286,"x":0.14286,"p":[[0,2,0.0,0.14286,0.23145,0.0,0.0,0.2857,0.0,0.85714,19,0,14,19,0,4,0,0,6,0,0,0,0,0,1,0,0,0,0,0,2,0,0]]},{"b":5,"e":0.85714,"k":"volatile","v":0.08036,"x":0.31697,"p":[[0,3,0.0,0.08036,0.12846,0.0,0.0,0.14286,0.0,0.42857,22,0,19,22,0,3,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.31697,0.24675,0.25,0.28571,0.42857,0.0,1.0,6,2,6,6,0,2,0,0,15,0,0,4,0,0,2,0,0,1,0,0,0,0,2]]}]},{"i":"14fd6f124190eb87","q":"Solve in positive integers the equation  $10^{a}+2^{b}-3^{c}=1997$ .","t":[{"b":0,"e":0.14286,"k":"flat","v":0.10259,"x":0.29909,"p":[[0,48,0.0,0.25,0.11845,0.14286,0.2857,0.28571,0.0,0.57143,2,0,0,2,0,9,0,0,17,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[4,48,0.0833,0.27232,0.1488,0.14286,0.28571,0.28571,0.0,0.71429,3,0,0,3,0,6,0,0,17,0,0,4,0,0,1,0,0,1,0,0,0,0,0],[8,48,0.1667,0.24107,0.14914,0.14286,0.2857,0.28571,0.0,0.57143,4,0,0,4,0,10,0,0,12,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[12,48,0.25,0.29909,0.20313,0.14286,0.28571,0.32143,0.0,0.857,4,0,0,4,0,6,0,0,14,0,0,2,0,0,4,0,0,1,0,0,1,0,0],[16,48,0.3333,0.19634,0.12246,0.14286,0.21428,0.28571,0.0,0.4286,6,0,0,6,0,10,0,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,48,0.4167,0.18303,0.18293,0.14286,0.14286,0.2857,0.0,1.0,7,1,0,7,0,15,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[24,48,0.5,0.1964,0.1504,0.14286,0.14286,0.2857,0.0,0.571,7,0,0,7,0,11,0,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[28,48,0.5833,0.20981,0.13821,0.14286,0.14288,0.2857,0.0,0.57143,4,0,0,4,0,14,0,0,11,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[32,48,0.6667,0.17839,0.15157,0.0,0.14286,0.28571,0.0,0.4286,10,0,0,10,0,9,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[36,48,0.75,0.17857,0.11845,0.14286,0.14286,0.2857,0.0,0.57143,4,0,0,4,0,19,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[40,48,0.8333,0.16071,0.09279,0.14286,0.14286,0.14287,0.0,0.42857,4,0,0,4,0,21,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,48,0.9167,0.10259,0.08913,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,48,1.0,0.1517,0.07088,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.20982,"x":0.38393,"p":[[0,39,0.0,0.29911,0.15303,0.2857,0.28571,0.32143,0.0,0.71429,3,0,0,3,0,3,0,0,18,0,0,5,0,0,2,0,0,1,0,0,0,0,0],[4,39,0.1026,0.24089,0.14925,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,11,0,0,14,0,0,2,0,0,1,0,0,1,0,0,0,0,0],[8,39,0.2051,0.2723,0.11491,0.2857,0.28571,0.28571,0.0,0.571,2,0,0,2,0,5,0,0,20,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[12,39,0.3077,0.25,0.13832,0.14286,0.2857,0.28571,0.0,0.57143,3,0,0,3,0,9,0,0,15,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,39,0.4103,0.23661,0.15815,0.14286,0.2857,0.28571,0.0,0.85714,4,0,0,4,0,9,0,0,16,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[20,39,0.5128,0.28123,0.13589,0.2857,0.28571,0.28571,0.0,0.57143,4,0,0,4,0,1,0,0,21,0,0,4,0,0,2,0,0,0,0,0,0,0,0],[24,39,0.6154,0.26777,0.14624,0.14286,0.28571,0.28571,0.0,0.57143,3,0,0,3,0,7,0,0,16,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[28,39,0.7179,0.20982,0.13356,0.14286,0.2857,0.28571,0.0,0.4286,7,0,0,7,0,6,0,0,16,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[32,39,0.8205,0.37719,0.19739,0.28571,0.42857,0.57143,0.0,0.57143,4,0,0,4,0,3,0,0,5,0,1,7,0,0,12,0,0,0,0,0,0,0,0],[36,39,0.9231,0.38393,0.18013,0.28571,0.42857,0.57143,0.0,0.57143,2,0,0,2,0,5,0,0,5,0,0,9,0,0,11,0,0,0,0,0,0,0,0],[39,39,1.0,0.29451,0.18878,0.14286,0.2857,0.42857,0.0,0.57143,4,0,0,4,0,8,0,0,9,0,0,4,0,0,7,0,0,0,0,0,0,0,0]]}]},{"i":"6c663ca7492157ad","q":"Let $d(k)$ denote the number of positive integer divisors of $k$. For example, $d(6)=4$ since 6 has 4 positive divisors, namely, $1,2,3$, and 6 . Prove that for all positive integers $n$,\n\n$$\nd(1)+d(3)+d(5)+\\cdots+d(2 n-1) \\leq d(2)+d(4)+d(6)+\\cdots+d(2 n)\n$$","t":[{"b":1,"e":0.85714,"k":"flat","v":0.57589,"x":0.76337,"p":[[0,14,0.0,0.57589,0.30406,0.28571,0.57143,0.85714,0.0,1.0,2,4,0,2,0,0,0,0,9,0,0,5,0,0,0,0,0,4,0,0,8,0,4],[4,14,0.2857,0.65176,0.33109,0.39286,0.71429,1.0,0.0,1.0,2,9,0,2,0,3,0,0,3,0,0,2,0,0,3,0,0,4,0,0,6,0,9],[8,14,0.5714,0.72318,0.27419,0.57132,0.85714,0.85714,0.0,1.0,1,7,0,1,0,2,0,0,1,0,0,2,0,0,4,0,0,3,0,0,12,0,7],[12,14,0.8571,0.76337,0.13652,0.71429,0.85707,0.85714,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,4,0,0,8,0,0,17,0,1],[14,14,1.0,0.70979,0.18031,0.57132,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,5,0,0,4,0,0,7,0,0,14,0,1]]},{"b":2,"e":0.85714,"k":"flat","v":0.57589,"x":0.75893,"p":[[0,17,0.0,0.62944,0.30275,0.39286,0.71429,0.85714,0.0,1.0,2,6,1,2,0,0,0,0,6,0,0,4,0,0,3,0,0,3,0,0,8,0,6],[4,17,0.2353,0.57589,0.29555,0.28571,0.64286,0.85714,0.0,1.0,2,4,0,2,0,1,0,0,6,0,0,6,0,0,1,0,0,6,0,0,6,0,4],[8,17,0.4706,0.59821,0.3102,0.28571,0.71429,0.85714,0.0,1.0,1,4,0,1,0,5,0,0,4,0,0,1,0,0,3,0,0,6,0,0,8,0,4],[12,17,0.7059,0.74553,0.24152,0.67857,0.85714,0.85714,0.0,1.0,1,6,0,1,0,1,0,0,0,0,0,3,0,0,3,0,0,5,0,0,13,0,6],[16,17,0.9412,0.75893,0.18363,0.57143,0.85714,0.85714,0.42857,1.0,0,6,0,0,0,0,0,0,0,0,0,3,0,0,8,0,0,3,0,0,12,0,6],[17,17,1.0,0.74552,0.20744,0.57143,0.85712,0.85714,0.2857,1.0,0,7,0,0,0,0,0,0,1,0,0,4,0,0,6,0,0,4,0,0,10,0,7]]}]},{"i":"7bacb78f518a8142","q":"Show that there is no continuous function  $f:\\mathbb{R}\\rightarrow \\mathbb{R}$  such that for every real number  $x$ \\[f(x-f(x)) = \\dfrac x2.\\]","t":[{"b":2,"e":0.2857,"k":"rising","v":0.16964,"x":0.51339,"p":[[0,23,0.0,0.16964,0.17655,0.0,0.14286,0.17857,0.0,0.57143,11,0,0,11,0,13,0,0,1,0,0,5,0,0,2,0,0,0,0,0,0,0,0],[4,23,0.1739,0.23661,0.21902,0.0,0.14286,0.42858,0.0,0.57143,9,0,0,9,0,11,0,0,1,0,0,4,0,0,7,0,0,0,0,0,0,0,0],[8,23,0.3478,0.28571,0.32731,0.0,0.14286,0.57111,0.0,1.0,12,3,0,12,0,7,0,0,2,0,0,2,0,0,5,0,0,0,0,0,1,0,3],[12,23,0.5217,0.22771,0.20475,0.0,0.14286,0.42857,0.0,0.71429,9,0,0,9,0,9,0,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[16,23,0.6957,0.39283,0.1923,0.14289,0.42857,0.4642,0.14286,1.0,0,1,0,0,0,9,0,0,1,0,0,14,0,0,7,0,0,0,0,0,0,0,1],[20,23,0.8696,0.48212,0.21052,0.42857,0.4286,0.57143,0.0,1.0,1,1,0,1,0,3,0,0,2,0,0,12,0,0,9,0,0,2,0,0,2,0,1],[23,23,1.0,0.51339,0.13296,0.42857,0.50001,0.57143,0.2857,0.85714,0,0,0,0,0,0,0,0,3,0,0,13,0,0,11,0,0,4,0,0,1,0,0]]},{"b":3,"e":0.14286,"k":"falling","v":0.05357,"x":0.31686,"p":[[0,12,0.0,0.31686,0.29614,0.105,0.21429,0.4642,0.0,1.0,8,1,0,8,0,8,0,0,3,0,0,5,0,0,3,0,0,1,0,0,3,0,1],[4,12,0.3333,0.2008,0.22266,0.0,0.14286,0.2857,0.0,1.0,9,1,0,9,0,14,0,0,3,0,0,2,0,0,3,0,0,0,0,0,0,0,1],[8,12,0.6667,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.11606,0.13566,0.0,0.14286,0.14286,0.0,0.571,13,0,0,13,0,16,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"8a5611b8819c1fc6","q":"Pentagon \\(ABCDE\\) is inscribed in a circle. Its diagonals \\(AC\\) and \\(BD\\) intersect at \\(F\\). The bisectors of \\(\\angle BAC\\) and \\(\\angle CDB\\) intersect at \\(G\\). Let \\(AG\\) intersect \\(BD\\) at \\(H\\), let \\(DG\\) intersect \\(AC\\) at \\(I\\), and let \\(EG\\) intersect \\(AD\\) at \\(J\\). If \\(FHGI\\) is cyclic and \\[JA \\cdot FC \\cdot GH = JD \\cdot FB \\cdot GI,\\] prove that \\(G\\), \\(F\\) and \\(E\\) are collinear.","t":[{"b":4,"e":0.0,"k":"flat","v":0.00893,"x":0.07589,"p":[[0,10,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,1,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.07589,0.15146,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,1,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.04464,0.12079,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,36,0.0,0.05804,0.13767,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.04911,0.12169,0.0,0.0,0.0,0.0,0.42857,27,0,0,27,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.04911,0.11071,0.0,0.0,0.0,0.0,0.42857,26,0,0,26,0,2,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.04464,0.11539,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,0,2,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"554fe1b0370e2445","q":"Show that in the decimal representation of $\\sqrt[3]{3}$, there is a digit different from 2 between the 1000000th and 3141592nd decimal place.","t":[{"b":1,"e":0.85714,"k":"rising","v":0.64286,"x":0.87054,"p":[[0,6,0.0,0.64286,0.38132,0.28571,0.85714,1.0,0.0,1.0,6,10,0,6,0,1,0,0,2,0,0,1,0,0,1,0,0,4,0,0,7,0,10],[4,6,0.6667,0.87054,0.06546,0.85714,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,25,0,5],[6,6,1.0,0.85714,0.07143,0.85714,0.85714,0.85714,0.71429,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,24,0,4]]},{"b":6,"e":1.0,"k":"volatile","v":0.30804,"x":0.96875,"p":[[0,19,0.0,0.45536,0.3719,0.0,0.42857,0.85714,0.0,1.0,9,3,1,9,0,1,0,0,5,0,0,4,0,0,0,0,0,2,0,0,8,0,3],[4,19,0.2105,0.30804,0.38318,0.0,0.07143,0.71429,0.0,1.0,16,4,0,16,0,1,0,0,6,0,0,0,0,0,0,0,0,2,0,0,3,0,4],[8,19,0.4211,0.41964,0.42847,0.0,0.28571,0.85714,0.0,1.0,15,5,3,15,0,0,0,0,2,0,0,1,0,0,0,0,0,2,0,0,7,0,5],[12,19,0.6316,0.95536,0.07523,0.85714,1.0,1.0,0.71429,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,23],[16,19,0.8421,0.92411,0.10092,0.85714,1.0,1.0,0.71429,1.0,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,9,0,19],[19,19,1.0,0.96875,0.05907,1.0,1.0,1.0,0.857,1.0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,25]]}]},{"i":"c65b7f33091a0534","q":"Let $ABC$ be a triangle with all angles acute.\nThe altitudes $\\left[A A_{1}\\right],\\left[B B_{1}\\right]$, and $\\left[C C_{1}\\right]$ intersect at point $H$. Let $A_{2}$ be the symmetric point of $A$ with respect to $\\left(B_{1} C_{1}\\right)$, and let $O$ be the center of the circumcircle of $ABC$.\na) Prove that the points $O, A_{2}, B_{1}, C$ are concyclic.\nb) Prove that $O, H, A_{1}, A_{2}$ are concyclic.","t":[{"b":1,"e":1.0,"k":"rising","v":0.17857,"x":1.0,"p":[[0,28,0.0,0.17857,0.36422,0.0,0.0,0.0,0.0,1.0,25,4,3,25,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,4],[4,28,0.1429,0.37054,0.44872,0.0,0.0,1.0,0.0,1.0,17,10,4,17,0,1,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,10],[8,28,0.2857,0.64731,0.27891,0.42857,0.64286,1.0,0.0,1.0,1,9,1,1,0,0,0,0,5,0,0,4,0,0,6,0,0,6,0,0,1,0,9],[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],[28,28,1.0,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":7,"e":1.0,"k":"volatile","v":0.05804,"x":1.0,"p":[[0,24,0.0,0.09822,0.25364,0.0,0.0,0.0,0.0,1.0,26,2,4,26,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[4,24,0.1667,0.15178,0.31931,0.0,0.0,0.0,0.0,1.0,25,3,5,25,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,3],[8,24,0.3333,0.05804,0.20159,0.0,0.0,0.0,0.0,1.0,29,1,8,29,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,24,0.5,0.86161,0.26119,0.96429,1.0,1.0,0.2857,1.0,0,24,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,1,0,0,1,0,24],[16,24,0.6667,0.60714,0.32143,0.28571,0.57143,1.0,0.0,1.0,1,11,0,1,0,0,0,0,11,0,0,2,0,0,4,0,0,3,0,0,0,0,11],[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],[24,24,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"5a45eb0aeee85792","q":"Prove that, if every three consecutive vertices of a convex  $n{}$ -gon,  $n\\geqslant 4$ , span a triangle of area at least 1, then the area of the  $n{}$ -gon is (strictly) greater than  $(n\\log_2 n)/4-1/2.$ *Radu Bumb\u0103cea & C\u0103lin Popescu*","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,19,0.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,17,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"bb498ff7f8d5e605","q":"The bisector of angle  $A$  of triangle  $ABC$  ( $AB > AC$ ) meets its circumcircle at point  $P$ . The perpendicular to  $AC$  from  $C$  meets the bisector of angle  $A$  at point  $K$ . A c\u1eebcle with center  $P$   and radius  $PK$  meets the minor arc  $PA$  of the circumcircle at point  $D$ . Prove that the quadrilateral  $ABDC$  is circumscribed.","t":[{"b":2,"e":1.0,"k":"rising","v":0.76786,"x":0.99554,"p":[[0,24,0.0,0.76786,0.31693,0.5,1.0,1.0,0.14286,1.0,0,18,0,0,0,2,0,0,6,0,0,0,0,0,1,0,0,2,0,0,3,0,18],[4,24,0.1667,0.7991,0.2942,0.67857,1.0,1.0,0.14286,1.0,0,19,0,0,0,2,0,0,3,0,0,2,0,0,1,0,0,2,0,0,3,0,19],[8,24,0.3333,0.97768,0.05187,1.0,1.0,1.0,0.85714,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,27],[12,24,0.5,0.95982,0.06423,0.85714,1.0,1.0,0.85714,1.0,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,23],[16,24,0.6667,0.96429,0.06186,0.96429,1.0,1.0,0.85714,1.0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,24],[20,24,0.8333,0.94643,0.06916,0.85714,1.0,1.0,0.85714,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,20],[24,24,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":3,"e":0.57143,"k":"falling","v":0.54464,"x":0.86161,"p":[[0,27,0.0,0.86161,0.22156,0.71429,1.0,1.0,0.2857,1.0,0,20,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,5,0,0,3,0,20],[4,27,0.1481,0.77232,0.30064,0.57143,1.0,1.0,0.14286,1.0,0,17,0,0,0,2,0,0,4,0,0,1,0,0,3,0,0,1,0,0,4,0,17],[8,27,0.2963,0.54464,0.36846,0.24999,0.42857,1.0,0.0,1.0,2,9,2,2,0,6,0,0,8,0,0,0,0,0,1,0,0,3,0,0,3,0,9],[12,27,0.4444,0.76338,0.28708,0.57143,0.92857,1.0,0.14286,1.0,0,16,0,0,0,2,0,0,3,0,0,1,0,0,4,0,0,4,0,0,2,0,16],[16,27,0.5926,0.58032,0.08703,0.57143,0.57143,0.57143,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,28,0,0,1,0,0,0,0,1],[20,27,0.7407,0.55355,0.04723,0.57143,0.57143,0.57143,0.42857,0.57143,0,0,0,0,0,0,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0],[24,27,0.8889,0.55349,0.0592,0.571,0.57143,0.57143,0.42857,0.71429,0,0,0,0,0,0,0,0,0,0,0,5,0,0,26,0,0,1,0,0,0,0,0],[27,27,1.0,0.55354,0.07783,0.57143,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,1,0,0,4,0,0,25,0,0,2,0,0,0,0,0]]}]},{"i":"a56fda2303efe8dc","q":"Let $f(x)$ and $g(x)$ be given by\n\n$$\nf(x)=\\frac{1}{x}+\\frac{1}{x-2}+\\frac{1}{x-4}+\\cdots+\\frac{1}{x-2018}\n$$\n\nand\n\n$$\ng(x)=\\frac{1}{x-1}+\\frac{1}{x-3}+\\frac{1}{x-5}+\\cdots+\\frac{1}{x-2017} .\n$$\n\nProve that\n\n$$\n|f(x)-g(x)|>2\n$$\n\nfor any non-integer real number $x$ satisfying $0<x<2018$.","t":[{"b":3,"e":0.0,"k":"falling","v":0.10268,"x":0.28122,"p":[[0,7,0.0,0.28122,0.23,0.10714,0.2857,0.42858,0.0,0.71429,8,0,0,8,0,7,0,0,4,0,0,6,0,0,5,0,0,2,0,0,0,0,0],[4,7,0.5714,0.10268,0.14827,0.0,0.07143,0.14286,0.0,0.71429,16,0,0,16,0,13,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[7,7,1.0,0.12054,0.14334,0.0,0.14286,0.14286,0.0,0.57143,13,0,0,13,0,15,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.57143,"k":"rising","v":0.20982,"x":0.49995,"p":[[0,10,0.0,0.20982,0.2172,0.0,0.14286,0.42857,0.0,0.71429,13,0,0,13,0,5,0,0,5,0,0,5,0,0,3,0,0,1,0,0,0,0,0],[4,10,0.4,0.30802,0.23174,0.14286,0.28571,0.57111,0.0,0.71429,6,0,0,6,0,7,0,0,7,0,0,3,0,0,6,0,0,3,0,0,0,0,0],[8,10,0.8,0.49995,0.14723,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,1,0,0,2,0,0,7,0,0,19,0,0,2,0,0,0,0,0],[10,10,1.0,0.47765,0.15405,0.42857,0.57143,0.57143,0.0,0.71429,1,0,0,1,0,2,0,0,2,0,0,8,0,0,18,0,0,1,0,0,0,0,0]]}]},{"i":"3f91676728c3df71","q":"Suppose that a sequence $a_{1}, a_{2}, \\ldots$ of positive real numbers satisfies  $$ a_{k+1} \\geqslant \\frac{k a_{k}}{a_{k}^{2}+(k-1)} $$  for every positive integer $k$. Prove that $a_{1}+a_{2}+\\cdots+a_{n} \\geqslant n$ for every $n \\geqslant 2$.","t":[{"b":4,"e":1.0,"k":"rising","v":0.26777,"x":1.0,"p":[[0,47,0.0,0.44196,0.35778,0.14286,0.35714,0.78571,0.0,1.0,2,8,0,2,0,12,0,0,2,0,0,6,0,0,1,0,0,1,0,0,0,0,8],[4,47,0.0851,0.37499,0.38423,0.14286,0.14286,0.78571,0.0,1.0,3,8,0,3,0,19,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,8],[8,47,0.1702,0.40172,0.3561,0.14286,0.14286,0.60682,0.0,1.0,1,7,0,1,0,18,0,0,0,0,0,2,0,0,3,0,0,1,0,0,0,0,7],[12,47,0.2553,0.30349,0.34399,0.14286,0.14286,0.2857,0.0,1.0,4,6,0,4,0,19,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,6],[16,47,0.3404,0.52679,0.41869,0.14286,0.42857,1.0,0.0,1.0,4,13,0,4,0,10,0,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,13],[20,47,0.4255,0.26777,0.31495,0.0,0.14286,0.42857,0.0,1.0,9,4,0,9,0,12,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,4],[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],[32,47,0.6809,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[36,47,0.766,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[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],[44,47,0.9362,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[47,47,1.0,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31]]},{"b":5,"e":0.2857,"k":"falling","v":0.17384,"x":0.58481,"p":[[0,20,0.0,0.58481,0.40777,0.14286,0.64264,1.0,0.0,1.0,3,14,0,3,0,8,0,0,2,0,0,2,0,0,1,0,0,1,0,0,1,0,14],[4,20,0.2,0.29456,0.33874,0.14286,0.14286,0.2857,0.0,1.0,5,5,0,5,0,18,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,5],[8,20,0.4,0.17384,0.21943,0.105,0.14286,0.14286,0.0,1.0,8,1,0,8,0,20,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[12,20,0.6,0.17402,0.17031,0.14286,0.14286,0.14286,0.0,1.0,4,1,0,4,0,23,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[16,20,0.8,0.38825,0.27237,0.14286,0.42857,0.57143,0.0,1.0,4,1,0,4,0,8,0,0,2,0,0,7,0,0,4,0,0,5,0,0,1,0,1],[20,20,1.0,0.38837,0.27251,0.14289,0.28571,0.57143,0.0,1.0,4,1,0,4,0,5,0,0,8,0,0,5,0,0,4,0,0,2,0,0,3,0,1]]}]},{"i":"0331ffc5d770885f","q":"We denote $S(k)$ as the sum of the digits of an integer $k$. We say that an integer $a$ is of order $n$ if there exists a sequence of integers $a_{0}, a_{1}, \\ldots, a_{n}$ such that $a_{n}=a$ and $a_{i+1}=a_{i}-S\\left(a_{i}\\right)$ for all $i=0,1, \\ldots, n-1$. Show that for every integer $n \\geqslant 1$ there exists an integer $b$ which is of order $n$ but not of order $n+1$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,8,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,12,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,21,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0133,"x":0.0133,"p":[[0,5,0.0,0.0133,0.04136,0.0,0.0,0.0,0.0,0.14286,29,0,15,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a2a8cbc643494048","q":"The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle  $ABC$ )  $F$  is marked in triangle  $ABC$ .  $A_1$  is on the side  $BC$  such that  $AA_1$  is the altitude of triangle  $ABC$ . Prove that the line symmetric to  $FA_1$  with respect to  $BC$  is perpendicular to  $IO$ , where  $O$  is the center of the circumcircle of the triangle  $ABC$  and  $I$  is the center of its incircle.","t":[{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.17847,"p":[[0,111,0.0,0.14713,0.15762,0.0,0.14143,0.28571,0.0,0.571,14,0,0,14,0,7,0,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,111,0.036,0.11607,0.18363,0.0,0.0,0.2857,0.0,0.71429,21,0,0,21,0,2,0,0,5,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[8,111,0.0721,0.06016,0.09952,0.0,0.0,0.14071,0.0,0.2857,22,0,0,22,1,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,111,0.1081,0.125,0.17405,0.0,0.0,0.2857,0.0,0.5714,19,0,0,19,0,4,0,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[16,111,0.1441,0.1249,0.15042,0.0,0.07,0.2857,0.0,0.571,16,0,0,16,0,7,0,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,111,0.1802,0.08929,0.13716,0.0,0.0,0.14287,0.0,0.4286,21,0,1,21,0,4,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,111,0.2162,0.09375,0.13175,0.0,0.0,0.1786,0.0,0.42857,20,0,0,20,0,4,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[28,111,0.2523,0.17847,0.1956,0.0,0.14143,0.28571,0.0,0.57143,15,0,0,15,0,2,0,0,11,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[32,111,0.2883,0.08026,0.1425,0.0,0.0,0.14286,0.0,0.571,22,0,1,22,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[36,111,0.3243,0.07588,0.15142,0.0,0.0,0.03571,0.0,0.571,24,0,0,24,0,3,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[40,111,0.3604,0.08036,0.13333,0.0,0.0,0.17857,0.0,0.42857,23,0,1,23,0,1,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,111,0.3964,0.06697,0.11286,0.0,0.0,0.14289,0.0,0.28571,23,0,2,23,0,3,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,111,0.4324,0.08481,0.1466,0.0,0.0,0.14286,0.0,0.571,22,0,3,22,0,4,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[52,111,0.4685,0.07142,0.15563,0.0,0.0,0.0,0.0,0.57143,25,0,3,25,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[56,111,0.5045,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,111,0.5405,0.06241,0.13798,0.0,0.0,0.0,0.0,0.57143,25,0,3,25,0,3,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[64,111,0.5766,0.07813,0.16304,0.0,0.0,0.03571,0.0,0.64286,24,0,8,24,0,3,0,0,3,0,0,0,0,0,1,1,0,0,0,0,0,0,0],[68,111,0.6126,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,111,0.6486,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[76,111,0.6847,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,5,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[80,111,0.7207,0.01777,0.04701,0.0,0.0,0.0,0.0,0.14286,28,0,5,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[84,111,0.7568,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,4,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[88,111,0.7928,0.02679,0.09062,0.0,0.0,0.0,0.0,0.4286,29,0,4,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[92,111,0.8288,0.04464,0.10374,0.0,0.0,0.0,0.0,0.28571,27,0,2,27,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[96,111,0.8649,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,4,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[100,111,0.9009,0.11607,0.15746,0.0,0.0,0.2857,0.0,0.57143,19,0,1,19,0,3,0,0,8,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[104,111,0.9369,0.10268,0.1439,0.0,0.0,0.1786,0.0,0.57143,19,0,1,19,0,5,0,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[108,111,0.973,0.07589,0.13825,0.0,0.0,0.14286,0.0,0.42857,23,0,1,23,0,4,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[111,111,1.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.12054,"x":0.2232,"p":[[0,15,0.0,0.16964,0.16145,0.0,0.2857,0.28571,0.0,0.57143,14,0,1,14,0,1,0,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,15,0.2667,0.2232,0.22565,0.0,0.14295,0.28571,0.0,0.71429,11,0,0,11,0,7,0,0,7,0,0,1,0,0,4,0,0,2,0,0,0,0,0],[8,15,0.5333,0.1741,0.19797,0.0,0.07145,0.28571,0.0,0.57143,16,0,0,16,0,2,0,0,8,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[12,15,0.8,0.15179,0.17105,0.0,0.07143,0.28571,0.0,0.5714,16,0,0,16,0,3,0,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0],[15,15,1.0,0.12054,0.13415,0.0,0.0,0.2857,0.0,0.28571,17,0,0,17,0,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"33f47e0f1b370309","q":"Say that an ordered pair $(x, y)$ of integers is an irreducible lattice point if $x$ and $y$ are relatively prime. For any finite set $S$ of irreducible lattice points, show that there is a homogenous polynomial in two variables, $f(x, y)$, with integer coefficients, of degree at least 1 , such that $f(x, y)=1$ for each $(x, y)$ in the set $S$.  Note: A homogenous polynomial of degree $n$ is any nonzero polynomial of the form  $$ f(x, y)=a_{0} x^{n}+a_{1} x^{n-1} y+a_{2} x^{n-2} y^{2}+\\cdots+a_{n-1} x y^{n-1}+a_{n} y^{n} . $$","t":[{"b":1,"e":0.07143,"k":"flat","v":0.01554,"x":0.03996,"p":[[0,20,0.0,0.01562,0.03885,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,3,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.02679,0.05283,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,2,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.0267,0.06606,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.03996,0.06138,0.0,0.0,0.08857,0.0,0.14286,22,0,0,22,2,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.01554,0.0425,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.02897,0.05312,0.0,0.0,0.0175,0.0,0.1429,24,0,0,24,3,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.1429,"k":"flat","v":0.01786,"x":0.12277,"p":[[0,25,0.0,0.01786,0.03993,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,4,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,25,0.16,0.04,0.06141,0.0,0.0,0.08857,0.0,0.1429,22,0,0,22,2,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.06241,0.07723,0.0,0.0,0.14286,0.0,0.28571,18,0,0,18,2,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.1158,0.08876,0.05357,0.14286,0.14286,0.0,0.42857,8,0,0,8,2,20,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.05134,0.07626,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,1,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,25,0.8,0.12045,0.05183,0.14214,0.14286,0.14286,0.0,0.21429,4,0,0,4,3,24,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.12277,0.04803,0.14286,0.14286,0.14286,0.0,0.1429,4,0,0,4,1,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.11366,0.05598,0.14,0.14286,0.14286,0.0,0.1429,6,0,0,6,1,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f6a4d22fb8eff207","q":"Prove that for any positive integers  $x, y, z$  with  $xy-z^2 = 1$  one can find non-negative integers  $a, b, c, d$  such that  $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$ .\nSet  $z = (2q)!