diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/bootstrap/math_encoded_rollouts.jsonl b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/bootstrap/math_encoded_rollouts.jsonl
new file mode 100644
index 0000000000000000000000000000000000000000..be90691d687dba7d9ba9e7086423db11f9f35ef0
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/bootstrap/math_encoded_rollouts.jsonl
@@ -0,0 +1,216 @@
+{"group_id": "math_002", "origin": "budgie", "answer": "1. **Understanding the problem:** We have a race with 100 meters, and the runners are running at constant speeds. We need to determine how far behind Carol was when Anna finished.\n\n2. **Relative Speed and Time:** Since the runners are running at constant speeds, we can assume that the time taken by each runner to complete the race is constant. This means we can use relative speed and time to find out how much distance each runner covers in the time it takes for them to complete the race.\n\n3. **First runner's Time:** When Anna finishes the race, the first runner has already run for 100 meters. Since they are running at constant speeds, the first runner will cover 100 meters in the same amount of time.\n\n4. **Second runner's Time:** When Anna finishes the race, the second runner has only 90 meters left to run. Since they are running at constant speeds, the second runner will cover 90 meters in the same amount of time.\n\n5. **Distance Calculation for Anna's dye run:** When Anna finishes the race, the second runner has already run for 90 meters. Since they are running at constant speeds, the second runner covers a distance of 90 meters in the same amount of time. Therefore, the distance between Carol and Anna is 100 meters minus 90 meters, which is 10 meters.\n\nIn summary, when Anna finishes, Carol is 10 meters behind.\n\nFinal answer:\n#### 10", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 49164, 30, 16435, 28, 20259, 16422, 284, 6606, 1658, 281, 253, 1885, 49160, 49159, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 4677, 30, 1550, 16435, 27668, 28, 20259, 16422, 314, 1885, 49160, 49165, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 284, 645, 20259, 16422, 27668, 28, 6606, 314, 1885, 49161, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 30, 378, 5901, 1658, 418, 3836, 10870, 2126, 260, 4677, 30, 1073, 1869, 2893, 436, 6606, 645, 16435, 7721, 47, 198, 49, 1885, 49162, 49166, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 50, 1885, 49163, 49160, 3814, 8898, 5167, 107, 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+{"group_id": "math_002", "origin": "budgie", "answer": "1. **Identify the speeds of Anna and Bridgit:**\n   - Since Anna finishes the race in $37 \\mathrm{~m}$, her speed is $37 \\mathrm{~m/ \\text{min}}$.\n   - Bridgit finishes the race in $41 \\mathrm{~m}$, so her speed is $41 \\mathrm{~m/ \\text{min}}$.\n\n2. **Determine the time taken by each girl to finish:**\n   - Time taken by Anna: $37 \\mathrm{~m/ \\text{min}}$.\n   - Time taken by Bridgit: $41 \\mathrm{~m/ \\text{min}}$.\n\n3. **Calculate the distance each girl covers after the start of the race:**\n   - Distance covered by Anna: $37 \\mathrm{~m/ \\text{min}} \\times 30 \\mathrm{min} = 1110 \\mathrm{ \\text{ m}}$.\n   - Distance covered by Bridgit: $41 \\mathrm{~m/ \\text{min}} \\times 30 \\mathrm{min} = 1230 \\mathrm{ \\text{ m}}$.\n\n4. **Determine the time taken by Carol to finish:**\n   - Sits for $100 \\mathrm{ \\text{ m}}$ at a constant speed.\n   - Carol's speed: $100 \\mathrm{ \\text{ m}} \\times \\frac{1}{37} \\mathrm{ \\text{ min}} = \\frac{100}{37} \\mathrm{ \\text{ min}} = \\frac{100}{37} \\times \\frac{1}{3} \\mathrm", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 49164, 30, 16435, 28, 20259, 16422, 284, 6606, 1658, 281, 253, 1885, 49160, 49159, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 4677, 30, 1550, 16435, 27668, 28, 20259, 16422, 314, 1885, 49160, 49165, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 284, 645, 20259, 16422, 27668, 28, 6606, 314, 1885, 49161, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 30, 378, 5901, 1658, 418, 3836, 10870, 2126, 260, 4677, 30, 1073, 1869, 2893, 436, 6606, 645, 16435, 7721, 47, 198, 49, 1885, 49162, 49166, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 50, 1885, 49163, 49160, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 51, 1885, 49164, 49159, 3814, 8898, 5167, 107, 110, 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+{"group_id": "math_002", "origin": "verified_reference", "answer": "25. A Carol finishes 25 metres behind Bridgit, so she travels 75 metres while Bridgit runs 100 metres. Therefore she runs 3 metres for every 4 metres Bridgit runs. When Anna finishes, Bridgit has run 84 metres, so that at that time Carol has run $\\frac{3}{4} \\times 84$ metres $=63$ metres. Hence Carol finishes $(100-63)$ metres $=37$ metres behind Anna.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 49164, 30, 16435, 28, 20259, 16422, 284, 6606, 1658, 281, 253, 1885, 49160, 49159, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 4677, 30, 1550, 16435, 27668, 28, 20259, 16422, 314, 1885, 49160, 49165, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 284, 645, 20259, 16422, 27668, 28, 6606, 314, 1885, 49161, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 30, 378, 5901, 1658, 418, 3836, 10870, 2126, 260, 4677, 30, 1073, 1869, 2893, 436, 6606, 645, 16435, 7721, 47, 198, 49, 1885, 49162, 49166, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 50, 1885, 49163, 49160, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 51, 1885, 49164, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 52, 1885, 49164, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 53, 1885, 49165, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 49154, 49158, 198, 49161, 49164, 30, 330, 6606, 27668, 216, 49161, 49164, 13897, 2893, 20259, 16422, 28, 588, 1041, 12517, 216, 49166, 49164, 13897, 979, 20259, 16422, 7313, 216, 49160, 49159, 49159, 13897, 30, 4882, 1041, 7313, 216, 49162, 13897, 327, 897, 216, 49163, 13897, 20259, 16422, 7313, 30, 1550, 16435, 27668, 28, 20259, 16422, 553, 1658, 216, 49167, 49163, 13897, 28, 588, 338, 418, 338, 655, 6606, 553, 1658, 19573, 18169, 107, 49162, 15009, 49163, 109, 3814, 14602, 216, 49167, 49163, 20, 13897, 1885, 45, 49165, 49162, 20, 13897, 30, 10987, 6606, 27668, 46408, 49160, 49159, 49159, 29, 49165, 49162, 38224, 13897, 1885, 45, 49162, 49166, 20, 13897, 2893, 16435, 30, 49154], "prefix_len": 173, "old_logprobs": [-2.4253787994384766, -0.21279913187026978, -0.0605345219373703, -5.797369956970215, -11.757119178771973, -3.399350881576538, -0.8138746619224548, -1.701750636100769, -0.17213009297847748, -9.585885047912598, -0.9990986585617065, -7.616662502288818, -3.276886224746704, -2.155653476715088, -1.7828266620635986, -2.3982431888580322, -6.556173324584961, -0.3374185562133789, -6.223995685577393, -0.14264428615570068, -1.4982037544250488, -4.522729396820068, -1.346922516822815, -2.68030047416687, -4.66306734085083, -0.7837975025177002, -1.6714181900024414, -2.1646103858947754, -0.12729915976524353, -0.49660807847976685, -0.24436023831367493, -4.100385665893555, -7.9015679359436035, -2.658106565475464, -0.5397435426712036, -2.163759231567383, -10.774786949157715, -5.409459590911865, -2.4475653171539307, -0.434548556804657, -3.2349119186401367, -3.6678080558776855, -4.68051815032959, -2.774183988571167, -0.27711769938468933, -0.26706257462501526, -3.8077964782714844, -1.9413135051727295, -0.1424228549003601, -0.46229642629623413, -0.5820859670639038, -2.832719087600708, -1.7813196182250977, -0.29981404542922974, -0.14988763630390167, -4.299286365509033, -2.5988945960998535, -0.49550122022628784, -0.7137501835823059, -1.2265372276306152, -5.838071346282959, -9.68909740447998, -3.0647103786468506, -1.1420059204101562, -3.3198444843292236, -0.9523718357086182, -0.0818081945180893, -7.95733642578125, -0.34137916564941406, -0.012337321415543556, -1.0691508054733276, -0.18798594176769257, -0.1195061132311821, -0.315854012966156, -0.1612364500761032, -0.1644655168056488, -0.025299500674009323, -0.5588744878768921, -0.007398467976599932, -3.2329370975494385, -1.079887866973877, -9.250434875488281, -0.28762415051460266, -3.934783458709717, -1.977260947227478, -0.12869928777217865, -0.2151605188846588, -0.20074860751628876, -4.286519527435303, -1.8321847915649414, -4.4094648361206055, -7.663887977600098, -2.674865245819092, -0.14342664182186127, -0.0030125256162136793, -0.8097465634346008, -0.27315112948417664, -0.025270558893680573, -3.4341368675231934, -3.2909789085388184, -1.9462978839874268, -0.2908145785331726, -1.7154597043991089, -0.0015464978059753776, -0.029404832050204277, -0.3377261161804199, -0.061109308153390884, -0.5189911127090454, -0.38879287242889404, -4.2782111167907715]}
+{"group_id": "math_004", "origin": "budgie", "answer": "The best answer is 12.\n\nFinal answer:\n#### 12", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49165, 378, 3372, 282, 253, 476, 4638, 29, 1110, 3133, 18, 20570, 3283, 42, 1985, 260, 1885, 94, 20, 29, 373, 1106, 28, 253, 5437, 1251, 325, 18221, 1885, 94, 20, 1711, 30, 1094, 260, 2028, 282, 260, 2876, 6238, 429, 623, 1885, 94, 20, 20587, 314, 2852, 670, 1885, 49161, 21990, 94, 24362, 28, 260, 1106, 314, 2500, 4180, 30, 11984, 42, 198, 24, 49160, 25, 1812, 314, 260, 5428, 1230, 282, 1825, 253, 1055, 416, 1301, 281, 451, 3133, 47, 198, 24, 49161, 25, 1812, 314, 260, 8746, 338, 384, 416, 1301, 260, 808, 1296, 1825, 15233, 316, 1320, 47, 198, 24, 8928, 42, 330, 5437, 314, 253, 8281, 20636, 351, 2876, 22266, 1885, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 28, 49165, 20, 335, 624, 7975, 30, 378, 1230, 282, 2876, 335, 260, 2715, 338, 2874, 614, 990, 13549, 260, 5437, 314, 260, 966, 282, 260, 5089, 2677, 49154, 49158, 198, 504, 1450, 2988, 314, 216, 49160, 49161, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49161, 49154], "prefix_len": 185, "old_logprobs": [-1.2357776165008545, -4.043502330780029, -0.17622926831245422, -0.06694753468036652, -0.9411191940307617, -1.3033380508422852, -2.1477341651916504, -0.6913499236106873, -0.3729572892189026, -0.034664541482925415, -0.11858867108821869, -0.0007474012672901154, -0.0005296258023008704, -0.0447004958987236, -0.0042093489319086075, -0.003246871754527092, -0.0006475735572166741, -0.0013892533024773002, -0.00043084874050691724]}
+{"group_id": "math_004", "origin": "budgie", "answer": "The minimum possible outcome is rolling a 1 on one of the dice, resulting in 1 point. The maximum possible outcome is rolling an 8 on one of the dice, resulting in 8 points.\n\nThe probability of rolling a 1 on one die is $\\frac{1}{6}$, the probability of rolling an 8 on one die is $\\frac{1}{6}$. The probability of rolling an 8 on one die and then rolling a 1 on the next roll is $\\frac{1}{6} \\times \\frac{1}{6} = \\frac{1}{36}$.\n\nTherefore, the maximum number of levels is 6, and the probability that a person can pass the first three levels consecutively is $\\frac{1}{36} \\times \\frac{1}{6} = \\frac{1}{216}$.\n\nFinal answer:\n#### \\frac{1}{216}", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49165, 378, 3372, 282, 253, 476, 4638, 29, 1110, 3133, 18, 20570, 3283, 42, 1985, 260, 1885, 94, 20, 29, 373, 1106, 28, 253, 5437, 1251, 325, 18221, 1885, 94, 20, 1711, 30, 1094, 260, 2028, 282, 260, 2876, 6238, 429, 623, 1885, 94, 20, 20587, 314, 2852, 670, 1885, 49161, 21990, 94, 24362, 28, 260, 1106, 314, 2500, 4180, 30, 11984, 42, 198, 24, 49160, 25, 1812, 314, 260, 5428, 1230, 282, 1825, 253, 1055, 416, 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+{"group_id": "math_004", "origin": "verified_reference", "answer": "Solution: Since the dice is a uniform cube, the possibility of each number appearing after a roll is equal.\n(1) Because the maximum number that can appear on a single dice is 6, and $6 \\times 1 > 2^4, 6 \\times 5 < 2$.\nTherefore, when $n \\geqslant 5$, the sum of the numbers that appear after rolling the dice $n$ times cannot be greater than 2, which means this is an impossible event, and the probability of passing is 0.\nSo, the maximum number of consecutive levels that can be passed is 4.\n(2) Let event $A_{n}$ be \"failing the $n$-th level\", then the complementary event $\\overline{A_{n}}$ is \"passing the $n$-th level\". In the $n$-th level game, the total number of basic events is $6^n$.\nLevel 1: The number of basic events contained in event $A_{1}$ is 2 (i.e., the cases where the number is 1 or 2),\nSo the probability of passing this level is: $P\\left(\\overline{A_{1}}\\right)=1-P\\left(A_{1}\\right)=1-\\frac{2}{6}=\\frac{2}{3}$\nLevel 2: The number of basic events contained in event $A_{2}$ is the sum of the number of positive integer solutions to the equation $x+y=a$ when $a$ takes the values $2,3,4$. That is, $C_{1}^{3}+C_{3}^{1}+C_{3}^{1}=1+2+3=6$ (cases).\nSo the probability of passing this level is $P\\left(\\bar{A}_{2}\\right)=1-P\\left(A_{2}\\right)=1-\\frac{6}{6^{2}}=\\frac{5}{6}$\nLevel 3: The number of basic events contained in event $A_{3}$ is the sum of the number of positive integer solutions to the equation $x+y+z=a$ when $a$ takes the values $3,4,5,6,7,8$. That is, $C_{2}^{3}+C_{3}^{3}+C_{4}^{2}+C_{5}^{3}+C_{6}^{3}+C_{5}^{2}=1+3+6+10+15+21=56$ (cases).\nSo the probability of passing this level is: $P\\left(\\bar{A}_{3}\\right)=1-P\\left(A_{3}\\right)=1-\\frac{56}{6^{3}}=\\frac{20}{27}$\nTherefore, the probability of passing the first 3 levels consecutively is: $P\\left(\\bar{A}_{1}\\right) \\cdot P\\left(\\bar{A}_{2}\\right) \\cdot P\\left(\\bar{A}_{3}\\right)=\\frac{2}{3} \\times \\frac{5}{6} \\times \\frac{20}{27}=\\frac{100}{243}$", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49165, 378, 3372, 282, 253, 476, 4638, 29, 1110, 3133, 18, 20570, 3283, 42, 1985, 260, 1885, 94, 20, 29, 373, 1106, 28, 253, 5437, 1251, 325, 18221, 1885, 94, 20, 1711, 30, 1094, 260, 2028, 282, 260, 2876, 6238, 429, 623, 1885, 94, 20, 20587, 314, 2852, 670, 1885, 49161, 21990, 94, 24362, 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+{"group_id": "math_005", "origin": "budgie", "answer": "To find the probability that a randomly placed point falls within one of the five smaller spheres, we can approach the problem by considering the scenario in which the point $P$ is placed in the center of the largest sphere. Here's how we can break it down:\n\n1. The largest sphere has a radius of 1. This is because if we place a point in this sphere, the maximum distance to any other point on the sphere is 1.