$  to deduce that for any prime number  $p = 4q + 1$ ,  $p$  can be represented as the sum of squares of two integers.","t":[{"b":2,"e":1.0,"k":"falling","v":0.53566,"x":0.90176,"p":[[0,16,0.0,0.90176,0.14483,0.857,1.0,1.0,0.571,1.0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,4,0,0,5,0,20],[4,16,0.25,0.8973,0.10249,0.857,0.85714,1.0,0.71429,1.0,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,13,0,14],[8,16,0.5,0.79461,0.1595,0.71429,0.857,0.89286,0.42857,1.0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,9,0,0,9,0,8],[12,16,0.75,0.66065,0.16657,0.571,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,3,0,0,8,0,0,12,0,0,6,0,1],[16,16,1.0,0.53566,0.18209,0.42857,0.571,0.57143,0.14286,0.85714,0,0,0,0,0,1,0,0,3,0,0,11,0,0,10,0,0,2,0,0,5,0,0]]},{"b":5,"e":1.0,"k":"flat","v":0.95089,"x":0.99107,"p":[[0,3,0.0,0.95089,0.11071,1.0,1.0,1.0,0.57143,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,2,0,26],[3,3,1.0,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30]]}]},{"i":"58215654f9b2521c","q":"Let $a$ and $b$ be distinct positive integers such that $3^{a}+2$ is divisible by $3^{b}+2$. Prove that $a>b^{2}$ \u3002","t":[{"b":0,"e":0.1429,"k":"volatile","v":0.02232,"x":0.34357,"p":[[0,8,0.0,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,8,0.5,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,0,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.34357,0.32697,0.14214,0.14286,0.57143,0.0,1.0,7,2,0,7,0,11,0,0,1,0,0,2,0,0,4,0,0,2,0,0,3,0,2]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.06249,"p":[[0,19,0.0,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,1,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,19,0.2105,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,2,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.06249,0.19861,0.0,0.0,0.0,0.0,1.0,27,1,4,27,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,15,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6e904fbb4cd341a2","q":"Let $G$ be the centroid of a right-angled triangle $A B C$ with $\\angle B C A=90^{\\circ}$. Let $P$ be the point on ray $A G$ such that $\\angle C P A=\\angle C A B$, and let $Q$ be the point on ray $B G$ such that $\\angle C Q B=\\angle A B C$. Prove that the circumcircles of triangles $A Q G$ and $B P G$ meet at a point on side $A B$.","t":[{"b":3,"e":0.0,"k":"falling","v":0.0,"x":0.17857,"p":[[0,43,0.0,0.17411,0.29609,0.0,0.0,0.28571,0.0,1.0,21,2,4,21,0,2,0,0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,2],[4,43,0.093,0.17857,0.26964,0.0,0.0,0.2857,0.0,0.85714,18,0,1,18,0,5,0,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,0],[8,43,0.186,0.16518,0.26027,0.0,0.0,0.17857,0.0,0.71429,20,0,0,20,0,4,0,0,1,0,0,1,0,0,2,0,0,4,0,0,0,0,0],[12,43,0.2791,0.15179,0.21998,0.0,0.0,0.17857,0.0,0.71429,17,0,2,17,0,7,0,0,3,0,0,2,0,0,0,0,0,3,0,0,0,0,0],[16,43,0.3721,0.08473,0.17442,0.0,0.0,0.14286,0.0,0.71429,21,0,1,21,0,9,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],[20,43,0.4651,0.06697,0.18552,0.0,0.0,0.0,0.0,0.71429,27,0,1,27,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[24,43,0.5581,0.03125,0.12234,0.0,0.0,0.0,0.0,0.57143,30,0,5,30,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,43,0.6512,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,43,0.8372,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,43,0.9302,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[43,43,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.42857,"k":"rising","v":0.15179,"x":0.45982,"p":[[0,31,0.0,0.15179,0.25489,0.0,0.0,0.14286,0.0,0.71429,20,0,6,20,0,5,0,0,2,0,0,0,0,0,0,0,0,5,0,0,0,0,0],[4,31,0.129,0.15179,0.23402,0.0,0.0,0.14286,0.0,0.85714,17,0,3,17,0,8,0,0,3,0,0,0,0,0,1,0,0,2,0,0,1,0,0],[8,31,0.2581,0.21427,0.30092,0.0,0.0,0.32143,0.0,1.0,17,1,1,17,0,5,0,0,2,0,0,1,0,0,2,0,0,3,0,0,1,0,1],[12,31,0.3871,0.45982,0.17762,0.42857,0.42857,0.57143,0.0,0.85714,2,0,0,2,0,1,0,0,1,0,0,16,0,0,10,0,0,0,0,0,2,0,0],[16,31,0.5161,0.44643,0.15043,0.39286,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,2,0,0,6,0,0,13,0,0,9,0,0,1,0,0,1,0,0],[20,31,0.6452,0.38393,0.16536,0.2857,0.42857,0.4286,0.0,0.71429,2,0,0,2,0,3,0,0,6,0,0,14,0,0,6,0,0,1,0,0,0,0,0],[24,31,0.7742,0.44194,0.13531,0.39286,0.42857,0.57143,0.0,0.57143,1,0,0,1,0,0,0,0,7,0,0,11,0,0,13,0,0,0,0,0,0,0,0],[28,31,0.9032,0.43524,0.14099,0.28571,0.42857,0.57111,0.14286,0.85714,0,0,0,0,0,1,0,1,7,0,0,13,0,0,9,0,0,0,0,0,1,0,0],[31,31,1.0,0.41517,0.12035,0.39286,0.42857,0.4286,0.14286,0.71429,0,0,0,0,0,2,0,0,6,0,0,18,0,0,5,0,0,1,0,0,0,0,0]]}]},{"i":"45fb8069bcc65c17","q":"Let $n$ be an even positive integer, and let $G$ be an $n$-vertex (simple) graph with exactly $\\frac{n^{2}}{4}$ edges. An unordered pair of distinct vertices $\\{x, y\\}$ is said to be amicable if they have a common neighbor (there is a vertex $z$ such that $x z$ and $y z$ are both edges). Prove that $G$ has at least $2\\binom{n / 2}{2}$ pairs of vertices which are amicable.","t":[{"b":1,"e":0.0,"k":"flat","v":0.01786,"x":0.03125,"p":[[0,12,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.01786,0.06916,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.03125,"x":0.08482,"p":[[0,22,0.0,0.08482,0.13767,0.0,0.0,0.14286,0.0,0.57143,21,0,0,21,0,5,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,22,0.1818,0.04018,0.08917,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.05804,0.10012,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.05357,0.09279,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bf07c8f4d7b2286f","q":"One side of a square sheet of paper is colored red, the other - in blue. On both sides, the sheet is divided into  $n^2$  identical square cells. In each of these  $2n^2$  cells is written a number from  $1$  to  $k$ . Find the smallest  $k$ ,for which the following properties hold simultaneously:\n (i) on the red side, any two numbers in different rows are distinct;\n (ii) on the blue side, any two numbers in different columns are different;\n (iii) for each of the  $n^2$  squares of the partition, the number on the blue side is not equal to the number on the red side.","t":[{"b":2,"e":0.0,"k":"falling","v":0.07143,"x":0.50892,"p":[[0,24,0.0,0.49107,0.33108,0.14286,0.57143,0.71429,0.0,1.0,7,2,1,7,0,2,0,0,2,0,0,1,0,0,8,0,0,5,0,0,5,0,2],[4,24,0.1667,0.32589,0.38004,0.0,0.0,0.60714,0.0,1.0,17,2,4,17,0,1,0,0,0,0,0,1,0,0,5,0,0,1,0,0,5,0,2],[8,24,0.3333,0.50892,0.35881,0.0,0.57143,0.85714,0.0,0.85714,10,0,2,10,0,0,0,0,0,0,0,0,0,0,7,0,0,4,0,0,11,0,0],[12,24,0.5,0.19642,0.29395,0.0,0.0,0.57111,0.0,0.85714,20,0,1,20,0,3,0,0,0,0,0,0,0,0,6,0,0,1,0,0,2,0,0],[16,24,0.6667,0.09374,0.22189,0.0,0.0,0.0,0.0,1.0,26,1,0,26,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,1],[20,24,0.8333,0.08482,0.19516,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,2,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[24,24,1.0,0.07143,0.19562,0.0,0.0,0.0,0.0,0.85714,26,0,0,26,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0]]},{"b":6,"e":0.71429,"k":"flat","v":0.25893,"x":0.61159,"p":[[0,15,0.0,0.50892,0.3272,0.21429,0.57143,0.71429,0.0,1.0,8,1,0,8,0,0,0,0,1,0,0,2,0,0,7,0,0,7,0,0,6,0,1],[4,15,0.2667,0.3616,0.35711,0.0,0.42857,0.60714,0.0,1.0,14,2,5,14,0,0,0,0,1,0,0,5,0,0,4,0,0,2,0,0,4,0,2],[8,15,0.5333,0.25893,0.29974,0.0,0.0,0.57143,0.0,0.85714,17,0,1,17,0,1,0,0,0,0,0,4,0,0,7,0,0,1,0,0,2,0,0],[12,15,0.8,0.26786,0.31894,0.0,0.0,0.57143,0.0,0.85714,18,0,2,18,0,0,0,0,0,0,0,3,0,0,7,0,0,1,0,0,3,0,0],[15,15,1.0,0.61159,0.11971,0.57143,0.57143,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,19,0,0,5,0,0,4,0,0]]}]},{"i":"1dace20be8d2f9b4","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 be numbered in such a way that the first spy watches on the second, the second watches on the third, .., the tenth watches on the first. Prove that any 11 spies can also be numbered in a similar manner.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,34,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,34,0.1176,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,34,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,34,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,34,0.5882,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,34,0.8235,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,34,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,22,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,3,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"eedf059b974b612f","q":"Let points $A$ and $B$ be on circle $\\omega$ centered at $O$. Suppose that $\\omega_{A}$ and $\\omega_{B}$ are circles not containing $O$ which are internally tangent to $\\omega$ at $A$ and $B$, respectively. Let $\\omega_{A}$ and $\\omega_{B}$ intersect at $C$ and $D$ such that $D$ is inside triangle $A B C$. Suppose that line $B C$ meets $\\omega$ again at $E$ and let line $E A$ intersect $\\omega_{A}$ at $F$. If $F C \\perp C D$, prove that $O, C$, and $D$ are collinear.","t":[{"b":6,"e":0.14,"k":"flat","v":0.14241,"x":0.17857,"p":[[0,29,0.0,0.16509,0.06301,0.14286,0.14286,0.1429,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.17857,0.06186,0.14286,0.14286,0.1786,0.14286,0.28571,0,0,0,0,0,24,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.16045,0.04735,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.16054,0.04732,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.14241,0.00104,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.15161,0.03463,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.1517,0.0346,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.15616,0.04167,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,29,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.14277,"x":0.16491,"p":[[0,44,0.0,0.15598,0.06552,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,28,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.16491,0.06307,0.14286,0.14286,0.1429,0.0,0.28571,1,0,0,1,0,25,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,26,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.15152,0.03466,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.15179,0.04971,0.14286,0.14286,0.14286,0.0,0.28571,1,0,0,1,0,28,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,44,0.4545,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,44,0.5455,0.16071,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,44,0.7273,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.1429,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f0255cf00b8a5677","q":"Let $n \\geqslant 1$ be an integer. Morgane writes on the board, in base 10, the numbers $2023, 2023 \\times 2, \\ldots, 2023 \\times n$. For every digit $c$ between 1 and 9, she then notes $\\mathrm{d}_{c}(n)$ as the number of occurrences of the digit $c$ on the board. For example, if $n=3$, she writes the numbers 2023, 4046, and 6069, so $\\mathrm{d}_{1}(3)=\\mathrm{d}_{5}(3)=\\mathrm{d}_{7}(3)=\\mathrm{d}_{8}(3)=0, \\mathrm{~d}_{3}(3)=\\mathrm{d}_{9}(3)=1$, $\\mathrm{d}_{2}(3)=\\mathrm{d}_{4}(3)=2$ and $\\mathrm{d}_{6}(3)=3$; these nine numbers thus take exactly four values.\nProve that there are infinitely many integers $n \\geqslant 1$ for which the nine numbers $\\mathrm{d}_{1}(n), \\mathrm{d}_{2}(n), \\ldots, \\mathrm{d}_{9}(n)$ take exactly two values.","t":[{"b":1,"e":0.28571,"k":"flat","v":0.02902,"x":0.09821,"p":[[0,73,0.0,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,13,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,73,0.0548,0.0625,0.11259,0.0,0.0,0.14286,0.0,0.42857,22,0,10,22,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,73,0.1096,0.04241,0.07332,0.0,0.0,0.08929,0.0,0.28571,23,0,17,23,1,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,73,0.1644,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,17,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,73,0.2192,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,15,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,73,0.274,0.02902,0.05607,0.0,0.0,0.0,0.0,0.14286,25,0,19,25,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,73,0.3288,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,13,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,73,0.3836,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,18,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,73,0.4384,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.2857,24,0,19,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,73,0.4932,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.2857,23,0,12,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,73,0.5479,0.0625,0.09407,0.0,0.0,0.14286,0.0,0.28571,21,0,8,21,0,8,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,73,0.6027,0.08697,0.09228,0.0,0.10571,0.14286,0.0,0.28571,15,0,7,15,1,13,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,73,0.6575,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,13,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,73,0.7123,0.07143,0.10102,0.0,0.0,0.14286,0.0,0.28571,20,0,2,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,73,0.7671,0.0625,0.07087,0.0,0.0,0.14286,0.0,0.14286,18,0,0,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,73,0.8219,0.0892,0.08559,0.0,0.14286,0.14286,0.0,0.28571,14,0,1,14,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,73,0.8767,0.09821,0.10374,0.0,0.14286,0.14286,0.0,0.28571,15,0,4,15,0,12,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02232,"x":0.05804,"p":[[0,49,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,16,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,49,0.0816,0.05804,0.1177,0.0,0.0,0.03571,0.0,0.4286,24,0,8,24,0,5,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,49,0.1633,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,13,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,49,0.2449,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,17,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,49,0.3265,0.05357,0.08564,0.0,0.0,0.14286,0.0,0.28571,22,0,16,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,49,0.4082,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,16,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,49,0.4898,0.05116,0.08401,0.0,0.0,0.14,0.0,0.28571,22,0,5,22,1,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,49,0.5714,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,2,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,49,0.6531,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,12,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,49,0.7347,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,16,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,49,0.8163,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.28571,21,0,7,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.05804,0.09354,0.0,0.0,0.14286,0.0,0.28571,22,0,13,22,0,7,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b33567c99f6d6d58","q":"Let $n$ be a positive integer and consider a square with dimensions $2^{n} \\times 2^{n}$. We cover this square with a number (at least 2) of non-overlapping rectangles, such that each rectangle has integer dimensions and a power of two as its area. Prove that two of the rectangles in the covering have the same dimensions. (Two rectangles have the same dimensions if they have the same width and the same height, where they cannot be rotated.)","t":[{"b":6,"e":0.14286,"k":"flat","v":0.1383,"x":0.29454,"p":[[0,34,0.0,0.2007,0.13771,0.14286,0.14286,0.17857,0.0,0.57143,2,0,1,2,0,22,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[4,34,0.1176,0.29454,0.22002,0.14286,0.14286,0.42857,0.14,0.85714,0,0,0,0,0,18,0,0,5,0,0,4,0,0,1,0,0,2,0,0,2,0,0],[8,34,0.2353,0.18295,0.0961,0.14286,0.14286,0.14286,0.0,0.42857,1,0,1,1,0,24,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,34,0.3529,0.26777,0.1981,0.14286,0.14286,0.42857,0.14,1.0,0,1,0,0,0,20,0,0,2,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[16,34,0.4706,0.28576,0.21432,0.14286,0.21428,0.32143,0.14286,1.0,0,2,0,0,0,16,0,0,8,0,0,6,0,0,0,0,0,0,0,0,0,0,2],[20,34,0.5882,0.21857,0.11848,0.14286,0.14286,0.28571,0.14,0.4286,0,0,0,0,0,22,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[24,34,0.7059,0.28124,0.19391,0.14286,0.14286,0.42857,0.0,1.0,1,1,0,1,0,16,0,0,3,0,0,10,0,0,1,0,0,0,0,0,0,0,1],[28,34,0.8235,0.19187,0.12172,0.14286,0.14286,0.14286,0.0,0.57143,1,0,1,1,0,25,0,0,1,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[32,34,0.9412,0.14732,0.02486,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[34,34,1.0,0.1383,0.06666,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"flat","v":0.13393,"x":0.38393,"p":[[0,36,0.0,0.2232,0.12336,0.14286,0.14286,0.2857,0.0,0.571,1,0,0,1,0,18,0,0,8,0,0,4,0,0,1,0,0,0,0,0,0,0,0],[4,36,0.1111,0.22311,0.1819,0.14286,0.14286,0.17857,0.14,1.0,0,1,0,0,0,24,0,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,1],[8,36,0.2222,0.32124,0.26737,0.14286,0.14286,0.42857,0.0,1.0,1,3,0,1,0,16,0,0,4,0,0,6,0,0,1,0,0,1,0,0,0,0,3],[12,36,0.3333,0.2767,0.21116,0.14286,0.14286,0.32143,0.14,1.0,0,1,0,0,0,19,0,0,5,0,0,4,0,0,1,0,0,2,0,0,0,0,1],[16,36,0.4444,0.38393,0.25374,0.14286,0.42857,0.42895,0.14,1.0,0,2,0,0,0,12,0,0,3,0,0,11,0,0,0,0,0,3,0,0,1,0,2],[20,36,0.5556,0.17402,0.07775,0.14286,0.14286,0.14286,0.14,0.4286,0,0,0,0,0,27,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[24,36,0.6667,0.18304,0.08171,0.14286,0.14286,0.14292,0.14286,0.42857,0,0,0,0,0,25,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.17839,0.16755,0.14286,0.14286,0.14286,0.0,1.0,3,1,3,3,0,24,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[32,36,0.8889,0.14268,0.00069,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,36,1.0,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ffcc3242507fc28e","q":"Let an acute-angled triangle  $ABC$  with  $AB<AC<BC$ , inscribed in circle  $c(O,R)$ . The angle bisector  $AD$  meets  $c(O,R)$  at  $K$ . The circle  $c_1(O_1,R_1)$ (which passes from  $A,D$  and has its center  $O_1$  on  $OA$ ) meets  $AB$  at  $E$  and  $AC$  at  $Z$ . If  $M,N$  are the midpoints of  $ZC$  and  $BE$  respectively, prove that:**a)**the lines  $ZE,DM,KC$  are concurrent at one point  $T$ .**b)**the lines  $ZE,DN,KB$  are concurrent at one point  $X$ .**c)** $OK$  is the perpendicular bisector of  $TX$ .","t":[{"b":3,"e":0.571,"k":"flat","v":0.40175,"x":0.49328,"p":[[0,17,0.0,0.43746,0.19862,0.2857,0.4998,0.57143,0.0,0.71429,2,0,1,2,0,2,0,0,8,0,0,4,0,0,12,0,0,4,0,0,0,0,0],[4,17,0.2353,0.45527,0.22138,0.2857,0.571,0.57143,0.0,0.71429,4,0,0,4,0,0,0,0,7,0,0,1,0,0,15,0,0,5,0,0,0,0,0],[8,17,0.4706,0.42185,0.18501,0.2857,0.42859,0.57143,0.0,0.71429,1,0,0,1,0,1,0,1,12,0,0,5,0,0,6,2,0,4,0,0,0,0,0],[12,17,0.7059,0.49328,0.20621,0.39286,0.57141,0.58929,0.0,0.85714,2,0,0,2,0,1,0,0,5,0,0,5,0,0,11,1,0,6,0,0,1,0,0],[16,17,0.9412,0.40175,0.14909,0.28571,0.42857,0.4642,0.0,0.71429,1,0,0,1,0,0,0,0,13,0,0,10,0,0,6,0,0,2,0,0,0,0,0],[17,17,1.0,0.41066,0.21939,0.2857,0.4286,0.57143,0.0,0.71429,4,0,0,4,0,2,0,0,6,0,0,6,0,0,10,0,0,4,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.09374,"x":0.41064,"p":[[0,43,0.0,0.40617,0.24766,0.2857,0.42857,0.57111,0.0,1.0,4,1,0,4,0,3,0,0,8,0,0,3,0,0,9,0,0,4,0,0,0,0,1],[4,43,0.093,0.40173,0.21256,0.2857,0.42857,0.57143,0.0,0.71429,4,0,1,4,0,0,0,0,11,0,0,4,0,0,9,0,0,4,0,0,0,0,0],[8,43,0.186,0.37049,0.24704,0.14289,0.42857,0.57143,0.0,0.71429,7,0,1,7,0,2,0,0,6,0,0,2,0,0,12,0,0,3,0,0,0,0,0],[12,43,0.2791,0.41064,0.24414,0.24999,0.571,0.57143,0.0,0.857,6,0,0,6,0,2,0,0,4,0,0,0,0,0,19,0,0,0,0,0,1,0,0],[16,43,0.3721,0.32138,0.28117,0.0,0.4286,0.57143,0.0,0.85714,13,0,3,13,0,0,0,0,2,0,0,2,0,0,14,0,0,0,0,0,1,0,0],[20,43,0.4651,0.27673,0.23935,0.0,0.2857,0.571,0.0,0.57143,11,0,1,11,0,2,0,0,8,0,0,0,0,0,11,0,0,0,0,0,0,0,0],[24,43,0.5581,0.21875,0.24996,0.0,0.0,0.46431,0.0,0.57143,17,0,1,17,0,1,0,0,2,0,0,4,0,0,8,0,0,0,0,0,0,0,0],[28,43,0.6512,0.22331,0.25236,0.0,0.0,0.571,0.0,0.57143,17,0,1,17,0,0,0,0,4,0,0,2,0,0,9,0,0,0,0,0,0,0,0],[32,43,0.7442,0.09374,0.1943,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0],[36,43,0.8372,0.17855,0.202,0.0,0.07143,0.2857,0.0,0.57143,16,0,0,16,0,1,0,0,10,0,0,1,0,0,4,0,0,0,0,0,0,0,0]]}]},{"i":"1fdf262c17137e1f","q":"Let $A_{1} B_{1} C_{1} D_{1} E_{1}$ be a regular pentagon. For $2 \\leq n \\leq 11$,\n\nlet $A_{n} B_{n} C_{n} D_{n} E_{n}$ be the pentagon whose vertices are the midpoints of the sides of the pentagon $A_{n-1} B_{n-1} C_{n-1} D_{n-1} E_{n-1}$. All the 5 vertices of each of the 11 pentagons are arbitrarily coloured red or blue. Prove that four points among these 55 points have the same colour and form the vertices of a cyclic quadrilateral.","t":[{"b":4,"e":0.0,"k":"falling","v":0.08919,"x":0.25893,"p":[[0,33,0.0,0.25893,0.27994,0.0,0.14286,0.46429,0.0,0.85714,12,0,5,12,0,6,0,0,5,0,0,1,0,0,2,0,0,5,0,0,1,0,0],[4,33,0.1212,0.08919,0.17763,0.0,0.0,0.14286,0.0,0.857,22,0,4,22,0,5,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[8,33,0.2424,0.20089,0.21387,0.0,0.14286,0.28571,0.0,1.0,9,1,0,9,0,12,0,0,6,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[12,33,0.3636,0.21866,0.1515,0.14286,0.14286,0.28571,0.0,0.85714,2,0,0,2,0,17,0,0,10,0,0,2,0,0,0,0,0,0,0,0,1,0,0],[16,33,0.4848,0.16062,0.12243,0.105,0.14286,0.2857,0.0,0.42857,8,0,0,8,0,14,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.16947,0.13574,0.105,0.14286,0.2857,0.0,0.57143,8,0,0,8,0,13,0,0,9,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[24,33,0.7273,0.22321,0.19541,0.14286,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,10,0,0,10,0,0,3,0,0,0,0,0,1,0,0,1,0,0],[28,33,0.8485,0.12045,0.14772,0.0,0.07,0.14287,0.0,0.4286,16,0,0,16,0,9,0,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[32,33,0.9697,0.15615,0.19018,0.0,0.14286,0.2857,0.0,0.857,13,0,0,13,0,10,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[33,33,1.0,0.10259,0.1758,0.0,0.0,0.14286,0.0,0.85714,19,0,0,19,0,8,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.08027,"x":0.17857,"p":[[0,20,0.0,0.14722,0.25624,0.0,0.0,0.14287,0.0,1.0,18,2,3,18,0,7,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,2],[4,20,0.2,0.12946,0.16506,0.0,0.07143,0.2857,0.0,0.71429,16,0,8,16,0,7,0,0,7,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,20,0.4,0.17857,0.24223,0.0,0.14286,0.28571,0.0,1.0,15,1,1,15,0,5,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,1],[12,20,0.6,0.1339,0.17828,0.0,0.0,0.17857,0.0,0.571,17,0,2,17,0,7,0,0,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[16,20,0.8,0.08027,0.08696,0.0,0.07,0.14286,0.0,0.28571,16,0,1,16,0,14,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.0892,0.09274,0.0,0.14143,0.14286,0.0,0.28571,15,0,0,15,0,14,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"62c2f23496325eaf","q":"The acute triangle  $ABC$  satisfies  $\\overline {AB}<\\overline {BC}<\\overline {CA}$ . Let  $H$  a orthocenter of  $ABC$ ,   $D$  a intersection point of  $AH$  and  $BC$ ,  $E$  a intersection point of  $BH$  and  $AC$ , and  $M$  a midpoint of segment  $BC$ . \nA circle with center  $E$  and radius  $AE$  intersects the segment  $AC$  at point  $F$ ( $\\neq A$ ), and circumcircle of triangle  $BFC$  intersects the segment  $AM$  at point  $S$ . \nLet  $P$ ( $\\neq D$ ),  $Q$ ( $\\neq F$ ) a intersection point of circumcircle of triangle  $ASD$  and  $DF$ , circumcircle of triangle  $ASF$  and  $DF$  respectively. Also, define  $R$  as a intersection point of circumcircles of triangle  $AHQ$  and  $AEP$ . Prove that  $R$  lies on line  $DF$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,40,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,40,0.3,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,40,0.6,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,40,0.7,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,40,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,40,0.9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,40,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,30,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c814330becafdd06","q":"Let $a, b, c, d, e$ be distinct positive integers such that $a^{4}+b^{4}=c^{4}+d^{4}=e^{5}$. Show that $a c+b d$ is a composite number.","t":[{"b":2,"e":0.0,"k":"flat","v":0.02232,"x":0.03572,"p":[[0,8,0.0,0.03572,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05357,"p":[[0,13,0.0,0.03562,0.07974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"c13b0eff7f9a7231","q":"The bisector of  $\\angle BAD$  of a parallelogram  $ABCD$  meets  $BC$  at  $K$ . The point  $L$  lies on  $AB$  such that  $AL=CK$ . The lines  $AK$  and  $CL$  meet at  $M$ . Let  $(ALM)$  meet  $AD$  after  $D$  at  $N$ . Prove that  $\\angle CNL=90^{o}$","t":[{"b":0,"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]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,23,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,23,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"56d29bda086fac56","q":"The circle inscribed in the triangle  $ABC$  with center  $I$  touches the side  $BC$  at the point  $D$ . The line  $DI$  intersects the side  $AC$  at the point  $M$ . The tangent from  $M$  to the inscribed circle, different from  $AC$ , intersects the side  $AB$  at the point  $N$ . The line  $NI$  intersects the side  $BC$  at the point  $P$ . Prove that  $AB = BP$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.04009,"x":0.24544,"p":[[0,8,0.0,0.13838,0.1493,0.0,0.14286,0.2857,0.0,0.571,15,0,0,15,0,5,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,8,0.5,0.24544,0.14833,0.14289,0.2857,0.28571,0.0,0.71429,5,0,0,5,0,4,0,0,21,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[8,8,1.0,0.04009,0.08907,0.0,0.0,0.0,0.0,0.28571,26,0,3,26,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0758,"x":0.20534,"p":[[0,19,0.0,0.2007,0.19518,0.0,0.2143,0.2857,0.0,0.71429,11,0,1,11,0,5,0,0,13,0,0,0,0,0,1,0,0,2,0,0,0,0,0],[4,19,0.2105,0.20534,0.15123,0.0,0.2857,0.28571,0.0,0.571,10,0,0,10,0,1,0,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,19,0.4211,0.16964,0.1357,0.0,0.2143,0.28571,0.0,0.42857,11,0,1,11,0,5,0,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.0758,0.107,0.0,0.0,0.14286,0.0,0.28571,20,0,0,20,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.13384,0.12845,0.0,0.14286,0.2857,0.0,0.28571,14,0,0,14,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[19,19,1.0,0.11161,0.11143,0.0,0.14286,0.1429,0.0,0.28571,14,0,0,14,0,11,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2b8cb0bfea984800","q":"Let $n$ be a positive integer. Consider $n$ points in the plane such that no three of them are collinear and no two of the distances between them are equal. One by one, we connect each point to the two points nearest to it by line segments (if there are already other line segments drawn to this point, we do not erase these). Prove that there is no point from which line segments will be drawn to more than 11 