\n\n2. The remaining smaller spheres are arranged in a regular tetrahedron with two inscribed spheres. Each face of the tetrahedron is a sphere, and there are four smaller spheres between each pair of faces. Since the tetrahedron is regular, each face has a smaller sphere (tangent to the center of the face) and a center sphere (radius 1).\n\n3. Since there are five smaller spheres, and each face of the tetrahedron is a sphere, one of these smaller spheres is inscribed in the circumscribed sphere (radius 1), and the other four are circumscribed (radius 1).\n\n4. Each of these smaller spheres is tangent to the center of the face of the tetrahedron. Therefore, a randomly placed point on the circumscribed sphere (radius 1) will fall within one of these smaller spheres.\n\nNow, let's consider the total number of possible outcomes. Each of the five smaller spheres has a radius of 1, and there are four of them. Therefore, the total number of possible outcomes for the tetrahedron is $5^4$.\n\nHowever, we need to be careful not to overcount the cases where a point falls within a smaller sphere. For example, if a point is placed in the center of a smaller sphere but not on the edge of the tetrahedron, then it falls within the smaller sphere.\n\nTo find the probability, we need to divide the number", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49168, 49168, 49168, 1547, 4264, 3056, 14810, 17884, 25, 330, 2649, 29889, 39513, 553, 827, 21684, 28, 582, 30036, 284, 582, 4584, 45115, 30, 4454, 28, 665, 359, 1876, 3764, 21684, 826, 971, 2715, 282, 260, 29889, 39513, 284, 260, 4584, 45115, 13510, 28, 971, 37153, 288, 260, 3712, 282, 260, 2715, 30, 8779, 1885, 64, 20, 314, 253, 1225, 2972, 260, 4584, 45115, 13510, 30, 378, 8746, 338, 1885, 64, 20, 8227, 1127, 582, 282, 260, 2531, 3764, 21684, 314, 13039, 288, 365, 15165, 198, 49, 30, 216, 49159, 198, 50, 30, 216, 49159, 30, 49160, 198, 51, 30, 216, 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+{"group_id": "math_005", "origin": "budgie", "answer": "There are $24$ possible outcomes for the tetrahedron, and only one of them has the point $P$ inside. The probability that $P$ is one of the five smaller spheres is $\\frac{1}{5}$.\n\nSo, the answer is 0.1.\n\nFinal answer: B", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49168, 49168, 49168, 1547, 4264, 3056, 14810, 17884, 25, 330, 2649, 29889, 39513, 553, 827, 21684, 28, 582, 30036, 284, 582, 4584, 45115, 30, 4454, 28, 665, 359, 1876, 3764, 21684, 826, 971, 2715, 282, 260, 29889, 39513, 284, 260, 4584, 45115, 13510, 28, 971, 37153, 288, 260, 3712, 282, 260, 2715, 30, 8779, 1885, 64, 20, 314, 253, 1225, 2972, 260, 4584, 45115, 13510, 30, 378, 8746, 338, 1885, 64, 20, 8227, 1127, 582, 282, 260, 2531, 3764, 21684, 314, 13039, 288, 365, 15165, 198, 49, 30, 216, 49159, 198, 50, 30, 216, 49159, 30, 49160, 198, 51, 30, 216, 49159, 30, 49161, 198, 52, 30, 216, 49159, 30, 49162, 198, 53, 30, 216, 49159, 30, 49163, 49154, 49158, 198, 2122, 359, 1885, 49161, 49163, 20, 1636, 4530, 327, 260, 29889, 39513, 28, 284, 805, 582, 282, 601, 553, 260, 1225, 1885, 64, 20, 2972, 30, 378, 8746, 338, 1885, 64, 20, 314, 582, 282, 260, 2531, 3764, 21684, 314, 19573, 18169, 107, 49160, 15009, 49164, 24362, 30, 198, 198, 2931, 28, 260, 2988, 314, 216, 49159, 30, 49160, 30, 198, 198, 29288, 2988, 42, 389, 49154], "prefix_len": 151, "old_logprobs": [-3.278898000717163, -0.13734246790409088, -3.2542598247528076, -2.033658504486084, -2.994133472442627, -0.2270866483449936, -2.9858627319335938, -1.2006670236587524, -0.8743399381637573, -0.4853042960166931, -0.5512608289718628, -0.04467564448714256, -1.3101723194122314, -1.5326091051101685, -2.0393378734588623, -0.6318479180335999, -0.2827099859714508, -0.342880517244339, -2.889920711517334, -1.7568949460983276, -2.5293686389923096, -0.16359692811965942, -0.003510267473757267, -0.05464000254869461, -0.32643088698387146, -2.690767765045166, -2.0274410247802734, -0.2802373468875885, -0.3368476629257202, -0.21558339893817902, -0.0004991239402443171, -0.0006609162664972246, -2.3114943504333496, -2.591273069381714, -0.0004602803383022547, -0.07188691198825836, -0.2643154263496399, -0.028727667406201363, -0.006216712296009064, -0.09859368205070496, -2.596771478652954, -0.07623772323131561, -0.010312250815331936, -0.05624609813094139, -0.007632735185325146, -0.925724983215332, -0.4367166757583618, -0.250544935464859, -0.3638399839401245, -0.04707839712500572, -3.2555441856384277, -0.4908730685710907, -0.04638521745800972, -0.09929360449314117, -0.0015575670404359698, -4.2626142501831055, -0.2678670883178711, -0.02668517827987671, -0.8097407817840576, -0.3648257553577423, -0.04547082632780075, -0.02375580556690693, -0.13372793793678284, -0.0007489498239010572, -0.004175753798335791, -0.17980630695819855, -0.010958842933177948]}
+{"group_id": "math_005", "origin": "verified_reference", "answer": "6. C. Reason: Let the edge length of the regular tetrahedron be \\( a \\). It is known that the radius of its inscribed sphere is \\( R_{1} = \\frac{\\sqrt{6}}{12} a \\), and the radius of its circumscribed sphere is \\( R_{2} = \\frac{\\sqrt{6}}{4} a \\).\n\nIf the radius of the sphere that is tangent to the circumscribed sphere and the faces of the tetrahedron is \\( r \\), then\n\\[\nr = \\frac{R_{2} - R_{1}}{2} = \\frac{\\sqrt{6}}{12} a \\text{.}\n\\]\n\nThus, the required probability is\n\\[\n\\frac{\\frac{4}{3} \\pi R_{1}^{3} + 4 \\cdot \\frac{4}{3} \\pi r^{3}}{\\frac{4}{3} \\pi R_{2}^{3}} = \\frac{5 \\left( \\frac{1}{12} \\right)^{3}}{\\left( \\frac{1}{4} \\right)^{3}} = \\frac{5}{27} \\approx 0.19.\n\\]", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49168, 49168, 49168, 1547, 4264, 3056, 14810, 17884, 25, 330, 2649, 29889, 39513, 553, 827, 21684, 28, 582, 30036, 284, 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+{"group_id": "math_007", "origin": "budgie", "answer": "First, we need to understand the properties of the circumscribed circle of a triangle and the specific geometric constructions involved.\n\nIn a triangle, the circumscribed circle (circumcircle) is the circle that passes through all three vertices of the triangle. This circle is the circumcircle of the triangle.\n\nGiven that $B C = B E$, and considering the circumcircle properties in a triangle (which is a special case of a cyclic quadrilateral), we can infer several things:\n\n1. Since $B C = B E$, it follows that $B E = C E$. This means that $B, E, C$ are collinear, and angle $B E C$ is a straight line.\n\n2. The circumscribed circle's center (circumcenter) must be equidistant from the vertices of the triangle, meaning $E$ must be the circumcenter of triangle $A B C$.\n\n3. In a triangle, the circumcenter is effectively the center of the circumcircle, without the need for the circumcenter to be the circumcenter itself.\n\nThe key insight here is recognizing that because $B E = C E$, it directly follows from the properties of circumcircle in a triangle that $B E$ is the circumradius of triangle $B C D$. This is because the circumradius is the distance from the circumcenter to any vertex, and since $E$ is the circumcenter, it's the unique point that is equidistant from the other vertices of the triangle. Therefore, $\\angle B E C$ is a straight line, making it a right angle.\n\nThus, the angle $\\angle A E D$ is a right angle because it is the complement of $\\angle B E C$ (or its vertical angle, depending on the direction of the line segment $B E$), making it a right angle.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49164, 30, 365, 49162, 2876, 25, 198, 198, 788, 635, 31999, 1633, 1885, 49, 389, 340, 422, 26265, 260, 15070, 2103, 1885, 50, 340, 20, 314, 4037, 288, 260, 24570, 1885, 50, 422, 28422, 1985, 260, 3764, 13210, 1885, 49, 389, 20, 282, 260, 4584, 45115, 7842, 282, 14973, 1885, 49, 389, 340, 26265, 253, 1225, 1885, 53, 20, 314, 5958, 715, 338, 1885, 50, 340, 45, 50, 414, 28422, 7985, 260, 7256, 19573, 6935, 330, 414, 422, 28422, 49154, 49158, 198, 5345, 28, 392, 737, 288, 1044, 260, 3849, 282, 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+{"group_id": "math_007", "origin": "budgie", "answer": "To solve this problem, we will use the properties of the circumscribed circle and inscribed angles.\n\n1. Since $B C = B E$, the angle $\\angle B C E = 2 \\angle B E C$.\n2. The arc $A B$ of the circumscribed circle of triangle $A B C$ is equal to the arc $A C$. Therefore, the angle $\\angle A E C$ is twice the angle $\\angle B E C$.\n3. Since $\\angle B E C = \\frac{1}{2} \\angle B C E$ and $\\angle B C E = 2 \\angle B E C$, we have $\\angle A E C = 2 \\angle B E C = 2 \\cdot 2 \\angle B E C = \\frac{1}{2} \\angle B E C$.\n4. Substituting this value into (2), we find that $\\angle A E C = 2 \\angle B E C = 2 \\cdot 2 \\angle B E C = 4 \\angle B E C$.\n5. Since $\\angle B C E = \\frac{1}{2} \\angle B E C$, we can find $\\angle A E C = 2 \\angle B E C$.\n6. Also, since the sum of angles in a triangle is $180^\\circ$, we have $\\angle A D C + \\angle B E D = 180^\\circ - \\angle B E C$.\n7. Therefore, $\\angle A E D + 2 \\angle B E E = 180^\\circ - \\angle B E C$.\n8. 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+{"group_id": "math_007", "origin": "verified_reference", "answer": "Answer: $90^{\\circ}$.\n\n## Solution:\n\nFrom the isosceles triangle $B C D$ and the parallelism of $A B$ and $C D$, we get $\\angle B C D = \\angle B D C = \\angle D B A = \\alpha$.\n\nLet the line $C D$ intersect the circumcircle of triangle $A B C$ at point $F$. Then $B C F A$ is an inscribed, i.e., isosceles, trapezoid, from which the arcs $B C, B E$, and $A F$ are equal. Hence, $\\angle B C E = \\angle A C F = \\beta$.\n\n$\\angle E B A = \\angle E C A = \\gamma$, since these angles subtend the same arc.\n\n$\\angle B C A = \\angle B C E + \\angle E C A = \\beta + \\gamma$.\n\n$\\angle E B D = \\angle E B A + \\angle D B A = \\gamma + \\alpha$. Therefore, in the isosceles triangle $E B D$, the equality $\\angle B D E = \\angle B E D = \\frac{180^{\\circ} - \\alpha - \\gamma}{2}$ holds.\n\nMoreover, $\\alpha = \\angle B C D = \\angle B C F = \\angle B C A + \\angle A C F = 2 \\beta + \\gamma$.\n\n$\\angle A E D = \\angle B E A - \\angle B E D = (180^{\\circ} - \\angle B C A) - \\frac{180^{\\circ} - \\alpha - \\gamma}{2} = 180^{\\circ} - \\beta - \\gamma - 90^{\\circ} + \\frac{2 \\beta + 2 \\gamma}{2} = 90^{\\circ}$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49164, 30, 365, 49162, 2876, 25, 198, 198, 788, 635, 31999, 1633, 1885, 49, 389, 340, 422, 26265, 260, 15070, 2103, 1885, 50, 340, 20, 314, 4037, 288, 260, 24570, 1885, 50, 422, 28422, 1985, 260, 3764, 13210, 1885, 49, 389, 20, 282, 260, 4584, 45115, 7842, 282, 14973, 1885, 49, 389, 340, 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can determine the value of the expression.\n## Step 8:  Finally, using the value of the expression, we can calculate the value of the expression and subsequently the value of the third side of the triangle.\n\nThe final answer is: $\\boxed{50}$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 365, 30129, 216, 49160, 595, 198, 198, 788, 260, 1048, 14973, 1885, 35542, 42, 3814, 6935, 12121, 50, 45, 49168, 49159, 21990, 76, 21386, 6663, 12121, 45, 49165, 28, 7636, 45, 49163, 28422, 1985, 260, 1761, 1885, 6190, 26265, 253, 1225, 1885, 52, 365, 6998, 46, 11526, 38224, 314, 6319, 715, 338, 19573, 6935, 49132, 45, 49163, 49164, 21990, 76, 21386, 24362, 30, 1985, 260, 1761, 1885, 3907, 26265, 253, 1225, 1885, 53, 20, 314, 6319, 715, 338, 260, 25096, 282, 14973, 1885, 51, 19698, 20, 314, 260, 13341, 1636, 30, 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+{"group_id": "math_008", "origin": "verified_reference", "answer": "# Solution:\n\n$C D=A C=6 \\Rightarrow B D=B_{1} D=6-4=2$. Since triangles $\\mathrm{ACE}$ and $\\mathrm{B}_{1} \\mathrm{ED}$\n\nare similar ( $\\angle A E C=\\angle B_{1} E D$ - vertical angles and $\\angle E A C=\\angle B_{1} D E=45^{\\circ}$ ), then\n\n$E C: B_{1} E=A C: D B_{1}=A E: E D=6: 2=3: 1$\n\nLet $E H \\perp D C \\Rightarrow$ right triangles ACD and EHD are similar ( $\\angle A C D-$ common), then $C D: H D=A C: E H=A D: E D=4: 1 \\Rightarrow E H=H D=1.5 \\Rightarrow C H=4.5$.\n\nRight triangles FHE and $\\mathrm{FCA}_{1}$ are similar $\\left(\\angle E F H=\\angle A_{1} F C\\right.$ vertical angles) $)$\n\nthen $C F: H F=A_{1} F: E F=A_{1} C: E H=6: 1.5=4: 1 ; C H=4.5 \\Rightarrow C F=0.8 \\cdot 4.5=3.6$\n\nAnswer: 3.6 .", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 365, 30129, 216, 49160, 595, 198, 198, 788, 260, 1048, 14973, 1885, 35542, 42, 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+{"group_id": "math_009", "origin": "budgie", "answer": "Let's start by considering the geometry of the graph from the provided image. The function is a parabola opening upwards, with its vertex at the origin. The equation is $\\frac{1}{12} x^2 + ax + b = 0$, which we can rewrite as $y = -\\frac{1}{12} x^2 - ax + b$. This is a downward-opening parabola, and its axis of symmetry is the x-axis.\n\nThe point T with coordinates $(3, 3)$ lies on the axis of symmetry. Since the parabola is symmetric about this line, the point T is also on the axis of symmetry. Therefore, the axis of symmetry of the parabola is the $y$-axis. This implies that the equation of the axis of symmetry is $y = b$.\n\nNow, we need to find the intersection points of the graph with the $O y$ axis. The point $A$ is the intersection of the parabola with the $y$ axis, which is the line $x = 0$. Therefore, we can substitute $x = 0$ into the equation $y = b$ to find the value of $b$.\n\n$$b = \\boxed{3}$$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49160, 49159, 30, 49166, 30, 378, 3670, 282, 260, 1517, 1885, 86, 24, 104, 25, 24814, 18169, 107, 49160, 15009, 49160, 49161, 109, 1792, 21990, 49161, 109, 27, 81, 1792, 27, 82, 20, 42993, 260, 1885, 63, 1792, 20, 6867, 418, 2876, 1885, 49, 20, 284, 1885, 51, 26265, 284, 260, 1885, 63, 329, 20, 6867, 418, 1225, 1885, 50, 26265, 347, 3057, 281, 260, 4110, 30, 657, 4642, 578, 338, 327, 260, 1225, 1885, 68, 20, 351, 12957, 46408, 49162, 8544, 216, 49162, 25, 26265, 260, 2619, 1885, 68, 330, 45, 68, 389, 45, 68, 340, 20, 314, 14090, 30, 7985, 1885, 82, 28422, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 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+{"group_id": "math_009", "origin": "budgie", "answer": "Since the graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, we have the following system of equations:\n$$\n\\begin{cases}\nf(A)=\\frac{1}{12} a+b=0 \\\\\nf(C)=\\frac{1}{12} a+b=0 \\\\\nf(B)=\\frac{1}{12} a+b=0\n\\end{cases}\n$$\n\nSimplifying the equations, we obtain:\n$$\n\\begin{cases}\na=\\frac{1}{3} \\\\\nb=-\\frac{12}{3}\n\\end{cases}\n$$\n\nThus, $a=\\boxed{\\frac{1}{3}}$.\n\nFor the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ holds, which is satisfied.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49160, 49159, 30, 49166, 30, 378, 3670, 282, 260, 1517, 1885, 86, 24, 104, 25, 24814, 18169, 107, 49160, 15009, 49160, 49161, 109, 1792, 21990, 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+{"group_id": "math_009", "origin": "verified_reference", "answer": "Answer: -6.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=359&width=614&top_left_y=600&top_left_x=420)\n\nFig. 11: to the solution of problem 10.7\n\nSolution. Let point $A$ have coordinates $\\left(x_{1} ; 0\\right)$, and point $C$ have coordinates $\\left(x_{2} ; 0\\right)$. From the condition, it is clear that $x_{1} < 0$ and $x_{2} > 0$. Since $x_{1}$ and $x_{2}$ are the roots of the quadratic trinomial $f(x)$, by Vieta's theorem, we have $x_{1} \\cdot x_{2} = 12b$, from which we get $b = \\frac{x_{1} \\cdot x_{2}}{12} < 0$.\n\nLet $H$ be the point with coordinates $(3 ; 0)$ (Fig. 11). Clearly, in the isosceles triangle $A T C$, the segment $T H$ is the height, and therefore it is also the median. Thus, $3 - x_{1} = A H = H C = x_{2} - 3$, from which we get $x_{1} = 6 - x_{2}$.\n\nLet $M$ be the point with coordinates $(0, 3)$. Since $T H = T M = 3$ and $T A = T B$, the right triangles $A T H$ and $B T M$ are equal by the leg and hypotenuse. Therefore, $3 - x_{1} = H A = M B = 3 - b$, that is, $x_{1} = b = \\frac{x_{1} \\cdot x_{2}}{12}$ (by Vieta's theorem), from which we find $x_{2} = 12$. Finally, $x_{1} = 6 - x_{2} = 6 - 12 = -6$ and $b = x_{1} = -6$.\n\nAnother solution. As in the previous solution, let the abscissas of points $A$ and $C$ be $x_{1}$ and $x_{2}$, respectively; we will also use the fact that point $B$ has coordinates $(0 ; b)$. Immediately, we understand that $O A = |x_{1}| = -x_{1}$, $O C = |x_{2}| = x_{2}$, and $O B = |b| = -b$.\n\nLet's find the second intersection of the circle with the y-axis, let this be point $D$ with coordinates $(0 ; d)$ (Fig. 12). The chords $A C$ and $B D$ of the circle intersect at the origin $O$; from the properties of the circle, we know that $O A \\cdot O C = O B \\cdot O D$. We get $-x_{1} \\cdot x_{2} = -b \\cdot d$, from which, replacing $x_{1} \\cdot x_{2}$ with $12b$ by Vieta's theorem, we get $d = 12$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-40.jpg?height=521&width=571&top_left_y=83&top_left_x=437)\n\nFig. 12: to the solution of problem 10.7\n\nIt remains to note that triangle $BTD$ is isosceles, and the midpoint of its base, point $M$, has coordinates $(0 ; 3)$. 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+{"group_id": "math_010", "origin": "budgie", "answer": "Let $x$ be the length of theosal-O-L in centimeters.\n\nWe can set up a proportion based on the given information:\n\\[ \\frac{10}{x} = \\frac{75}{x + 75} \\]\n\nCross-multiply and solve for $x$:\n\\[ 10(x + 75) = 75x \\]\n\n\\[ 10x + 750 = 75x \\]\n\nSubtract $75x$ from both sides:\n\\[ -65x + 750 = 0 \\]\n\n\\[ 65x = 750 \\]\n\n\\[ x = 750 : 65 = \\boxed{10} \\]\n\nFinal answer:\n#### 10", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49162, 30, 8948, 10203, 12920, 1885, 49167, 2272, 216, 49165, 49164, 20, 198, 198, 49, 35735, 314, 1036, 15326, 288, 253, 3901, 2530, 30, 1626, 2621, 260, 3519, 282, 260, 35735, 28, 7288, 16028, 429, 260, 8618, 282, 260, 35735, 288, 260, 1112, 284, 2933, 216, 49160, 49164, 3301, 365, 5528, 3519, 216, 49167, 49159, 19573, 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+{"group_id": "math_010", "origin": "budgie", "answer": "Peter walks from the tip of thebinding sign to the end of thedeliberation sign, which is 150 steps long.\n\nSince his speed is constant, he will take 150 steps to the end of the Deliberation sign.\n\nThis is also the length of the Garcia-Petersen sign.\n\nTherefore, the length of theWA sign is 150 steps.\n\nThe answer is: $\\boxed{150}$", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49162, 30, 8948, 10203, 12920, 1885, 49167, 2272, 216, 49165, 49164, 20, 198, 198, 49, 35735, 314, 1036, 15326, 288, 253, 3901, 2530, 30, 1626, 2621, 260, 3519, 282, 260, 35735, 28, 7288, 16028, 429, 260, 8618, 282, 260, 35735, 288, 260, 1112, 284, 2933, 216, 49160, 49164, 3301, 365, 5528, 3519, 216, 49167, 49159, 19573, 8898, 5167, 107, 6946, 24362, 595, 198, 198, 2068, 4549, 1056, 429, 260, 1112, 288, 260, 8618, 282, 260, 35735, 979, 260, 4445, 314, 4138, 418, 253, 3836, 3547, 28, 384, 1923, 216, 49166, 49164, 3301, 418, 260, 1142, 6139, 3547, 30, 198, 198, 2020, 986, 314, 260, 35735, 47, 49154, 49158, 198, 29300, 16028, 429, 260, 8618, 282, 260, 26057, 914, 288, 260, 1112, 282, 260, 11074, 505, 2497, 914, 28, 527, 314, 216, 49160, 49164, 49159, 3301, 986, 30, 198, 198, 7230, 650, 3547, 314, 3836, 28, 384, 523, 1188, 216, 49160, 49164, 49159, 3301, 288, 260, 1112, 282, 260, 6374, 505, 2497, 914, 30, 198, 198, 1348, 314, 597, 260, 3519, 282, 260, 36217, 29, 64, 2922, 264, 914, 30, 198, 198, 17308, 28, 260, 3519, 282, 260, 15543, 914, 314, 216, 49160, 49164, 49159, 3301, 30, 198, 198, 504, 2988, 314, 42, 19573, 4810, 277, 107, 49160, 49164, 49159, 24362, 49154], "prefix_len": 138, "old_logprobs": [-1.529862642288208, -0.9334662556648254, -1.2981688976287842, -0.040169086307287216, -0.13930290937423706, -0.052779633551836014, -0.00245496048592031, -10.089473724365234, -7.112239837646484, -0.10736431181430817, -0.02028902806341648, -0.22162280976772308, -0.8292356729507446, -0.017735816538333893, -9.609851837158203, -3.571889877319336, -0.5101991295814514, -0.5783768892288208, -1.1507925987243652, -1.002791404724121, -0.5246886014938354, -0.3390176594257355, -0.7253702878952026, -0.22637566924095154, -1.1971886157989502, -1.4512660503387451, -1.1169061660766602, -0.485227108001709, -1.5278570652008057, -0.2769743800163269, -1.875618577003479, -3.418984889984131, -0.5003563165664673, -0.09346194565296173, -0.32593414187431335, -0.13046498596668243, -0.861644446849823, -1.6131227016448975, -0.6465285420417786, -0.32608410716056824, -0.8811635375022888, -0.07976793497800827, -0.1984155774116516, -0.38221266865730286, -1.0937529802322388, -5.303661823272705, -0.7923413515090942, -0.391355037689209, -0.07085540890693665, -2.1764492988586426, -0.024696078151464462, -0.041062138974666595, -0.0032298800069838762, -0.6901053786277771, -0.18792203068733215, -0.03515230491757393, -4.010353088378906, -2.481473922729492, -1.7644774913787842, -1.157983660697937, -2.7035489082336426, -0.04470505565404892, -0.5561898350715637, -10.625822067260742, -2.2919774055480957, -1.6565078496932983, -0.6162632703781128, -0.694914698600769, -2.468756675720215, -1.145898699760437, -0.10378701239824295, -0.007659352850168943, -1.3342466354370117, -0.011962460353970528, -0.3417043089866638, -0.5564751625061035, -0.002112539717927575, -0.09829550981521606, -10.26856803894043, -1.2196836471557617, -0.12456151098012924, -0.4014105498790741, -0.3481045365333557, -0.04279879853129387, -0.02802141383290291, -0.9259229302406311, -2.4354748725891113, -0.024871328845620155, -0.013503755442798138, -0.7680684924125671, -0.6343242526054382, -0.0038171776104718447, -0.8864362835884094, -3.210083484649658, -0.0001392267586197704, -5.960462772236497e-07, -0.0019972631707787514, -0.03017488494515419, -0.013067089952528477, -0.0019840572495013475, -0.08322035521268845, -0.1170370876789093]}
+{"group_id": "math_010", "origin": "verified_reference", "answer": "With $v=$ speed, $s=$ distance, $t_{1}=$ time for 15 steps, $t_{2}=$ time for 75 steps, and $l=$ crane length, we have $l-v t_{1}=12 \\mathrm{~m}, l+v t_{2}=60 \\mathrm{~m}$.\n\nIt holds that $\\frac{t_{1}}{t_{2}}=\\frac{1}{5} \\rightarrow t_{1}=\\frac{1}{5} t_{2}$.\n\nThus, $l-v \\cdot \\frac{1}{5} t_{2}=12 \\mathrm{~m}$ and $l+v t_{2}=60 \\mathrm{~m}$. Multiplying the first equation by 5 and adding both equations yields $6 l=120 \\mathrm{~m}, l=20 \\mathrm{~m}$. The length of the crane is $20 \\mathrm{~m}$.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49162, 30, 8948, 10203, 12920, 1885, 49167, 2272, 216, 49165, 49164, 20, 198, 198, 49, 35735, 314, 1036, 15326, 288, 253, 3901, 2530, 30, 1626, 2621, 260, 3519, 282, 260, 35735, 28, 7288, 16028, 429, 260, 8618, 282, 260, 35735, 288, 260, 1112, 284, 2933, 216, 49160, 49164, 3301, 365, 5528, 3519, 216, 49167, 49159, 19573, 8898, 5167, 107, 6946, 24362, 595, 198, 198, 2068, 4549, 1056, 429, 260, 1112, 288, 260, 8618, 282, 260, 35735, 979, 260, 4445, 314, 4138, 418, 253, 3836, 3547, 28, 384, 1923, 216, 49166, 49164, 3301, 418, 260, 1142, 6139, 3547, 30, 198, 198, 2020, 986, 314, 260, 35735, 47, 49154, 49158, 198, 3825, 1885, 102, 45, 20, 3547, 28, 1885, 99, 45, 20, 4052, 28, 1885, 100, 11300, 49160, 109, 45, 20, 655, 327, 216, 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+{"group_id": "math_011", "origin": "budgie", "answer": "The distance from $C$ to $F$ is:\n\\[ CF = \\sqrt{(s - 0)^2 + (s - 0)^2} = \\sqrt{s^2 + s^2} = s\\sqrt{2} \\]\n\nThe only possible value of $s$ is 1, but this would mean that $A = B$ or $A = C$ or $A = D$ or $B = D$ or $F = A$ and $E = C$. However, this would mean that $A = F$ and $E = D$, which is a contradiction, so $s = 1$ is the only possible solution.\n\nThe distance from $F$ to $G$ is:\n\\[ FG = \\sqrt{(s - 0)^2 + (s - 0)^2} = s\\sqrt{2} \\]\n\nThe possible values of $s$ are 1 and 2, and the distances from $C$ to $F$ and $FG$ are the same. In this case, the area $AEFG$ is:\n\\[ AEFG = \\frac{1}{2} \\cdot 1 \\cdot s\\sqrt{2} = \\frac{s\\sqrt{2}}{2} \\]\n\nThe area of the figure $AEFG$ is $2AEFG = 2 \\cdot \\frac{s\\sqrt{2}}{2} = s\\sqrt{2}$.\n\nThe final answer is:\n#### \\frac{\\sqrt{2}s}{2}", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 1777, 344, 261, 494, 389, 29, 49160, 30, 49161, 30, 198, 198, 26062, 11998, 89, 393, 273, 20598, 1437, 8026, 7494, 45945, 13450, 273, 1932, 2231, 316, 30, 1885, 49, 389, 340, 422, 414, 20, 2056, 501, 102, 344, 6095, 1885, 49, 389, 426, 452, 28422, 17120, 455, 663, 283, 90, 259, 101, 501, 26747, 1885, 54, 330, 422, 28422, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49162, 49159, 79, 84, 49163, 49164, 49164, 721, 1992, 85, 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+{"group_id": "math_011", "origin": "budgie", "answer": "The problem mentions a rectangle in which $A B F G$ is a square. We need to determine the all possible positions of the square.\n\nTo start solving this problem, let's consider the different steps in the process.\n\nFirst, we need to identify the vertices of the square. The side length of the square is equal to the side length of the rectangle.\n\nLet's denote the side length of the rectangle as $x$. Then, the side length of the square is $x$.\n\nNow, we have a square with a side length of $x$, so its top vertex is $x$ units above the bottom face of the rectangle.\n\nNext, we can consider the other vertices of the square. The bottom face of the rectangle is a square with a side length of $x$. The top vertices of this square are $x$ units above the bottom face of the rectangle.\n\nNow, we have a square with a side length of $x$, so its bottom face is a square with a side length of $x$. The top vertices of this square are $x$ units above the bottom face of the rectangle.\n\nNext, we can consider the other vertices of the square. The left face of the rectangle is a square with a side length of $x$. The bottom vertices of this square are $x$ units above the left face of the rectangle.\n\nFinally, we can consider the other vertices of the square. The right face of the rectangle is a square with a side length of $x$. The top vertices of this square are $x$ units above the right face of the rectangle.\n\nNow, we have a square with a side length of $x$, so its top face is a square with a side length of $x$. 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+{"group_id": "math_011", "origin": "verified_reference", "answer": "## Rješenje.\n\nZbroj mjera unutrašnjih kutova pravilnog peterokuta iznosi $(5-2) 180^{\\circ}=540^{\\circ}$ pa je mjera unutrašnjeg kuta pravilnog peterokuta jednaka $\\frac{540^{\\circ}}{5}=108^{\\circ}$.\n\nBudući da je trokut $A E D$ jednakokračan, $\\varangle D A E=\\frac{180^{\\circ}-108^{\\circ}}{2}=36^{\\circ}$\n\nKut $\\varangle B A F=45^{\\circ}$ jer je to kut između dijagonale i stranice kvadrata.\n\nTada je $\\varangle F A D=\\varangle B A E-\\varangle B A F-\\varangle D A E=108^{\\circ}-45^{\\circ}-36^{\\circ}=27^{\\circ}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 1777, 344, 261, 494, 389, 29, 49160, 30, 49161, 30, 198, 198, 26062, 11998, 89, 393, 273, 20598, 1437, 8026, 7494, 45945, 13450, 273, 1932, 2231, 316, 30, 1885, 49, 389, 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+{"group_id": "math_012", "origin": "budgie", "answer": "Since $x, y \\le 1$, we can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nWe can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nSince $n$ is a positive integer, we can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nWe can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\frac{x^2+y^2}{4n^2} \\le 1 + \\frac{2 - \\frac{x^2+y^2}{4n^2-4n+1}}{4n^2} \\]\n\nSimplifying the inequality to:\n\\[ 1 \\le \\frac{x^2+y^2}{4n^2} + \\frac{2 - \\frac{x^2+y^2}{4n^2-4n", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 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+{"group_id": "math_012", "origin": "budgie", "answer": "1. **Identify the Function:**\n   The function to analyze is $f(x) = \\min_{1 \\leqslant n \\leqslant 10} \\max\\left(\\frac{x^2 + y^2}{4n^2}, \\, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right)$.\n\n2. **Simplify the Function:**\n   We can simplify $f(x)$ by considering the expressions $\\frac{x^2 + y^2}{4n^2}$ and $\\frac{x^2 + y^2}{4n^2 - 4n + 1}$. Let's denote these as $p(x, y)$ for $x, y \\in \\mathbb{R}$.