points.","t":[{"b":6,"e":0.0,"k":"flat","v":0.00893,"x":0.03562,"p":[[0,7,0.0,0.03562,0.07129,0.0,0.0,0.0,0.0,0.2857,25,0,4,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.01339,"x":0.04465,"p":[[0,30,0.0,0.03116,0.06887,0.0,0.0,0.0,0.0,0.2857,26,0,1,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.03098,0.05856,0.0,0.0,0.0,0.0,0.1429,25,0,3,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.04009,0.07337,0.0,0.0,0.035,0.0,0.2857,24,0,3,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,1,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.03563,0.0713,0.0,0.0,0.0,0.0,0.2857,25,0,7,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.01786,0.05923,0.0,0.0,0.0,0.0,0.2857,29,0,3,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.03562,0.07974,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"030ef061450902cb","q":"The bisector of  $\\angle BAC$  in  $\\triangle ABC$  intersects  $BC$  in point  $L$ . The external bisector of  $\\angle ACB$  intersects  $\\overrightarrow{BA}$  in point  $K$ . If the length of  $AK$  is equal to the perimeter of  $\\triangle ACL$ ,  $LB=1$ , and  $\\angle ABC=36^\\circ$ , find the length of  $AC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00893,"x":0.01339,"p":[[0,8,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,8,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"842d8b0fc784a127","q":"Let $ABC$ be a triangle, and $M$ the midpoint of $[BC]$. We denote $I_{b}$ and $I_{c}$ as the centers of the incircles of $AMB$ and $AMC$. Show that the second point of intersection of the circumcircles of triangles $ABI_{b}$ and $ACI_{c}$ lies on the line $(AM)$.","t":[{"b":1,"e":0.0,"k":"falling","v":0.00893,"x":0.18304,"p":[[0,21,0.0,0.18304,0.34485,0.0,0.0,0.14286,0.0,1.0,20,4,0,20,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[4,21,0.1905,0.16964,0.30186,0.0,0.0,0.14286,0.0,1.0,18,2,0,18,0,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,2],[8,21,0.381,0.07589,0.18205,0.0,0.0,0.14286,0.0,1.0,22,1,1,22,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[12,21,0.5714,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"rising","v":0.10259,"x":1.0,"p":[[0,29,0.0,0.14286,0.28571,0.0,0.0,0.14286,0.0,1.0,19,3,2,19,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[4,29,0.1379,0.11161,0.26422,0.0,0.0,0.14286,0.0,1.0,23,2,2,23,0,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2],[8,29,0.2759,0.10259,0.18975,0.0,0.0,0.14286,0.0,1.0,18,1,0,18,0,11,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,29,0.4138,0.18741,0.31021,0.0,0.07,0.14286,0.0,1.0,16,3,3,16,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,3],[16,29,0.5517,0.95982,0.17582,1.0,1.0,1.0,0.0,1.0,1,29,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[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":"4cd715db3890d03c","q":"Prove that for every positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with the following two properties:\n(i) $a^{2}+b=n$,\n(ii) $a+b$ is a power of two, i.e., there is an integer $k \\geq 0$ with $a+b=2^{k}$.","t":[{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.16071,"p":[[0,20,0.0,0.02232,0.08828,0.0,0.0,0.0,0.0,0.42857,30,0,0,30,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.07589,0.2082,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[8,20,0.4,0.09375,0.14987,0.0,0.0,0.17857,0.0,0.4286,22,0,0,22,0,2,0,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.11605,0.20953,0.0,0.0,0.14287,0.0,0.71429,23,0,0,23,0,2,0,0,1,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[16,20,0.8,0.16071,0.29826,0.0,0.0,0.2857,0.0,1.0,22,2,0,22,0,1,0,0,5,0,0,0,0,0,0,0,0,1,0,0,1,0,2],[20,20,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.05357,"x":0.07589,"p":[[0,7,0.0,0.07589,0.14719,0.0,0.0,0.03571,0.0,0.57143,24,0,0,24,0,2,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,7,0.5714,0.05357,0.13716,0.0,0.0,0.0,0.0,0.57143,27,0,0,27,0,1,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"1fbd927ab4f67942","q":"The altitudes of the acute-angled non-isosceles triangle  $ ABC $  intersect at the point  $ H $ . On the segment  $ C_1H $ , where  $ CC_1 $  is the altitude of the triangle, the point  $ K $  is marked. Points  $ L $  and  $ M $  are the feet of perpendiculars from point  $ K $  on straight lines  $ AC $  and  $ BC $ , respectively. The lines  $ AM $  and  $ BL $  intersect at  $ N $ . Prove that  $ \\angle ANK = \\angle HNL $ .","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.03125,"p":[[0,28,0.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.01777,0.05904,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,30,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.04464,0.09062,0.0,0.0,0.0,0.0,0.28571,25,0,2,25,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.03571,0.07986,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"8ee2c167014355a1","q":"We call an infinite set  $S\\subseteq\\mathbb{N}$  good if for all parwise different integers  $a,b,c\\in S$ , all positive divisors of  $\\frac{a^c-b^c}{a-b}$  are in  $S$ . for all positive integers  $n>1$ , prove that there exists a good set  $S$  such that  $n \\not \\in S$ .\n\nProposed by Seyed Reza Hosseini Dolatabadi","t":[{"b":2,"e":1.0,"k":"rising","v":0.45536,"x":0.99107,"p":[[0,34,0.0,0.45536,0.43218,0.0,0.28571,1.0,0.0,1.0,13,9,11,13,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,1,0,9],[4,34,0.1176,0.57589,0.43372,0.0,0.71429,1.0,0.0,1.0,10,13,10,10,0,0,0,0,2,0,0,1,0,0,0,0,0,5,0,0,1,0,13],[8,34,0.2353,0.63839,0.41185,0.21429,0.71429,1.0,0.0,1.0,8,14,6,8,0,0,0,0,2,0,0,0,0,0,0,0,0,7,0,0,1,0,14],[12,34,0.3529,0.66964,0.39679,0.28571,0.85714,1.0,0.0,1.0,6,15,6,6,0,1,0,0,2,0,0,1,0,0,0,0,0,5,0,0,2,0,15],[16,34,0.4706,0.54018,0.46803,0.0,0.71429,1.0,0.0,1.0,13,14,10,13,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,1,0,14],[20,34,0.5882,0.95982,0.09606,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,27],[24,34,0.7059,0.95982,0.08917,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,26],[28,34,0.8235,0.99107,0.03458,1.0,1.0,1.0,0.85714,1.0,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,30],[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],[34,34,1.0,0.94643,0.11152,1.0,1.0,1.0,0.71429,1.0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,26]]},{"b":5,"e":0.0,"k":"flat","v":0.59374,"x":0.65624,"p":[[0,8,0.0,0.59374,0.42123,0.0,0.71429,1.0,0.0,1.0,9,13,9,9,0,0,0,0,2,0,0,1,0,0,1,0,0,5,0,0,1,0,13],[4,8,0.5,0.65624,0.39425,0.28571,0.85714,1.0,0.0,1.0,5,16,2,5,0,1,0,0,4,0,0,2,0,0,0,0,0,4,0,0,0,0,16]]}]},{"i":"05ba37b02f3b39bd","q":"Let $a_{0}, b_{0}, c_{0}$ be complex numbers, and define  $$ \\begin{aligned} a_{n+1} & =a_{n}^{2}+2 b_{n} c_{n} \\\\ b_{n+1} & =b_{n}^{2}+2 c_{n} a_{n} \\\\ c_{n+1} & =c_{n}^{2}+2 a_{n} b_{n} \\end{aligned} $$  for all nonnegative integers $n$. Suppose that $\\max \\left\\{\\left|a_{n}\\right|,\\left|b_{n}\\right|,\\left|c_{n}\\right|\\right\\} \\leq 2022$ for all $n \\geq 0$. Prove that  $$ \\left|a_{0}\\right|^{2}+\\left|b_{0}\\right|^{2}+\\left|c_{0}\\right|^{2} \\leq 1 $$","t":[{"b":0,"e":0.14286,"k":"falling","v":0.0267,"x":0.45536,"p":[[0,13,0.0,0.39732,0.39887,0.0,0.14286,0.71429,0.0,1.0,12,6,0,12,0,5,0,0,0,0,0,0,0,0,3,0,0,6,0,0,0,0,6],[4,13,0.3077,0.45536,0.34523,0.14286,0.42857,0.71429,0.0,1.0,6,4,0,6,0,5,0,0,3,0,0,4,0,0,1,0,0,7,0,0,2,0,4],[8,13,0.6154,0.06696,0.14279,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,6,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,13,0.9231,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":1.0,"k":"rising","v":0.48437,"x":0.99107,"p":[[0,22,0.0,0.5,0.40089,0.10714,0.42857,0.89286,0.0,1.0,8,8,0,8,0,4,0,0,1,0,0,4,0,0,0,0,0,4,0,0,3,0,8],[4,22,0.1818,0.48437,0.42059,0.0,0.53571,1.0,0.0,1.0,11,9,0,11,0,2,0,0,1,0,0,1,0,1,1,0,0,5,0,0,1,0,9],[8,22,0.3636,0.94643,0.14617,1.0,1.0,1.0,0.28571,1.0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,26],[12,22,0.5455,0.96875,0.11143,1.0,1.0,1.0,0.42857,1.0,0,29,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,29],[16,22,0.7273,0.98661,0.04164,1.0,1.0,1.0,0.85714,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,29],[20,22,0.9091,0.97321,0.06622,1.0,1.0,1.0,0.71429,1.0,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,4,0,27],[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":"a50b4ca594cd5ede","q":"Prove that for some positive integer $n$ the remainder of $3^{n}$ when divided by $2^{n}$ is greater than $10^{2021}$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.02232,"x":0.06697,"p":[[0,31,0.0,0.03571,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.06697,0.08737,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.04911,0.09182,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,31,0.3871,0.02232,0.06298,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,31,0.5161,0.03554,0.07116,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0267,0.05557,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04464,"p":[[0,10,0.0,0.04464,0.08328,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"cd42c830bbc135a9","q":"Prove or disprove the following hypotheses.\n\na) For all $k \\geq 2$, each sequence of $k$ consecutive positive integers contains a number that is not divisible by any prime number less than $k$.\n\nb) For all $k \\geq 2$, each sequence of $k$ consecutive positive integers contains a number that is relatively prime to all other members of the sequence.","t":[{"b":0,"e":0.14286,"k":"flat","v":0.13366,"x":0.28562,"p":[[0,44,0.0,0.2366,0.09181,0.24999,0.2857,0.28571,0.0,0.28571,3,0,3,3,0,5,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,44,0.0909,0.16955,0.09743,0.14286,0.14286,0.2857,0.0,0.4286,4,0,4,4,0,19,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.14723,0.07563,0.14286,0.14286,0.14287,0.0,0.28571,4,0,4,4,0,23,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,44,0.2727,0.17402,0.09271,0.14286,0.14286,0.2857,0.0,0.42857,3,0,3,3,0,20,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,44,0.3636,0.28562,0.27669,0.14286,0.14286,0.28571,0.0,1.0,3,3,3,3,0,15,0,0,8,0,0,2,0,0,0,0,0,0,0,0,1,0,3],[20,44,0.4545,0.17857,0.12372,0.14286,0.14286,0.17857,0.0,0.71429,3,0,3,3,0,21,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[24,44,0.5455,0.14268,0.08748,0.14214,0.14286,0.1429,0.0,0.28571,6,0,6,6,0,20,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.19625,0.13251,0.14286,0.14286,0.2857,0.0,0.71429,2,0,2,2,0,21,0,0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,0],[32,44,0.7273,0.1517,0.07088,0.14286,0.14286,0.14286,0.0,0.28571,3,0,3,3,0,24,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.16063,0.06919,0.14286,0.14286,0.14286,0.0,0.4286,1,0,1,1,0,27,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.13366,0.07084,0.14214,0.14286,0.14286,0.0,0.28571,5,0,5,5,0,24,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.14723,0.06668,0.14286,0.14286,0.14286,0.0,0.4286,2,0,2,2,0,28,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.13795,"x":0.25,"p":[[0,16,0.0,0.25,0.11294,0.2857,0.28571,0.28571,0.0,0.42857,5,0,5,5,0,0,0,0,25,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.15126,0.03473,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.14714,0.04353,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.13822,0.02483,0.14286,0.14286,0.14286,0.0,0.1429,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.13795,0.0248,0.14286,0.14286,0.14286,0.0,0.14286,1,0,1,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"5d5adcddcf79aa83","q":"The circumcentre  $O$  of a given cyclic quadrilateral  $ABCD$  lies inside the quadrilateral but not on the diagonal  $AC$ .  The diagonals of the quadrilateral intersect at  $I$ .  The circumcircle of the triangle  $AOI$  meets the sides  $AD$  and  $AB$  at points  $P$  and  $Q$ , respectively; the circumcircle of the triangle  $COI$  meets the sides  $CB$  and  $CD$  at points  $R$  and  $S$ , respectively.  Prove that  $PQRS$  is a parallelogram.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.05795,"p":[[0,15,0.0,0.05795,0.11207,0.0,0.0,0.035,0.0,0.42857,24,0,1,24,0,4,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,15,0.2667,0.04464,0.11538,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,15,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,15,0.8,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,1,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[15,15,1.0,0.05357,0.09943,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.04018,"x":0.0892,"p":[[0,16,0.0,0.04464,0.11538,0.0,0.0,0.0,0.0,0.57143,26,0,2,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,16,0.25,0.0892,0.16266,0.0,0.0,0.14286,0.0,0.71429,22,0,0,22,0,4,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,16,0.5,0.06696,0.11836,0.0,0.0,0.14286,0.0,0.42857,23,0,0,23,0,4,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.08911,0.198,0.0,0.0,0.14,0.0,0.85714,23,0,0,23,0,5,0,0,2,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[16,16,1.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ff9747ed4cee43e6","q":"Let $n \\geqslant 6$ be an integer. In the plane, we have arranged $n$ pairwise disjoint disks $D_{1}, D_{2}, \\ldots, D_{n}$ with radii $r_{1} \\geqslant r_{2} \\geqslant \\ldots \\geqslant r_{n}$. For every integer $i \\leqslant n$, consider a point $P_{i}$ inside the disk $D_{i}$. Finally, let $A$ be any point in the plane. Prove that\n\n$$\nA P_{1}+A P_{2}+\\ldots+A P_{n} \\geqslant r_{6}+r_{7}+\\ldots+r_{n}\n$$","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,30,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"fbe25353ae93c2f1","q":"Suppose  $f : \\mathbb{R} \\longrightarrow \\mathbb{R}$  be a function such that\n\\[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\\]\nfor all real  $x$ . Find  $f (2)$  and all possible values of  $f (1)$ . For each value of  $f (1)$ , construct a function achieving it and satisfying the given equation.","t":[{"b":4,"e":0.71429,"k":"flat","v":0.60713,"x":0.74106,"p":[[0,23,0.0,0.70087,0.16506,0.57143,0.71429,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,7,0,0,9,0,0,11,0,1],[4,23,0.1739,0.72767,0.16507,0.67857,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,13,0,0,7,0,4],[8,23,0.3478,0.69642,0.13244,0.57143,0.71429,0.75,0.42857,1.0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,9,0,0,13,0,0,7,0,1],[12,23,0.5217,0.6875,0.1729,0.57143,0.71429,0.75,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,3,0,0,4,0,0,15,0,0,6,0,2],[16,23,0.6957,0.60713,0.20203,0.42857,0.57143,0.75,0.2857,1.0,0,2,0,0,0,0,0,0,2,0,0,11,0,0,6,0,0,5,0,0,6,0,2],[20,23,0.8696,0.71429,0.08748,0.71429,0.71429,0.71429,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,23,0,0,3,0,1],[23,23,1.0,0.74106,0.1208,0.71429,0.71429,0.71429,0.571,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,20,0,0,3,0,4]]},{"b":5,"e":0.85714,"k":"rising","v":0.65625,"x":0.87054,"p":[[0,105,0.0,0.65625,0.15916,0.57143,0.71429,0.71429,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,6,0,0,9,0,0,11,0,0,4,0,2],[4,105,0.0381,0.66964,0.18707,0.57143,0.71429,0.75,0.0,1.0,1,2,1,1,0,0,0,0,0,0,0,2,0,0,11,0,0,10,0,0,6,0,2],[8,105,0.0762,0.75446,0.1394,0.71429,0.71429,0.85714,0.42857,1.0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,14,0,0,8,0,4],[12,105,0.1143,0.70535,0.19212,0.67857,0.71429,0.85714,0.2857,1.0,0,3,0,0,0,0,0,0,3,0,0,2,0,0,3,0,0,13,0,0,8,0,3],[16,105,0.1524,0.68304,0.15458,0.57143,0.71429,0.85714,0.28571,0.85714,0,0,0,0,0,0,0,0,1,0,0,4,0,0,5,0,0,13,0,0,9,0,0],[20,105,0.1905,0.68303,0.18808,0.57143,0.71429,0.85714,0.28571,1.0,0,2,0,0,0,0,0,0,2,0,0,4,0,0,6,0,0,9,0,0,9,0,2],[24,105,0.2286,0.77687,0.10069,0.71429,0.71429,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,16,0,0,12,0,2],[28,105,0.2667,0.77679,0.10677,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,17,0,0,10,0,3],[32,105,0.3048,0.77232,0.11214,0.71429,0.71429,0.85714,0.57143,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,16,0,0,10,0,3],[36,105,0.3429,0.79464,0.11259,0.71429,0.78571,0.85714,0.57143,1.0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,14,0,0,12,0,4],[40,105,0.381,0.87054,0.09006,0.85714,0.85714,0.89286,0.71429,1.0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,19,0,8],[44,105,0.419,0.82143,0.08748,0.71429,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,3],[48,105,0.4571,0.85268,0.09771,0.82143,0.85714,0.85714,0.71429,1.0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,17,0,7],[52,105,0.4952,0.84375,0.11495,0.71429,0.85714,1.0,0.71429,1.0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,11,0,9],[56,105,0.5333,0.76339,0.21902,0.71429,0.85714,0.85714,0.0,1.0,2,3,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,10,0,0,16,0,3],[60,105,0.5714,0.7991,0.07873,0.71429,0.85714,0.85714,0.57143,0.85714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,11,0,0,20,0,0],[64,105,0.6095,0.79464,0.07936,0.71429,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,16,0,1],[68,105,0.6476,0.82142,0.08748,0.71429,0.85714,0.85714,0.71429,1.0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,18,0,3],[72,105,0.6857,0.83482,0.09523,0.71429,0.85714,0.85714,0.71429,1.0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,17,0,5],[76,105,0.7238,0.76786,0.10564,0.71429,0.71429,0.85714,0.42857,1.0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,19,0,0,10,0,2],[80,105,0.7619,0.79911,0.08645,0.71429,0.85714,0.85714,0.71429,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,15,0,2],[84,105,0.8,0.79464,0.09407,0.71429,0.85707,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,14,0,0,15,0,2],[88,105,0.8381,0.79464,0.08702,0.71429,0.85714,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,13,0,0,17,0,1],[92,105,0.8762,0.8125,0.07523,0.71429,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,20,0,1],[96,105,0.9143,0.80357,0.09942,0.71429,0.85714,0.85714,0.57143,1.0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,10,0,0,18,0,2],[100,105,0.9524,0.79018,0.11285,0.71429,0.78571,0.85714,0.42857,1.0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,0,0,13,0,3],[104,105,0.9905,0.78125,0.08737,0.71429,0.71429,0.85714,0.57143,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,0,0,14,0,1],[105,105,1.0,0.80803,0.07667,0.71429,0.85714,0.85714,0.71429,1.0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,19,0,1]]}]},{"i":"789d19097908bd4a","q":"Quadrilateral  $ ABCD$  has an inscribed circle which centered at  $ O$  with radius  $ r$ .  $ AB$  intersects  $ CD$  at  $ P$ ;  $ AD$  intersects  $ BC$  at  $ Q$  and the diagonals  $ AC$  and  $ BD$  intersects each other at  $ K$ . If the distance from  $ O$  to the line  $ PQ$  is  $ k$ , prove that  $ OK\\cdot\\ k \\equal{} r^2$ .","t":[{"b":2,"e":0.85714,"k":"flat","v":0.82589,"x":0.97321,"p":[[0,21,0.0,0.82589,0.27137,0.71429,1.0,1.0,0.0,1.0,2,19,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,19],[4,21,0.1905,0.90625,0.13175,0.71429,1.0,1.0,0.71429,1.0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,1,0,21],[8,21,0.381,0.92411,0.2448,1.0,1.0,1.0,0.0,1.0,2,28,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,28],[12,21,0.5714,0.96875,0.17399,1.0,1.0,1.0,0.0,1.0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[16,21,0.7619,0.90178,0.2911,1.0,1.0,1.0,0.0,1.0,3,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,28],[20,21,0.9524,0.96429,0.17496,1.0,1.0,1.0,0.0,1.0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,30],[21,21,1.0,0.97321,0.14914,1.0,1.0,1.0,0.14286,1.0,0,31,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31]]},{"b":3,"e":0.71429,"k":"flat","v":0.72768,"x":0.8482,"p":[[0,22,0.0,0.81696,0.27019,0.71429,1.0,1.0,0.0,1.0,2,18,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,0,18],[4,22,0.1818,0.8482,0.23675,0.71429,1.0,1.0,0.0,1.0,1,20,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,8,0,0,0,0,20],[8,22,0.3636,0.80357,0.19149,0.71429,0.71429,1.0,0.42857,1.0,0,14,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,13,0,0,0,0,14],[12,22,0.5455,0.77232,0.22263,0.71429,0.71429,1.0,0.0,1.0,1,12,0,1,0,0,0,0,0,0,0,2,0,0,2,0,0,15,0,0,0,0,12],[16,22,0.7273,0.72768,0.27976,0.67857,0.71429,1.0,0.0,1.0,2,12,0,2,0,0,0,0,2,0,0,1,0,0,3,0,0,12,0,0,0,0,12],[20,22,0.9091,0.81249,0.25863,0.71429,1.0,1.0,0.0,1.0,2,17,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,11,0,0,0,0,17],[22,22,1.0,0.82589,0.28735,0.71429,1.0,1.0,0.0,1.0,2,21,0,2,0,0,0,0,1,0,0,1,0,0,2,0,0,5,0,0,0,0,21]]}]},{"i":"c36eec9fe6a97905","q":"Let \\( ABC \\) be a scalene triangle. Let \\( E \\) and \\( F \\) be the midpoints of sides \\( AC \\) and \\( AB \\), respectively, and let \\( D \\) be any point on segment \\( BC \\). The circumcircles of triangles \\( BDF \\) and \\( CDE \\) intersect line \\( EF \\) at points \\( K  \\neq F \\), and \\( L \\neq E \\), respectively, and intersect at points \\( X \\neq D \\). The point \\( Y \\) is on line \\( DX \\) such that \\( AY \\) is parallel to \\( BC \\). Prove that points \\( K \\), \\( L \\), \\( X \\), and \\( Y \\) lie on the same circle.","t":[{"b":0,"e":0.0,"k":"flat","v":0.04,"x":0.06688,"p":[[0,20,0.0,0.04911,0.06786,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,20,0.2,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,20,0.4,0.04902,0.06773,0.0,0.0,0.14286,0.0,0.1429,21,0,1,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.04,0.06395,0.0,0.0,0.14,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.06688,0.0712,0.0,0.0,0.14286,0.0,0.1429,17,0,0,17,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.03116,"x":0.07125,"p":[[0,18,0.0,0.04902,0.07657,0.0,0.0,0.14286,0.0,0.2857,22,0,1,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,1,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.03116,0.05889,0.0,0.0,0.0,0.0,0.1429,25,0,0,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.07125,0.0797,0.0,0.0,0.14286,0.0,0.2857,17,0,0,17,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.2857,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"97a2f25cb9869a3a","q":"Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is 46 , there is a set of 10 students in which no group is properly contained.","t":[{"b":2,"e":0.28571,"k":"falling","v":0.17402,"x":0.79018,"p":[[0,44,0.0,0.6875,0.40945,0.2857,1.0,1.0,0.0,1.0,3,20,0,3,0,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[4,44,0.0909,0.79018,0.36067,0.75,1.0,1.0,0.0,1.0,2,23,2,2,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0,1,0,23],[8,44,0.1818,0.69634,0.39581,0.28571,1.0,1.0,0.0,1.0,1,20,0,1,0,6,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[12,44,0.2727,0.67402,0.42605,0.14286,1.0,1.0,0.0,1.0,4,20,2,4,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[16,44,0.3636,0.5625,0.44455,0.14286,0.64286,1.0,0.0,1.0,6,16,5,6,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[20,44,0.4545,0.62946,0.37263,0.28571,0.64286,1.0,0.14286,1.0,0,16,0,0,0,3,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,16],[24,44,0.5455,0.52223,0.40353,0.14286,0.28571,1.0,0.0,1.0,3,13,1,3,0,6,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,13],[28,44,0.6364,0.66964,0.43218,0.14286,1.0,1.0,0.0,1.0,5,20,1,5,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[32,44,0.7273,0.53572,0.42258,0.14286,0.28571,1.0,0.0,1.0,4,14,1,4,0,8,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,14],[36,44,0.8182,0.58928,0.39082,0.2857,0.28571,1.0,0.0,1.0,1,15,0,1,0,5,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,15],[40,44,0.9091,0.2991,0.25344,0.14286,0.2857,0.28571,0.0,1.0,3,3,0,3,0,9,0,0,15,0,0,1,0,0,1,0,0,0,0,0,0,0,3],[44,44,1.0,0.17402,0.11145,0.14286,0.14286,0.2857,0.0,0.42857,6,0,0,6,0,14,0,0,11,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.28571,"k":"falling","v":0.27232,"x":0.66964,"p":[[0,9,0.0,0.62946,0.42985,0.28571,1.0,1.0,0.0,1.0,6,18,5,6,0,1,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,18],[4,9,0.4444,0.66964,0.43366,0.24999,1.0,1.0,0.0,1.0,6,20,2,6,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,20],[8,9,0.8889,0.27232,0.25843,0.14286,0.28571,0.28571,0.0,1.0,6,3,0,6,0,6,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[9,9,1.0,0.28125,0.22442,0.14286,0.28571,0.28571,0.0,1.0,5,2,0,5,0,4,0,0,19,0,0,1,0,0,1,0,0,0,0,0,0,0,2]]}]},{"i":"925c1382bd8ccd2d","q":"Show that for positive integers $n_{1}, n_{2}$ and $d$,\n\n$$\nf\\left(n_{1} n_{2}, d\\right) \\leq f\\left(n_{1}, d\\right)+n_{1}\\left(f\\left(n_{2}, d\\right)-1\\right)\n$$","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.0,"p":[[0,17,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.0,"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],[4,19,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,19,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":"6dd1309d63516a84","q":"Show that there are infinitely many positive integers $n$ such that the largest prime factor of $n^{4}+n^{2}+1$ is equal to the largest prime factor of $(n+1)^{4}+(n+1)^{2}+1$.","t":[{"b":0,"e":0.42857,"k":"rising","v":0.11589,"x":0.37052,"p":[[0,40,0.0,0.13393,0.10677,0.0,0.14286,0.14286,0.0,0.42857,9,0,2,9,0,17,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,40,0.1,0.11589,0.09735,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,16,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,40,0.2,0.14285,0.1184,0.10714,0.14286,0.14287,0.0,0.571,8,0,0,8,0,18,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,40,0.3,0.13393,0.10062,0.0,0.14286,0.14286,0.0,0.28571,9,0,1,9,0,16,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,40,0.4,0.14286,0.10101,0.14286,0.14286,0.14286,0.0,0.42857,7,0,0,7,0,19,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,40,0.5,0.34373,0.13765,0.2857,0.28571,0.42857,0.14286,0.57143,0,0,0,0,0,5,0,0,15,0,0,6,0,0,6,0,0,0,0,0,0,0,0],[24,40,0.6,0.3125,0.12078,0.2857,0.28571,0.42857,0.0,0.57143,1,0,0,1,0,4,0,0,17,0,0,8,0,0,2,0,0,0,0,0,0,0,0],[28,40,0.7,0.2991,0.10326,0.2857,0.28571,0.28571,0.0,0.57143,1,0,0,1,0,2,0,0,24,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[32,40,0.8,0.33929,0.13716,0.28571,0.28571,0.42858,0.0,0.57143,1,0,0,1,0,2,0,0,19,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[36,40,0.9,0.32589,0.09606,0.28571,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,1,0,0,24,0,0,4,0,0,3,0,0,0,0,0,0,0,0],[40,40,1.0,0.37052,0.11767,0.28571,0.28571,0.42858,0.2857,0.57143,0,0,0,0,0,0,0,0,20,0,0,5,0,0,7,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.28571,"k":"flat","v":0.15607,"x":0.17857,"p":[[0,6,0.0,0.15625,0.09006,0.14286,0.14286,0.17857,0.0,0.28571,5,0,0,5,0,19,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.17857,0.08748,0.14286,0.14286,0.28571,0.0,0.28571,3,0,1,3,0,18,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.15607,0.10328,0.14,0.14286,0.2857,0.0,0.28571,7,0,0,7,0,15,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3d6ba6fee93084c0","q":"Let $n \\geqslant 3$ be an integer, and let $x_{1}, x_{2}, \\ldots, x_{n}$ be real numbers in the interval $[0,1]$. Let $s=x_{1}+x_{2}+\\ldots+x_{n}$, and assume that $s \\geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \\leqslant i<j \\leqslant n$ such that  $$ 2^{j-i} x_{i} x_{j}>2^{s-3} $$  (Trinidad and Tobago)","t":[{"b":3,"e":0.2857,"k":"flat","v":0.03572,"x":0.0625,"p":[[0,23,0.0,0.04018,0.08917,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.04463,0.11532,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,23,0.3478,0.04911,0.09181,0.0,0.0,0.03571,0.0,0.28571,24,0,2,24,0,5,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.0625,0.11812,0.0,0.0,0.14286,0.0,0.4286,23,0,1,23,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.05134,0.09819,0.0,0.0,0.01786,0.0,0.28571,24,0,0,24,1,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.03572,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,1,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.03571,"x":0.1116,"p":[[0,36,0.0,0.09811,0.18702,0.0,0.0,0.14286,0.0,0.85714,21,0,0,21,0,6,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[4,36,0.1111,0.05804,0.15093,0.0,0.0,0.0,0.0,0.71429,26,0,0,26,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,36,0.2222,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,1,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.03571,0.10101,0.0,0.0,0.0,0.0,0.42857,28,0,1,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.06694,0.15957,0.0,0.0,0.0,0.0,0.571,26,0,0,26,0,2,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[20,36,0.5556,0.09374,0.17713,0.0,0.0,0.14287,0.0,0.71429,23,0,0,23,0,2,0,0,5,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[24,36,0.6667,0.06241,0.11806,0.0,0.0,0.14071,0.0,0.4286,23,0,0,23,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,36,0.7778,0.0982,0.16914,0.0,0.0,0.14286,0.0,0.57143,20,0,0,20,0,8,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[32,36,0.8889,0.1116,0.16259,0.0,0.0,0.14286,0.0,0.57143,18,0,0,18,0,8,0,0,3,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[36,36,1.0,0.05357,0.13716,0.0,0.0,0.0,0.0,0.71429,25,0,0,25,0,5,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"573af7d0b6ad676f","q":"Let $m$ and $n$ be two positive integers. Prove that the integer $m^{2}+\\left\\lceil\\frac{4 m^{2}}{n}\\right\\rceil$ is not a perfect square.