\n\n3. **Analyze the Sums:**\n   The function $f(x)$ can be rewritten as:\n   \\[\n   f(x) = \\min_{1 \\leqslant n \\leqslant 10} \\max\\left(\\frac{x^2 + y^2}{4n^2}, \\, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) = \\min_{n=1}^\\infty \\max\\left(\\frac{1}{4n^2}, \\, 2 - \\frac{1}{4n^2 - 4n + 1}\\right) = \\max\\left(\\frac{1}{4(1)}, \\, \\frac{2}{4(1) - 1}\\right) = \\max\\left(\\frac{1}{4}, \\frac{1}{3}\\right)\n   \\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 504, 1557, 282, 260, 2695, 281, 260, 1885, 8860, 20, 29, 12758, 19131, 260, 9724, 3814, 40561, 1659, 11300, 49160, 3814, 290, 304, 3814, 290, 216, 49160, 49159, 109, 3814, 3831, 76, 8842, 19407, 18169, 107, 104, 78, 49161, 27, 105, 78, 49161, 15009, 49163, 94, 78, 49161, 6663, 3814, 28, 216, 49161, 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+{"group_id": "math_012", "origin": "verified_reference", "answer": "To find the area of the region in the $xy$-plane satisfying the inequality \n\\[\n\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1,\n\\]\nwe need to analyze the given inequality step by step.\n\n1. **Understanding the Inequality:**\n   The inequality involves two expressions:\n   \\[\n   \\max\\left(\\frac{x^2 + y^2}{4n^2}, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) \\le 1.\n   \\]\n   This means that for some integer $n$ in the range $1 \\le n \\le 10$, the maximum of the two expressions must be less than or equal to 1.\n\n2. **Breaking Down the Inequality:**\n   For the maximum of the two expressions to be less than or equal to 1, both expressions must individually be less than or equal to 1:\n   \\[\n   \\frac{x^2 + y^2}{4n^2} \\le 1 \\quad \\text{and} \\quad 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1.\n   \\]\n\n3. **Simplifying Each Expression:**\n   - For the first expression:\n     \\[\n     \\frac{x^2 + y^2}{4n^2} \\le 1 \\implies x^2 + y^2 \\le 4n^2.\n     \\]\n   - For the second expression:\n     \\[\n     2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1 \\implies \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\ge 1 \\implies x^2 + y^2 \\ge 4n^2 - 4n + 1.\n     \\]\n\n4. **Combining the Conditions:**\n   Therefore, for some $n$ in the range $1 \\le n \\le 10$, the point $(x, y)$ must satisfy:\n   \\[\n   4n^2 - 4n + 1 \\le x^2 + y^2 \\le 4n^2.\n   \\]\n   This describes a ring (annulus) in the $xy$-plane with inner radius $\\sqrt{4n^2 - 4n + 1}$ and outer radius $2n$.\n\n5. **Calculating the Area of Each Annulus:**\n   The area of an annulus with inner radius $r_1$ and outer radius $r_2$ is given by:\n   \\[\n   \\pi (r_2^2 - r_1^2).\n   \\]\n   For our specific case:\n   \\[\n   r_2 = 2n \\quad \\text{and} \\quad r_1 = \\sqrt{4n^2 - 4n + 1}.\n   \\]\n   Therefore, the area of the annulus is:\n   \\[\n   \\pi \\left((2n)^2 - (4n^2 - 4n + 1)\\right) = \\pi \\left(4n^2 - (4n^2 - 4n + 1)\\right) = \\pi (4n - 1).\n   \\]\n\n6. **Summing the Areas for $n$ from 1 to 10:**\n   We need to sum the areas of the annuli for $n = 1, 2, \\ldots, 10$:\n   \\[\n   \\sum_{n=1}^{10} \\pi (4n - 1) = \\pi \\sum_{n=1}^{10} (4n - 1).\n   \\]\n   The sum inside the parentheses is:\n   \\[\n   \\sum_{n=1}^{10} (4n - 1) = 4 \\sum_{n=1}^{10} n - \\sum_{n=1}^{10} 1 = 4 \\left(\\frac{10 \\cdot 11}{2}\\right) - 10 = 4 \\cdot 55 - 10 = 220 - 10 = 210.\n   \\]\n\n7. **Final Area:**\n   Therefore, the total area is:\n   \\[\n   210\\pi.\n   \\]\n\nThe final answer is $\\boxed{210}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 504, 1557, 282, 260, 2695, 281, 260, 1885, 8860, 20, 29, 12758, 19131, 260, 9724, 3814, 40561, 1659, 11300, 49160, 3814, 290, 304, 3814, 290, 216, 49160, 49159, 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+{"group_id": "math_013", "origin": "budgie", "answer": "(1) For $n=1$, we have $a_{1}=2 a_{2}+3 a_{3}$, which simplifies to $2a_{2}=a_{3}$, therefore $a_{2}=a_{3}=\\frac{1}{2}$.\n\nFor $n\\geqslant 2$, the recursive formula $a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}$ can be simplified to $a_{n+1}=a_{n}$.\n\nHence, $a_{n+1}=a_{n}$.\n\nTherefore, for all $n\\in \\mathbb{N}_{+}$, $a_{n}=\\frac{1}{n}$.\n\n(2) For $n=1$, we have $a_{1}=\\frac{1}{2}$, and for all $n\\in \\mathbb{N}_{+}$, $b_{n}=1+\\frac{1}{a_{n}}(n \\in \\mathbb{N}_{+})$.\n\nWe can rewrite $b_{n}$ as follows:\n\n$$b_{n}=1+\\frac{1}{a_{n}}\\cdot n \\in \\mathbb{N}_{+}$$\n\nThe given inequality becomes:\n\n$$1+\\frac{1}{a_{1}}\\cdot\\left(1+\\frac{1}{a_{2}}\\cdot\\left(1+\\frac{1}{a_{3}}\\cdot\\left(1+\\frac{1}{a_{4}}\\cdot\\left(1+\\frac{1}{a_{5}}\\cdot\\left(1+\\frac{1}{a_{6}}\\cdot\\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 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+{"group_id": "math_013", "origin": "budgie", "answer": "(1) When $n=1$, $\\frac{1}{a_{1}}$, $a_{2}$, $a_{3}$, $\\ldots$ ;\nWhen $n=2$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $a_{3}$, $\\frac{1}{a_{3}}$, $\\ldots$;\nWhen $n=3$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $a_{4}$, $\\frac{1}{a_{4}}$, $\\ldots$;\nWhen $n=4$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$;\nWhen $n=5$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$;\n...\nWhen $n=49$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$...\n$\\end{aligned}$\n\n(2) Since for any positive integer $n(n \\geqslant 2)$, the inequality\n$\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n\nalways holds,\n\n$\\therefore$ The sequence $\\left\\{b_{n}\\right\\}$ is monotonically increasing,\n\n$\\therefore b_{1}<", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 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+{"group_id": "math_013", "origin": "verified_reference", "answer": "(1) From\n$$\n\\begin{array}{l}\na_{n}=2 a_{n} a_{n+1}+3 a_{n+1} \\\\\n\\Rightarrow a_{n+1}=\\frac{a_{n}}{2 a_{n}+3} \\Rightarrow a_{n}>0 . \\\\\n\\text { Hence } \\frac{1}{a_{n+1}}-\\frac{3}{a_{n}}=2 \\Rightarrow \\frac{1}{a_{n+1}}+1=3\\left(\\frac{1}{a_{n}}+1\\right) .\n\\end{array}\n$$\n\nTherefore, $\\left\\{\\frac{1}{a_{n}}+1\\right\\}$ is a geometric sequence with the first term 3 and common ratio 3.\nThus, $\\frac{1}{a_{n}}+1=3^{n} \\Rightarrow a_{n}=\\frac{1}{3^{n}-1}\\left(n \\in \\mathbf{N}_{+}\\right)$.\n(2) From (1), we get $b_{n}=1+\\frac{1}{a_{n}}=3^{n}$.\nTherefore, $\\log _{3} b_{k}=k(k=1,2, \\cdots)$.\nHence, $\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n$\\Leftrightarrow \\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}>\\frac{m}{24}$.\nLet $f(n)=\\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}$. 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+{"group_id": "math_015", "origin": "budgie", "answer": "a) To show that this could never have happened before and that, after the next birthday of\"{ parent_id }\", it will never happen again, we need to find a counterexample.\n\nConsider the following scenario:\n\n- The oldest daughter is 49 years old.\n- The youngest daughter is 24 years old.\n- The oldest daughter is 48 years old.\n- The youngest daughter is 24 years old.\n\nAt the next birthday of their father, as mentioned before, each daughter's age at that time is halved, and in 10 years, this will occur two times.\n\n- The oldest daughter will be 14 years old, and the youngest daughter will be 8 years old.\n- The oldest daughter will be 12 years old, and the youngest daughter will be 6 years old.\n- The youngest daughter will be 6 years old, and the oldest daughter will be 4 years old.\n\nIn this case, the oldest daughter is 6 years old.\n\nb) To determine if it is possible for the oldest daughter to be over 30 years old, we can see that she will be 30 years old only if she is 30 - 24 = 6 years old.\n\nSince the oldest daughter is 6 years old, we know that the youngest daughter is 24 - 6 = 18 years old.\n\nTherefore, Máiding will be 24 - 6 = 18 years old, and his son will be 18 + 6 = 24 years old.\n\nIn conclusion, there is a misunderstanding in the problem, as the statement \"All three daughters were born on March 18\" is not true. 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+{"group_id": "math_015", "origin": "budgie", "answer": "a) 10 years ago, Alina was 36 years old.\n2014 - 10 = 2004.\n2004 / 3 = 668.\nAlina's age is now double the age of Valentina, who was 36 years old 10 years ago.\nA 10-year-old daughter, the product of socks and the age of each sock is 10! The socks are 6 feet, 2 feet each.\nHence, Alina is 6 times 2 feet old.\nAlina is now 36 * 2 = 72 years old.\nIn 10 years, Alina will be 72 + 10 = 82 years old.\nThus, in 10 years, Traditional will be 82 x 3 = 216 years old.\nThe oldest of the daughters will be 216 - 10 = 206 years old in 10 years.\nThis fact can never have happened before and it can never happen again.\nb) \nThe oldest of the daughters is over 30 years old...\n#### 30", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 37004, 16462, 553, 216, 49162, 16924, 42, 1065, 1787, 28, 22488, 1787, 28, 284, 31492, 10945, 30, 1117, 3684, 1027, 314, 338, 511, 1296, 16924, 592, 3988, 335, 3903, 216, 49160, 49167, 30, 4104, 28, 3903, 216, 49160, 49167, 28, 216, 49161, 49159, 49160, 49163, 28, 314, 480, 15208, 30, 3349, 274, 1372, 7436, 1027, 28, 10463, 1921, 42, 198, 198, 29, 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+{"group_id": "math_015", "origin": "verified_reference", "answer": "Solution\n\na) It is clear that Alina is older than Valentina. Let's call the difference in age between Alina and Valentina $d$. Since they were born on the same day of the year, $d$ does not vary (note that this would not be true if they were born on different days, as in that case, the difference would change on the birthday of each one).\n\nIf Valentina's age today is $\\alpha$, then Alina's age must be $2 \\alpha$, so the age difference between the two is given by:\n\n$$\nd=2 \\alpha-\\alpha\n$$\n\nWe conclude that $d=\\alpha$, so Alina is $2 d$ years old and Valentina is $d$ years old.\n\nSuch a relationship between the ages of Alina and Valentina could not have happened before, as Valentina was younger than $d$ years old. It also will not happen again, as Valentina will be older than $d$ years.\n\nb) Let's call the age of the youngest daughter $A$, the age of the middle daughter $B$, and the age of the oldest daughter $C$. Let $x=B-A$ be the age difference between the two youngest, and $y=C-B$ be the age difference between the two oldest daughters. The age difference between the youngest and the oldest is therefore $C-A=x+y$. Since they were born on the same day of the year, these differences do not vary.\n\nIt is known that in each of the years 2004, 2014, and 2024, the age of one of the daughters was, is, or will be double the age of one of the others. For this, there are only 3 possibilities:\n\n(i) $B$ is double $A$.\n\n(ii) $C$ is double $B$.\n\n(iii) $C$ is double $A$.\n\nBy the reasoning used in part a), we know that each situation can only occur in one of the three years. Therefore, the three situations: (i), (ii), and (iii) must happen in the three years: 2004, 2014, and 2024, but we do not yet know in what order. Also, from part a), we know that:\n\n- When (i) happens, $B$ must be equal to $2 x$ and $A$ must be equal to $x$. Moreover, $C$ is equal to $B+y$, that is, $C$ must be equal to $2 x+y$.\n- When (ii) happens, $B=y, C=2 y$, and therefore, $A=y-x$.\n- When (iii) happens, $C=2(x+y), A=x+y$, and therefore, $B=2 x+y$.\n\nObserve that the ages of $A$ in cases (i), (ii), and (iii) are\n\n$$\nx, \\quad y-x \\quad \\text { and } \\quad x+y\n$$\n\nrespectively. Since $x<x+y$ and $y-x<x+y$, we conclude that $x+y$ is the largest of the three ages and, therefore, (iii) must happen in the year 2024. Only remains to see in which of the two years 2004 or 2014 each of the events in (i) and (ii) happens. Let's assume that (i) and (ii) happen in the years 2014 and 2004, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 would be $y-x, x$, and $x+y$, respectively. Thus, we would have the equations\n\n$$\ny-x+10=x \\quad \\text { and } \\quad x+10=x+y\n$$\n\nSolving, we would have $x=10$ and $y=10$. But in that case, $C$ would be today equal to $2 x+y=30$. But the oldest daughter is more than 30 years old, as stated in the problem. We discard, then, this possibility.\n\nWe know then that (i) and (ii) happen in the years 2004 and 2014, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 are $x, y-x$, and $x+y$, respectively. Therefore, we have that\n\n$$\nx+10=y-x \\quad \\text { and } \\quad y-x+10=x+y .\n$$\n\nSolving, we get $x=5$ and $y=20$. Finally, the ages of Francisco's daughters today are equal to\n\n$$\ny-x=15, \\quad y=20 \\quad \\text { and } \\quad 2 y=40\n$$\n\nSince Alina's age must be double Valentina's age, the ages of Alina and Valentina must be 40 and 20, respectively. 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+{"group_id": "math_016", "origin": "budgie", "answer": "From the problem, we know that $a_{1}=0$ and $a_{2}=1$.\n\nFor $n \\geqslant 3$, we have the following condition:\n$$a_{n} = a_{1} + (n-2)d$$\n\nwhere $d$ is the common difference.\n\nGiven that $a_{n}$ must be the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence, we know that $d < a_{n-1}$.\n\nTherefore, we have the following inequalities:\n\\begin{align*}\na_{n} &= a_{1} + (n-2)d \\\\\na_{n-1} &= a_{1} + (n-3)d \\\\\na_{n-2} &= a_{1} + (n-4)d\n\\end{align*}\n\nAdding these two inequalities together, we get:\n$$a_{n} + a_{n-2} &= (a_{1} + (n-2)d) + (a_{1} + (n-4)d) \\\\\n&= 2a_{1} + (n-3)d$$\n\nSince $a_{1} = 0$, we have:\n$$2a_{1} + (n-3)d = 0$$\n\nDividing both sides by 2, we get:\n$$a_{1} + \\frac{n-3}{2}d = 0$$\n\nSince $a_{1} = 0$, we have:\n$$0 + \\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 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+{"group_id": "math_016", "origin": "budgie", "answer": "Assume, by the method of differences, that $a_{n-1} = b_{n-1} - b_{n-2}$, where $b_{n-1}$ is the largest index such that $a_{n-1} > b_{n-2}$.\n\nThen, $a_{n} = b_{n-1} + b_{n-2}$, and the subsequence $a_{n-1}, a_{n-2}, \\cdots$ is the arithmetic sequence $b_{n-1} - b_{n-2}, b_{n-2} - b_{n-1}, \\cdots$.\n\nTherefore, $a_{n-1} > a_{n-2} > b_{n-3} > b_{n-4} \\cdots$, which contradicts $a_{n} = b_{n-1} + b_{n-2}$.\n\nHence, $a_{n-1} = b_{1} + b_{2} + \\cdots + b_{n-n}$.\n\nSince $a_{1} = 0$, the sequence $\\{a_{n}\\}$ is a geometric sequence with a common ratio of $b_{1} + b_{2} + \\cdots + b_{n-n}$.\n\nWe know that $a_{n-1} = b_{1} + b_{2} + \\cdots + b_{n-n-1}$.\n\nBy the property of geometric sequences, we have:\n\n$$a_{n} = b_{1} + (b_{1} + b_{2} + \\cdots + b_{n-n}) = b_{1} + (b_{1} + (b_{1} +", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 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+{"group_id": "math_016", "origin": "verified_reference", "answer": "4. First, prove a lemma using mathematical induction.\n\nLemma A non-negative integer appears in the sequence if and only if its ternary expansion contains only 0 and 1.\nProof It is obvious that the proposition holds for 0.