\n(Here, $\\lceil x\\rceil$ denotes the smallest integer not less than $x$.)","t":[{"b":1,"e":0.14286,"k":"flat","v":0.11161,"x":0.14732,"p":[[0,24,0.0,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.4286,3,0,0,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[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],[12,24,0.5,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,3,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,24,0.6667,0.14732,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,1,0,1,1,0,29,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.13839,0.05629,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,27,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.10705,"x":0.12054,"p":[[0,9,0.0,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,1,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.10705,0.06181,0.105,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.12054,0.10779,0.0,0.14286,0.14286,0.0,0.57143,9,0,0,9,0,21,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[9,9,1.0,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,1,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"14a206ffa66fb9f2","q":"Let $n$ be a positive integer. Show that in a set $A$ of $2^{n}$ strictly positive numbers, one can choose a subset $B$ of size $n+1$ such that the sum of any two different numbers in $B$ is never in $A$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.01786,"x":0.05795,"p":[[0,30,0.0,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.04018,0.08171,0.0,0.0,0.0,0.0,0.28571,25,0,3,25,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,2,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.02679,0.10374,0.0,0.0,0.0,0.0,0.57143,29,0,4,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,30,0.5333,0.02679,0.05577,0.0,0.0,0.0,0.0,0.143,26,0,3,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.01786,0.05922,0.0,0.0,0.0,0.0,0.2857,29,0,5,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.02679,0.07524,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.05795,0.08636,0.0,0.0,0.14286,0.0,0.2857,21,0,3,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00446,"x":0.08482,"p":[[0,22,0.0,0.07143,0.11294,0.0,0.0,0.14286,0.0,0.42857,21,0,1,21,0,7,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.05348,0.0927,0.0,0.0,0.14071,0.0,0.28571,23,0,0,23,0,6,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.08482,0.1984,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,22,0.5455,0.03116,0.06888,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"76d7e153c6e76b02","q":"Suppose $S$ tiles the natural numbers $\\mathbf{N}$. Show that $S$ tiles the set $\\{1,2, \\ldots, k\\}$ for some positive integer $k$.","t":[{"b":1,"e":0.42857,"k":"flat","v":0.39731,"x":0.4687,"p":[[0,3,0.0,0.39731,0.20432,0.24999,0.42857,0.57143,0.0,0.71429,2,0,0,2,0,6,0,0,5,0,0,5,0,0,12,0,0,2,0,0,0,0,0],[3,3,1.0,0.4687,0.13936,0.42857,0.4286,0.57143,0.14286,0.71429,0,0,0,0,0,2,0,0,4,0,0,11,0,0,13,0,0,2,0,0,0,0,0]]},{"b":7,"e":0.1429,"k":"flat","v":0.25,"x":0.4731,"p":[[0,22,0.0,0.42856,0.22867,0.2857,0.42857,0.57143,0.0,1.0,3,1,0,3,0,3,0,0,4,0,0,9,0,0,10,0,0,1,0,0,1,0,1],[4,22,0.1818,0.4731,0.22723,0.28571,0.42859,0.57143,0.14,1.0,0,2,0,0,0,5,0,0,5,0,0,7,0,0,10,0,0,2,0,0,1,0,2],[8,22,0.3636,0.33928,0.19805,0.2857,0.28571,0.42857,0.0,1.0,1,1,0,1,0,5,0,0,17,0,0,4,0,0,3,0,0,0,0,0,1,0,1],[12,22,0.5455,0.28115,0.16169,0.14286,0.2143,0.42857,0.14,0.57143,0,0,0,0,0,16,0,0,6,0,0,5,0,0,5,0,0,0,0,0,0,0,0],[16,22,0.7273,0.25,0.08748,0.14286,0.28571,0.28571,0.14286,0.4286,0,0,0,0,0,11,0,0,18,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.25893,0.12078,0.14286,0.28571,0.28571,0.14286,0.57143,0,0,0,0,0,13,0,0,14,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[22,22,1.0,0.28125,0.15765,0.14286,0.28571,0.32143,0.14286,0.85714,0,0,0,0,0,13,0,0,11,0,0,6,0,0,1,0,0,0,0,0,1,0,0]]}]},{"i":"a184b21fdd13f195","q":"Let points  $A_1$ ,  $A_2$  and  $A_3$  lie on the circle  $\\Gamma$  in a 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 rotation is equial to the angle at vertex  $A_i$  in  $\\triangle A_1A_2A_3$ . Further, define  $P_i$  to be the point  $\\tau_{i+2}(\\tau_{i}(\\tau_{i+1}(P)))$ , where the 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_1P_2P_3$  is at most the radius of  $\\Gamma$ .\n\n\n*Proposed by Anant Mudgal*","t":[{"b":1,"e":0.42857,"k":"flat","v":0.38839,"x":0.42857,"p":[[0,17,0.0,0.38839,0.1439,0.28571,0.42857,0.42857,0.0,1.0,1,1,1,1,0,0,0,0,10,0,0,20,0,0,0,0,0,0,0,0,0,0,1],[4,17,0.2353,0.38839,0.08918,0.42857,0.42857,0.42857,0.0,0.4286,1,0,1,1,0,0,0,0,6,0,0,25,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.41964,0.03458,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.42857,1e-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],[16,17,0.9412,0.41964,0.03458,0.42857,0.42857,0.42857,0.2857,0.4286,0,0,0,0,0,0,0,0,2,0,0,30,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.42857,0.0,0.42857,0.42857,0.42857,0.42857,0.4286,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.42857,"k":"flat","v":0.35268,"x":0.39732,"p":[[0,24,0.0,0.35714,0.11294,0.28571,0.42857,0.42857,0.0,0.4286,2,0,2,2,0,0,0,0,10,0,0,20,0,0,0,0,0,0,0,0,0,0,0],[4,24,0.1667,0.37946,0.09182,0.28571,0.42857,0.42857,0.0,0.4286,1,0,1,1,0,0,0,0,8,0,0,23,0,0,0,0,0,0,0,0,0,0,0],[8,24,0.3333,0.37054,0.11769,0.28571,0.42857,0.42857,0.0,0.57143,2,0,2,2,0,0,0,0,8,0,0,21,0,0,1,0,0,0,0,0,0,0,0],[12,24,0.5,0.36161,0.10092,0.28571,0.42857,0.42857,0.0,0.57143,1,0,1,1,0,0,0,0,13,0,0,17,0,0,1,0,0,0,0,0,0,0,0],[16,24,0.6667,0.35268,0.07129,0.28571,0.28571,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,17,0,0,15,0,0,0,0,0,0,0,0,0,0,0],[20,24,0.8333,0.36161,0.07129,0.28571,0.42857,0.42857,0.2857,0.42857,0,0,0,0,0,0,0,0,15,0,0,17,0,0,0,0,0,0,0,0,0,0,0],[24,24,1.0,0.39732,0.05906,0.42857,0.42857,0.42857,0.28571,0.4286,0,0,0,0,0,0,0,0,7,0,0,25,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b2fa93fa5f91df0b","q":"Let $n \\geq 2$ be an integer. Let $x_{1} \\geq x_{2} \\geq \\cdots \\geq x_{n}$ and $y_{1} \\geq y_{2} \\geq \\cdots \\geq y_{n}$ be $2 n$ real numbers such that  $$ \\begin{aligned} 0 & =x_{1}+x_{2}+\\cdots+x_{n}=y_{1}+y_{2}+\\cdots+y_{n} \\\\ \\text { and } \\quad 1 & =x_{1}^{2}+x_{2}^{2}+\\cdots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\\cdots+y_{n}^{2} \\end{aligned} $$  Prove that  $$ \\sum_{i=1}^{n}\\left(x_{i} y_{i}-x_{i} y_{n+1-i}\\right) \\geq \\frac{2}{\\sqrt{n-1}} $$","t":[{"b":2,"e":0.0,"k":"flat","v":0.00893,"x":0.12946,"p":[[0,12,0.0,0.12946,0.23517,0.0,0.0,0.17857,0.0,0.85714,23,0,3,23,0,1,0,0,2,0,0,3,0,0,1,0,0,1,0,0,1,0,0],[4,12,0.3333,0.01777,0.0777,0.0,0.0,0.0,0.0,0.4286,30,0,4,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,0,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.125,"x":0.16965,"p":[[0,17,0.0,0.15178,0.24983,0.0,0.0,0.2857,0.0,0.85714,20,0,5,20,0,3,0,0,4,0,0,2,0,0,0,0,0,1,0,0,2,0,0],[4,17,0.2353,0.13393,0.24206,0.0,0.0,0.14287,0.0,1.0,20,1,1,20,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[8,17,0.4706,0.13839,0.22442,0.0,0.0,0.17857,0.0,1.0,19,1,0,19,0,5,0,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,1],[12,17,0.7059,0.125,0.15465,0.0,0.0,0.2857,0.0,0.42857,17,0,0,17,0,6,0,0,5,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.125,0.13243,0.0,0.14286,0.28571,0.0,0.42857,15,0,0,15,0,7,0,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.16965,0.1448,0.0,0.14286,0.2857,0.0,0.4286,10,0,0,10,0,10,0,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"22c200c62edb747d","q":"Prove that, for every positive integer $n$, there exists an integer $m$ such that $2^{m}+m$ is divisible by $n$.  (Estonia)","t":[{"b":1,"e":0.14286,"k":"flat","v":0.16061,"x":0.31692,"p":[[0,18,0.0,0.31692,0.28508,0.0,0.2857,0.571,0.0,0.85714,11,0,2,11,0,2,0,0,5,0,0,4,0,0,5,0,0,3,0,0,2,0,0],[4,18,0.2222,0.26784,0.26902,0.0,0.14286,0.46418,0.0,0.85714,9,0,1,9,0,9,0,0,5,0,0,1,0,0,5,0,0,0,0,0,3,0,0],[8,18,0.4444,0.16061,0.17403,0.0,0.14286,0.2857,0.0,0.57143,13,0,0,13,0,9,0,0,5,0,0,3,0,0,2,0,0,0,0,0,0,0,0],[12,18,0.6667,0.29012,0.21266,0.14286,0.21431,0.42858,0.0,0.71429,4,0,0,4,0,12,0,0,4,0,0,5,0,0,5,0,0,2,0,0,0,0,0],[16,18,0.8889,0.21873,0.21715,0.0,0.14288,0.28571,0.0,0.85714,9,0,0,9,0,11,0,0,5,0,0,2,0,0,4,0,0,0,0,0,1,0,0],[18,18,1.0,0.22322,0.14698,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,18,0,0,6,0,0,5,0,0,0,0,0,1,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.19195,"x":0.34374,"p":[[0,22,0.0,0.30782,0.23457,0.14286,0.2857,0.4286,0.0,0.85714,4,0,0,4,0,10,0,0,7,0,0,4,0,0,4,0,0,1,0,0,2,0,0],[4,22,0.1818,0.25444,0.24148,0.10714,0.14286,0.4286,0.0,0.857,8,0,0,8,0,11,0,0,3,0,0,4,0,0,3,0,0,2,0,0,1,0,0],[8,22,0.3636,0.34374,0.29202,0.14286,0.2857,0.60714,0.0,0.85714,7,0,2,7,0,7,0,0,5,0,0,3,0,0,2,0,0,5,0,0,3,0,0],[12,22,0.5455,0.32142,0.28792,0.14286,0.1429,0.60682,0.0,0.85714,7,0,0,7,0,10,0,0,2,0,0,4,0,0,1,0,0,6,0,0,2,0,0],[16,22,0.7273,0.24098,0.14039,0.14286,0.2857,0.28571,0.0,0.71429,3,0,0,3,0,10,0,0,15,0,0,3,0,0,0,0,0,1,0,0,0,0,0],[20,22,0.9091,0.20972,0.1287,0.14286,0.1429,0.2857,0.0,0.571,4,0,0,4,0,13,0,0,12,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[22,22,1.0,0.19195,0.21007,0.0,0.14286,0.28571,0.0,0.857,11,0,0,11,0,9,0,0,8,0,0,1,0,0,1,0,0,1,0,0,1,0,0]]}]},{"i":"67a7f6be02ecc2e7","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  $$ a_{n+1}=\\left\\{\\begin{array}{ll} a_{n} b_{n}+a_{n-1}, & \\text { if } b_{n-1}=1 \\\\ a_{n} b_{n}-a_{n-1}, & \\text { if } b_{n-1}>1 \\end{array} \\quad \\text { for } n=1,2, \\ldots\\right. $$  Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to 2017 . (Australia)","t":[{"b":3,"e":0.0,"k":"flat","v":0.03125,"x":0.16955,"p":[[0,20,0.0,0.16516,0.18592,0.0,0.14286,0.2857,0.0,0.85714,11,0,1,11,0,12,0,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,20,0.2,0.16955,0.14034,0.0,0.14286,0.28571,0.0,0.57143,9,0,0,9,0,11,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,20,0.4,0.12045,0.11353,0.0,0.14286,0.1786,0.0,0.28571,13,0,0,13,0,11,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.15616,0.18337,0.0,0.14286,0.2857,0.0,0.71429,13,0,0,13,0,10,0,0,6,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[16,20,0.8,0.13393,0.15947,0.0,0.14286,0.14286,0.0,0.71429,12,0,0,12,0,15,0,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0],[20,20,1.0,0.03125,0.06902,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.2857,"k":"flat","v":0.10715,"x":0.14277,"p":[[0,7,0.0,0.10715,0.11845,0.0,0.14286,0.1429,0.0,0.42857,15,0,0,15,0,11,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.13392,0.12335,0.0,0.14286,0.14287,0.0,0.571,9,0,0,9,0,19,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[7,7,1.0,0.14277,0.06186,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7469858daab1a280","q":"Let $x$ and $y$ be positive integers. If $x^{2^{n}}-1$ is divisible by $2^{n} y+1$ for every positive integer $n$, prove that $x=1$.","t":[{"b":1,"e":0.0,"k":"flat","v":0.03125,"x":0.09822,"p":[[0,30,0.0,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,4,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.2857,15,0,0,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.03125,0.05906,0.0,0.0,0.0,0.0,0.14286,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0625,0.07936,0.0,0.0,0.14286,0.0,0.28571,19,0,0,19,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.06251,0.07087,0.0,0.0,0.14286,0.0,0.1429,18,0,1,18,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.09375,0.11071,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,30,0.9333,0.09822,0.07523,0.0,0.14286,0.14286,0.0,0.2857,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.02679,"x":0.04911,"p":[[0,41,0.0,0.04465,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,41,0.0976,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,1,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,41,0.1951,0.03116,0.08541,0.0,0.0,0.0,0.0,0.42857,27,0,2,27,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,41,0.2927,0.04018,0.06423,0.0,0.0,0.14286,0.0,0.14286,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,41,0.3902,0.03572,0.06186,0.0,0.0,0.03571,0.0,0.1429,24,0,0,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,41,0.4878,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,41,0.5854,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,41,0.6829,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,41,0.7805,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,41,0.878,0.04009,0.06409,0.0,0.0,0.14071,0.0,0.1429,23,0,1,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"ed8307a60cdcfac4","q":"One thousand people are in a tennis tournament where each person plays against each other person exactly once, and there are no ties. Prove that it is possible to put all the competitors in a line so that each of the 998 people who are not at an end of the line either defeated both their neighbors or lost to both their neighbors","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,30,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,29,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,29,0.1379,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,6,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.00884,0.03424,0.0,0.0,0.0,0.0,0.1429,30,0,8,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,14,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,29,0.6897,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,8,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,6,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6b0575b9730b8dd9","q":"Let a triangle  $ABC$  inscribed in circle  $c(O,R)$  and  $D$  an arbitrary point on  $BC$ (different from the midpoint).The circumscribed circle of  $BOD$ ,which is  $(c_1)$ , meets  $c(O,R)$  at  $K$  and  $AB$  at  $Z$ .The circumscribed circle of  $COD$   $(c_2)$ ,meets  $c(O,R)$  at  $M$  and  $AC$  at  $E$ .Finally, the circumscribed circle of  $AEZ$   $(c_3)$ ,meets  $c(O,R)$  at  $N$ .Prove that  $\\triangle{ABC}=\\triangle{KMN}.$","t":[{"b":2,"e":0.0,"k":"flat","v":0.02679,"x":0.20982,"p":[[0,11,0.0,0.08034,0.19209,0.0,0.0,0.0,0.0,0.85714,25,0,1,25,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,11,0.3636,0.06688,0.10699,0.0,0.0,0.14286,0.0,0.42857,21,0,0,21,0,8,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.20982,0.32337,0.0,0.07143,0.1786,0.0,1.0,16,3,0,16,0,8,0,0,2,0,0,1,0,0,0,0,0,1,0,0,1,0,3],[11,11,1.0,0.02679,0.06622,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.11152,"p":[[0,10,0.0,0.03572,0.08749,0.0,0.0,0.0,0.0,0.42857,26,0,2,26,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.11152,0.22226,0.0,0.0,0.14286,0.0,1.0,21,1,0,21,0,6,0,0,2,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,10,0.8,0.07143,0.18558,0.0,0.0,0.03571,0.0,1.0,24,1,0,24,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[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":"dd425cdf2fc1a671","q":"The circle  $\\Gamma$  and the line  $\\ell$  have no common points. Let  $AB$  be the diameter of  $\\Gamma$  perpendicular to  $\\ell$ , with  $B$  closer to  $\\ell$  than  $A$ . An arbitrary point  $C\\not= A$ ,  $B$  is chosen on  $\\Gamma$ . The line  $AC$  intersects  $\\ell$  at  $D$ . The line  $DE$  is tangent to  $\\Gamma$  at  $E$ , with  $B$  and  $E$  on the same side of  $AC$ . Let  $BE$  intersect  $\\ell$  at  $F$ , and let  $AF$  intersect  $\\Gamma$  at  $G\\not= A$ . Let  $H$  be the reflection of  $G$  in  $AB$ . Show that  $F,C$ , and  $H$  are collinear.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00447,"p":[[0,9,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"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],[8,20,0.4,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,20,0.6,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,20,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,0,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,20,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"85ef54cad05b41e2","q":"Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n))=f(n)+1$ and $g(f(n))=$ $g(n)+1$ hold for all positive integers. Prove that $f(n)=g(n)$ for all positive integer $n$. (Germany)","t":[{"b":1,"e":0.0,"k":"flat","v":0.01339,"x":0.09375,"p":[[0,25,0.0,0.09375,0.17715,0.0,0.0,0.14286,0.0,0.857,21,0,0,21,0,6,0,0,3,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,25,0.16,0.05353,0.11007,0.0,0.0,0.07036,0.0,0.42857,23,0,1,23,2,5,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,25,0.32,0.04009,0.07116,0.0,0.0,0.07143,0.0,0.2857,23,0,1,23,2,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,25,0.48,0.04464,0.0974,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,25,0.64,0.05348,0.11703,0.0,0.0,0.035,0.0,0.57143,24,0,0,24,0,6,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[20,25,0.8,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,25,0.96,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[25,25,1.0,0.0267,0.06606,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.02901,"x":0.09598,"p":[[0,18,0.0,0.09598,0.19034,0.0,0.0,0.14286,0.0,0.85714,19,0,2,19,1,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0],[4,18,0.2222,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,1,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.02901,0.08349,0.0,0.0,0.0,0.0,0.4286,27,0,0,27,1,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,18,0.6667,0.04911,0.07668,0.0,0.0,0.14286,0.0,0.28571,22,0,0,22,0,9,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,18,0.8889,0.07143,0.17126,0.0,0.0,0.03571,0.0,0.857,24,0,0,24,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[18,18,1.0,0.08036,0.11259,0.0,0.0,0.14286,0.0,0.42857,19,0,0,19,0,9,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d2dd388da8f181bd","q":"Let $n$ be a positive integer and let $W=\\ldots x_{-1} x_{0} x_{1} x_{2} \\ldots$ be an infinite periodic word consisting of the letters $a$ and $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^{n}$.  A finite nonempty word $U$ is said to appear in $W$ if there exist indices $k \\leq \\ell$ such that $U=x_{k} x_{k+1} \\ldots x_{\\ell}$. A finite word $U$ is called ubiquitous if the four words $U a, U b, a U$, and $b U$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.00884,"p":[[0,66,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,66,0.0606,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,66,0.1212,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,66,0.1818,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,66,0.2424,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,66,0.303,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[28,66,0.4242,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,66,0.4848,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,4,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,66,0.5455,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,66,0.6061,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,66,0.6667,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,66,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,66,0.7879,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,66,0.8485,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,66,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,66,0.9697,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[66,66,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,32,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[8,32,0.25,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[16,32,0.5,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,32,0.625,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,32,0.75,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,32,0.875,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c1f8201cafd36fb5","q":"Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from  $1$  to  $2024$  that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by  $2024$  loses. Who wins if every player wants to win?\n\n*Proposed by Mykhailo Shtandenko*","t":[{"b":3,"e":0.0,"k":"flat","v":0.01786,"x":0.42411,"p":[[0,43,0.0,0.18304,0.35756,0.0,0.0,0.14286,0.0,1.0,22,5,0,22,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,5],[4,43,0.093,0.26339,0.4317,0.0,0.0,0.57143,0.0,1.0,23,8,4,23,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8],[8,43,0.186,0.42411,0.4864,0.0,0.0,1.0,0.0,1.0,18,13,3,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13],[12,43,0.2791,0.31697,0.46116,0.0,0.0,1.0,0.0,1.0,21,10,6,21,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10],[16,43,0.3721,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,8,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6],[20,43,0.4651,0.13839,0.33405,0.0,0.0,0.0,0.0,1.0,27,4,6,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[24,43,0.5581,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,6,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[28,43,0.6512,0.01786,0.04725,0.0,0.0,0.0,0.0,0.1429,28,0,6,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,43,0.7442,0.10714,0.29233,0.0,0.0,0.0,0.0,1.0,27,3,18,27,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[36,43,0.8372,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,17,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,43,0.9302,0.1875,0.39031,0.0,0.0,0.0,0.0,1.0,26,6,17,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6]]},{"b":6,"e":0.0,"k":"falling","v":0.00893,"x":0.2,"p":[[0,5,0.0,0.2,0.35992,0.0,0.0,0.14286,0.0,1.0,21,5,0,21,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,5],[4,5,0.8,0.05804,0.22548,0.0,0.0,0.0,0.0,1.0,30,1,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[5,5,1.0,0.00893,0.04971,0.0,0.0,0.0,0.0,0.28571,31,0,1,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"61c9a0ae62419b5a","q":"Suppose that a positive integer $n$ has the property that $n, 2 n, 3 n, \\ldots, 9 n$ are all palindromes. Prove that the decimal digits of $n$ are all zeros or ones.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.14277,"x":0.38394,"p":[[0,46,0.0,0.31249,0.20023,0.14286,0.28571,0.42857,0.0,0.85714,2,0,0,2,0,10,0,0,9,0,0,5,0,0,4,0,0,1,0,0,1,0,0],[4,46,0.087,0.38394,0.25614,0.14286,0.28571,0.57143,0.14286,1.0,0,1,0,0,0,11,0,0,9,0,0,2,0,0,3,0,0,4,0,0,2,0,1],[8,46,0.1739,0.2589,0.18008,0.14286,0.2857,0.28571,0.0,0.71429,4,0,0,4,0,10,0,0,12,0,0,1,0,0,4,0,0,1,0,0,0,0,0],[12,46,0.2609,0.30784,0.21768,0.14286,0.28571,0.28571,0.0,1.0,1,1,0,1,0,11,0,0,13,0,0,2,0,0,2,0,0,1,0,0,1,0,1],[16,46,0.3478,0.31696,0.19474,0.14286,0.28571,0.28571,0.14286,0.85714,0,0,0,0,0,10,0,0,15,0,0,2,0,0,2,0,0,1,0,0,2,0,0],[20,46,0.4348,0.34375,0.19516,0.24999,0.28571,0.42857,0.14286,1.0,0,1,0,0,0,8,0,0,13,0,0,6,0,0,2,0,0,2,0,0,0,0,1],[24,46,0.5217,0.32579,0.19317,0.2857,0.28571,0.28571,0.0,1.0,1,1,0,1,0,5,0,0,19,0,0,3,0,0,2,0,0,0,0,0,1,0,1],[28,46,0.6087,0.23652,0.07679,0.14286,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,9,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,46,0.6957,0.18732,0.07534,0.14286,0.14286,0.28571,0.0,0.28571,1,0,0,1,0,20,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,46,0.7826,0.18294,0.06428,0.14286,0.14286,0.2857,0.14,0.28571,0,0,0,0,0,23,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,46,0.8696,0.14724,0.02487,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,46,0.9565,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[46,46,1.0,0.14286,1e-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":3,"e":0.85714,"k":"flat","v":0.1741,"x":0.39277,"p":[[0,11,0.0,0.29907,0.2153,0.14286,0.28571,0.42858,0.0,0.85714,5,0,0,5,0,8,0,0,7,0,0,5,0,0,6,0,0,0,0,0,1,0,0],[4,11,0.3636,0.39277,0.31346,0.14286,0.28571,0.5,0.0,1.0,2,3,1,2,0,11,0,0,6,0,0,5,0,0,0,0,0,1,0,0,4,0,3],[8,11,0.7273,0.27222,0.20631,0.14286,0.28571,0.28571,0.0,0.85714,4,0,0,4,0,10,0,0,11,0,0,3,0,0,1,0,0,2,0,0,1,0,0],[11,11,1.0,0.1741,0.2251,0.0,0.14286,0.2857,0.0,0.85714,15,0,0,15,0,8,0,0,2,0,0,3,0,0,3,0,0,0,0,0,1,0,0]]}]},{"i":"33dc4446148abe03","q":"The circles $\\Gamma_{1}$ and $\\Gamma_{2}$ intersect at $D$ and $P$. The common tangent of the two circles closest to point $D$ touches $\\Gamma_{1}$ at $A$ and $\\Gamma_{2}$ at $B$. The line $A D$ intersects $\\Gamma_{2}$ again at $C$. Let $M$ be the midpoint of segment $B C$.