\nAssume the proposition holds for all non-negative integers less than \\( N \\), and consider \\( N \\).\n\nIf the ternary expansion of \\( N \\) contains the digit 2, replace all occurrences of 2 with 0 to get the number \\( N_{0} \\); replace all occurrences of 2 with 1 to get the number \\( N_{1} \\). Thus, the ternary expansions of \\( N_{0} \\) and \\( N_{1} \\) do not contain the digit 2.\nBy the induction hypothesis, the numbers \\( N_{0} \\) and \\( N_{1} \\) are in the sequence.\nSince \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence, \\( N \\) does not appear in the sequence. If the ternary expansion of \\( N \\) does not contain the digit 2, then we need to prove that \\( N \\) must appear in the sequence. If not, then there exist terms \\( N_{0} \\) and \\( N_{1} \\) in the sequence such that \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence. Let the common difference be \\( d \\), and \\( 3^{k} \\| d \\). Then the ternary expansions of these three numbers have the same lowest \\( k-1 \\) digits, and the \\( k \\)-th digit is different for each pair, so one of the numbers must have a 2 in the \\( k \\)-th digit, which contradicts the assumption.\nReturning to the original problem.\nWe only need to find the 2014th non-negative integer whose ternary expansion does not contain the digit 2.\n\nNotice that under this restriction, the ternary carry-over method is equivalent to binary, so we just need to write 2013 (note \\( a_{1}=0 \\)) in binary \\((11111011101)_{2}\\) and then convert it to ternary \\( a_{2014}=88327 \\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 2959, 260, 5972, 19573, 8842, 76, 107, 81, 11300, 94, 17243, 2771, 76, 24362, 15765, 1885, 81, 11300, 49160, 109, 45, 49159, 28, 253, 11300, 49161, 109, 45, 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+{"group_id": "math_017", "origin": "budgie", "answer": "To determine the smallest positive integer $d$ for which the two numbers $m+1947n$ and $n+1947m$ share a common prime divisor $d > 9$, we need to examine the properties of the greatest common divisor (GCD) of the two numbers.\n\nLet's express the two numbers in terms of their GCD $d$. We can write:\n\\[ m = d \\times a \\]\n\\[ n = d \\times b \\]\nwhere $a$ and $b$ are integers and $a$ and $b$ are relatively prime (i.e., their GCD is 1).\n\nNow, we can express $m+1947n$ and $n+1947m$ in terms of $a$ and $b$:\n\\[ m+1947n = d(a+b) \\]\n\\[ n+1947m = d(a+b) \\]\n\nSince $a$ and $b$ are relatively prime, we can write $a = 1$ and $b = 1$, as they are relatively prime.\n\nSubstituting these values into the equations, we get:\n\\[ m+1947n = d(1+1) = 2d \\]\n\\[ n+1947m = d(1+1) = d \\]\n\nNow, we can see that $d$ must be equal to either $2$ or $d = 2$.\n\nIf $d = 2$, then $m+1947n$ and $n+1947m$ share a common divisor $2$ with $d=2$, which is $2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 657, 314, 1343, 338, 327, 634, 2731, 4078, 98, 556, 2966, 1885, 93, 20, 284, 1885, 94, 26265, 260, 2966, 1885, 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+{"group_id": "math_017", "origin": "budgie", "answer": "1. Let's denote $k$ as the common prime divisor of $m$ and $n$. We can express $m+1947 n$ and $n+1947 m$ in terms of $k$.\n\n\\[ m+1947 n = k(m+n) \\]\n\n\\[ n+1947 m = k(n+m) \\]\n\nSince $m$ and $n$ are coprime, $k$ is a positive integer. Also, since $m$ and $n$ are positive integers, $k$ must also be a positive integer.\n\nWe have the following system of equations:\n\n\\[ m+n = kn \\]\n\n\\[ m+n = kn+km \\]\n\nAdding these two equations, we get:\n\n\\[ 2mn = 2kn + 2k \\cdot kn \\]\n\n\\[ (m+n)^2 = 2kn(k+1) \\]\n\nSince $m+n$ and $n+m$ are both coprime, $k(k+1)$ is a prime number. Therefore, $2kn$ must be a prime number as well.\n\nSince $k$ is a positive integer, $2n$ must also be a prime number. This implies that $n$ is a prime number.\n\nNow, since $m$ and $n$ are coprime, $m$ and $n$ must each be a prime number. Let $m = p$ and $n = p$, where $p$ is a positive integer.\n\nSubstituting $m = p$ and $n = p$ into the first equation, we get:\n\n\\[ p + p", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 657, 314, 1343, 338, 327, 634, 2731, 4078, 98, 556, 2966, 1885, 93, 20, 284, 1885, 94, 26265, 260, 2966, 1885, 93, 27, 49160, 49168, 49163, 49166, 304, 20, 284, 1885, 94, 27, 49160, 49168, 49163, 49166, 283, 20, 457, 253, 1273, 9552, 3044, 40031, 1885, 84, 46, 49168, 28422, 7985, 260, 13341, 1636, 1685, 282, 260, 1230, 1885, 84, 20, 656, 623, 1920, 30, 49154, 49158, 198, 49160, 30, 2959, 506, 25832, 1885, 91, 20, 347, 260, 1273, 9552, 3044, 40031, 282, 1885, 93, 20, 284, 1885, 94, 28422, 1046, 416, 2667, 1885, 93, 27, 49160, 49168, 49163, 49166, 304, 20, 284, 1885, 94, 27, 49160, 49168, 49163, 49166, 283, 20, 281, 2656, 282, 1885, 91, 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+{"group_id": "math_017", "origin": "verified_reference", "answer": "Answer: $d_{\\min }=139$.\n\nFor example,\n\n$$\n\\begin{aligned}\n& m=138, n=1 \\rightarrow 1947 m+n=1947 \\cdot 138+1=139 \\cdot 1933 \\\\\n& m+1947 n=138+1947=139 \\cdot 15\n\\end{aligned}\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 657, 314, 1343, 338, 327, 634, 2731, 4078, 98, 556, 2966, 1885, 93, 20, 284, 1885, 94, 26265, 260, 2966, 1885, 93, 27, 49160, 49168, 49163, 49166, 304, 20, 284, 1885, 94, 27, 49160, 49168, 49163, 49166, 283, 20, 457, 253, 1273, 9552, 3044, 40031, 1885, 84, 46, 49168, 28422, 7985, 260, 13341, 1636, 1685, 282, 260, 1230, 1885, 84, 20, 656, 623, 1920, 30, 49154, 49158, 198, 21350, 42, 1885, 84, 11300, 76, 1659, 4029, 45, 49160, 49162, 49168, 28422, 198, 198, 2193, 1183, 28, 198, 198, 9212, 198, 76, 21083, 107, 33270, 109, 198, 22, 283, 45, 49160, 49162, 49167, 28, 304, 45, 49160, 3814, 2771, 5281, 216, 49160, 49168, 49163, 49166, 283, 27, 94, 45, 49160, 49168, 49163, 49166, 3814, 41591, 216, 49160, 49162, 49167, 27, 49160, 45, 49160, 49162, 49168, 3814, 41591, 216, 49160, 49168, 49162, 49162, 28372, 198, 22, 283, 27, 49160, 49168, 49163, 49166, 304, 45, 49160, 49162, 49167, 27, 49160, 49168, 49163, 49166, 45, 49160, 49162, 49168, 3814, 41591, 216, 49160, 49164, 198, 76, 486, 107, 33270, 109, 198, 9212, 49154], "prefix_len": 96, "old_logprobs": [-8.255396842956543, -0.06686603277921677, -3.6822292804718018, -0.7856916189193726, -6.256600856781006, -1.5826562643051147, -1.8330450057983398, -3.076355457305908, -0.7772067785263062, -1.0350090265274048, -2.794403076171875, -3.0637598037719727, -2.408771514892578, -0.6834844350814819, -0.024125806987285614, -5.4597272872924805, -1.680654764175415, -0.1114504411816597, -4.3503570556640625, -0.9929459095001221, -0.4503622055053711, -2.9832370281219482, -0.8010879158973694, -0.09334076941013336, -0.0006517431465908885, -1.5331660509109497, -0.0023445994593203068, -0.1352905035018921, -0.5642392635345459, -1.4848933219909668, -0.5996435880661011, -0.9302833080291748, -1.6878135204315186, -2.821399450302124, -1.037066102027893, -1.2712712287902832, -0.03892925754189491, -0.7761404514312744, -4.901301383972168, -3.8517258167266846, -0.00037353215157054365, -1.8723373413085938, -0.2766309976577759, -3.080369472503662, -0.44148731231689453, -0.19885674118995667, -5.6482253074646, -1.317383885383606, -2.94870924949646, -0.3810924291610718, -0.7201460599899292, -0.7355027198791504, -0.19103261828422546, -0.1093345582485199, -1.5462276935577393, -1.1664506196975708, -0.018977854400873184, -0.025431297719478607, -1.869305968284607, -0.09163767099380493, -0.7983227372169495, -0.002969144145026803, -0.7642041444778442, -2.384397506713867, -4.388052940368652, -1.2131673097610474, -4.79478645324707, -0.6617344617843628, -0.1591063141822815, -0.25342801213264465, -0.9685747623443604, -3.695786476135254, -3.1779181957244873, -1.5064858198165894, -0.13558658957481384, -0.021712806075811386, -2.0364482402801514, -0.6807119846343994, -0.7646220326423645, -0.02001470886170864, -0.003829765599220991, -0.002971640322357416, -0.07327891141176224, -0.3972114324569702, -0.08197635412216187, -0.5401096940040588, -0.44117647409439087, -0.17422930896282196, -0.010364635847508907, -0.17060358822345734, -0.01253335177898407, -0.004531471990048885, -1.666386604309082, -3.81386661529541, -2.830246686935425, -1.4262957572937012, -7.138075828552246, -0.27800849080085754, -0.02956560254096985, -0.08346811681985855, -4.835251808166504, -4.22920560836792, -0.06690104305744171, -0.001206861576065421, -5.709961988031864e-05, -0.005302887409925461, -0.016429614275693893, -0.0033410938922315836, -0.0029131616465747356, -3.315094470977783]}
+{"group_id": "math_018", "origin": "budgie", "answer": "To solve this problem, we need to find the length of $AE$ and $ED$ and use them to calculate the area of quadrilateral $A E G D$. We can do this by using the properties of similar triangles and the given values.\n\nLet's denote $AE = x$, $EB = y$, $BC = z$, $DF = w$, $DF$ intersects $AC$ at $G$. By the properties of similar triangles, we have:\n\n$$\\frac{AE}{EB} = \\frac{DF}{BC}$$\n\nSubstitute the given values for $AE$, $EB$, $BC$, and $DF$:\n\n$$\\frac{x}{y} = \\frac{w}{z}$$\n\nNow, we can simplify this equation:\n\n$$xz = yw$$\n\nWe also know that $x = \\frac{2}{3}y$, $y = \\frac{1}{2}z$, and $w = \\frac{3}{4}z$.\n\nSubstitute these values into the equation:\n\n$$\\frac{2}{3}y\\cdot\\frac{3}{4}z = y\\cdot\\frac{1}{2}z\\cdot\\frac{3}{4}z$$\n\nSimplify this equation:\n\n$$\\frac{2}{3}\\cdot\\frac{3}{4}z\\cdot\\frac{3}{4}z = y\\cdot\\frac{y}{2}z$$\n\n$z\\cdot z = y^2$\n\nSince the area of square $ABCDE$ is $840$ square centimeters, we can write the area of quadrilateral $A E G D$ as:\n\n$840 = x^2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49159, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 260, 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+{"group_id": "math_018", "origin": "budgie", "answer": "Let $x=AE$. Then $BE=x-10$. Since $BE=200$, we have $x=200+10=210$. Therefore, $DE=210-x=210-210=0$. The area of quadrilateral $A E G D$ is the sum of the areas of square $A B C D$ and triangles $A B E$ and $A E G D$ and quarter-circle $F C E$. We have $[A B C D] + [A B E] + [A AE G] + [A DE G] + [F C E] = 840 + [A E G D] + 1900 + 3400 + [210*(10-210)/2].$ Therefore, the area of the region is $x^2=10000-3318000=14200$. So, the area of quadrilateral $A E G D$ is $\\boxed{14200}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49159, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 1885, 49, 389, 340, 422, 20, 314, 216, 49167, 49163, 49159, 5222, 25556, 28, 1885, 49, 414, 45, 53, 389, 28, 389, 426, 45, 49161, 426, 340, 28, 422, 426, 20, 42993, 1885, 53, 340, 20, 418, 1885, 55, 28422, 3875, 260, 1557, 282, 20378, 13293, 1885, 49, 414, 452, 422, 20, 314, 5222, 25556, 30, 49154, 49158, 198, 4239, 1885, 104, 45, 25331, 28422, 3875, 1885, 19698, 45, 104, 29, 49160, 49159, 28422, 4311, 1885, 19698, 45, 49161, 49159, 49159, 26265, 392, 457, 1885, 104, 45, 49161, 49159, 49159, 27, 49160, 49159, 45, 49161, 49160, 49159, 28422, 4882, 28, 1885, 6279, 45, 49161, 49160, 49159, 29, 104, 45, 49161, 49160, 49159, 29, 49161, 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+{"group_id": "math_018", "origin": "verified_reference", "answer": "510\n【Analysis】This figure easily leads one to think about finding the areas of triangles $BCE$, $CDF$, and $CGF$. The most challenging part is finding the area of $\\triangle CGF$. According to the given conditions, we should draw $GM \\perp BC$ with the intersection point at $M$. This forms two sets of similar triangles corresponding to $\\triangle BCE$ and $\\triangle CDF$. By using the property that the ratio of the areas of similar triangles is equal to the square of the similarity ratio, we can find the area of $\\triangle CGF$. Finally, based on the figure, we can calculate the area of the required shape.\n\n【Solution】Draw $GM \\perp BC$ with the intersection point at $M$,\n$$\n\\begin{array}{l}\n\\therefore \\triangle FMG \\sim \\triangle FCD \\Rightarrow FM: FC = MG: CD, \\\\\n\\because BF = 2FC \\Rightarrow BC = 3FC, \\\\\n\\therefore MG = 3FM, \\\\\n\\because \\triangle CGM: \\triangle CEB \\Rightarrow CM: CB = GM: BE, \\\\\nBC = 2BE, \\\\\n\\therefore GM = \\frac{1}{2} CM = 3FM \\Rightarrow CM = 6FM, \\\\\n\\therefore FM: FC = 1: 7, \\\\\nCM: CB = 2: 7, \\\\\nS \\triangle BCE = \\frac{1}{4} \\times \\square ABCD = 210, \\\\\nS \\triangle CGM = 4 \\div 49 \\times 210 = 840 \\div 49 = 120 \\div 7, \\\\\nS \\triangle CDF = S \\square ABCD \\div 6 = 140, \\\\\nS \\triangle MGF = 140 \\times 1 \\div 49 = 140 \\div 49 = 20 \\div 7, \\\\\nS \\triangle CGM + \\triangle MGF = 120 \\div 7 + 20 \\div 7 = 20, \\\\\n840 - 210 - 140 + 20 = 510 \\text{ (square centimeters). }\n\\end{array}\n$$\n\nTherefore, the area of 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+{"group_id": "math_019", "origin": "budgie", "answer": "The volume of a tetrahedron is given by the formula $V = \\frac{1}{3} \\times \\text{area of base} \\times \\text{height}$.\n\nSince $ABCD$ is a triangle, the height is the distance from the vertex $D$ to the line $CD$. We can find the area of triangle $ABC$ using the formula for the area of a triangle: $\\frac{1}{2} \\times \\text{base} \\times \\text{height}$.\n\nSince the distance from $D$ to $CD$ is 2, the base of the triangle is 2. Therefore, the area of triangle $ABC$ is:\n$$\\frac{1}{2} \\times 2 \\times 2 = 2$$\n\nNow, we need to find the volume of the tetrahedron. We can do this by multiplying the area of triangle $ABC$ by the height from $D$ to line $CD$.\n\n$$V = \\frac{1}{3} \\times 2 \\times 2 = \\frac{4}{3}$$\n\nTherefore, the volume of tetrahedron $ABCD$ is $P = \\frac{4}{3}$.\n\nThe answer is:\n\\[ \\boxed{\\frac{4}{3}} \\]\nFinal answer:\n#### \\frac{4}{3}", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 533, 29889, 39513, 1885, 5431, 6998, 26265, 1303, 1885, 5431, 45, 49160, 26265, 1885, 6998, 24814, 16166, 107, 49162, 24362, 28, 260, 4052, 826, 3204, 1885, 5431, 20, 284, 1885, 6998, 20, 314, 216, 49161, 28, 284, 260, 7256, 826, 601, 314, 19573, 18169, 26754, 9020, 15009, 49162, 24362, 30, 3875, 260, 4981, 282, 29889, 39513, 1885, 5431, 6998, 20, 314, 365, 15165, 198, 24, 49161, 49159, 49159, 49162, 2071, 4264, 3056, 12406, 15232, 25, 198, 49, 30, 19573, 18169, 26754, 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+{"group_id": "math_019", "origin": "budgie", "answer": "1. Let $P$ be the foot of the perpendicular dropped from point $A$ to the plane of tetrahedron $ABCD$. Since the height of the tetrahedron from $A$ to plane $ABCD$ is the perpendicular distance between $AB$ and the plane of $ABCD$, we have $AP = AB = 1$.\n2. Now, let $|AP| = 1$, $|BP| = 2$, and $|CD| = \\sqrt{3}$. Then, by the Pythagorean theorem, we have $|BP| = \\sqrt{|BP|^2 + |BP|^2} = \\sqrt{4} = 2$.\n3. The volume of a tetrahedron is given by the formula $V = \\frac{1}{3} \\times \\text{area of base} \\times \\text{height}$. Here, the area of the base is the area of triangle $ABP$, and the height is the distance from $P$ to the plane $ABCD$.\n4. Therefore, the volume of tetrahedron $ABCD$ is $V = \\frac{1}{3} \\times 1 \\times 1 \\times \\frac{\\sqrt{3}}{2} = \\boxed{\\frac{\\sqrt{3}}{6}}$.\nFinal answer:\n#### \\frac{\\sqrt{3}}{6}", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 533, 29889, 39513, 1885, 5431, 6998, 26265, 1303, 1885, 5431, 45, 49160, 26265, 1885, 6998, 24814, 16166, 107, 49162, 24362, 28, 260, 4052, 826, 3204, 1885, 5431, 20, 284, 1885, 6998, 20, 314, 216, 49161, 28, 284, 260, 7256, 826, 601, 314, 19573, 18169, 26754, 9020, 15009, 49162, 24362, 30, 3875, 260, 4981, 282, 29889, 39513, 1885, 5431, 6998, 20, 314, 365, 15165, 198, 24, 49161, 49159, 49159, 49162, 2071, 4264, 3056, 12406, 15232, 25, 198, 49, 30, 19573, 18169, 26754, 16166, 107, 49162, 109, 15009, 49161, 24362, 198, 50, 30, 19573, 18169, 107, 49160, 15009, 49161, 24362, 198, 51, 30, 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+{"group_id": "math_019", "origin": "verified_reference", "answer": "1. Choose B. Reason: The area of the parallelogram formed by $\\overrightarrow{A B}$ and $\\overrightarrow{C D}$ is $|\\overrightarrow{A B}| \\cdot|\\overrightarrow{C D}| \\cdot \\sin 60^{\\circ}=\\frac{3}{2}$. If a line $C E \\mathbb{E}$ is drawn through point $C$ parallel to $A B$, then $S_{\\triangle C D E}=\\frac{1}{2} \\cdot \\frac{3}{2}=\\frac{3}{4}$, and the volume of the prism $A B F-E C D$ with $\\triangle C D E$ as the base and $B C$ as the side edge is 3 times that of $V_{A B C D}$, and $V_{A B F-E C D}=S_{\\triangle C D E} \\cdot 2=\\frac{3}{2}$, hence $V_{A B C D}=\\frac{1}{2}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 533, 29889, 39513, 1885, 5431, 6998, 26265, 1303, 1885, 5431, 45, 49160, 26265, 1885, 6998, 24814, 16166, 107, 49162, 24362, 28, 260, 4052, 826, 3204, 1885, 5431, 20, 284, 1885, 6998, 20, 314, 216, 49161, 28, 284, 260, 7256, 826, 601, 314, 19573, 18169, 26754, 9020, 15009, 49162, 24362, 30, 3875, 260, 4981, 282, 29889, 39513, 1885, 5431, 6998, 20, 314, 365, 15165, 198, 24, 49161, 49159, 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+{"group_id": "math_022", "origin": "budgie", "answer": "Okay, let me think about this. I need to find the largest natural number where all digits are different and adjacent digits differ by 6 or 7. Hmm. So, the idea is that this number has to have as many digits as possible, with all digits unique and the adjacency condition being 6 or 7. \n\nAlright, so first, let's understand the problem. We need a number that uses digits, with all different and adjacent digits differing by 6 or 7. Let's denote the digits as a, b, c, d, e, where a ≠ b, a ≠ c, a ≠ d, b ≠ c, b ≠ d, c ≠ d, and all digits are different. The question is asking for the maximum possible number under these constraints.\n\nSo, the problem is about constructing such a number with as many digits as possible, with no two adjacent digits differing by 6 or 7. To maximize the number, we want to use as many 6s as possible and 7s as much as possible. However, we have a restriction on the digits: they can't repeat digits. So, we need to find a way to arrange the digits while keeping the adjacency rule and all digits different.\n\nLet me think about how to approach this. One way to maximize the number is to use as many 6s as possible, but with some remainder. For example, if we have two 6s next to each other, we can have 6s with one digit in between, but then the digits would have to be different. However, the next digit should have to be different from both 6s. But if we have two 6s next to each other, the digits would have to be the same, which is not allowed.\n\nAlternatively, maybe we", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 216, 49164, 30, 49160, 30, 7985, 260, 3995, 1782, 1230, 281, 527, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 49154, 49158, 198, 39122, 28, 1303, 549, 1510, 563, 451, 30, 339, 737, 288, 1042, 260, 3995, 1782, 1230, 837, 511, 18638, 359, 896, 284, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 41827, 30, 1644, 28, 260, 2359, 314, 338, 451, 1230, 553, 288, 457, 347, 800, 18638, 347, 1636, 28, 351, 511, 18638, 2116, 284, 260, 7408, 31986, 2619, 1036, 216, 49165, 355, 216, 49166, 30, 3717, 198, 2219, 2771, 28, 588, 808, 28, 1303, 506, 1044, 260, 1732, 30, 1046, 737, 253, 1230, 338, 2925, 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+{"group_id": "math_022", "origin": "budgie", "answer": "To find the largest natural number where all digits are different and any two adjacent digits differ by either 6 or 7, we can approach this by considering the constraints imposed by the digits and the divisibility rules.\n\nA natural number has the form $abcdefg$, where each digit is a single digit from 0 to 9. We need to satisfy the following conditions:\n1. All digits are different.\n2. Any two adjacent digits differ by either 6 or 7.\n3. The number is as large as possible.\n\nConsidering the constraints, we can start by finding the largest possible number with the given constraints. Since the digits must all be different, the largest possible number would be a six-digit number with the first digit being 9.\n\nLet's consider the constraints again:\n\n1. The first digit cannot be 0, so it has 9 choices (1 through 9).\n2. The second digit cannot be the same as the first digit, so it has 8 choices (since one digit is already used).\n3. The third digit cannot be the same as the second digit, so it has 7 choices (since one digit is already used).\n4. The fourth digit cannot be the same as the third digit, so it has 6 choices (since two digits are already used).\n5. The fifth digit cannot be the same as the fourth digit, so it has 5 choices (since one digit is already used).\n6. The sixth digit cannot be the same as the fifth digit, so it has 4 choices (since two digits are already used).\n\nWe can initially iterate through all possible combinations of the digits and then check if they satisfy the conditions.\n\nHere is a starting point:\n\nFor the first digit, we have 9 choices.\nFor the", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 216, 49164, 30, 49160, 30, 7985, 260, 3995, 1782, 1230, 281, 527, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 49154, 49158, 198, 2068, 1042, 260, 3995, 1782, 1230, 837, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 2267, 216, 49165, 355, 216, 49166, 28, 392, 416, 2424, 451, 411, 6361, 260, 10195, 12083, 411, 260, 18638, 284, 260, 3044, 271, 1968, 3372, 30, 198, 198, 49, 1782, 1230, 553, 260, 910, 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+{"group_id": "math_022", "origin": "verified_reference", "answer": "Answer: 60718293.\n\n## Solution.\n\nWe will map each digit from 0 to 9 to a vertex in a graph and connect the vertices with an edge if the corresponding digits differ by 6 or 7.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8a1d5b347fbc679ac0eeg-5.jpg?height=448&width=590&top_left_y=524&top_left_x=819)\n\nWe see that the vertices corresponding to the digits 4 and 5 are not connected to anything, so the maximum number that can be obtained will be 8 digits long. The vertices corresponding to the digits 3 and 6 can only be the first or last in the number. Since we need to find the maximum number, it is uniquely determined by moving along the edges of the graph and will be equal to 60718293.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 216, 49164, 30, 49160, 30, 7985, 260, 3995, 1782, 1230, 281, 527, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 49154, 49158, 198, 21350, 42, 216, 49165, 49159, 49166, 49160, 49167, 49161, 49168, 49162, 30, 198, 198, 712, 19857, 30, 198, 198, 1882, 523, 4141, 971, 11403, 429, 216, 49159, 288, 216, 49168, 288, 253, 24121, 281, 253, 3670, 284, 2084, 260, 24796, 351, 354, 5595, 585, 260, 8030, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 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+{"group_id": "math_023", "origin": "budgie", "answer": "To determine the minimum value of $f(a, b, c, d)$, let's first consider each fraction in the expression $\\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$.\n\nNotice that each fraction can be rewritten as:\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$$\n\nSince the denominators are the same, we can combine the numerators:\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$$\n\nNow, let's simplify the expression by finding a common denominator. Since the denominators are the same, the common denominator is the product of the denominators.\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c} = \\frac{ad+bc+ab}{(b+c+d)(a+b+d)}$$\n\nNow, let's simplify the numerator. Notice that $(b+c+d)(a+b+d) = ab + bc + cd + ad + bd + a", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 49161, 10887, 1885, 81, 28, 278, 28, 265, 28, 287, 3814, 254, 3814, 33918, 107, 66, 41217, 27, 24362, 28, 1576, 288, 1042, 198, 20, 86, 24, 81, 28, 278, 28, 265, 28, 287, 25, 24814, 18169, 107, 81, 15009, 82, 27, 83, 27, 84, 109, 42639, 18169, 107, 82, 15009, 83, 27, 84, 27, 81, 109, 42639, 18169, 107, 83, 15009, 81, 27, 82, 27, 84, 109, 42639, 18169, 107, 84, 15009, 81, 27, 82, 27, 83, 24362, 260, 5869, 1685, 30, 49154, 49158, 198, 2068, 3346, 260, 5869, 1685, 282, 1885, 86, 24, 81, 28, 278, 28, 265, 28, 287, 25, 26265, 1303, 506, 808, 1771, 971, 10236, 281, 260, 4352, 19573, 18169, 107, 81, 15009, 82, 27, 83, 27, 84, 109, 1232, 3814, 18169, 107, 82, 15009, 83, 27, 84, 27, 81, 109, 1232, 3814, 18169, 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+{"group_id": "math_023", "origin": "budgie", "answer": "To find the minimum value of the given expression, we can use the method of Lagrange multipliers or recognize it as a known optimization problem.\n\nLet $f(x, y, z) = \\frac{x}{y+z+y} + \\frac{y}{z+x+z} + \\frac{z}{x+y+z}$, and $g(x, y, z) = \\frac{x}{y+z} + \\frac{y}{x+z} + \\frac{z}{x+y} + \\frac{w}{x+w}$, where $x, y, z, w > 0$ with $w > 0$.\n\nWe want to minimize $f(x, y, z, x) + g(x, y, z, y)$.\n\nSince $x, y, z, w > 0$, we can set $w = x + y + z$.\n\nNow, we have:\n\n$$f(x, y, z) = \\frac{x}{y+z+y} + \\frac{y}{z+x+z} + \\frac{z}{x+y+z} + \\frac{x+y+z}{x+y+z}$$\n\n$$g(x, y, z) = \\frac{x}{y+z} + \\frac{y}{x+z} + \\frac{z}{x+y} + \\frac{x+y+z}{x+y+z}$$\n\nTo minimize $f(x, y, z, x) + g(x, y, z, y)$, we want to minimize $f(x, y, z, x)$ and $g(x, y,", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 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+{"group_id": "math_023", "origin": "verified_reference", "answer": "Solution 1 Since directly eliminating the denominator is complicated, consider changing the problem.\nLet $b+c+d=A, c+d+a=B, d+a+b=C, a+b+c=D$.\nAdding these four equations, we get $a+b+c+d=\\frac{1}{3}(A+B+C+D)$.\nThus, we can find that $a=\\frac{1}{3}(B+C+D-2 A), b=\\frac{1}{3}(C+D+A-2 B)$,\n$$\nc=\\frac{1}{3}(D+A+B-2 C), d=\\frac{1}{3}(A+B+C-2 D).\n$$\n\nTherefore, the original expression $=\\frac{B+C+D-2 A}{3 A}+\\frac{C+D+A-2 B}{3 B}+\\frac{D+A+B-2 C}{3 C}+$\n$$\n\\frac{A+B+C-2 D}{3 D},\n$$\n\nwhich is $3 \\cdot f(a, b, c, d)+12$\n$$\n\\begin{array}{l}\n=\\frac{A+B+C+D}{A}+\\frac{A+B+C+D}{B}+\\frac{A+B+C+D}{C}+\\frac{A+B+C+D}{D} \\\\\n=(A+B+C+D) \\cdot\\left(\\frac{1}{A}+\\frac{1}{B}+\\frac{1}{C}+\\frac{1}{D}\\right) \\geqslant 16,\n\\end{array}\n$$\n\ni.e., $3 \\cdot f(a, b, c, d)+12 \\geqslant 16$, hence $f(a, b, c, d) \\geqslant \\frac{4}{3}$, where equality holds if and only if $A=$ $B=C=D$, i.e., $a=b=c=d$.\nTherefore, the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 2 Noting the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a^{2}}{a(b+c+d)}+\\frac{b^{2}}{b(a+c+d)}+\\frac{c^{2}}{c(a+b+d)}+\\frac{d^{2}}{d(a+b+c)} \\\\\n& \\geqslant \\frac{(a+b+c+d)^{2}}{2(a b+a c+a d+b c+b d+c d)} \\\\\n& =\\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)}\n\\end{aligned}\n$$\n\nand $a^{2}+b^{2}+c^{2}+d^{2}=\\frac{a^{2}}{1}+\\frac{b^{2}}{1}+\\frac{c^{2}}{1}+\\frac{d^{2}}{1} \\geqslant \\frac{(a+b+c+d)^{2}}{4}$, thus\n$$\nf(a, b, c, d) \\geqslant \\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\frac{1}{4}(a+b+c+d)^{2}}=\\frac{4}{3} .\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 3 Noting the Cauchy inequality or the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a+b+c+d}{b+c+d}+\\frac{a+b+c+d}{a+c+d}+\\frac{a+b+c+d}{a+b+d}+ \\\\\n& \\frac{a+b+c+d}{a+b+c}-4 \\\\\n& \\geqslant(a+b+c+d)\\left(\\frac{1}{b+c+d}+\\frac{1}{a+c+d}+\\frac{1}{a+b+d}+\\right. \\\\\n& \\left.\\frac{1}{a+b+c}\\right)-4 \\\\\n& \\geqslant(a+b+c+d) \\cdot \\frac{4^{2}}{3(a+b+c+d)}-4=\\frac{4}{3} .\n\\end{aligned}\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 49161, 10887, 1885, 81, 28, 278, 28, 265, 28, 287, 3814, 254, 3814, 33918, 107, 66, 41217, 27, 24362, 28, 1576, 288, 1042, 198, 20, 86, 24, 81, 28, 278, 28, 265, 28, 287, 25, 24814, 18169, 107, 81, 15009, 82, 27, 83, 27, 84, 109, 42639, 18169, 107, 82, 15009, 83, 27, 84, 27, 81, 109, 42639, 18169, 107, 83, 15009, 81, 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+{"group_id": "math_024", "origin": "budgie", "answer": "1. The expression is $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$.\n2. Simplify the numerator:\n$$\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31} = \\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{\\frac{31+0.31}{1}} = \\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{\\frac{31+0.31}{1}} = 2015 + 2015$$\n\nFinal answer:\n#### 2015", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 365, 49165, 2876, 25, 4104, 314, 4450, 216, 49162, 49160, 28, 216, 49161, 49159, 49160, 49164, 28, 10668, 288, 260, 216, 49161, 49159, 49160, 49164, 476, 32155, 23278, 18, 19238, 30, 378, 13310, 966, 282, 260, 4352, 19573, 18169, 26754, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49160, 109, 42639, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49159, 30, 49162, 49160, 109, 15009, 49160, 27, 49159, 30, 49162, 49160, 24362, 314, 49154, 49158, 198, 49160, 30, 378, 4352, 314, 19573, 18169, 26754, 18169, 107, 49161, 49159, 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+{"group_id": "math_024", "origin": "budgie", "answer": "2. (1 point) The problem involves calculating a complex expression.