\nProve that $\\angle D P M=\\angle B D C$.\n\n\u4fdd\u7559\u4e86\u6e90\u6587\u672c\u7684\u6362\u884c\u548c\u683c\u5f0f\uff0c\u76f4\u63a5\u8f93\u51fa\u4e86\u7ffb\u8bd1\u7ed3\u679c\u3002","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.07589,"p":[[0,10,0.0,0.07589,0.12869,0.0,0.0,0.14287,0.0,0.42857,23,0,0,23,0,2,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,22,0.0,0.03563,0.10092,0.0,0.0,0.0,0.0,0.4286,28,0,0,28,0,1,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.04018,0.10853,0.0,0.0,0.0,0.0,0.42857,28,0,0,28,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0067,0.02743,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"aa7bf3132ce1c330","q":"Let $c>2$, and let $a(1), a(2), \\ldots$ be a sequence of nonnegative real numbers such that  $$ a(m+n) \\leq 2 a(m)+2 a(n) \\text { for all } m, n \\geq 1 \\text {, } $$  and  $$ a\\left(2^{k}\\right) \\leq \\frac{1}{(k+1)^{c}} \\quad \\text { for all } k \\geq 0 $$  Prove that the sequence $a(n)$ is bounded.  (Croatia)","t":[{"b":0,"e":0.0,"k":"flat","v":0.05357,"x":0.05357,"p":[[0,6,0.0,0.05357,0.09942,0.0,0.0,0.14286,0.0,0.42857,23,0,16,23,0,7,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.07143,"x":0.07589,"p":[[0,13,0.0,0.07143,0.12372,0.0,0.0,0.14286,0.0,0.42857,23,0,17,23,0,3,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.07589,0.12364,0.0,0.0,0.14286,0.0,0.42857,21,0,13,21,0,7,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.07143,0.11845,0.0,0.0,0.14286,0.0,0.42857,22,0,3,22,0,5,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"458df2de3c8699bc","q":"Suppose we have a  $n$ -gon. Some  $n-3$  diagonals are coloured black and some other  $n-3$  diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of  $n$ .\n\n*Proposed by Alexander Ivanov, Bulgaria*","t":[{"b":5,"e":0.14286,"k":"flat","v":0.09813,"x":0.125,"p":[[0,47,0.0,0.09813,0.06616,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,47,0.0851,0.11607,0.06622,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,24,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,47,0.1702,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,47,0.2553,0.125,0.07784,0.14286,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,22,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,47,0.3404,0.12054,0.05187,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,47,0.4255,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.14286,4,0,0,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,47,0.5106,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.1429,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,47,0.5957,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,47,0.6809,0.11161,0.06902,0.10714,0.14286,0.14286,0.0,0.2857,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,47,0.766,0.12045,0.05183,0.14286,0.14286,0.14286,0.0,0.14286,5,0,0,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,47,0.8511,0.10714,0.06186,0.10714,0.14286,0.14286,0.0,0.14286,8,0,0,8,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,47,0.9362,0.11152,0.05901,0.14214,0.14286,0.14286,0.0,0.14286,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[47,47,1.0,0.10268,0.06423,0.0,0.14286,0.14286,0.0,0.14286,9,0,0,9,0,23,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.0892,"x":0.14286,"p":[[0,28,0.0,0.11152,0.06898,0.105,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,23,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,28,0.1429,0.10269,0.08919,0.0,0.14286,0.14286,0.0,0.286,12,0,1,12,0,17,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,28,0.2857,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.1429,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,28,0.4286,0.0892,0.06909,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,0,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.09812,0.07518,0.0,0.14286,0.14286,0.0,0.28571,11,0,0,11,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,28,0.7143,0.14286,0.06186,0.14286,0.14286,0.14286,0.0,0.28571,3,0,0,3,0,26,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.12483,0.05918,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.11161,0.08552,0.0,0.14286,0.14286,0.0,0.42857,9,0,0,9,0,22,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"27fb11e2d0d3e4c0","q":"Points  $C_1, B_1$  on sides  $AB, AC$  respectively of triangle  $ABC$  are such that  $BB_1 \\perp  CC_1$ . Point  $X$  lying inside the triangle is such that  $\\angle XBC = \\angle B_1BA, \\angle XCB = \\angle C_1CA$ . Prove that  $\\angle B_1XC_1 =90^o- \\angle A$ .\n\n(A. Antropov, A. Yakubov)","t":[{"b":1,"e":0.14286,"k":"flat","v":0.11161,"x":0.19196,"p":[[0,44,0.0,0.16518,0.09523,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,28,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,44,0.0909,0.19196,0.08458,0.14286,0.14286,0.2857,0.14286,0.42857,0,0,0,0,0,23,0,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,44,0.1818,0.19195,0.12678,0.14286,0.14286,0.2857,0.0,0.57143,3,0,0,3,0,19,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0],[12,44,0.2727,0.15616,0.12557,0.14214,0.14286,0.14287,0.0,0.57143,7,0,0,7,0,18,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,44,0.3636,0.15179,0.11258,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,23,0,0,2,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,44,0.4545,0.15179,0.10677,0.14286,0.14286,0.14286,0.0,0.57143,5,0,0,5,0,22,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[24,44,0.5455,0.11161,0.07771,0.0,0.14286,0.14286,0.0,0.28571,9,0,0,9,0,21,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,44,0.6364,0.125,0.13243,0.0,0.14286,0.14286,0.0,0.71429,10,0,0,10,0,19,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[32,44,0.7273,0.16071,0.04724,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,28,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,44,0.8182,0.15179,0.07936,0.14286,0.14286,0.14286,0.0,0.42857,3,0,0,3,0,25,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,44,0.9091,0.18304,0.08918,0.14286,0.14286,0.14286,0.14286,0.4286,0,0,0,0,0,26,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[44,44,1.0,0.1517,0.0346,0.14286,0.14286,0.14286,0.14,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.14714,"x":0.20089,"p":[[0,8,0.0,0.20089,0.13767,0.14286,0.14286,0.2857,0.0,0.57143,2,0,0,2,0,21,0,0,6,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[4,8,0.5,0.17411,0.09268,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,25,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,8,1.0,0.14714,0.0249,0.14286,0.14286,0.14286,0.14,0.2857,0,0,0,0,0,31,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bbb19fa1df56d869","q":"Now suppose again that $n$ is odd. Prove that\n\n$$\n\\sigma(1)\\left\\lfloor\\log _{2} n\\right\\rfloor+\\sigma(3)\\left\\lfloor\\log _{2}(n / 3)\\right\\rfloor+\\sigma(5)\\left\\lfloor\\log _{2}(n / 5)\\right\\rfloor+\\cdots+\\sigma(n)\\left\\lfloor\\log _{2}(n / n)\\right\\rfloor<n^{2} / 8\n$$","t":[{"b":2,"e":0.28571,"k":"falling","v":0.09821,"x":0.30804,"p":[[0,5,0.0,0.30804,0.35734,0.14286,0.14286,0.28571,0.0,1.0,6,5,0,6,0,18,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,5],[4,5,0.8,0.09821,0.09062,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.11607,0.11538,0.0,0.14286,0.14286,0.0,0.42857,12,0,0,12,0,16,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.02233,"x":0.25438,"p":[[0,12,0.0,0.25438,0.3004,0.105,0.14286,0.32142,0.0,1.0,8,3,1,8,0,15,0,0,1,0,0,3,0,0,0,0,0,2,0,0,0,0,3],[4,12,0.3333,0.20536,0.33681,0.0,0.14286,0.14286,0.0,1.0,15,4,0,15,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,4],[8,12,0.6667,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.02233,0.05188,0.0,0.0,0.0,0.0,0.143,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f629186de0cdf434","q":"On a $5 \\times 5$ board, $n$ white markers are positioned, each marker in a distinct $1 \\times 1$ square. A smart child got an assignment to recolor in black as many markers as possible, in the following manner: a white marker is taken from the board; it is colored in black, and then put back on the board on an empty square such that none of the neighboring squares contains a white marker (two squares are called neighboring if they contain a common side). If it is possible for the child to succeed in coloring all the markers black, we say that the initial positioning of the markers was good.\n\na) Prove that if $n=20$, then a good initial positioning exists.\n\nb) Prove that if $n=21$, then a good initial positioning does not exist.","t":[{"b":0,"e":0.0,"k":"flat","v":0.09803,"x":0.09803,"p":[[0,3,0.0,0.09803,0.11531,0.0,0.0,0.14286,0.0,0.28571,17,0,15,17,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"volatile","v":0.10259,"x":0.28572,"p":[[0,4,0.0,0.10259,0.10849,0.0,0.14143,0.14286,0.0,0.28571,15,0,10,15,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.28572,5e-05,0.2857,0.28571,0.28571,0.2857,0.286,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6ab06c51bb421c03","q":"Show that for all real numbers $x_{1}, \\ldots, x_{n}$ the following inequality holds:  $$ \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\sqrt{\\left|x_{i}-x_{j}\\right|} \\leqslant \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\sqrt{\\left|x_{i}+x_{j}\\right|} $$","t":[{"b":2,"e":0.0,"k":"volatile","v":0.01786,"x":0.38397,"p":[[0,10,0.0,0.16964,0.36498,0.0,0.0,0.0,0.0,1.0,26,5,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5],[4,10,0.4,0.38397,0.44384,0.0,0.0,0.89286,0.0,1.0,17,8,0,17,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,2,0,8],[8,10,0.8,0.01786,0.09942,0.0,0.0,0.0,0.0,0.57143,31,0,0,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[10,10,1.0,0.07589,0.16746,0.0,0.0,0.0,0.0,0.57143,26,0,0,26,0,1,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.2857,"p":[[0,17,0.0,0.22321,0.36059,0.0,0.0,0.46429,0.0,1.0,22,2,0,22,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,0,3,0,2],[4,17,0.2353,0.2857,0.40247,0.0,0.0,0.71429,0.0,1.0,20,5,0,20,0,1,0,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,5],[8,17,0.4706,0.25893,0.36672,0.0,0.0,0.46431,0.0,1.0,20,3,0,20,0,0,0,0,1,0,0,3,0,0,1,0,0,2,0,0,2,0,3],[12,17,0.7059,0.13839,0.28456,0.0,0.0,0.0,0.0,0.85714,25,0,0,25,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0,0],[16,17,0.9412,0.02679,0.14914,0.0,0.0,0.0,0.0,0.85714,31,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[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]]}]},{"i":"939d7499dd012217","q":"Let $N \\geqslant 2$ be an integer. $N(N+1)$ soccer players, no two of the same height, stand in a row in some order. Coach Ralph wants to remove $N(N-1)$ people from this row so that in the remaining row of $2 N$ players, no one stands between the two tallest ones, no one stands between the third and the fourth tallest ones, ..., and finally no one stands between the two shortest ones. Show that this is always possible. (Russia)","t":[{"b":1,"e":0.0,"k":"flat","v":0.00446,"x":0.04018,"p":[[0,14,0.0,0.04018,0.10248,0.0,0.0,0.0,0.0,0.42857,27,0,22,27,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,20,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,21,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,22,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.03571,"x":0.09821,"p":[[0,6,0.0,0.03571,0.07986,0.0,0.0,0.0,0.0,0.28571,26,0,22,26,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.09821,0.13092,0.0,0.0,0.17857,0.0,0.42857,19,0,13,19,0,5,0,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"09fbb61dd2f0ac10","q":"Let $m$ and $n$ be positive integers such that $m>n$. Define $x_{k}=(m+k) /(n+k)$ for $k=$ $1,2, \\ldots, n+1$. Prove that if all the numbers $x_{1}, x_{2}, \\ldots, x_{n+1}$ are integers, then $x_{1} x_{2} \\cdots x_{n+1}-1$ is divisible by an odd prime. (Austria)","t":[{"b":0,"e":0.14286,"k":"falling","v":0.05357,"x":0.29911,"p":[[0,12,0.0,0.25445,0.31284,0.0,0.14286,0.42857,0.0,1.0,11,3,0,11,0,11,0,0,1,0,0,3,0,0,1,0,0,2,0,0,0,0,3],[4,12,0.3333,0.29911,0.38525,0.0,0.14286,0.42857,0.0,1.0,11,7,1,11,0,12,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,7],[8,12,0.6667,0.11607,0.24074,0.0,0.0,0.14286,0.0,1.0,19,2,0,19,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[12,12,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.1429,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.16964,"x":0.43749,"p":[[0,10,0.0,0.43749,0.43439,0.0,0.21429,1.0,0.0,1.0,11,10,0,11,0,5,0,0,2,0,0,0,0,0,2,0,0,1,0,0,1,0,10],[4,10,0.4,0.37053,0.37941,0.14286,0.14286,0.85704,0.0,1.0,5,6,1,5,0,15,0,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,6],[8,10,0.8,0.24107,0.21261,0.0,0.2857,0.42857,0.0,1.0,9,1,3,9,0,5,0,0,9,0,0,8,0,0,0,0,0,0,0,0,0,0,1],[10,10,1.0,0.16964,0.14914,0.0,0.14286,0.28571,0.0,0.42857,11,0,0,11,0,8,0,0,9,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0721a8cecb30f019","q":"Let ABCD be a convex quadrilateral and O a point inside it. Let the parallels to the lines BC, AB, DA, CD through the point O meet the sides AB, BC, CD, DA of the quadrilateral ABCD at the points E, F, G, H, respectively. Then, prove that  $ \\sqrt {\\left|AHOE\\right|} \\plus{} \\sqrt {\\left|CFOG\\right|}\\leq\\sqrt {\\left|ABCD\\right|}$ , where  $ \\left|P_1P_2...P_n\\right|$  is an abbreviation for the non-directed area of an arbitrary polygon  $ P_1P_2...P_n$ .","t":[{"b":1,"e":0.1429,"k":"flat","v":0.03125,"x":0.0759,"p":[[0,17,0.0,0.0759,0.09439,0.0,0.0,0.14286,0.0,0.28571,18,0,2,18,0,11,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.07134,0.10095,0.0,0.0,0.14286,0.0,0.28571,20,0,3,20,0,8,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.1429,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.03125,0.06902,0.0,0.0,0.0,0.0,0.2857,26,0,1,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.04464,0.07523,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.05357,0.07784,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.11607,"p":[[0,7,0.0,0.11607,0.10972,0.0,0.14286,0.1429,0.0,0.28571,13,0,0,13,0,12,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,7,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[7,7,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f0821dcfd26ed4f8","q":"Let $n$ be a positive integer. Prove that the equation\n\n$$\nx+y+\\frac{1}{x}+\\frac{1}{y}=3 n\n$$\n\ndoes not have solutions in positive rational numbers.","t":[{"b":0,"e":0.0,"k":"flat","v":0.16963,"x":0.29017,"p":[[0,5,0.0,0.16963,0.32228,0.0,0.0,0.14286,0.0,1.0,22,3,3,22,0,4,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,3],[4,5,0.8,0.29017,0.40639,0.0,0.0,0.60714,0.0,1.0,18,6,2,18,0,4,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,6]]},{"b":4,"e":0.0,"k":"falling","v":0.00446,"x":0.24107,"p":[[0,22,0.0,0.15625,0.30797,0.0,0.0,0.14286,0.0,1.0,21,3,2,21,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,3],[4,22,0.1818,0.12054,0.28818,0.0,0.0,0.14286,0.0,1.0,23,3,4,23,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3],[8,22,0.3636,0.24107,0.3787,0.0,0.0,0.17857,0.0,1.0,18,5,2,18,0,6,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,5],[12,22,0.5455,0.16062,0.26667,0.0,0.0,0.14286,0.0,1.0,17,1,5,17,0,9,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,1],[16,22,0.7273,0.19197,0.28033,0.0,0.14286,0.14286,0.0,1.0,15,2,7,15,0,10,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,2],[20,22,0.9091,0.12482,0.20437,0.0,0.14143,0.14286,0.0,1.0,14,1,1,14,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[22,22,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"a93691f2aac71609","q":"Let $n \\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2 n$ positive integers satisfying the following property: For every $m=2,3, \\ldots, n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.  (Iceland)","t":[{"b":0,"e":0.14286,"k":"flat","v":0.18304,"x":0.31695,"p":[[0,43,0.0,0.23213,0.3149,0.0,0.0,0.46418,0.0,1.0,18,2,11,18,0,2,0,0,2,0,0,2,0,0,4,0,0,2,0,0,0,0,2],[4,43,0.093,0.31695,0.35306,0.0,0.14286,0.57143,0.0,1.0,13,4,7,13,0,5,0,0,1,0,0,2,0,0,5,0,0,2,0,0,0,0,4],[8,43,0.186,0.24112,0.27997,0.0,0.07143,0.42893,0.0,0.85714,16,0,5,16,0,1,0,0,4,0,0,4,0,0,4,0,0,1,0,0,2,0,0],[12,43,0.2791,0.18304,0.25313,0.0,0.0,0.42857,0.0,1.0,18,1,8,18,0,3,0,0,2,0,0,5,0,0,3,0,0,0,0,0,0,0,1],[16,43,0.3721,0.21875,0.28344,0.0,0.07143,0.32143,0.0,1.0,16,1,7,16,0,3,0,0,5,0,0,3,0,0,1,0,0,2,0,0,1,0,1],[20,43,0.4651,0.26782,0.24931,0.0,0.28571,0.42857,0.0,0.85714,12,0,4,12,0,2,0,0,5,0,0,7,0,0,4,0,0,1,0,0,1,0,0],[24,43,0.5581,0.26784,0.27139,0.0,0.2857,0.4286,0.0,1.0,14,1,10,14,0,0,0,0,5,0,0,6,0,0,5,0,0,1,0,0,0,0,1],[28,43,0.6512,0.27231,0.2509,0.0,0.2857,0.42857,0.0,0.85714,12,0,4,12,0,2,0,0,4,0,0,8,0,0,4,0,0,1,0,0,1,0,0],[32,43,0.7442,0.22765,0.21971,0.0,0.14288,0.42857,0.0,0.57143,12,0,2,12,0,5,0,0,5,0,0,4,0,0,6,0,0,0,0,0,0,0,0],[36,43,0.8372,0.25891,0.18705,0.14286,0.28571,0.42857,0.0,0.71429,6,0,3,6,0,8,0,0,8,0,0,7,0,0,2,0,0,1,0,0,0,0,0],[40,43,0.9302,0.19195,0.18763,0.0,0.14286,0.42857,0.0,0.571,13,0,8,13,0,5,0,0,5,0,0,8,0,0,1,0,0,0,0,0,0,0,0],[43,43,1.0,0.2009,0.15916,0.0,0.28571,0.28571,0.0,0.4286,10,0,4,10,0,5,0,0,11,0,0,6,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.27675,"p":[[0,57,0.0,0.16518,0.22619,0.0,0.0,0.28571,0.0,1.0,18,1,11,18,0,1,0,0,7,0,0,5,0,0,0,0,0,0,0,0,0,0,1],[4,57,0.0702,0.27675,0.27646,0.0,0.21428,0.4642,0.0,1.0,11,1,5,11,0,5,0,0,5,0,0,3,0,0,4,0,0,3,0,0,0,0,1],[8,57,0.1404,0.17411,0.28287,0.0,0.0,0.17857,0.0,1.0,20,1,10,20,0,4,0,0,1,0,0,1,0,0,2,0,0,3,0,0,0,0,1],[12,57,0.2105,0.24107,0.28669,0.0,0.14286,0.42857,0.0,1.0,15,1,9,15,0,3,0,0,4,0,0,3,0,0,4,0,0,1,0,0,1,0,1],[16,57,0.2807,0.16518,0.24513,0.0,0.0,0.32143,0.0,1.0,19,1,11,19,0,3,0,0,2,0,0,5,0,0,2,0,0,0,0,0,0,0,1],[20,57,0.3509,0.12045,0.19919,0.0,0.0,0.1786,0.0,0.57143,22,0,13,22,0,2,0,0,2,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[24,57,0.4211,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[28,57,0.4912,0.07143,0.20825,0.0,0.0,0.0,0.0,1.0,27,1,0,27,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[32,57,0.5614,0.02232,0.10171,0.0,0.0,0.0,0.0,0.57143,30,0,4,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[36,57,0.6316,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,57,0.7018,0.01786,0.09942,0.0,0.0,0.0,0.0,0.5714,31,0,2,31,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[44,57,0.7719,0.02679,0.12595,0.0,0.0,0.0,0.0,0.71429,30,0,1,30,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[48,57,0.8421,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,4,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,57,0.9123,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,57,0.9825,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"02cdafba78557682","q":"Rays  $l$  and  $m$  forming an angle of  $a$  are drawn from the same point. Let  $P$  be a fixed point on  $l$ . For each circle  $C$  tangent to  $l$  at  $P$  and intersecting  $m$  at  $Q$  and  $R$ , let  $T$  be the intersection point of the bisector of angle  $QPR$  with  $C$ . Describe the locus of  $T$  and justify your answer.","t":[{"b":1,"e":0.57143,"k":"flat","v":0.33032,"x":0.39281,"p":[[0,23,0.0,0.39281,0.23686,0.14289,0.4286,0.57143,0.0,0.71429,4,0,1,4,0,5,0,0,5,0,0,4,0,0,9,0,0,5,0,0,0,0,0],[4,23,0.1739,0.33032,0.2407,0.10714,0.35714,0.4642,0.0,0.71429,8,0,1,8,0,2,0,0,6,0,0,8,0,0,4,0,0,4,0,0,0,0,0],[8,23,0.3478,0.36157,0.17486,0.2857,0.28571,0.4642,0.0,0.71429,3,0,0,3,0,1,0,0,13,0,0,7,0,0,7,0,0,1,0,0,0,0,0],[12,23,0.5217,0.36153,0.17843,0.2857,0.28571,0.571,0.0,0.857,2,0,0,2,0,1,0,0,18,0,0,2,0,0,8,0,0,0,0,0,1,0,0],[16,23,0.6957,0.34371,0.14217,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,1,0,0,21,0,0,3,0,0,5,0,0,1,0,0,0,0,0],[20,23,0.8696,0.38834,0.17576,0.2857,0.28571,0.57111,0.0,0.71429,2,0,0,2,0,0,0,0,16,0,0,3,0,0,9,0,0,2,0,0,0,0,0],[23,23,1.0,0.34372,0.20156,0.2857,0.28571,0.4286,0.0,1.0,3,1,0,3,0,1,0,0,19,0,0,2,0,0,5,0,0,1,0,0,0,0,1]]},{"b":6,"e":0.1429,"k":"flat","v":0.10268,"x":0.36158,"p":[[0,27,0.0,0.30352,0.2307,0.10717,0.28571,0.4642,0.0,0.85714,8,0,1,8,0,1,0,0,13,0,0,2,0,0,6,0,0,1,0,0,1,0,0],[4,27,0.1481,0.36158,0.2672,0.10714,0.42857,0.57111,0.0,1.0,8,1,1,8,0,1,0,0,6,0,0,7,0,0,5,0,0,4,0,0,0,0,1],[8,27,0.2963,0.23654,0.22467,0.0,0.2857,0.42858,0.0,0.57143,13,0,1,13,0,1,0,0,9,0,0,2,0,0,7,0,0,0,0,0,0,0,0],[12,27,0.4444,0.10268,0.1439,0.0,0.0,0.14286,0.0,0.4286,19,0,0,19,0,6,0,0,4,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[16,27,0.5926,0.14285,0.22585,0.0,0.0,0.1429,0.0,1.0,17,1,0,17,0,9,0,0,1,0,0,2,0,0,2,0,0,0,0,0,0,0,1],[20,27,0.7407,0.13838,0.15351,0.0,0.14286,0.2857,0.0,0.571,14,0,0,14,0,9,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[24,27,0.8889,0.12492,0.20747,0.0,0.0,0.1429,0.0,0.8571,17,0,0,17,0,11,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0],[27,27,1.0,0.15626,0.13534,0.0,0.14286,0.17868,0.0,0.4286,9,0,0,9,0,15,0,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"b56a07bcfcbb15dd","q":"Let $\\mathrm{n} \\geqslant 1$ be an integer.\nConsider a set of $4 \\mathrm{n}+5$ points in the plane, no three of which are collinear, and each colored either red or blue.\n\nProve that there exist $n$ triangles whose vertices are all of the same color (the same for all triangles), and whose respective interiors are pairwise disjoint and do not contain any colored points.","t":[{"b":5,"e":0.0,"k":"flat","v":0.00438,"x":0.03125,"p":[[0,17,0.0,0.01116,0.0362,0.0,0.0,0.0,0.0,0.14286,29,0,2,29,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[17,17,1.0,0.03125,0.06901,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.02455,"p":[[0,11,0.0,0.02455,0.06341,0.0,0.0,0.0,0.0,0.2857,27,0,1,27,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,2,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.00438,0.02436,0.0,0.0,0.0,0.0,0.14,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[11,11,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"14cd98502aca9a19","q":"Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove  $K$  computers so that there are two computers, which can not exchange data, or there is one computer left. Let  $k$  be the minimum value of  $K$ . Let's remove  $L$  cable from original network so that there are two computers, which can not exchange data. Let  $l$  be the minimum value of  $L$ . Show that  $k=l$ .","t":[{"b":4,"e":0.571,"k":"falling","v":0.34373,"x":0.71874,"p":[[0,17,0.0,0.71874,0.2868,0.5354,0.71429,1.0,0.0,1.0,1,13,1,1,0,0,0,0,4,0,0,3,0,0,3,0,0,7,0,0,1,0,13],[4,17,0.2353,0.50433,0.27671,0.28571,0.42859,0.71429,0.14,1.0,0,3,0,0,0,4,0,0,11,0,0,2,0,0,3,0,0,6,0,0,3,0,3],[8,17,0.4706,0.50891,0.24727,0.28571,0.42857,0.71429,0.14286,1.0,0,3,0,0,0,2,0,0,11,0,0,4,0,0,3,0,0,9,0,0,0,0,3],[12,17,0.7059,0.4107,0.20123,0.28571,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,4,0,0,10,0,0,6,0,0,5,0,0,6,0,0,0,0,0],[16,17,0.9412,0.34373,0.18849,0.2857,0.2857,0.42857,0.14286,0.85714,0,0,0,0,0,7,0,0,16,0,0,3,0,0,2,0,0,3,0,0,1,0,0],[17,17,1.0,0.38834,0.15244,0.2857,0.28571,0.571,0.14286,0.71429,0,0,0,0,0,2,0,0,16,0,0,5,0,0,7,0,0,2,0,0,0,0,0]]},{"b":7,"e":1.0,"k":"rising","v":0.62944,"x":1.0,"p":[[0,23,0.0,0.62944,0.33286,0.28571,0.71429,1.0,0.0,1.0,1,11,0,1,0,4,0,0,5,0,0,1,0,0,3,0,0,7,0,0,0,0,11],[4,23,0.1739,0.63392,0.3387,0.28571,0.71429,1.0,0.14286,1.0,0,12,0,0,0,4,0,0,8,0,0,1,0,0,1,0,0,5,0,0,1,0,12],[8,23,0.3478,0.98214,0.05923,1.0,1.0,1.0,0.71429,1.0,0,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,29],[12,23,0.5217,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32],[16,23,0.6957,0.99554,0.02486,1.0,1.0,1.0,0.85714,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[20,23,0.8696,0.99553,0.02488,1.0,1.0,1.0,0.857,1.0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,31],[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":"9de76f8164afbab9","q":"Let $p$ be an odd prime number. For every integer $a$, define the number  $$ S_{a}=\\frac{a}{1}+\\frac{a^{2}}{2}+\\cdots+\\frac{a^{p-1}}{p-1} $$  Let $m$ and $n$ be integers such that  $$ S_{3}+S_{4}-3 S_{2}=\\frac{m}{n} $$  Prove that $p$ divides $m$.","t":[{"b":4,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,6,0.0,0.02232,0.06297,0.0,0.0,0.0,0.0,0.2857,28,0,0,28,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.04455,"p":[[0,33,0.0,0.03572,0.08748,0.0,0.0,0.0,0.0,0.28571,27,0,0,27,0,2,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,33,0.1212,0.04455,0.1765,0.0,0.0,0.0,0.0,1.0,28,1,0,28,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[8,33,0.2424,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,33,0.3636,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,33,0.4848,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,33,0.6061,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,33,0.7273,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,33,0.8485,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"a5317b5c7459f5fd","q":"Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls. A move consists of taking all of the golf balls from one of the boxes and placing them into the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove that if the next move always starts with the box where the last ball of the previous move was placed, then after some number of moves, we get back to the initial distribution of golf balls in the boxes.","t":[{"b":0,"e":0.57143,"k":"falling","v":0.35714,"x":0.63838,"p":[[0,7,0.0,0.63838,0.1988,0.57132,0.71429,0.75,0.14286,0.85714,0,0,0,0,0,2,0,0,1,0,0,4,0,0,6,0,0,11,0,0,8,0,0],[4,7,0.5714,0.50446,0.22011,0.28571,0.57143,0.71429,0.0,0.85714,1,0,1,1,0,1,0,0,9,0,0,3,0,0,8,0,0,7,0,0,3,0,0],[7,7,1.0,0.35714,0.15972,0.2857,0.28571,0.46431,0.14286,0.71429,0,0,0,0,0,6,0,0,13,0,0,5,0,0,7,0,0,1,0,0,0,0,0]]},{"b":2,"e":0.71429,"k":"rising","v":0.57142,"x":0.75,"p":[[0,26,0.0,0.57142,0.19562,0.42857,0.57143,0.71429,0.14286,1.0,0,1,0,0,0,1,0,0,4,0,0,6,0,0,9,0,0,8,0,0,3,0,1],[4,26,0.1538,0.61607,0.19045,0.53572,0.64286,0.71429,0.28571,0.85714,0,0,0,0,0,0,0,0,5,0,0,3,0,0,8,0,0,9,0,0,7,0,0],[8,26,0.3077,0.62279,0.13854,0.571,0.64286,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,5,0,1,9,0,0,13,0,0,3,0,0],[12,26,0.4615,0.66518,0.14555,0.57143,0.71429,0.71429,0.2857,1.0,0,1,0,0,0,0,0,0,2,0,0,1,0,0,8,0,0,17,0,0,3,0,1],[16,26,0.6154,0.66516,0.13176,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,9,0,0,15,0,0,5,0,0],[20,26,0.7692,0.67854,0.14727,0.57143,0.71429,0.85704,0.2857,0.85714,0,0,0,0,0,0,0,0,1,0,0,2,0,0,10,0,0,10,0,0,9,0,0],[24,26,0.9231,0.68303,0.12745,0.57143,0.71429,0.71429,0.42857,0.85714,0,0,0,0,0,0,0,0,0,0,0,4,0,0,5,0,0,17,0,0,6,0,0],[26,26,1.0,0.75,0.14285,0.71429,0.78564,0.85714,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,9,0,0,15,0,1]]}]},{"i":"830fecd98b7eae3b","q":"Let $a_{1}<a_{2}<a_{3}<\\cdots$ be positive integers such that $a_{k+1}$ divides $2\\left(a_{1}+a_{2}+\\cdots+a_{k}\\right)$ for every $k \\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_{k}$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_{k}$. (Netherlands)","t":[{"b":1,"e":0.71429,"k":"volatile","v":0.30803,"x":0.6696,"p":[[0,5,0.0,0.30803,0.37983,0.0,0.14286,0.71429,0.0,1.0,15,4,2,15,0,4,0,0,3,0,0,0,0,0,1,0,0,3,0,0,2,0,4],[4,5,0.8,0.6696,0.13095,0.57143,0.71429,0.71429,0.28571,1.0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,7,0,0,19,0,0,2,0,1],[5,5,1.0,0.64286,0.12877,0.57143,0.71429,0.71429,0.2857,0.85714,0,0,0,0,0,0,0,0,2,0,0,2,0,0,7,0,0,20,0,0,1,0,0]]},{"b":7,"e":0.0,"k":"falling","v":0.0,"x":0.46873,"p":[[0,21,0.0,0.46873,0.38834,0.0,0.49979,0.85714,0.0,1.0,10,5,2,10,0,1,0,0,4,0,0,1,0,0,1,0,0,6,0,0,4,0,5],[4,21,0.1905,0.33927,0.33834,0.0,0.28571,0.60682,0.0,1.0,12,2,0,12,0,2,0,0,4,0,0,5,0,0,1,0,0,3,0,0,3,0,2],[8,21,0.381,0.29462,0.33679,0.0,0.14286,0.60714,0.0,1.0,15,1,0,15,0,3,0,0,2,0,0,1,0,0,3,0,0,5,0,0,2,0,1],[12,21,0.5714,0.1116,0.25685,0.0,0.0,0.0,0.0,0.857,26,0,0,26,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,2,0,0],[16,21,0.7619,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"41548da91b7ab614","q":"Nine journalists from different countries attend a press conference. None of these speaks more than three\nlanguages, and each pair of the journalists share a common language. Show that there are at least five journalists sharing a common language.","t":[{"b":1,"e":0.14286,"k":"falling","v":0.17408,"x":0.33927,"p":[[0,21,0.0,0.33927,0.27139,0.14286,0.2857,0.57111,0.0,1.0,6,1,1,6,0,7,0,0,5,0,0,5,0,0,4,0,0,3,0,0,1,0,1],[4,21,0.1905,0.20094,0.21091,0.0,0.14286,0.2857,0.0,0.71429,11,0,1,11,0,9,0,0,6,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[8,21,0.381,0.26338,0.24511,0.14286,0.14286,0.42857,0.0,0.85714,7,0,1,7,0,12,0,0,3,0,0,4,0,0,2,0,0,3,0,0,1,0,0],[12,21,0.5714,0.20973,0.20823,0.0,0.14286,0.28571,0.0,0.71429,10,0,0,10,0,9,0,0,7,0,0,2,0,0,2,0,0,2,0,0,0,0,0],[16,21,0.7619,0.17408,0.20428,0.0,0.14286,0.28571,0.0,0.57143,15,0,2,15,0,6,0,0,4,0,0,3,0,0,4,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.2857,"k":"flat","v":0.31247,"x":0.46875,"p":[[0,18,0.0,0.34374,0.25965,0.14286,0.28571,0.57143,0.0,0.85714,6,0,1,6,0,6,0,0,6,0,0,4,0,0,4,0,0,5,0,0,1,0,0],[4,18,0.2222,0.31247,0.26827,0.14286,0.28571,0.42858,0.0,1.0,7,1,1,7,0,6,0,0,7,0,0,6,0,0,2,0,0,1,0,0,2,0,1],[8,18,0.4444,0.46875,0.28175,0.2857,0.42857,0.71429,0.0,1.0,1,2,0,1,0,5,0,0,9,0,0,5,0,0,1,0,0,5,0,0,4,0,2],[12,18,0.6667,0.34822,0.20181,0.24999,0.28571,0.42893,0.0,0.85714,1,0,0,1,0,7,0,0,13,0,0,4,0,0,3,0,0,3,0,0,1,0,0],[16,18,0.8889,0.35713,0.17856,0.14286,0.28571,0.4642,0.14286,0.71429,0,0,0,0,0,9,0,0,8,0,0,7,0,0,6,0,0,2,0,0,0,0,0],[18,18,1.0,0.35714,0.18558,0.2857,0.28571,0.46429,0.14286,0.71429,0,0,0,0,0,7,0,0,14,0,0,3,0,0,4,0,0,4,0,0,0,0,0]]}]},{"i":"579fc7b3e971db0a","q":"Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo's tetrahedron turn out to coincide with the four numbers written on Juliet's tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo's tetrahedron are identical to the four numbers assigned to the vertices of Juliet's tetrahedron?","t":[{"b":4,"e":0.0,"k":"flat","v":0.08482,"x":0.11161,"p":[[0,12,0.0,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.1429,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.11161,0.05906,0.14286,0.14286,0.14286,0.0,0.1429,7,0,0,7,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.09375,0.06785,0.0,0.14286,0.14286,0.0,0.14286,11,0,0,11,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.09822,0.06622,0.0,0.14286,0.14286,0.0,0.1429,10,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.08482,"x":0.09375,"p":[[0,5,0.0,0.09375,0.07668,0.0,0.14286,0.14286,0.0,0.28571,12,0,0,12,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.08482,0.07016,0.0,0.14286,0.14286,0.0,0.14286,13,0,0,13,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.08696,0.06847,0.0,0.14286,0.14286,0.0,0.14286,12,0,0,12,1,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"c79e76a78d9a8075","q":"Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $\\operatorname{gcd}\\left(a_{i}, a_{i+1}\\right)>a_{i-1}$. Prove that $a_{n} \\geq 2^{n}$ for all $n \\geq 0$.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,9,0.0,0.04018,0.07349,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,7,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,6,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,17,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.00884,"x":0.01786,"p":[[0,6,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,4,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,6,0.6667,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,3,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.00884,0.03424,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e472dd5624f5af99","q":"On a rectangular board with $m \\times n$ squares ($m, n \\geq 3$), dominoes ($2 \\times 1$ or $1 \\times 2$ tiles) are placed such that they do not overlap and do not extend beyond the board. Each domino covers exactly two squares of the board. Assume that the tiling with dominoes has the property that no additional domino can be placed on the board, and that not all four corner squares of the board are empty. Prove that at least $\\frac{2}{3}$ of the squares of the board are covered by dominoes.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.13393,"x":0.14286,"p":[[0,8,0.0,0.1383,0.04351,0.14286,0.14286,0.14286,0.0,0.28571,2,0,1,2,0,29,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,8,0.5,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.11607,"x":0.15179,"p":[[0,42,0.0,0.12036,0.0518,0.14286,0.14286,0.14286,0.0,0.14286,5,0,2,5,0,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,42,0.0952,0.13384,0.07086,0.14286,0.14286,0.14286,0.0,0.42857,4,0,0,4,0,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,42,0.1905,0.13839,0.02486,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.11607,0.05576,0.14286,0.14286,0.14286,0.0,0.14286,6,0,0,6,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,42,0.381,0.14286,0.0,0.14286,0.14286,0.14286,0.14286,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,42,0.4762,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.1429,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.13393,0.03458,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,42,0.6667,0.15179,0.06121,0.14286,0.14286,0.14286,0.0,0.42857,1,0,0,1,0,29,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[32,42,0.7619,0.13384,0.03456,0.14286,0.14286,0.14286,0.0,0.14286,2,0,0,2,0,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,42,0.8571,0.13821,0.02483,0.14286,0.14286,0.14286,0.0,0.14286,1,0,0,1,0,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[42,42,1.0,0.14277,0.0005,0.14286,0.14286,0.14286,0.14,0.14286,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d6e3e052b32163a7","q":"Let $n \\geq 1$ be an integer and let $t_{1}<t_{2}<\\ldots<t_{n}$ be positive integers. In a group of $t_{n}+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following conditions to hold at the same time:\ni) The number of games played by each person is one of $t_{1}, t_{2}, \\ldots, t_{n}$,\nii) For every $i$ with $1 \\leq i \\leq n$, there is someone who has played exactly $t_{i}$ games of chess.\n\nGerhard W\u00f6ginger, Luxembourg\n\n## Comment\n\nIn graph theory terms the problem is to prove that for any finite nonempty set $\\mathcal{T}$ of positive integers there exists a graph of size $\\max \\mathcal{T}+1$ such that the degree set of the graph is equal to $\\mathcal{T}$.\n\nAmong graph theory specialists a generalization of this problem is known [1]. Nevertheless, the problem still suited the contest.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.13831,"x":0.16723,"p":[[0,9,0.0,0.14732,0.23003,0.0,0.14286,0.14286,0.0,1.0,11,2,2,11,0,19,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[4,9,0.4444,0.16723,0.22206,0.0,0.14286,0.14286,0.0,1.0,9,1,2,9,1,18,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1],[8,9,0.8889,0.15616,0.12557,0.14214,0.14286,0.14286,0.0,0.57143,7,0,1,7,0,18,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[9,9,1.0,0.13831,0.06666,0.14286,0.14286,0.14286,0.0,0.42857,3,0,1,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.14286,"k":"flat","v":0.11813,"x":0.28786,"p":[[0,20,0.0,0.17402,0.26902,0.0,0.14286,0.14286,0.0,1.0,13,2,5,13,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,2],[4,20,0.2,0.28786,0.36968,0.0,0.14286,0.32142,0.0,1.0,10,6,1,10,1,12,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,6],[8,20,0.4,0.23652,0.30644,0.0,0.14286,0.14286,0.0,1.0,9,2,1,9,0,16,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,0,2],[12,20,0.6,0.1383,0.19719,0.0,0.14143,0.14286,0.0,0.71429,15,0,3,15,0,12,0,0,1,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[16,20,0.8,0.15179,0.20497,0.0,0.14286,0.14287,0.0,1.0,13,1,1,13,0,12,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,1],[20,20,1.0,0.11813,0.05809,0.14214,0.14286,0.14286,0.0,0.21429,6,0,2,6,0,25,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3b2eb9d75ae7b63a","q":"Prove that for any  $\\lambda > 3$  there is a number  $x$  for which   $$ \\sin x + \\sin (\\lambda  x) \\ge  1.8. $$","t":[{"b":0,"e":0.42857,"k":"rising","v":0.3281,"x":0.52008,"p":[[0,13,0.0,0.3281,0.32675,0.0,0.24999,0.60682,0.0,1.0,10,1,6,10,0,5,0,1,5,0,0,1,0,0,2,0,0,3,0,0,4,0,1],[4,13,0.3077,0.38392,0.32229,0.14286,0.28571,0.71429,0.0,1.0,5,2,4,5,0,9,0,0,4,0,0,4,0,0,1,0,0,3,0,0,4,0,2],[8,13,0.6154,0.43523,0.14324,0.357,0.42857,0.571,0.14286,0.85714,0,0,0,0,0,2,0,1,4,0,2,12,0,2,8,0,0,0,0,0,1,0,0],[12,13,0.9231,0.49329,0.17437,0.39286,0.5355,0.60714,0.14286,0.71429,0,0,0,0,0,2,0,0,6,0,0,7,0,1,8,0,0,8,0,0,0,0,0],[13,13,1.0,0.52008,0.13626,0.42857,0.5355,0.57143,0.28571,0.85714,0,0,0,0,0,0,0,0,3,0,0,12,0,1,10,0,0,5,0,0,1,0,0]]},{"b":4,"e":0.85714,"k":"volatile","v":0.22768,"x":0.68302,"p":[[0,16,0.0,0.30131,0.34612,0.0,0.14286,0.46418,0.0,1.0,12,4,8,12,0,5,0,1,4,0,0,2,0,0,2,0,0,1,0,0,1,0,4],[4,16,0.25,0.47982,0.33141,0.28571,0.39279,0.85714,0.0,1.0,4,4,3,4,0,2,0,0,9,0,1,4,0,0,1,0,0,1,0,0,6,0,4],[8,16,0.5,0.22768,0.24964,0.0,0.14288,0.28571,0.0,0.85714,12,0,8,12,0,6,0,0,7,0,0,2,0,0,1,0,0,3,0,0,1,0,0],[12,16,0.75,0.68302,0.16263,0.57143,0.71429,0.85714,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0,0,7,0,0,11,0,0],[16,16,1.0,0.65178,0.16342,0.57143,0.64286,0.75,0.14286,0.85714,0,0,0,0,0,1,0,0,0,0,0,3,0,0,12,0,0,8,0,0,8,0,0]]}]},{"i":"5935bd9aad0ca24a","q":"The angle bisectors at  $A$  and  $C$  in a non-isosceles triangle  $ABC$  with incenter  $I$  intersect its circumcircle  $k$  at  $A_0$  and  $C_0$ , respectively. The line through  $I$ , parallel to  $AC$ , intersects  $A_0C_0$  at  $P$ . Prove that  $PB$  is tangent to  $k$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,17,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,12,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,17,0.4706,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,13,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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":3,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,26,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,9,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,26,0.4615,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,26,0.6154,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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],[24,26,0.9231,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[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]]}]},{"i":"1a4095da4c2a4534","q":"Point  $P$  is chosen in the plane of triangle  $ABC$  such that  $\\angle{ABP}$  is congruent to  $\\angle{ACP}$  and  $\\angle{CBP}$  is congruent to  $\\angle{CAP}$ . Show  $P$  is the orthocentre.","t":[{"b":1,"e":0.0,"k":"volatile","v":0.3482,"x":0.71426,"p":[[0,21,0.0,0.71426,0.36943,0.57132,0.85714,1.0,0.0,1.0,5,14,0,5,0,1,0,0,1,0,0,0,0,0,2,0,0,3,0,0,6,0,14],[4,21,0.1905,0.70977,0.33406,0.571,0.85714,1.0,0.0,1.0,3,11,0,3,0,2,0,0,1,0,0,0,0,0,6,0,0,0,0,0,9,0,11],[8,21,0.381,0.3482,0.38121,0.0,0.2143,0.57111,0.0,1.0,11,7,0,11,0,5,0,0,5,0,0,2,0,0,2,0,0,0,0,0,0,0,7]]},{"b":4,"e":1.0,"k":"rising","v":0.49103,"x":1.0,"p":[[0,28,0.0,0.6607,0.39407,0.24999,0.85714,1.0,0.0,1.0,5,14,1,5,0,3,0,0,1,0,0,1,0,0,2,0,0,2,0,0,4,0,14],[4,28,0.1429,0.76785,0.34947,0.67857,1.0,1.0,0.0,1.0,2,18,0,2,0,4,0,0,0,0,0,1,0,0,1,0,0,1,0,0,5,0,18],[8,28,0.2857,0.67855,0.37965,0.28571,0.85714,1.0,0.0,1.0,3,14,1,3,0,4,0,0,2,0,0,2,0,0,1,0,0,0,0,0,6,0,14],[12,28,0.4286,0.49103,0.38288,0.14286,0.35714,0.89286,0.0,1.0,5,8,0,5,0,6,0,0,5,0,0,1,0,0,3,0,0,1,0,0,3,0,8],[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],[20,28,0.7143,0.96428,0.15153,1.0,1.0,1.0,0.14286,1.0,0,29,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,29],[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":"c96d1ac445c3826b","q":"Let \\( L \\) be a positive integer. There are  $2024$  balls with distinct weights placed on a table. Initially, player  $A$  knows the weight of each ball, but player  $B$  knows nothing. To determine the weight of every ball, player  $B$  asks player  $A$  a series of questions. In each question, player  $B$  may specify any set of at most \\( L \\) balls, and player  $A$  will only respond with the set of weights of the specified balls (i.e., without revealing which weight corresponds to which ball).  \n\nPlayer  $B$  aims to ensure that after \\( m \\) questions, they can deduce the weight of every ball.  \n(1) If \\( L = 5 \\), find the minimum possible value of \\( m \\);  \n(2) If \\( L = 100 \\), find the minimum possible value of \\( m \\).  \n\nProposed by *Wang Bin*","t":[{"b":5,"e":0.571,"k":"flat","v":0.25894,"x":0.41061,"p":[[0,33,0.0,0.29018,0.20973,0.14286,0.2857,0.42857,0.0,0.71429,7,0,5,7,0,4,0,0,8,0,0,10,0,0,0,0,0,3,0,0,0,0,0],[4,33,0.1212,0.25894,0.17655,0.14286,0.28571,0.42857,0.0,0.71429,5,0,3,5,0,9,0,0,8,0,0,8,0,0,1,0,0,1,0,0,0,0,0],[8,33,0.2424,0.40624,0.23175,0.2857,0.42857,0.57111,0.0,1.0,2,1,2,2,0,5,0,0,6,0,0,10,0,0,4,0,0,3,0,0,1,0,1],[12,33,0.3636,0.35713,0.22587,0.2857,0.28571,0.42857,0.0,1.0,1,2,0,1,0,6,0,0,13,0,0,8,0,0,1,0,0,0,0,0,1,0,2],[16,33,0.4848,0.41061,0.21064,0.25,0.42857,0.57143,0.0,0.71429,1,0,0,1,0,7,0,0,4,0,0,9,0,0,5,0,0,6,0,0,0,0,0],[20,33,0.6061,0.40622,0.17534,0.28571,0.42857,0.571,0.0,0.85714,1,0,0,1,0,1,0,0,13,0,0,8,0,0,6,0,0,2,0,0,1,0,0],[24,33,0.7273,0.3839,0.1836,0.2857,0.42857,0.42857,0.14286,0.857,0,0,0,0,0,6,0,0,9,0,0,11,0,0,2,0,0,3,0,0,1,0,0],[28,33,0.8485,0.2857,0.13829,0.14286,0.2857,0.42857,0.0,0.571,1,0,0,1,0,11,0,0,8,0,0,11,0,0,1,0,0,0,0,0,0,0,0],[32,33,0.9697,0.33926,0.18468,0.14286,0.42857,0.42858,0.0,0.71429,2,0,0,2,0,8,0,0,5,0,0,12,0,0,3,0,0,2,0,0,0,0,0],[33,33,1.0,0.40175,0.20021,0.28571,0.28586,0.571,0.14286,1.0,0,1,0,0,0,4,0,0,13,0,0,6,0,0,6,0,0,1,0,0,1,0,1]]},{"b":6,"e":0.0,"k":"falling","v":0.00446,"x":0.29455,"p":[[0,42,0.0,0.29455,0.23134,0.105,0.28571,0.42857,0.0,0.85714,8,0,5,8,0,4,0,0,6,0,0,10,0,0,1,0,0,2,0,0,1,0,0],[4,42,0.0952,0.07143,0.13363,0.0,0.0,0.03571,0.0,0.42857,24,0,8,24,0,2,0,0,4,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,42,0.1905,0.05348,0.11703,0.0,0.0,0.0,0.0,0.4286,25,0,12,25,0,4,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,42,0.2857,0.04463,0.12073,0.0,0.0,0.0,0.0,0.571,27,0,11,27,0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,42,0.381,0.04018,0.12993,0.0,0.0,0.0,0.0,0.71429,27,0,7,27,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[20,42,0.4762,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,9,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,42,0.5714,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,6,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"f98efaa1664836a0","q":"Prove that there exist infinitely many positive integers $n$ such that $n^{2}+1$ has a prime divisor greater than $2 n+\\sqrt{2 n}$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.05357,"x":0.10705,"p":[[0,6,0.0,0.10268,0.22084,0.0,0.0,0.14286,0.0,1.0,22,1,4,22,0,6,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,6,0.6667,0.10705,0.20823,0.0,0.0,0.14286,0.0,1.0,21,1,3,21,0,5,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,6,1.0,0.05357,0.09942,0.0,0.0,0.03571,0.0,0.28571,24,0,0,24,0,4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.0,"x":0.16964,"p":[[0,6,0.0,0.16964,0.32818,0.0,0.0,0.1786,0.0,1.0,22,4,4,22,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,6,0.6667,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[6,6,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"e690ba7170cdd0cd","q":"Let $p \\geqslant 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\\{0,1, \\ldots, p-1\\}$ that was not chosen before by either of the two players and then chooses an element $a_{i}$ of the set $\\{0,1,2,3,4,5,6,7,8,9\\}$. Eduardo has the first move. The game ends after all the indices $i \\in\\{0,1, \\ldots, p-1\\}$ have been chosen. Then the following number is computed:  $$ M=a_{0}+10 \\cdot a_{1}+\\cdots+10^{p-1} \\cdot a_{p-1}=\\sum_{j=0}^{p-1} a_{j} \\cdot 10^{j} $$  The goal of Eduardo is to make the number $M$ divisible by $p$, and the goal of Fernando is to prevent this.  Prove that Eduardo has a winning strategy. (Morocco)","t":[{"b":6,"e":0.14286,"k":"rising","v":0.15179,"x":0.44192,"p":[[0,12,0.0,0.16964,0.1357,0.14286,0.14286,0.28571,0.0,0.57143,7,0,1,7,0,16,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[4,12,0.3333,0.15179,0.11811,0.10714,0.14286,0.1786,0.0,0.4286,8,0,0,8,0,16,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.16964,0.10374,0.14286,0.14286,0.2857,0.0,0.28571,6,0,1,6,0,14,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.44192,0.16503,0.28571,0.42857,0.57143,0.14286,0.71429,0,0,0,0,0,3,0,0,7,0,0,10,0,0,8,0,0,4,0,0,0,0,0]]},{"b":7,"e":0.14286,"k":"falling","v":0.05357,"x":0.24552,"p":[[0,14,0.0,0.24552,0.17212,0.14286,0.14286,0.28571,0.0,0.71429,2,0,0,2,0,17,0,0,6,0,0,3,0,0,3,0,0,1,0,0,0,0,0],[4,14,0.2857,0.16509,0.11906,0.14286,0.14286,0.28571,0.0,0.57143,6,0,1,6,0,17,0,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,14,0.5714,0.08036,0.07936,0.0,0.14286,0.14286,0.0,0.28571,15,0,5,15,0,16,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.08915,0.1318,0.0,0.07143,0.14286,0.0,0.71,16,0,8,16,0,15,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[14,14,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,14,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1523b75b9210b422","q":"Problem 5. Consider the integer number n>2. Let a_1,a_2,\u2026,a_n and b_1,b_2,\u2026,b_n be two permutations of 0,1,2,\u2026,n-1. Prove that there exist some i\u2260j such that:\nn|a_i b_i-a_j b_j\n\n<span style=\"color:#00f\">Moved to HSO. ~ oVlad</span>","t":[{"b":3,"e":0.14286,"k":"falling","v":0.02679,"x":0.34821,"p":[[0,22,0.0,0.21875,0.19228,0.0,0.14288,0.28571,0.0,0.71429,9,0,1,9,0,8,0,0,8,0,0,4,0,0,2,0,0,1,0,0,0,0,0],[4,22,0.1818,0.30802,0.19267,0.14289,0.28571,0.42857,0.0,0.71429,3,0,1,3,0,7,0,0,12,0,0,5,0,0,2,0,0,3,0,0,0,0,0],[8,22,0.3636,0.23212,0.21647,0.10714,0.14286,0.28571,0.0,0.85714,8,0,0,8,0,10,0,0,7,0,0,3,0,0,2,0,0,1,0,0,1,0,0],[12,22,0.5455,0.28115,0.19066,0.14286,0.28571,0.42857,0.0,0.71429,5,0,1,5,0,7,0,0,9,0,0,8,0,0,1,0,0,2,0,0,0,0,0],[16,22,0.7273,0.34821,0.18536,0.2857,0.28571,0.42857,0.0,0.71429,1,0,0,1,0,6,0,0,13,0,0,6,0,0,2,0,0,4,0,0,0,0,0],[20,22,0.9091,0.19643,0.15046,0.14286,0.14286,0.2857,0.0,0.57143,6,0,4,6,0,14,0,0,8,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[22,22,1.0,0.02679,0.06621,0.0,0.0,0.0,0.0,0.2857,27,0,0,27,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"falling","v":0.05348,"x":0.28123,"p":[[0,13,0.0,0.26786,0.23891,0.0,0.2857,0.42857,0.0,0.71429,9,0,1,9,0,6,0,0,7,0,0,4,0,0,2,0,0,4,0,0,0,0,0],[4,13,0.3077,0.20982,0.21124,0.0,0.14286,0.42857,0.0,0.71429,10,0,1,10,0,11,0,0,2,0,0,6,0,0,1,0,0,2,0,0,0,0,0],[8,13,0.6154,0.28123,0.22438,0.14286,0.21428,0.46418,0.0,0.71429,7,0,1,7,0,9,0,0,3,0,0,5,0,0,7,0,0,1,0,0,0,0,0],[12,13,0.9231,0.20536,0.19212,0.0,0.14286,0.28571,0.0,0.71429,11,0,0,11,0,6,0,0,8,0,0,5,0,0,1,0,0,1,0,0,0,0,0],[13,13,1.0,0.05348,0.08555,0.0,0.0,0.14286,0.0,0.2857,22,0,0,22,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"57c7a29ac452d825","q":"Let $a_{1}<a_{2}<a_{3}<a_{4}<\\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence  $$ \\frac{a_{1}}{1}, \\frac{a_{2}}{2}, \\frac{a_{3}}{3}, \\frac{a_{4}}{4}, \\ldots $$","t":[{"b":0,"e":0.0,"k":"flat","v":0.04902,"x":0.10268,"p":[[0,11,0.0,0.10268,0.1931,0.0,0.0,0.14286,0.0,0.85714,21,0,10,21,0,6,0,0,2,0,0,1,0,0,1,0,0,0,0,0,1,0,0],[4,11,0.3636,0.04902,0.08449,0.0,0.0,0.14071,0.0,0.28571,23,0,14,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.06473,"x":0.09598,"p":[[0,11,0.0,0.09598,0.1245,0.0,0.03571,0.14286,0.0,0.42857,16,0,12,16,3,8,0,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,11,0.3636,0.06473,0.0934,0.0,0.0,0.14286,0.0,0.42857,19,0,16,19,1,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.08258,0.1349,0.0,0.0,0.14286,0.0,0.571,20,0,15,20,0,8,0,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"29195f554813752d","q":"Several numbers are written in a line. Thanima has the right to choose two adjacent numbers such that the number on the left is strictly greater than the number on the right, she then swaps these two numbers and multiplies them by 2. Show that Thanima can only perform a finite number of such operations.","t":[{"b":5,"e":0.85714,"k":"rising","v":0.33917,"x":0.56692,"p":[[0,6,0.0,0.33917,0.35132,0.0,0.14286,0.60714,0.0,1.0,11,3,1,11,0,6,0,0,2,0,0,2,0,0,3,0,0,3,0,0,2,0,3],[4,6,0.6667,0.37941,0.3146,0.14286,0.28571,0.57111,0.0,1.0,7,3,0,7,0,5,0,0,5,0,0,3,0,0,6,0,0,2,0,0,1,0,3],[6,6,1.0,0.56692,0.30195,0.28571,0.64286,0.74996,0.0,1.0,1,4,0,1,0,6,0,0,3,0,0,1,0,0,5,0,0,8,0,0,4,0,4]]},{"b":6,"e":0.0,"k":"falling","v":0.05357,"x":0.45531,"p":[[0,13,0.0,0.38389,0.36144,0.10714,0.14288,0.71429,0.0,1.0,8,4,0,8,0,9,0,0,1,0,0,0,0,0,5,0,0,3,0,0,2,0,4],[4,13,0.3077,0.38838,0.32188,0.14286,0.28571,0.60714,0.0,1.0,6,2,0,6,0,7,0,0,4,0,0,4,0,0,3,0,0,2,0,0,4,0,2],[8,13,0.6154,0.45531,0.35432,0.14286,0.49979,0.74996,0.0,1.0,6,4,0,6,0,6,0,0,3,0,0,1,0,0,5,0,0,3,0,0,4,0,4],[12,13,0.9231,0.32131,0.32929,0.0,0.14286,0.60682,0.0,1.0,10,1,1,10,0,7,0,0,4,0,0,1,0,0,2,0,0,3,0,0,4,0,1],[13,13,1.0,0.05357,0.06916,0.0,0.0,0.14286,0.0,0.14286,20,0,0,20,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6868bbde4e5a04db","q":"Let $m$ be a positive integer. Show that there exists a positive integer $n$ such that each of the $2 m+1$ integers\n\n$$\n2^{n}-m, 2^{n}-(m-1), \\ldots, 2^{n}+(m-1), 2^{n}+m\n$$\n\nis positive and composite.","t":[{"b":3,"e":0.14286,"k":"flat","v":0.15625,"x":0.28125,"p":[[0,28,0.0,0.23205,0.19154,0.14286,0.14286,0.28571,0.0,1.0,4,1,0,4,0,15,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[4,28,0.1429,0.18304,0.18293,0.0,0.14286,0.28571,0.0,0.85714,10,0,0,10,0,10,0,0,8,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[8,28,0.2857,0.28125,0.23003,0.14286,0.2857,0.32143,0.0,1.0,3,2,0,3,0,12,0,0,9,0,0,5,0,0,1,0,0,0,0,0,0,0,2],[12,28,0.4286,0.15625,0.11495,0.14286,0.14286,0.17857,0.0,0.42857,7,0,2,7,0,17,0,0,6,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[16,28,0.5714,0.2008,0.21087,0.14286,0.14286,0.2857,0.0,1.0,6,1,0,6,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,1],[20,28,0.7143,0.19196,0.11633,0.14286,0.14286,0.2857,0.0,0.42857,4,0,0,4,0,16,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[24,28,0.8571,0.20536,0.10677,0.14286,0.14286,0.28571,0.0,0.42857,3,0,0,3,0,14,0,0,13,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[28,28,1.0,0.2008,0.13771,0.14286,0.14286,0.28571,0.0,0.42857,6,0,0,6,0,12,0,0,9,0,0,5,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.14286,"k":"flat","v":0.1383,"x":0.20982,"p":[[0,16,0.0,0.18304,0.1684,0.14286,0.14286,0.14286,0.0,1.0,3,1,0,3,0,23,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,16,0.25,0.17412,0.12746,0.10714,0.14286,0.28571,0.0,0.42857,8,0,0,8,0,11,0,0,11,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.19643,0.09279,0.14286,0.14286,0.28571,0.0,0.42857,2,0,0,2,0,17,0,0,12,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.20982,0.17852,0.14286,0.14286,0.2857,0.0,1.0,4,1,0,4,0,16,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[16,16,1.0,0.1383,0.12103,0.0,0.14286,0.2857,0.0,0.4286,11,0,0,11,0,12,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"2ca923ffb8da6469","q":"Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than 1 and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^{k}-p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by 2018 .","t":[{"b":1,"e":0.57143,"k":"rising","v":0.09374,"x":0.33929,"p":[[0,10,0.0,0.13391,0.19538,0.0,0.0,0.17857,0.0,0.71429,18,0,6,18,0,6,0,0,4,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[4,10,0.4,0.13393,0.19212,0.0,0.0,0.17857,0.0,0.85714,17,0,9,17,0,7,0,0,4,0,0,3,0,0,0,0,0,0,0,0,1,0,0],[8,10,0.8,0.09374,0.1943,0.0,0.0,0.03571,0.0,0.71429,24,0,14,24,0,3,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,0,0],[10,10,1.0,0.33929,0.18813,0.14286,0.28571,0.42857,0.14286,0.85714,0,0,0,0,0,10,0,0,10,0,0,5,0,0,5,0,0,1,0,0,1,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.13392,"x":0.24107,"p":[[0,8,0.0,0.24107,0.34889,0.0,0.0,0.42858,0.0,1.0,19,3,10,19,0,1,0,0,3,0,0,2,0,0,2,0,0,0,0,0,2,0,3],[4,8,0.5,0.13392,0.18874,0.0,0.0,0.17857,0.0,0.57143,18,0,10,18,0,6,0,0,3,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[8,8,1.0,0.20973,0.14721,0.14286,0.14286,0.28571,0.0,0.857,2,0,0,2,0,18,0,0,10,0,0,1,0,0,0,0,0,0,0,0,1,0,0]]}]},{"i":"561e1b4d0c0a2c3b","q":"Prove that there exists a triangle which can be cut into 2005 congruent triangles.","t":[{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.02232,"p":[[0,16,0.0,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,16,0.25,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,16,0.5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,7,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,16,0.75,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,1,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,16,1.0,0.02232,0.05187,0.0,0.0,0.0,0.0,0.1429,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.00893,"x":0.00893,"p":[[0,5,0.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,4,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"9d316024bf0bab32","q":"Sasha has  $10$  cards with numbers  $1, 2, 4, 8,\\ldots, 512$ . He writes the number  $0$  on the board and invites Dima to play a game. Dima tells the integer  $0 < p < 10, p$  can vary from round to round. Sasha chooses  $p$  cards before which he puts a \u201c $+$ \u201d sign, and before the other cards he puts a \u201c $-$ \" sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha\u2019s moves.","t":[{"b":0,"e":0.14286,"k":"falling","v":0.28562,"x":0.54909,"p":[[0,49,0.0,0.54909,0.2969,0.42859,0.64286,0.71429,0.0,0.85714,6,0,4,6,0,0,0,0,1,0,0,2,0,0,7,0,0,9,0,0,7,0,0],[4,49,0.0816,0.28562,0.22873,0.14286,0.28571,0.28571,0.0,1.0,5,1,4,5,0,6,0,0,15,0,0,2,0,0,1,0,0,1,0,0,1,0,1],[8,49,0.1633,0.37053,0.22263,0.2857,0.42857,0.42857,0.0,0.85714,4,0,4,4,0,2,0,0,9,0,0,12,0,0,0,0,0,3,0,0,2,0,0],[12,49,0.2449,0.37054,0.20473,0.28571,0.42857,0.42857,0.0,0.85714,3,0,2,3,0,2,0,0,10,0,0,13,0,0,0,0,0,2,0,0,2,0,0],[16,49,0.3265,0.35268,0.20512,0.28571,0.35714,0.42857,0.0,0.85714,4,0,1,4,0,2,0,0,10,0,0,12,0,0,0,0,0,3,0,0,1,0,0],[20,49,0.4082,0.29464,0.14258,0.2857,0.28571,0.28571,0.0,0.71429,2,0,1,2,0,4,0,0,20,0,0,3,0,0,2,0,0,1,0,0,0,0,0],[24,49,0.4898,0.31254,0.11543,0.2857,0.28571,0.42857,0.0,0.57143,1,0,1,1,0,4,0,0,16,0,0,10,0,0,1,0,0,0,0,0,0,0,0],[28,49,0.5714,0.28571,0.14725,0.2857,0.28571,0.42857,0.0,0.57143,5,0,1,5,0,1,0,0,16,0,0,9,0,0,1,0,0,0,0,0,0,0,0],[32,49,0.6531,0.33927,0.17403,0.28571,0.28571,0.42857,0.0,0.85714,4,0,2,4,0,0,0,0,13,0,0,12,0,0,2,0,0,0,0,0,1,0,0],[36,49,0.7347,0.34822,0.15126,0.28571,0.42857,0.42857,0.0,0.85714,2,0,1,2,0,2,0,0,11,0,0,16,0,0,0,0,0,0,0,0,1,0,0],[40,49,0.8163,0.31696,0.10555,0.2857,0.28571,0.42857,0.0,0.4286,1,0,1,1,0,3,0,0,16,0,0,12,0,0,0,0,0,0,0,0,0,0,0],[44,49,0.898,0.28571,0.11294,0.2857,0.28571,0.28571,0.0,0.42857,3,0,1,3,0,1,0,0,21,0,0,7,0,0,0,0,0,0,0,0,0,0,0],[48,49,0.9796,0.33036,0.12596,0.2857,0.28571,0.42857,0.0,0.71429,2,0,1,2,0,0,0,0,18,0,0,11,0,0,0,0,0,1,0,0,0,0,0],[49,49,1.0,0.32589,0.09606,0.2857,0.28571,0.42857,0.0,0.4286,1,0,0,1,0,1,0,0,18,0,0,12,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.57143,"k":"flat","v":0.41518,"x":0.58927,"p":[[0,26,0.0,0.51339,0.33855,0.10714,0.57143,0.74996,0.0,1.0,8,1,7,8,0,1,0,0,1,0,0,0,0,0,7,0,0,7,0,0,7,0,1],[4,26,0.1538,0.41518,0.31615,0.0,0.57143,0.71429,0.0,0.85714,11,0,10,11,0,0,0,0,1,0,0,1,0,0,8,0,0,10,0,0,1,0,0],[8,26,0.3077,0.58927,0.14174,0.57143,0.57143,0.71429,0.14286,0.85714,0,0,0,0,0,1,0,0,1,0,0,3,0,0,17,0,0,8,0,0,2,0,0],[12,26,0.4615,0.43749,0.16341,0.28571,0.4286,0.57143,0.0,0.71429,1,0,1,1,0,2,0,0,6,0,0,10,0,0,11,0,0,2,0,0,0,0,0],[16,26,0.6154,0.49996,0.14283,0.42857,0.57141,0.57143,0.14286,0.71429,0,0,0,0,0,1,0,0,5,0,0,7,0,0,15,0,0,4,0,0,0,0,0],[20,26,0.7692,0.45522,0.16549,0.42857,0.42859,0.57111,0.0,0.71429,1,0,1,1,0,2,0,0,3,0,0,14,0,0,8,0,0,4,0,0,0,0,0],[24,26,0.9231,0.44646,0.12751,0.28571,0.42857,0.57143,0.14286,0.57143,0,0,0,0,0,1,0,0,8,0,0,9,0,0,14,0,0,0,0,0,0,0,0],[26,26,1.0,0.49105,0.1181,0.42857,0.57143,0.57143,0.2857,0.71429,0,0,0,0,0,0,0,0,6,0,0,7,0,0,18,0,0,1,0,0,0,0,0]]}]},{"i":"37861da711a1d097","q":"Suppose that  $ \\tau$  is a metrizable topology on a set  $ X$  of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on  $ X$  that is coarser that  $ \\tau$ . \r\n\r\n*L. Juhasz*","t":[{"b":4,"e":0.0,"k":"flat","v":0.16965,"x":0.20536,"p":[[0,19,0.0,0.20536,0.13333,0.10714,0.28571,0.28571,0.0,0.42857,8,0,6,8,0,4,0,0,18,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,19,0.2105,0.18741,0.11542,0.14214,0.28571,0.28571,0.0,0.28571,7,0,5,7,0,8,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,19,0.4211,0.17857,0.12877,0.0,0.2857,0.28571,0.0,0.28571,10,0,7,10,0,4,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,19,0.6316,0.16965,0.13092,0.0,0.143,0.28571,0.0,0.42857,10,0,6,10,0,7,0,0,14,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,19,0.8421,0.18294,0.17217,0.0,0.14286,0.28571,0.0,0.71429,11,0,7,11,0,6,0,0,13,0,0,0,0,0,1,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.2857,"k":"flat","v":0.20527,"x":0.26785,"p":[[0,22,0.0,0.20527,0.12344,0.14214,0.28571,0.28571,0.0,0.4286,7,0,6,7,0,5,0,0,19,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.20535,0.12339,0.10714,0.2857,0.28571,0.0,0.28571,8,0,6,8,0,2,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.21875,0.11285,0.14286,0.2857,0.28571,0.0,0.42857,5,0,2,5,0,6,0,0,20,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.24107,0.09061,0.2857,0.28571,0.28571,0.0,0.28571,3,0,2,3,0,4,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.23651,0.09191,0.14286,0.2857,0.28571,0.0,0.42857,2,0,1,2,0,8,0,0,21,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.26785,0.04724,0.2857,0.28571,0.28571,0.14286,0.28571,0,0,0,0,0,4,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.26339,0.06298,0.2857,0.28571,0.28571,0.0,0.28571,1,0,0,1,0,3,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0b627c20061d58e6","q":"Points $P$ and $Q$ lie inside parallelogram $A B C D$ and are such that triangles $A B P$ and $B C Q$ are equilateral. Prove that the line through $P$ perpendicular to $D P$ and the line through $Q$ perpendicular to $D Q$ meet on the altitude from $B$ in triangle $A B C$.","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.12946,"p":[[0,35,0.0,0.11607,0.16146,0.0,0.0,0.28571,0.0,0.4286,20,0,0,20,0,2,0,0,6,0,0,4,0,0,0,0,0,0,0,0,0,0,0],[4,35,0.1143,0.12946,0.15714,0.0,0.0,0.28571,0.0,0.42857,18,0,0,18,0,2,0,0,9,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,35,0.2286,0.04911,0.10479,0.0,0.0,0.0,0.0,0.28571,26,0,1,26,0,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,35,0.3429,0.08473,0.12803,0.0,0.0,0.14286,0.0,0.4286,21,0,1,21,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[16,35,0.4571,0.04911,0.09852,0.0,0.0,0.0,0.0,0.28571,25,0,0,25,0,3,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,35,0.5714,0.05357,0.11152,0.0,0.0,0.0,0.0,0.42857,25,0,0,25,0,3,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,35,0.8,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,35,0.9143,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0267,"x":0.125,"p":[[0,31,0.0,0.11143,0.13232,0.0,0.07,0.1429,0.0,0.4286,16,0,1,16,0,9,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,31,0.129,0.11607,0.1448,0.0,0.0,0.2857,0.0,0.4286,18,0,0,18,0,4,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[8,31,0.2581,0.07134,0.15148,0.0,0.0,0.0,0.0,0.57143,25,0,0,25,0,2,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,31,0.3871,0.125,0.15872,0.0,0.0,0.2857,0.0,0.57143,17,0,1,17,0,6,0,0,6,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[16,31,0.5161,0.08482,0.12807,0.0,0.0,0.14286,0.0,0.42857,21,0,1,21,0,4,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,31,0.6452,0.02678,0.07523,0.0,0.0,0.0,0.0,0.2857,28,0,1,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,31,0.7742,0.02679,0.07523,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,31,0.9032,0.0267,0.05558,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[31,31,1.0,0.03116,0.06887,0.0,0.0,0.0,0.0,0.2857,26,0,0,26,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"0e3141d8aab3fcb5","q":"Quadrilateral  $ABCD$  is inscribed into a circle with center  $O$ . The bisectors of its angles form a cyclic quadrilateral with circumcenter  $I$ , and its external bisectors form a cyclic quadrilateral with circumcenter  $J$ . Prove that  $O$  is the midpoint of  $IJ$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.01339,"x":0.05803,"p":[[0,23,0.0,0.02232,0.0724,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,23,0.1739,0.03125,0.08552,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,1,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,23,0.3478,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,0,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,23,0.5217,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,0,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,23,0.6957,0.05357,0.11152,0.0,0.0,0.0,0.0,0.28571,26,0,0,26,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,23,0.8696,0.05803,0.11214,0.0,0.0,0.0,0.0,0.28571,25,0,2,25,0,1,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[23,23,1.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.2857,30,0,0,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.02678,"x":0.0625,"p":[[0,5,0.0,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,1,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.0625,0.11811,0.0,0.0,0.0,0.0,0.28571,25,0,2,25,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.02678,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,0,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"12ec05517ec41c50","q":"Let $f$ be a function from the set of real numbers to itself that satisfies  $$ f(x+y) \\leq y f(x)+f(f(x)) $$  for all real numbers $x$ and $y$. Prove that $f(x)=0$ for all $x \\leq 0$.","t":[{"b":1,"e":0.2857,"k":"flat","v":0.13384,"x":0.17411,"p":[[0,11,0.0,0.13384,0.15125,0.0,0.14286,0.14287,0.0,0.71429,12,0,1,12,0,14,0,0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[4,11,0.3636,0.14286,0.11845,0.0,0.14286,0.2857,0.0,0.42857,10,0,2,10,0,13,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,11,0.7273,0.17411,0.19799,0.0,0.14286,0.2857,0.0,0.71429,12,0,2,12,0,10,0,0,6,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[11,11,1.0,0.14286,0.10101,0.10714,0.14286,0.17857,0.0,0.28571,8,0,0,8,0,16,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.09822,"x":0.1875,"p":[[0,65,0.0,0.16518,0.17536,0.0,0.14286,0.1786,0.0,0.57143,11,0,4,11,0,13,0,0,3,0,0,2,0,0,3,0,0,0,0,0,0,0,0],[4,65,0.0615,0.16071,0.18472,0.0,0.14286,0.2857,0.0,0.71429,12,0,2,12,0,11,0,0,6,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[8,65,0.1231,0.11161,0.15458,0.0,0.07143,0.14286,0.0,0.71429,16,0,4,16,0,11,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[12,65,0.1846,0.14732,0.16164,0.0,0.14286,0.2857,0.0,0.57143,13,0,2,13,0,10,0,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,65,0.2462,0.125,0.15465,0.0,0.07143,0.1786,0.0,0.57143,16,0,1,16,0,8,0,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[20,65,0.3077,0.1875,0.1729,0.14286,0.14286,0.28571,0.0,0.71429,7,0,2,7,0,15,0,0,7,0,0,1,0,0,0,0,0,2,0,0,0,0,0],[24,65,0.3692,0.16963,0.17287,0.0,0.14286,0.17857,0.0,0.71429,9,0,3,9,0,15,0,0,5,0,0,0,0,0,2,0,0,1,0,0,0,0,0],[28,65,0.4308,0.18294,0.17581,0.0,0.14286,0.28571,0.0,0.71429,9,0,2,9,0,13,0,0,5,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[32,65,0.4923,0.09822,0.14914,0.0,0.0,0.14286,0.0,0.57143,19,0,4,19,0,8,0,0,2,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[36,65,0.5538,0.13839,0.10998,0.0,0.14286,0.2857,0.0,0.28571,10,0,4,10,0,13,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,65,0.6154,0.16964,0.12078,0.14286,0.14286,0.28571,0.0,0.42857,7,0,2,7,0,14,0,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[44,65,0.6769,0.1517,0.09408,0.14286,0.14286,0.1786,0.0,0.28571,6,0,0,6,0,18,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,65,0.7385,0.1383,0.09094,0.14214,0.14286,0.14286,0.0,0.28571,7,0,0,7,0,19,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,65,0.8,0.125,0.09279,0.10714,0.14286,0.14286,0.0,0.42857,8,0,1,8,0,21,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[56,65,0.8615,0.17402,0.09271,0.14286,0.14286,0.2857,0.0,0.42857,3,0,1,3,0,20,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[60,65,0.9231,0.18303,0.1197,0.14286,0.14286,0.2857,0.0,0.57143,5,0,0,5,0,15,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[64,65,0.9846,0.12054,0.09523,0.0,0.14286,0.14286,0.0,0.28571,10,0,0,10,0,17,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[65,65,1.0,0.15616,0.13055,0.14286,0.14286,0.14286,0.0,0.71429,6,0,0,6,0,20,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"a640f940aaa35057","q":"Let \\( ABC \\) be a scalene triangle with smallest side \\( BC \\), and \\( D \\) be a point on the side \\( BC \\) such that \\( \\angle CAD = \\angle DAB \\). Let \\( \\omega_1 \\) and \\( \\omega_2 \\) be the circumcircles of \\( \\triangle ABD \\) and \\( \\triangle ACD \\) respectively. The line \\( AC \\) meets \\( \\omega_1 \\) again at \\( F \\), and the line \\( AB \\) meets \\( \\omega_2 \\) again at \\( E \\). The line \\( DE \\) meets \\( \\omega_1 \\) again at \\( G \\), and the line \\( DF \\) meets \\( \\omega_2 \\) again at \\( H \\). Prove that circumcircles of \\( \\triangle ABC \\), \\( \\triangle AEF \\) and \\( \\triangle AGH \\) have a common point other than \\( A \\).\n\n*Proposed by Bruno Bajo, Albania*","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.01339,"p":[[0,12,0.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,1,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.01339,0.04164,0.0,0.0,0.0,0.0,0.1429,29,0,1,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,12,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.14286,"k":"flat","v":0.00447,"x":0.03563,"p":[[0,22,0.0,0.03125,0.07771,0.0,0.0,0.0,0.0,0.2857,27,0,1,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,22,0.1818,0.03563,0.06171,0.0,0.0,0.035,0.0,0.1429,24,0,4,24,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,22,0.3636,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,4,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,22,0.5455,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,22,0.7273,0.01777,0.04701,0.0,0.0,0.0,0.0,0.1429,28,0,0,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,22,0.9091,0.02679,0.05576,0.0,0.0,0.0,0.0,0.1429,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"83d159c4c7568a7a","q":"Nancy shuffles a deck of  $52$  cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says \"stop.\"\n[list]**(a)** Can Andy guarantee that after he says \"stop,\" no card is in its initial spot?**(b)** Can Andy guarantee that after he says \"stop,\" the Queen of Spades is not adjacent to\nthe empty spot?[/list]","t":[{"b":0,"e":0.0,"k":"flat","v":0.03571,"x":0.05804,"p":[[0,14,0.0,0.04464,0.10972,0.0,0.0,0.0,0.0,0.42857,26,0,5,26,0,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.03571,0.07143,0.0,0.0,0.0,0.0,0.2857,25,0,9,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.05357,0.10564,0.0,0.0,0.03571,0.0,0.42857,24,0,2,24,0,5,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[14,14,1.0,0.05804,0.08645,0.0,0.0,0.14286,0.0,0.2857,21,0,2,21,0,9,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.00447,"x":0.06241,"p":[[0,14,0.0,0.06241,0.10671,0.0,0.0,0.14286,0.0,0.42857,22,0,4,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,25,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"37c1bd48c99f3e69","q":"Scalene triangle $A B C$ satisfies $\\angle A=60^{\\circ}$. Let the circumcenter of $A B C$ be $O$, the orthocenter be $H$, and the incenter be $I$. Let $D, T$ be the points where line $B C$ intersects the internal and external\nangle bisectors of $\\angle A$, respectively. Choose point $X$ on the circumcircle of $\\triangle I H O$ such that $H X \\| A I$. Prove that $O D \\perp T X$.","t":[{"b":2,"e":0.14286,"k":"flat","v":0.15625,"x":0.27677,"p":[[0,35,0.0,0.19188,0.18768,0.105,0.14286,0.2857,0.0,0.71429,8,0,0,8,0,14,0,0,6,0,0,1,0,0,1,0,0,2,0,0,0,0,0],[4,35,0.1143,0.25873,0.20031,0.14214,0.2857,0.42857,0.0,0.71429,6,0,0,6,0,9,0,0,8,0,0,5,0,0,2,0,0,2,0,0,0,0,0],[8,35,0.2286,0.27677,0.21108,0.14286,0.2143,0.42857,0.0,0.71429,4,0,1,4,0,12,0,0,7,0,0,3,0,0,3,0,0,3,0,0,0,0,0],[12,35,0.3429,0.24998,0.18553,0.14286,0.14288,0.28571,0.0,0.71429,3,0,0,3,0,15,0,0,7,0,0,3,0,0,2,0,0,2,0,0,0,0,0],[16,35,0.4571,0.17857,0.20202,0.0,0.14286,0.2857,0.0,0.857,11,0,1,11,0,12,0,0,4,0,0,2,0,0,2,0,0,0,0,0,1,0,0],[20,35,0.5714,0.16518,0.0724,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,23,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,35,0.6857,0.18304,0.10853,0.14286,0.14286,0.1786,0.0,0.57143,2,0,0,2,0,22,0,0,6,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[28,35,0.8,0.18286,0.10256,0.14286,0.14286,0.14286,0.0,0.57143,1,0,0,1,0,24,0,0,5,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[32,35,0.9143,0.15625,0.06546,0.14286,0.14286,0.14286,0.0,0.28571,2,0,0,2,0,25,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[35,35,1.0,0.16946,0.09745,0.14286,0.14286,0.14286,0.0,0.57143,2,0,0,2,0,24,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"falling","v":0.00893,"x":0.2319,"p":[[0,18,0.0,0.22756,0.22547,0.14214,0.14286,0.28571,0.0,0.85714,7,0,0,7,0,14,0,0,5,0,0,1,0,0,2,0,0,2,0,0,1,0,0],[4,18,0.2222,0.2319,0.21323,0.14214,0.14286,0.28571,0.0,0.71429,7,0,1,7,0,12,0,0,7,0,0,1,0,0,2,0,0,3,0,0,0,0,0],[8,18,0.4444,0.15613,0.19014,0.0,0.14286,0.1786,0.0,0.57143,14,0,0,14,0,10,0,0,3,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[12,18,0.6667,0.12045,0.15197,0.0,0.14286,0.14286,0.0,0.57143,14,0,0,14,0,14,0,0,1,0,0,1,0,0,2,0,0,0,0,0,0,0,0],[16,18,0.8889,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"38e122518410666c","q":"Prove that there are infinitely many integers $n$ such that $2^{2^{n}+1}+1$ is divisible by $n$, but $2^{n}+1$ is not.","t":[{"b":5,"e":0.0,"k":"flat","v":0.01339,"x":0.02679,"p":[[0,9,0.0,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,21,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,9,0.4444,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,14,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.01786,"x":0.08929,"p":[[0,17,0.0,0.03571,0.10714,0.0,0.0,0.0,0.0,0.42857,28,0,15,28,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.08929,0.1948,0.0,0.0,0.0,0.0,0.71429,26,0,18,26,0,0,0,0,1,0,0,3,0,0,1,0,0,1,0,0,0,0,0],[8,17,0.4706,0.01786,0.05923,0.0,0.0,0.0,0.0,0.28571,29,0,17,29,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.02679,0.08328,0.0,0.0,0.0,0.0,0.28571,29,0,22,29,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,17,0.9412,0.01786,0.06916,0.0,0.0,0.0,0.0,0.28571,30,0,16,30,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"d9b57f2bbda99be9","q":"The angles  $B$  and  $C$  of an acute-angled\u200b triangle  $ABC$  are greater than  $60^\\circ$ . Points  $P,Q$  are chosen on the sides  $AB,AC$  respectively so that the points  $A,P,Q$  are concyclic with the orthocenter  $H$  of the triangle  $ABC$ . Point  $K$  is the midpoint of  $PQ$ . Prove that  $\\angle BKC > 90^\\circ$ . \n\n*Proposed by A. Mudgal*","t":[{"b":4,"e":0.28571,"k":"flat","v":0.15177,"x":0.20965,"p":[[0,10,0.0,0.15177,0.2367,0.0,0.0,0.1786,0.0,0.857,19,0,2,19,0,5,0,0,2,0,0,2,0,0,2,0,0,1,0,0,1,0,0],[4,10,0.4,0.18293,0.18292,0.0,0.14286,0.28571,0.0,0.57143,11,0,2,11,0,10,0,0,5,0,0,3,0,0,3,0,0,0,0,0,0,0,0],[8,10,0.8,0.20965,0.13365,0.14286,0.14286,0.28571,0.0,0.4286,4,0,0,4,0,15,0,0,7,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[10,10,1.0,0.19188,0.16984,0.105,0.14286,0.2857,0.0,0.71429,8,0,1,8,0,12,0,0,8,0,0,2,0,0,1,0,0,1,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"volatile","v":0.02223,"x":0.19633,"p":[[0,5,0.0,0.19633,0.26183,0.0,0.14143,0.2857,0.0,1.0,15,1,4,15,0,5,0,0,7,0,0,1,0,0,1,0,0,1,0,0,1,0,1],[4,5,0.8,0.02223,0.05166,0.0,0.0,0.0,0.0,0.14286,27,0,4,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1e3754785b342eaa","q":"Players  $A$  and  $B$  play a game with  $N \\geq 2012$  coins and  $2012$  boxes arranged around a circle. Initially  $A$  distributes the coins among the boxes so that there is at least  $1$  coin in each box. Then the two of them make moves in the order  $B,A,B,A,\\ldots $  by the following rules:**(a)** On every move of his  $B$  passes  $1$  coin from every box to an adjacent box.**(b)** On every move of hers  $A$  chooses several coins that were *not* involved in  $B$ 's previous move and are in different boxes. She passes every coin to an adjacent box.\nPlayer  $A$ 's goal is to ensure at least  $1$  coin in each box after every move of hers, regardless of how  $B$  plays and how many moves are made. Find the least  $N$  that enables her to succeed.","t":[{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.03572,"p":[[0,5,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,22,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.03572,0.10714,0.0,0.0,0.0,0.0,0.57143,27,0,1,27,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[5,5,1.0,0.00893,0.03459,0.0,0.0,0.0,0.0,0.1429,30,0,3,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,21,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,17,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,21,0.1905,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,6,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,21,0.381,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,5,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,21,0.5714,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,21,0.7619,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,2,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,21,0.9524,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[21,21,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"aacf783c5b564845","q":"Nimatha and Thanima are playing a game on an $8 \\times 8$ chessboard. Taking turns, starting with Nimatha, each player chooses a cell that has not yet been chosen and colors it in their color (red for Nimatha, blue for Thanima). Show that Thanima can always ensure that Nimatha cannot color any $2 \\times 2$ square entirely in red.","t":[{"b":1,"e":0.42857,"k":"flat","v":0.18304,"x":0.47768,"p":[[0,43,0.0,0.36607,0.37786,0.0,0.28571,0.46431,0.0,1.0,12,7,11,12,0,1,0,0,5,0,0,6,0,0,1,0,0,0,0,0,0,0,7],[4,43,0.093,0.47768,0.36001,0.24999,0.42857,0.89286,0.0,1.0,6,8,6,6,0,2,0,0,4,0,0,10,0,0,0,0,0,1,0,0,1,0,8],[8,43,0.186,0.30357,0.33834,0.0,0.28571,0.42857,0.0,1.0,12,5,11,12,0,2,0,0,8,0,0,5,0,0,0,0,0,0,0,0,0,0,5],[12,43,0.2791,0.18304,0.22371,0.0,0.14286,0.28571,0.0,1.0,15,1,15,15,0,4,0,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,1],[16,43,0.3721,0.25446,0.22228,0.0,0.28571,0.42857,0.0,1.0,10,1,8,10,0,3,0,0,7,0,0,11,0,0,0,0,0,0,0,0,0,0,1],[20,43,0.4651,0.32143,0.33693,0.0,0.28571,0.42857,0.0,1.0,12,5,12,12,0,0,0,0,8,0,0,7,0,0,0,0,0,0,0,0,0,0,5],[24,43,0.5581,0.27678,0.29867,0.0,0.21428,0.42857,0.0,1.0,11,3,11,11,0,5,0,0,5,0,0,7,0,0,0,0,0,1,0,0,0,0,3],[28,43,0.6512,0.36161,0.22583,0.28571,0.42857,0.42857,0.0,1.0,5,2,4,5,0,0,0,0,9,0,0,15,0,0,1,0,0,0,0,0,0,0,2],[32,43,0.7442,0.33929,0.21053,0.2857,0.28571,0.42857,0.0,1.0,2,2,2,2,0,5,0,0,12,0,0,11,0,0,0,0,0,0,0,0,0,0,2],[36,43,0.8372,0.31696,0.21049,0.24999,0.28571,0.42857,0.0,1.0,5,1,5,5,0,3,0,0,11,0,0,10,0,0,1,0,0,1,0,0,0,0,1],[40,43,0.9302,0.45094,0.21162,0.28571,0.42857,0.42857,0.14286,1.0,0,3,0,0,0,2,0,0,7,0,0,18,0,0,0,0,0,2,0,0,0,0,3],[43,43,1.0,0.38393,0.0974,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,1,0,0,10,0,0,20,0,0,0,0,0,1,0,0,0,0,0]]},{"b":5,"e":0.4286,"k":"flat","v":0.27679,"x":0.37054,"p":[[0,9,0.0,0.27679,0.29653,0.0,0.28571,0.42857,0.0,1.0,13,3,13,13,0,1,0,0,5,0,0,10,0,0,0,0,0,0,0,0,0,0,3],[4,9,0.4444,0.37053,0.39425,0.0,0.28571,0.57143,0.0,1.0,12,8,12,12,0,2,0,0,5,0,0,5,0,0,0,0,0,0,0,0,0,0,8],[8,9,0.8889,0.35714,0.10715,0.28571,0.35714,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,14,0,0,15,0,0,0,0,0,1,0,0,0,0,0],[9,9,1.0,0.37054,0.1063,0.28571,0.42857,0.42857,0.14286,0.71429,0,0,0,0,0,2,0,0,11,0,0,18,0,0,0,0,0,1,0,0,0,0,0]]}]},{"i":"a4993ef8a988b728","q":"Points  $I_A, I_B, I_C$  are the centers of the excircles of  $ABC$  related to sides  $BC, AC$  and  $AB$  respectively. Perpendicular from  $I_A$  to  $AC$  intersects the perpendicular from  $I_B$  to  $B_C$  at point  $X_C$ . The points  $X_A$  and  $X_B$ . Prove that the lines  $I_AX_A, I_BX_B$  and  $I_CX_C$  intersect at the same