\n\nEliminating the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.\n\n$$\n\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+\\frac{2015}{0.31}} = \\frac{2015}{1} + 3030\n$$\n\nNext, we calculate the denominator of the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator, which is 1.\n\n$$\n1+\\frac{2015}{0.31} = 1 + 2015 = 2016\n$$\n\nNow that we have the full fraction, we can simplify it by dividing the numerator by 2015 and multiplying both the numerator and denominator by 1 (since 2015 is twice 1).\n\n$$\n\\frac{2015}{2015} + 3030 = \\frac{1}{2015} + 3030 = 3030 + 2015 = 5045\n$$\n\nThe result of the new problem is 5045.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 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+{"group_id": "math_024", "origin": "verified_reference", "answer": "【Solution】Solve: $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$\n$$\n\\begin{array}{l}\n=\\frac{\\left(\\frac{2015}{1}+\\frac{2015}{0.31}\\right) \\times 0.31}{(1+0.31) \\times 0.31} \\\\\n=\\frac{(1+0.31) \\times 2015}{(1+0.31) \\times 0.31} \\\\\n=6500 ;\n\\end{array}\n$$\n\nTherefore, the answer is: 6500.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 365, 49165, 2876, 25, 4104, 314, 4450, 216, 49162, 49160, 28, 216, 49161, 49159, 49160, 49164, 28, 10668, 288, 260, 216, 49161, 49159, 49160, 49164, 476, 32155, 23278, 18, 19238, 30, 378, 13310, 966, 282, 260, 4352, 19573, 18169, 26754, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49160, 109, 42639, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49159, 30, 49162, 49160, 109, 15009, 49160, 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+{"group_id": "math_026", "origin": "budgie", "answer": "1. To solve the given system of equations, we can start by rewriting the second equation in terms of $y$:\n\n$$\n\\log_{27} y - \\log_{4} x = \\frac{27}{6}\n$$\n\n2. Using the change of base formula, we can rewrite the second equation in terms of $x$:\n\n$$\n\\frac{1}{y} - \\frac{\\log_{27} x}{\\log_{4} 27} = \\frac{27}{6}\n$$\n\n3. Notice that $\\log_{27} x = \\log_{4} x$ because $27 = 4^2$. Substituting this into the second equation, we get:\n\n$$\n\\frac{1}{y} - \\frac{\\log_{4} x}{\\log_{4} 27} = \\frac{27}{6}\n$$\n\n4. Simplify the equation further by cancelling out $y$ and $\\log_{4} x$:\n\n$$\n\\frac{1}{\\log_{4} x} = \\frac{27}{6} - \\frac{\\log_{4} x}{\\log_{4} 27}\n$$\n\n5. Let $\\log_{4} 27 = a$, where $a > 0$ since $27 > 1$. Substitute this into the equation:\n\n$$\n\\frac{1}{\\log_{4} x} = \\frac{27}{6} - \\frac{a}{\\log_{4} x}\n$$\n\n6. Multiply both sides of the equation by $\\log_{4} x$ to clear the fractions:\n\n$$\n1 = \\frac{27}{6} \\log_{", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 378, 932, 282, 511, 1345, 1230, 9180, 46408, 104, 28, 329, 38224, 338, 15765, 19573, 8842, 76, 26754, 21083, 107, 4282, 15009, 92, 109, 49163, 21990, 29, 104, 109, 27, 49161, 49166, 21990, 29, 105, 109, 24814, 18169, 107, 49164, 15009, 49165, 109, 28372, 3814, 3107, 2483, 107, 49161, 49166, 109, 329, 46240, 3107, 2483, 107, 49163, 109, 1792, 3814, 361, 97, 7483, 403, 3814, 18169, 107, 49160, 15009, 49165, 109, 3814, 2692, 1662, 314, 4029, 28372, 216, 49161, 49166, 21990, 105, 39949, 49163, 21990, 104, 109, 3814, 290, 97, 7483, 403, 216, 49160, 76, 486, 107, 4282, 17243, 2771, 30, 20, 19573, 97, 32984, 20, 1673, 19573, 97, 32984, 20, 49154, 49158, 198, 49160, 30, 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+{"group_id": "math_026", "origin": "budgie", "answer": "To solve this problem, we first need to find the range of values for $x$ and $y$ such that the given equation holds true.\n\nGiven:\n\\[ \\frac{1}{4^x} + \\frac{1}{27^y} = \\frac{5}{6} \\]\n\nTo eliminate the fractions, we can multiply both sides by $6^x 4^y$. We know that $6^x 4^y = (2^2 \\cdot 3)^x 2^y = 2^x \\cdot 3^x \\cdot 2^y = 2^x \\cdot 3^x \\cdot 2^y$.\n\nSo, we have:\n\\[ 6^x 4^y \\cdot \\frac{1}{4^x} + 6^x \\cdot 4^y \\cdot \\frac{1}{27^y} = 6^x \\cdot 4^y \\cdot \\frac{5}{6} \\]\n\nNow, we can simplify the expressions on the left side of the equation:\n\\[ 6^x 4^y + 6^x \\cdot 4^y = 2^x \\cdot 3^x \\cdot 2^y \\]\n\nWe can further simplify the expression on the right side of the equation:\n\\[ 6^x \\cdot 4^y \\cdot \\frac{5}{6} = 2^x \\cdot 3^x \\cdot 2^y \\]\n\nNow, we can rewrite the equation as:\n\\[ 2^x \\cdot 3^x \\cdot 2^y + 6^x \\cdot 4^y \\cdot \\frac", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 378, 932, 282, 511, 1345, 1230, 9180, 46408, 104, 28, 329, 38224, 338, 15765, 19573, 8842, 76, 26754, 21083, 107, 4282, 15009, 92, 109, 49163, 21990, 29, 104, 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+{"group_id": "math_026", "origin": "verified_reference", "answer": "6. $\\left\\{\\left(\\frac{1}{2}, \\frac{1}{3}\\right)\\right\\}$ Detailed Explanation: Let $a=4^{x}, b=27^{y}$, then $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}, a>\\frac{6}{5}, b>\\frac{6}{5}$. From $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}$, we get $b=\\frac{6 a}{5 a-6}$, then $\\frac{6 a}{5 a-6}-a \\leqslant 1$, solving this yields $a \\geqslant 2$ or $a>\\frac{6}{5}$, so $a \\geqslant 2$. Similarly, we get $b \\leqslant 3$. Therefore, $x \\geqslant \\frac{1}{2}, y \\leqslant \\frac{1}{3} . \\log _{27} y-\\log _{4} x \\leqslant \\log _{27} \\frac{1}{3}-\\log _{4} \\frac{1}{2}=\\frac{1}{6}$. 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+{"group_id": "math_027", "origin": "budgie", "answer": "Let the number be $n$. We can represent this situation using modular arithmetic.\n\n\\[ n \\equiv 2 \\mod 3 \\]\n\\[ n \\equiv 3 \\mod 4 \\]\n\\[ n \\equiv 4 \\mod 5 \\]\n\nFirst, we need to find a number that satisfies all these conditions. We can do this by looking for a number that is 1 more than a multiple of 3.\n\n\\[ n = 3k + 1 \\]\n\nwhere $k$ is a positive integer.\n\nNow, we need to find a number that also satisfies the other two conditions:\n\n\\[ n \\equiv 2 \\mod 4 \\]\n\\[ n \\equiv 3 \\mod 4 \\]\n\nSubstitute $n = 3k + 1$ into the second congruence:\n\n\\[ 3k + 1 \\equiv 3 \\mod 4 \\]\n\nSubtract 1 from both sides:\n\n\\[ 3k \\equiv 2 \\mod 4 \\]\n\nSubtract 2 from both sides:\n\n\\[ 3k - 2 \\equiv 0 \\mod 4 \\]\n\nMultiply both sides by the multiplicative inverse of 3 modulo 4 (which is also 3 since 3 is a factor of 4):\n\n\\[ k \\equiv 2 \\times 3 \\mod 4 \\]\n\\[ k \\equiv 6 \\mod 4 \\]\n\\[ k \\equiv 2 \\mod 4 \\]\n\nSo, we have found that $k \\equiv 2 \\mod 4$. 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+{"group_id": "math_027", "origin": "budgie", "answer": "To solve this problem, we can use the Chinese Remainder Theorem (CRT). The CRT states that if we have a system of congruences\n\n$$\n\\begin{cases}\nx \\equiv 0 \\pmod{m} \\\\\nx \\equiv 2 \\pmod{m} \\\\\nx \\equiv 3 \\pmod{m} \\\\\nx \\equiv 4 \\pmod{m}\n\\end{cases}\n$$\n\nwhere $m$ is an integer, then there exists a unique solution modulo $m$.\n\nFirst, we need to find a solution modulo $m$ for each congruence. We can do this by setting the right-hand sides of each congruence equal to 0.\n\nFor the first congruence, we have $x \\equiv 0 \\pmod{3}$. This means that $x$ leaves a remainder of 0 when divided by 3.\n\nFor the second congruence, we have $2x \\equiv 2 \\pmod{4}$. Dividing both sides by 2 gives us $x \\equiv 1 \\pmod{2}$.\n\nFor the third congruence, we have $3x \\equiv 3 \\pmod{5}$. Dividing both sides by 3 gives us $x \\equiv 1 \\pmod{5}$.\n\nFor the fourth congruence, we have $4x \\equiv 4 \\pmod{6}$. Dividing both sides by 4 gives us $x \\equiv 1 \\pmod{3}$.\n\nNow, we need to find a solution modulo $m$ that satisfies all the congruences. We can do this by finding an integer $x$ that satisfies", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 365, 49165, 2876, 25, 330, 1230, 645, 6066, 411, 216, 49162, 3711, 253, 17867, 282, 216, 49161, 28, 645, 6066, 411, 216, 49163, 3711, 253, 17867, 282, 216, 49162, 28, 284, 645, 6066, 411, 216, 49164, 3711, 253, 17867, 282, 216, 49163, 30, 669, 1230, 314, 19573, 97, 32984, 20, 49154, 49158, 198, 2068, 5482, 451, 1732, 28, 392, 416, 722, 260, 4097, 3579, 402, 1223, 37050, 365, 9088, 68, 595, 378, 47281, 2496, 338, 585, 392, 457, 253, 817, 282, 37524, 1259, 198, 198, 9212, 198, 76, 21083, 107, 24407, 109, 198, 104, 3814, 2682, 380, 216, 49159, 3814, 96, 2172, 107, 93, 109, 28372, 198, 104, 3814, 2682, 380, 216, 49161, 3814, 96, 2172, 107, 93, 109, 28372, 198, 104, 3814, 2682, 380, 216, 49162, 3814, 96, 2172, 107, 93, 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+{"group_id": "math_027", "origin": "verified_reference", "answer": "【Analysis】Interpreting \"divided by 3 leaves a remainder of 2, divided by 4 leaves a remainder of 3, divided by 5 leaves a remainder of 4\" as being 1 less than when divided by 3, 1 less than when divided by 4, and 1 less than when divided by 5, means this number is at least 1 less than the least common multiple (LCM) of 3, 4, and 5. Since 3, 4, and 5 are pairwise coprime, the LCM of these three numbers is their product. Find the LCM of 3, 4, and 5, then subtract 1.\n\n【Solution】Solution: $3 \\times 4 \\times 5-1$,\n$$\n\\begin{array}{l}\n=60-1, \\\\\n=59 ;\n\\end{array}\n$$\n\nAnswer: The number is 59.\nTherefore, the answer is: 59.\n【Comment】This problem mainly examines the method of finding the least common multiple (LCM) when three numbers are pairwise coprime: If three numbers are pairwise coprime, their LCM is the product of these three numbers.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 365, 49165, 2876, 25, 330, 1230, 645, 6066, 411, 216, 49162, 3711, 253, 17867, 282, 216, 49161, 28, 645, 6066, 411, 216, 49163, 3711, 253, 17867, 282, 216, 49162, 28, 284, 645, 6066, 411, 216, 49164, 3711, 253, 17867, 282, 216, 49163, 30, 669, 1230, 314, 19573, 97, 32984, 20, 49154, 49158, 198, 12109, 234, 27931, 12109, 235, 8235, 1702, 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+{"group_id": "math_028", "origin": "budgie", "answer": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares.\n\nThe key is that the area of the rectangle is equal to the area of a square.\n\nThe area of the rectangle is $A = n^2$ (since it's a rectangle).\n\nThe area of the square is $A = (n + 76)^2$ (since it's a square).\n\nSince the area of the rectangle is equal to the area of a square, we have $A = A$.\n\nThis means $n^2 = (n + 76)^2$.\n\nExpanding the equation, we get $n^2 = n^2 + 152n + 5916$.\n\nSubtracting $n^2$ from both sides, we get $0 = 152n + 5915$.\n\nSubtracting $5915$ from both sides, we get $-5910 = 152n$.\n\nDividing both sides by $152$, we get $n = -5910/152$.\n\nSimplifying the right side, we get $n = -58.5785794...$.\n\nSince $n$ must be a positive integer, the only possible value of $n$ is $n = -5910/152 \\approx -41.4083$.\n\nHowever, since $n$ must be a non-negative integer, we consider the next value of $n$, which is $n = -5910/152 = -58$.\n\nThe only possible value of $n$", "reward": -1, "advantage": 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+{"group_id": "math_028", "origin": "budgie", "answer": "If a rectangle can be divided into $n$ equal squares, then the area of the rectangle must be a perfect square. Since the area is a perfect square, this means that the length and width of the rectangle must be powers of 2.\n\nLet $a$ and $b$ be the length and width of the rectangle, respectively. Then we have:\n\n\\[ a^2 = 2^n \\quad \\text{and} \\quad b^2 = 2^n + 76 \\]\n\nWe can rewrite these equations as:\n\n\\[ a = 2^x \\quad \\text{and} \\quad b = 2^y \\]\n\nwhere $x$ and $y$ are non-negative integers.\n\nSubstituting these expressions for $a$ and $b$, we get:\n\n\\[ (2^x)^2 = 2^n \\quad \\text{and} \\quad (2^y)^2 = 2^n + 76 \\]\n\nSimplifying these equations, we get:\n\n\\[ 2^{2x} = 2^n \\quad \\text{and} \\quad 2^{2y} = 2^n + 76 \\]\n\nDividing both sides of the first equation by $2^x$, we get:\n\n\\[ 2^{2x - x} = 2^n \\Rightarrow 2^x = 2^n \\Rightarrow x = n \\]\n\nSimilarly, dividing both sides of the second equation by $2^y$, we get:\n\n\\[ 2^{2y - y} = 2^n + 76 \\Rightarrow 2^y = 2^n +", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 49, 23710, 416, 325, 6066, 618, 1885, 94, 20, 4037, 16207, 30, 378, 1142, 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+{"group_id": "math_028", "origin": "verified_reference", "answer": "Answer: $n=324$.\n\nLet $a b=n$ and $c d=n+76$, where $a, b$ and $c, d$ are the numbers of squares in each direction for the partitioning of the rectangle into $n$ and $n+76$ squares, respectively. Then $\\frac{a}{c}=\\frac{b}{d}$, or $a d=b c$. Denote $u=\\operatorname{gcd}(a, c)$ and $v=\\operatorname{gcd}(b, d)$, then there exist positive integers $x$ and $y$ such that $\\operatorname{gcd}(x, y)=1, a=u x, c=u y$ and $b=v x, d=v y$. Hence we have\n\n$$\nc d-a b=u v\\left(y^{2}-x^{2}\\right)=u v(y-x)(y+x)=76=2^{2} \\cdot 19 .\n$$\n\nSince $y-x$ and $y+x$ are positive integers of the same parity and $\\operatorname{gcd}(x, y)=1$, we have $y-x=1$ and $y+x=19$ as the only possibility, yielding $y=10, x=9$ and $u v=4$. 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+{"group_id": "math_029", "origin": "budgie", "answer": "The maximum number of non-palindrome years is 999 years + 1 year = 1000 years.\n\nFinal answer:\n#### 1000", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 20, 75, 20, 933, 3780, 17516, 30, 11689, 13310, 28, 2686, 12879, 1885, 77, 20, 198, 198, 37835, 42, 13212, 383, 259, 431, 452, 30, 49, 1143, 38838, 10251, 12713, 422, 11171, 1116, 49161, 49159, 49159, 49161, 314, 253, 5008, 526, 4522, 713, 28, 527, 1530, 357, 12927, 260, 1142, 17301, 347, 3703, 30, 378, 2672, 5008, 526, 4522, 713, 436, 216, 49160, 49160, 929, 3685, 365, 49160, 49168, 49168, 49160, 595, 1812, 314, 260, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 338, 416, 1689, 15233, 316, 1320, 365, 18946, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 929, 14487, 49154, 49158, 198, 504, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 314, 216, 49168, 49168, 49168, 929, 1232, 216, 49160, 713, 446, 216, 49160, 49159, 49159, 49159, 929, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49159, 49159, 49159, 49154], "prefix_len": 135, "old_logprobs": [-0.7140514254570007, -1.5570822954177856, -0.21248187124729156, -0.009643280878663063, -0.11301965266466141, -0.006643115542829037, -0.0015851801726967096, -8.880697714630514e-05, -0.030584605410695076, -0.008943026885390282, -1.6076079607009888, -0.33844396471977234, -0.2769755423069, -0.25895941257476807, -0.10979203879833221, -0.6017013192176819, -1.5010671615600586, -0.038639362901449203, -0.4385890066623688, -0.5626175999641418, -0.05898374319076538, -0.07027687132358551, -0.2594640552997589, -0.05831923708319664, -0.0036374852061271667, -0.004357961006462574, -0.034328944981098175, -0.22865648567676544, -0.21582330763339996, -0.09211517125368118, -0.3602789342403412, -0.000337305391440168, -0.0008983152220025659, -0.017383594065904617, -0.00018845213344320655, -0.004211485851556063, -0.0029270683880895376, -0.00485574547201395, -0.001303061842918396, -0.00018463814922142774, -0.0016752745723351836]}