point.","t":[{"b":3,"e":0.0,"k":"flat","v":0.00446,"x":0.02232,"p":[[0,36,0.0,0.0134,0.0549,0.0,0.0,0.0,0.0,0.286,30,0,14,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,36,0.1111,0.02232,0.08073,0.0,0.0,0.0,0.0,0.4286,29,0,20,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,36,0.2222,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,17,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,36,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,19,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,36,0.4444,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,15,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,36,0.5556,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,25,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"flat","v":0.0,"x":0.04911,"p":[[0,53,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,14,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,53,0.0755,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,16,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,53,0.1509,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,18,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,53,0.2264,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,53,0.3019,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.1429,21,0,17,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,53,0.3774,0.04464,0.09062,0.0,0.0,0.03571,0.0,0.4286,24,0,17,24,0,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[24,53,0.4528,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,16,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,53,0.5283,0.03125,0.05906,0.0,0.0,0.0,0.0,0.1429,25,0,20,25,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,53,0.6038,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,27,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,53,0.6792,0.01786,0.04725,0.0,0.0,0.0,0.0,0.14286,28,0,26,28,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,53,0.7547,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,30,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"3f56e61536a54820","q":"Suppose that $a_{0}, a_{1}, \\ldots$ and $b_{0}, b_{1}, \\ldots$ are two sequences of positive integers satisfying $a_{0}, b_{0} \\geqslant 2$ and  $$ a_{n+1}=\\operatorname{gcd}\\left(a_{n}, b_{n}\\right)+1, \\quad b_{n+1}=\\operatorname{lcm}\\left(a_{n}, b_{n}\\right)-1 $$  for all $n \\geqslant 0$. Prove that the sequence $\\left(a_{n}\\right)$ is eventually periodic; in other words, there exist integers $N \\geqslant 0$ and $t>0$ such that $a_{n+t}=a_{n}$ for all $n \\geqslant N$. (France)","t":[{"b":0,"e":0.14,"k":"flat","v":0.125,"x":0.19643,"p":[[0,4,0.0,0.125,0.08564,0.10714,0.14286,0.14286,0.0,0.28571,8,0,0,8,0,20,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.19643,0.07783,0.14286,0.14286,0.2857,0.0,0.28571,1,0,0,1,0,18,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.14286,"k":"flat","v":0.12483,"x":0.19643,"p":[[0,9,0.0,0.12483,0.11707,0.0,0.14286,0.14286,0.0,0.57143,10,0,0,10,0,18,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,9,0.4444,0.125,0.06916,0.14286,0.14286,0.14286,0.0,0.2857,6,0,0,6,0,24,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,9,0.8889,0.19643,0.08564,0.14286,0.14286,0.2857,0.14286,0.4286,0,0,0,0,0,22,0,0,8,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[9,9,1.0,0.19643,0.06916,0.14286,0.14286,0.2857,0.14286,0.28571,0,0,0,0,0,20,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"24de856f3ef6a3df","q":"Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid $\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n1 & 1 & 1 & 1 & 1 & 0 \n\\hline\n1 & 1 & 1 & 1 & 0 & 0 \n\\hline\n1 & 1 & 1 & 0 & 0 & 0 \n\\hline\n1 & 1 & 0 & 0 & 0 & 1 \n\\hline\n1 & 0 & 0 & 0 & 1 & 0\n\\hline\n0 & 0 & 0 & 1 & 0 & 0\n\\hline\n\\end{tabular}$ and asked him to transform it to the new grid below $\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n1 & 2 & 18 &24 &28 &30\n\\hline\n21 & 3 & 4 &16 &22 &26\n\\hline\n23 &19 & 5 & 6 &14 &20\n\\hline\n32 &25 &17 & 7 & 8 &12\n\\hline\n33 &34 &27 &15 & 9 &10\n\\hline\n35 &31 &36 &29 &13 &11\n\\hline\n\\end{tabular}$ by only applying the following algorithm: $\\bullet$  At each step, Sheldon must choose either two rows or two columns. $\\bullet$  For two columns  $c_1, c_2$ , if  $a,b$  are entries in  $c_1, c_2$  respectively, then we say that  $a$  and  $b$  are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them.\n\nLeonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, \"But this is impossible!\". Was Sheldon right? Justify.","t":[{"b":0,"e":1.0,"k":"rising","v":0.84375,"x":1.0,"p":[[0,5,0.0,0.84375,0.36309,1.0,1.0,1.0,0.0,1.0,5,27,4,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27],[4,5,0.8,0.97768,0.12428,1.0,1.0,1.0,0.2857,1.0,0,31,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,31],[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":1,"e":1.0,"k":"rising","v":0.77232,"x":1.0,"p":[[0,5,0.0,0.77232,0.41166,0.92857,1.0,1.0,0.0,1.0,7,24,3,7,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,24],[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]]}]},{"i":"a6368f426ff5eaf6","q":"Prove that there exist two functions\n\n$$\nf, g: \\mathbb{R} \\rightarrow \\mathbb{R}\n$$\n\nsuch that $f \\circ g$ is strictly decreasing, while $g \\circ f$ is strictly increasing.\n(Poland) Andrzej KomisArsKi \\& Marcin Kuczma\n\n#","t":[{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.05804,"p":[[0,76,0.0,0.05804,0.16698,0.0,0.0,0.0,0.0,0.57143,28,0,22,28,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[4,76,0.0526,0.05804,0.21086,0.0,0.0,0.0,0.0,1.0,29,1,21,29,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[8,76,0.1053,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,28,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,76,0.1579,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,76,0.2105,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,76,0.2632,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,76,0.3158,0.05357,0.21053,0.0,0.0,0.0,0.0,1.0,30,1,29,30,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[28,76,0.3684,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,31,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,76,0.4211,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[36,76,0.4737,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[40,76,0.5263,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,76,0.5789,0.01339,0.05486,0.0,0.0,0.0,0.0,0.28571,30,0,30,30,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,76,0.6316,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,76,0.6842,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,76,0.7368,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,30,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[60,76,0.7895,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,76,0.8421,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,76,0.8947,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,31,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[72,76,0.9474,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.08929,"p":[[0,50,0.0,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,28,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,50,0.08,0.01777,0.0777,0.0,0.0,0.0,0.0,0.42857,30,0,29,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,50,0.16,0.01786,0.07784,0.0,0.0,0.0,0.0,0.42857,30,0,24,30,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[12,50,0.24,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,28,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,50,0.32,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,30,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[20,50,0.4,0.08929,0.27837,0.0,0.0,0.0,0.0,1.0,29,2,27,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2],[24,50,0.48,0.02232,0.1017,0.0,0.0,0.0,0.0,0.57143,30,0,24,30,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[28,50,0.56,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,27,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[32,50,0.64,0.0625,0.2141,0.0,0.0,0.0,0.0,1.0,29,1,23,29,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1],[36,50,0.72,0.02679,0.09062,0.0,0.0,0.0,0.0,0.42857,29,0,23,29,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[40,50,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,23,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[44,50,0.88,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,27,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[48,50,0.96,0.02677,0.10367,0.0,0.0,0.0,0.0,0.571,29,0,27,29,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]}]},{"i":"f648adc471f3acbb","q":"Let $k$ and $n$ be fixed positive integers. In the liar's guessing game, Amy chooses integers $x$ and $N$ with $1 \\leq x \\leq N$. She tells Ben what $N$ is, but not what $x$ is. Ben may then repeatedly ask Amy whether $x \\in S$ for arbitrary sets $S$ of integers. Amy will always answer with yes or no, but she might lie. The only restriction is that she can lie at most $k$ times in a row. After he has asked as many questions as he wants, Ben must specify a set of at most $n$ positive integers. If $x$ is in this set he wins; otherwise, he loses. Prove that:  a) If $n \\geq 2^{k}$ then Ben can always win.  b) For sufficiently large $k$ there exist $n \\geq 1.99^{k}$ such that Ben cannot guarantee a win.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,11,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,5,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,2,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.0,"x":0.09821,"p":[[0,13,0.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,17,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.02671,0.07513,0.0,0.0,0.0,0.0,0.286,28,0,1,28,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.09821,0.10971,0.0,0.07143,0.14286,0.0,0.28571,16,0,0,16,0,10,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"afde922afd60afcf","q":"Suppose  $H$  and  $O$  are orthocenter and circumcenter of triangle  $ABC$ .  $\\omega$  is circumcircle of  $ABC$ .  $AO$  intersects with  $\\omega$  at  $A_1$ .  $A_1H$  intersects with  $\\omega$  at  $A'$  and  $A''$  is the intersection point of  $\\omega$  and  $AH$ . We define points  $B',\\ B'',\\ C'$  and  $C''$  similiarly. Prove that  $A'A'',B'B''$  and  $C'C''$  are concurrent in a point on the Euler line of triangle  $ABC$ .","t":[{"b":0,"e":0.0,"k":"flat","v":0.00446,"x":0.12947,"p":[[0,26,0.0,0.02679,0.08328,0.0,0.0,0.0,0.0,0.42857,28,0,4,28,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,26,0.1538,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,5,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,26,0.3077,0.03571,0.13363,0.0,0.0,0.0,0.0,0.71429,29,0,7,29,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[12,26,0.4615,0.04018,0.1439,0.0,0.0,0.0,0.0,0.71429,29,0,2,29,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[16,26,0.6154,0.12947,0.28652,0.0,0.0,0.0,0.0,0.85714,25,0,1,25,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0],[20,26,0.7692,0.08481,0.21972,0.0,0.0,0.0,0.0,0.85714,27,0,1,27,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0],[24,26,0.9231,0.04464,0.15335,0.0,0.0,0.0,0.0,0.85714,27,0,0,27,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[26,26,1.0,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,3,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":1,"e":0.0,"k":"flat","v":0.0,"x":0.04018,"p":[[0,29,0.0,0.03571,0.12372,0.0,0.0,0.0,0.0,0.57143,29,0,6,29,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[4,29,0.1379,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,2,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,29,0.2759,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,3,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,29,0.4138,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,1,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,29,0.5517,0.04018,0.12492,0.0,0.0,0.0,0.0,0.57143,28,0,1,28,0,2,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[20,29,0.6897,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,29,0.8276,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,29,0.9655,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[29,29,1.0,0.0134,0.04166,0.0,0.0,0.0,0.0,0.143,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"50858703beb131b5","q":"The cells of a  $8 \\times 8$  table are initially white. Alice and Bob play a game. First Alice paints  $n$  of the fields in red. Then Bob chooses  $4$  rows and  $4$  columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of  $n$  such that Alice can win the game no matter how Bob plays.","t":[{"b":0,"e":0.0,"k":"flat","v":0.02679,"x":0.06696,"p":[[0,5,0.0,0.06696,0.17852,0.0,0.0,0.0,0.0,0.57143,28,0,14,28,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,0,0,0],[4,5,0.8,0.02679,0.10972,0.0,0.0,0.0,0.0,0.57143,30,0,18,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.0,"k":"flat","v":0.0,"x":0.11159,"p":[[0,22,0.0,0.02679,0.10974,0.0,0.0,0.0,0.0,0.57143,30,0,12,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[4,22,0.1818,0.11159,0.22225,0.0,0.0,0.0,0.0,0.57143,25,0,17,25,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0],[8,22,0.3636,0.08927,0.19801,0.0,0.0,0.0,0.0,0.57143,26,0,12,26,0,1,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,0],[12,22,0.5455,0.06697,0.16746,0.0,0.0,0.0,0.0,0.57143,27,0,16,27,0,1,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0],[16,22,0.7273,0.06247,0.17097,0.0,0.0,0.0,0.0,0.57143,28,0,19,28,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0],[20,22,0.9091,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,5,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[22,22,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,1,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"7ba7dfeda58ec46d","q":"Let us consider a convex hexagon ABCDEF. Let  $A_1, B_1,C_1, D_1, E_1, F_1$  be midpoints of the sides  $AB, BC, CD, DE, EF,FA$  respectively. Denote by  $p$  and  $p_1$ , respectively, the perimeter of the hexagon  $ A B C D E F $  and hexagon  $ A_1B_1C_1D_1E_1F_1 $ . Suppose that all inner angles of hexagon  $ A_1B_1C_1D_1E_1F_1 $  are equal. Prove that \\[ p \\geq \\frac{2 \\cdot \\sqrt{3}}{3} \\cdot p_1 .\\] When does equality hold ?","t":[{"b":0,"e":0.0,"k":"flat","v":0.125,"x":0.13839,"p":[[0,5,0.0,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,2,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,5,0.8,0.13839,0.06667,0.14286,0.14286,0.14286,0.0,0.42857,3,0,2,3,0,28,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.125,0.04725,0.14286,0.14286,0.14286,0.0,0.1429,4,0,2,4,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.14286,"k":"flat","v":0.12054,"x":0.15179,"p":[[0,4,0.0,0.12054,0.08073,0.14286,0.14286,0.14286,0.0,0.42857,7,0,6,7,0,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,4,1.0,0.15179,0.03458,0.14286,0.14286,0.14286,0.14286,0.28571,0,0,0,0,0,30,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"bdf212b3434ae07d","q":"Prove that there are infinitely many triples $(a, b, p)$ of integers, with $p$ prime and $0<a \\leq b<p$, for which $p^{5}$ divides $(a+b)^{p}-a^{p}-b^{p}$.","t":[{"b":0,"e":0.0,"k":"flat","v":0.0,"x":0.00446,"p":[[0,14,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,14,0.2857,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,19,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,14,0.5714,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,14,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,14,0.8571,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,20,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":6,"e":0.0,"k":"flat","v":0.0,"x":0.00893,"p":[[0,12,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,18,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,12,0.3333,0.00446,0.02486,0.0,0.0,0.0,0.0,0.14286,31,0,22,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,12,0.6667,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,20,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"1a202737b6c01c48","q":"Show that for each  $n \\ge 2$ , there is a set  $S$  of  $n$  integers such that  $(a-b)^2$  divides  $ab$  for every distinct  $a, b\\in S$ .","t":[{"b":2,"e":0.0,"k":"flat","v":0.0,"x":0.06696,"p":[[0,92,0.0,0.03571,0.09449,0.0,0.0,0.0,0.0,0.28571,28,0,28,28,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,92,0.0435,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,31,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,92,0.087,0.02677,0.10965,0.0,0.0,0.0,0.0,0.571,30,0,30,30,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[12,92,0.1304,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,92,0.1739,0.00893,0.04971,0.0,0.0,0.0,0.0,0.2857,31,0,31,31,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,92,0.2174,0.0625,0.24206,0.0,0.0,0.0,0.0,1.0,30,2,30,30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],[24,92,0.2609,0.06696,0.22011,0.0,0.0,0.0,0.0,1.0,29,1,29,29,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[28,92,0.3043,0.03125,0.17399,0.0,0.0,0.0,0.0,1.0,31,1,31,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[32,92,0.3478,0.02232,0.12428,0.0,0.0,0.0,0.0,0.71429,31,0,31,31,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[36,92,0.3913,0.03571,0.17496,0.0,0.0,0.0,0.0,1.0,30,1,29,30,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[40,92,0.4348,0.01339,0.07457,0.0,0.0,0.0,0.0,0.42857,31,0,31,31,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[44,92,0.4783,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[48,92,0.5217,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[52,92,0.5652,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[56,92,0.6087,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[60,92,0.6522,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[64,92,0.6957,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[68,92,0.7391,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,32,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":7,"e":0.28571,"k":"flat","v":0.05356,"x":0.17412,"p":[[0,5,0.0,0.05356,0.19801,0.0,0.0,0.0,0.0,1.0,29,1,28,29,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,5,0.8,0.17412,0.33261,0.0,0.0,0.28571,0.0,1.0,23,4,22,23,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,0,0,0,4],[5,5,1.0,0.17402,0.15042,0.0,0.2857,0.28571,0.0,0.42857,13,0,7,13,0,1,0,0,16,0,0,2,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"43fc630aea4312fb","q":"Some of the vertices of unit squares of an  $n\\times n$  chessboard are colored so that any  $k\\times k$  (  $1\\le k\\le n$ ) square consisting of these unit squares has a colored point on at least one of its sides. Let  $l(n)$  denote the minimum number of colored points required to satisfy this condition. Prove that  $\\underset{n\\to \\infty }{\\mathop \\lim }\\,\\frac{l(n)}{{{n}^{2}}}=\\frac{2}{7}$ .","t":[{"b":1,"e":0.0,"k":"flat","v":0.00893,"x":0.0625,"p":[[0,3,0.0,0.0625,0.11811,0.0,0.0,0.14286,0.0,0.42857,23,0,9,23,0,6,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[3,3,1.0,0.00893,0.03458,0.0,0.0,0.0,0.0,0.14286,30,0,0,30,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":2,"e":0.0,"k":"flat","v":0.01339,"x":0.02679,"p":[[0,13,0.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,4,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,13,0.3077,0.02232,0.08073,0.0,0.0,0.0,0.0,0.42857,29,0,3,29,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,13,0.6154,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,4,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,13,0.9231,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[13,13,1.0,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,1,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"6c5d7ac1686b5bdb","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)","t":[{"b":0,"e":0.0,"k":"flat","v":0.02232,"x":0.0625,"p":[[0,30,0.0,0.04464,0.06622,0.0,0.0,0.14286,0.0,0.14286,22,0,1,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,30,0.1333,0.0625,0.10677,0.0,0.0,0.14286,0.0,0.42857,22,0,0,22,0,7,0,0,2,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[8,30,0.2667,0.03571,0.07143,0.0,0.0,0.0,0.0,0.28571,25,0,1,25,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[12,30,0.4,0.04455,0.06609,0.0,0.0,0.14286,0.0,0.14286,22,0,0,22,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[16,30,0.5333,0.02232,0.05187,0.0,0.0,0.0,0.0,0.14286,27,0,0,27,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[20,30,0.6667,0.02679,0.05576,0.0,0.0,0.0,0.0,0.14286,26,0,0,26,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[24,30,0.8,0.04465,0.07524,0.0,0.0,0.14286,0.0,0.28571,23,0,0,23,0,8,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[28,30,0.9333,0.04,0.06395,0.0,0.0,0.14,0.0,0.14286,23,0,0,23,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[30,30,1.0,0.04911,0.06785,0.0,0.0,0.14286,0.0,0.14286,21,0,0,21,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":5,"e":0.0,"k":"flat","v":0.03125,"x":0.05804,"p":[[0,10,0.0,0.05804,0.13296,0.0,0.0,0.14286,0.0,0.71429,23,0,0,23,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[4,10,0.4,0.04911,0.08459,0.0,0.0,0.14286,0.0,0.28571,23,0,4,23,0,7,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,10,0.8,0.03125,0.07771,0.0,0.0,0.0,0.0,0.28571,27,0,10,27,0,3,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"02272b26f93d9fdf","q":"Let $a_{1}, a_{2}, \\ldots, a_{n}$ be integers whose greatest common divisor is 1 . Let $S$ be a set of integers with the following properties: (a) $a_{i} \\in S$ for $i=1, \\ldots, n$. (b) $a_{i}-a_{j} \\in S$ for $i, j=1, \\ldots, n$, not necessarily distinct. (c) If $x, y \\in S$ and $x+y \\in S$, then $x-y \\in S$ too.  Prove that $S=\\mathbb{Z}$.","t":[{"b":1,"e":0.14286,"k":"flat","v":0.09579,"x":0.20089,"p":[[0,17,0.0,0.18527,0.14601,0.125,0.14286,0.2857,0.0,0.42857,7,0,1,7,1,13,0,0,5,0,0,6,0,0,0,0,0,0,0,0,0,0,0],[4,17,0.2353,0.20089,0.13296,0.14286,0.14286,0.28571,0.0,0.71429,3,0,1,3,0,17,0,0,10,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[8,17,0.4706,0.19634,0.11156,0.14286,0.14288,0.2857,0.0,0.4286,4,0,1,4,0,14,0,0,12,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[12,17,0.7059,0.17402,0.12747,0.14214,0.14286,0.28571,0.0,0.57143,7,0,0,7,0,13,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[16,17,0.9412,0.2007,0.11219,0.14286,0.14286,0.28571,0.0,0.571,2,0,0,2,0,18,0,0,10,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[17,17,1.0,0.09579,0.08292,0.0,0.14286,0.14286,0.0,0.2857,12,0,0,12,1,17,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.85714,"k":"rising","v":0.18063,"x":0.36607,"p":[[0,18,0.0,0.18063,0.13367,0.14214,0.14286,0.2857,0.0,0.4286,6,0,2,6,0,16,0,1,4,0,0,5,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.18741,0.12598,0.14286,0.14286,0.28571,0.0,0.42857,6,0,3,6,0,13,0,0,10,0,0,3,0,0,0,0,0,0,0,0,0,0,0],[8,18,0.4444,0.19634,0.1777,0.14286,0.14286,0.1429,0.0,1.0,3,1,3,3,0,22,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,0,1],[12,18,0.6667,0.28795,0.13657,0.19643,0.28571,0.42857,0.0,0.57143,2,0,1,2,0,6,0,1,14,0,0,7,0,0,2,0,0,0,0,0,0,0,0],[16,18,0.8889,0.36596,0.18544,0.2857,0.35714,0.42858,0.14,1.0,0,1,0,0,0,7,0,0,9,0,0,11,0,0,3,0,0,1,0,0,0,0,1],[18,18,1.0,0.36607,0.21706,0.2857,0.42857,0.42857,0.0,0.85714,4,0,0,4,0,3,0,0,7,0,0,13,0,0,0,0,0,4,0,0,1,0,0]]}]},{"i":"3724df357ab29c3f","q":"Prove that for every positive integer  $n$  there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly  $n$  diffferent triangulations.\n\n(A *triangulation* is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon)","t":[{"b":2,"e":0.1429,"k":"flat","v":0.07572,"x":0.18286,"p":[[0,18,0.0,0.07572,0.07959,0.0,0.07,0.14286,0.0,0.2857,16,0,12,16,0,15,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[4,18,0.2222,0.12503,0.13722,0.0,0.14286,0.14286,0.0,0.571,13,0,11,13,0,13,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[8,18,0.4444,0.15159,0.1333,0.105,0.14286,0.14286,0.0,0.571,8,0,6,8,0,18,0,0,3,0,0,2,0,0,1,0,0,0,0,0,0,0,0],[12,18,0.6667,0.11151,0.12738,0.0,0.14286,0.14286,0.0,0.571,13,0,7,13,0,16,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[16,18,0.8889,0.12947,0.12556,0.0,0.14286,0.1429,0.0,0.4286,12,0,5,12,0,13,0,0,5,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[18,18,1.0,0.18286,0.13477,0.14286,0.14286,0.1429,0.0,0.571,4,0,0,4,0,21,0,0,2,0,0,4,0,0,1,0,0,0,0,0,0,0,0]]},{"b":3,"e":0.0,"k":"flat","v":0.08483,"x":0.12491,"p":[[0,10,0.0,0.08483,0.10013,0.0,0.07143,0.14286,0.0,0.42857,16,0,15,16,0,14,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[4,10,0.4,0.09811,0.1207,0.0,0.14143,0.14286,0.0,0.571,15,0,13,15,0,14,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[8,10,0.8,0.11606,0.14028,0.0,0.14286,0.14286,0.0,0.571,15,0,10,15,0,11,0,0,4,0,0,1,0,0,1,0,0,0,0,0,0,0,0],[10,10,1.0,0.12491,0.0592,0.14286,0.14286,0.14286,0.0,0.28571,5,0,0,5,0,26,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]},{"i":"23ef45c980db72d3","q":"Prove that among any 20 consecutive positive integers there exists an integer $d$ such that for each positive integer $n$ we have the inequality\n\n$$\nn \\sqrt{d}\\{n \\sqrt{d}\\}>\\frac{5}{2}\n$$\n\nwhere $\\{x\\}$ denotes the fractional part of the real number $x$. The fractional part of a real number $x$ is $x$ minus the greatest integer less than or equal to $x$.","t":[{"b":2,"e":0.0,"k":"falling","v":0.01339,"x":0.27679,"p":[[0,8,0.0,0.27679,0.3387,0.0,0.14286,0.28571,0.0,1.0,11,3,4,11,0,9,0,0,5,0,0,0,0,0,1,0,0,0,0,0,3,0,3],[4,8,0.5,0.05349,0.07774,0.0,0.0,0.14286,0.0,0.28571,21,0,0,21,0,10,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[8,8,1.0,0.01339,0.04164,0.0,0.0,0.0,0.0,0.14286,29,0,0,29,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]},{"b":4,"e":0.0,"k":"falling","v":0.0,"x":0.16518,"p":[[0,5,0.0,0.16518,0.22899,0.0,0.0,0.28571,0.0,0.85714,17,0,2,17,0,3,0,0,8,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[4,5,0.8,0.0,0.0,0.0,0.0,0.0,0.0,0.0,32,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[5,5,1.0,0.00447,0.02486,0.0,0.0,0.0,0.0,0.1429,31,0,0,31,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]]}]}]);