+{"group_id": "math_029", "origin": "budgie", "answer": "The maximum number of non-palindrome years between 1000 and 9999 that can occur consecutively is 9999 - 1000 + 1 = 9000 years.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 20, 75, 20, 933, 3780, 17516, 30, 11689, 13310, 28, 2686, 12879, 1885, 77, 20, 198, 198, 37835, 42, 13212, 383, 259, 431, 452, 30, 49, 1143, 38838, 10251, 12713, 422, 11171, 1116, 49161, 49159, 49159, 49161, 314, 253, 5008, 526, 4522, 713, 28, 527, 1530, 357, 12927, 260, 1142, 17301, 347, 3703, 30, 378, 2672, 5008, 526, 4522, 713, 436, 216, 49160, 49160, 929, 3685, 365, 49160, 49168, 49168, 49160, 595, 1812, 314, 260, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 338, 416, 1689, 15233, 316, 1320, 365, 18946, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 929, 14487, 49154, 49158, 198, 504, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 826, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 338, 416, 1689, 15233, 316, 1320, 314, 216, 49168, 49168, 49168, 49168, 731, 216, 49160, 49159, 49159, 49159, 1232, 216, 49160, 446, 216, 49168, 49159, 49159, 49159, 929, 30, 49154], "prefix_len": 135, "old_logprobs": [-0.7140514254570007, -1.5570822954177856, -0.21248187124729156, -0.009643280878663063, -0.11301965266466141, -0.006643115542829037, -0.0015851801726967096, -8.880697714630514e-05, -0.030584605410695076, -0.008943026885390282, -1.9826079607009888, -0.043660521507263184, -0.028725003823637962, -0.03302856534719467, -0.004643609281629324, -0.015606650151312351, -0.01720280759036541, -0.0006549597019329667, -0.006854236125946045, -0.00010239553375868127, -2.706014311115723e-05, -7.819823804311454e-05, -2.3928987979888916, -0.07833798974752426, -0.01675872877240181, -0.18078161776065826, -0.00021062063751742244, -0.001013361383229494, -0.1362125277519226, -0.5228233337402344, -0.2910989224910736, -0.22141991555690765, -0.08930355310440063, -0.595577597618103, -0.4740138351917267, -0.02962520532310009, -0.04225064441561699, -0.06289683282375336, -0.004007286392152309, -0.016442865133285522, -1.0021109580993652, -0.004009542521089315, -0.001444369088858366, -0.04972303658723831, -0.05039951205253601, -0.10613595694303513, -0.07420653849840164, -0.041313014924526215, -0.002894024597480893, -0.49600356817245483, -0.08759858459234238, -2.6232025623321533]}
+{"group_id": "math_029", "origin": "verified_reference", "answer": "Let the current year be a palindrome of the form $\\overline{a b b a}$. When will the next such year occur? Consider two cases:\n\na) $b=9$ (year of the form $\\overline{a 99 a}, a<9)$. Then, 11 years later, another palindromic year will occur: $\\overline{(a+1) 00(a+1)}$. For example, the years 3993 and 4004.\n\nb) $b<9$. In this case, the next palindromic year will occur 110 years later: $\\overline{a(b+1)(b+1) a}$. For example, the years 9339 and 9449. Therefore, the maximum number of consecutive non-palindromic years is 109.\n\n## Answer\n\n109 years.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 20, 75, 20, 933, 3780, 17516, 30, 11689, 13310, 28, 2686, 12879, 1885, 77, 20, 198, 198, 37835, 42, 13212, 383, 259, 431, 452, 30, 49, 1143, 38838, 10251, 12713, 422, 11171, 1116, 49161, 49159, 49159, 49161, 314, 253, 5008, 526, 4522, 713, 28, 527, 1530, 357, 12927, 260, 1142, 17301, 347, 3703, 30, 378, 2672, 5008, 526, 4522, 713, 436, 216, 49160, 49160, 929, 3685, 365, 49160, 49168, 49168, 49160, 595, 1812, 314, 260, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 338, 416, 1689, 15233, 316, 1320, 365, 18946, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 929, 14487, 49154, 49158, 198, 4239, 260, 1574, 713, 325, 253, 5008, 526, 4522, 282, 260, 910, 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+{"group_id": "math_030", "origin": "budgie", "answer": "First, we need to find the total number of students in the class. We can do this by adding up the number of students who prefer chocolate, apple, blueberry, and cherry pie.\n\n\\[ \\text{Total} = \\text{chocolate} + \\text{apple} + \\text{blueberry} + \\text{cherry} = 12 + 8 + 6 + 3 = 27 \\]\n\nNext, we need to find the total number of students who prefer cherry pie. We can do this by subtracting the total number of students who prefer apple and blueberry pie from the total number of students.\n\n\\[ \\text{Total cherry} = \\text{Total} - \\text{Total apple} - \\text{Total blueberry} = 27 - 5 - 6 = 16 \\]\n\nFinally, we need to find the fraction of the class that prefer cherry pie. We can do this by dividing the total number of students who prefer cherry pie by the total number of students in the class.\n\n\\[ \\text{Fraction} = \\frac{\\text{Total cherry}}{\\text{Total students}} = \\frac{16}{27} \\]\n\nNow, we can find the fraction of the class that prefer cherry pie by multiplying this fraction by the total number of students in the class.\n\n\\[ \\text{Fraction} = \\frac{16}{27} \\times 27 = 6 \\]\n\nSo, the fraction of the class that prefer cherry pie is 6.\n\nThe answer is:\n\\[ \\boxed{6} \\]\nFinal answer:\n#### 6", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 6228, 260, 216, 49162, 49165, 1058, 281, 25734, 2797, 290, 506, 1128, 28, 216, 49160, 49161, 4640, 9678, 2357, 28, 216, 49167, 4640, 11977, 28, 284, 216, 49165, 4640, 4461, 9500, 30, 22190, 282, 260, 6219, 1058, 4640, 23636, 2357, 284, 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+{"group_id": "math_030", "origin": "budgie", "answer": "First, we need to find out how many students prefer cherry pie. We can do this by subtracting the number of students who prefer other pie types from the total number of students.\n\n$ \\text{Number of students who prefer cherry pie} = \\text{Total number of students} - \\text{Number of students who prefer other pie types} $$ $= 36 - 12 $ $= 24 $\n\nNow that we have the number of students who prefer cherry pie, we can find the percentage of students who prefer cherry pie. We can do this by dividing the number of students who prefer cherry pie by the total number of students and then multiplying by 100.\n\n$ \\text{Percentage of students who prefer cherry pie} = \\frac{\\text{Number of students who prefer cherry pie}}{\\text{Total number of students}} $$ $= \\frac{24}{36} $ $= \\frac{2}{3} $\n\nSo, the answer is 2/3 or 60%.\n\nFinal answer:\n#### 60", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 6228, 260, 216, 49162, 49165, 1058, 281, 25734, 2797, 290, 506, 1128, 28, 216, 49160, 49161, 4640, 9678, 2357, 28, 216, 49167, 4640, 11977, 28, 284, 216, 49165, 4640, 4461, 9500, 30, 22190, 282, 260, 6219, 1058, 4640, 23636, 2357, 284, 2745, 4640, 11499, 30, 1068, 25734, 2797, 290, 506, 2357, 3670, 5369, 451, 940, 28, 638, 800, 5027, 868, 1041, 722, 327, 23636, 2357, 47, 3717, 198, 20, 3814, 2692, 107, 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+{"group_id": "math_030", "origin": "verified_reference", "answer": "1. First, we need to determine the number of students who prefer each type of pie. We know:\n   - 12 students prefer chocolate pie.\n   - 8 students prefer apple pie.\n   - 6 students prefer blueberry pie.\n\n2. Calculate the total number of students who prefer chocolate, apple, or blueberry pie:\n   \\[\n   12 + 8 + 6 = 26\n   \\]\n\n3. Subtract this number from the total number of students in the class to find the number of students who prefer either cherry or lemon pie:\n   \\[\n   36 - 26 = 10\n   \\]\n\n4. Since half of the remaining students prefer cherry pie and the other half prefer lemon pie, we divide the remaining students by 2:\n   \\[\n   \\frac{10}{2} = 5\n   \\]\n   Therefore, 5 students prefer cherry pie.\n\n5. To find the number of degrees Richelle should use for cherry pie in her pie graph, we need to calculate the proportion of students who prefer cherry pie and then convert this proportion to degrees. The proportion of students who prefer cherry pie is:\n   \\[\n   \\frac{5}{36}\n   \\]\n\n6. Multiply this proportion by the total number of degrees in a circle (360 degrees):\n   \\[\n   \\frac{5}{36} \\times 360 = 50\n   \\]\n\n7. 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+{"group_id": "math_031", "origin": "budgie", "answer": "The problem is based on the two-way circular track.\n\nThe key here is understanding the behavior of the runners and the time it takes for them to complete a lap.\n\nThe runners are moving at different constant speeds, and each runner has to cover the same distance around the track.\n\nAt the first meeting, the third runner will have to cover the distance that the second runner covered in the first 15 minutes.\n\nAfter 25 minutes, the second runner will have to cover the same distance that the first runner covered in the first 15 minutes.\n\nTherefore, the second runner will have to cover a distance that the first runner covered in 25 minutes, and the third runner will have to cover a distance that the first runner covered in 15 minutes.\n\nThe third runner will have to cover the same distance that the first runner covered in 25 minutes, and the second runner will have to cover a distance that the first runner covered in 15 minutes.\n\nSo, the distance that the third runner needs to cover is 25 minutes.\n\nLet's denote the distance that the first runner covers as $s$. At the time of the first meeting, the second runner will have to cover $s-15$ meters.\n\nThe first runner will have to cover $s+15$ meters in the same amount of time.\n\nWe know that the third runner will have to cover 25 meters in the same amount of time.\n\nTherefore, we have:\n\n$$s+15 = 25$$\n\nSolving for $s$ gives us:\n\n$$s = 25 - 15 = 10$$\n\nSo, the first runner will have to cover 10 meters in the time it takes the second runner to", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49165, 30, 365, 49162, 2876, 25, 198, 198, 16705, 25334, 359, 4138, 1683, 253, 11324, 4320, 418, 3836, 4037, 10870, 30, 1550, 827, 25334, 2220, 28, 502, 15881, 1607, 1130, 284, 1120, 4108, 281, 7011, 8939, 30, 198, 198, 3473, 634, 1225, 28, 260, 808, 23786, 11675, 260, 1796, 30, 2550, 216, 49160, 49164, 3487, 28, 260, 1796, 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+{"group_id": "math_031", "origin": "budgie", "answer": "The answer is 25 minutes.\n\nFinal answer:\n#### 25", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49165, 30, 365, 49162, 2876, 25, 198, 198, 16705, 25334, 359, 4138, 1683, 253, 11324, 4320, 418, 3836, 4037, 10870, 30, 1550, 827, 25334, 2220, 28, 502, 15881, 1607, 1130, 284, 1120, 4108, 281, 7011, 8939, 30, 198, 198, 3473, 634, 1225, 28, 260, 808, 23786, 11675, 260, 1796, 30, 2550, 216, 49160, 49164, 3487, 28, 260, 1796, 23786, 11675, 260, 3464, 327, 260, 808, 655, 30, 5839, 216, 49161, 49164, 3487, 1916, 28, 260, 3464, 23786, 11675, 260, 808, 327, 260, 808, 655, 30, 198, 198, 2020, 800, 3487, 1072, 357, 1188, 327, 582, 23786, 288, 2994, 260, 2415, 4320, 47, 49154, 49158, 198, 504, 2988, 314, 216, 49161, 49164, 3487, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49161, 49164, 49154], "prefix_len": 135, "old_logprobs": [-1.1664215326309204, -1.9958109855651855, -0.05354534834623337, -0.3489420413970947, -2.0087571144104004, -0.7786458730697632, -0.7872970104217529, -0.20889146625995636, -0.22673311829566956, -0.004990146495401859, -0.37720146775245667, -0.0009112972766160965, -0.0007460909546352923, -0.03270026668906212, -0.0009235645993612707, -0.0022326798643916845, -7.128461584215984e-05, -5.8053239627042785e-05, -0.00016080040950328112]}
+{"group_id": "math_031", "origin": "verified_reference", "answer": "# Answer: 80\n\n## Solution:\n\nLet the first runner meet the second, then after $a$ minutes the second runner meets the third for the first time, and after another $b$ minutes the third runner meets the first for the first time.\n\nLet the first and second runners meet at point $A$, the second and third at point $B$, and the first and third at point $C$. Additionally, let the first runner be at point $D$ at the moment the second and third runners meet.\n\nThen the points are arranged on the circle in the following order: $D, A, B, C$, where the first runner ran from $D$ to $A$ in $a$ minutes, the second runner ran from $A$ to $B$ in $a$ minutes, the third runner ran from $B$ to $C$ in $b$ minutes, and the first runner ran from $D$ to $C$ in $b$ minutes. These four segments together form the entire track, so it is covered in $2a + 2b$ minutes.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49165, 30, 365, 49162, 2876, 25, 198, 198, 16705, 25334, 359, 4138, 1683, 253, 11324, 4320, 418, 3836, 4037, 10870, 30, 1550, 827, 25334, 2220, 28, 502, 15881, 1607, 1130, 284, 1120, 4108, 281, 7011, 8939, 30, 198, 198, 3473, 634, 1225, 28, 260, 808, 23786, 11675, 260, 1796, 30, 2550, 216, 49160, 49164, 3487, 28, 260, 1796, 23786, 11675, 260, 3464, 327, 260, 808, 655, 30, 5839, 216, 49161, 49164, 3487, 1916, 28, 260, 3464, 23786, 11675, 260, 808, 327, 260, 808, 655, 30, 198, 198, 2020, 800, 3487, 1072, 357, 1188, 327, 582, 23786, 288, 2994, 260, 2415, 4320, 47, 49154, 49158, 198, 19, 19842, 42, 216, 49167, 49159, 198, 198, 712, 19857, 42, 198, 198, 4239, 260, 808, 23786, 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+{"group_id": "math_033", "origin": "budgie", "answer": "To find the minimum value of $k$ that satisfies the given condition, we need to analyze the inequality and identify the conditions under which a triangle with side lengths $a, b, c$ exists.\n\nGiven:\n\\[ k(a b + b c + c a) > 5(a^{2} + b^{2} + c^{2}) \\]\n\nWe can rewrite the inequality as:\n\\[ k(a b + b c + c a) > 5 a^{2} + 5 b^{2} + 5 c^{2} \\]\n\nTo simplify the analysis, divide both sides by $a^{2} + b^{2} + c^{2}$ (which is non-zero), yielding:\n\\[ \\frac{k(a b + b c + c a)}{a^{2} + b^{2} + c^{2}} > \\frac{5 a^{2} + 5 b^{2} + 5 c^{2}}{4} \\]\n\nWe want the left-hand side (LHS) to be greater than $k$ for some positive $a, b, c$. To achieve this, we must have:\n\\[ \\frac{5 a^{2} + 5 b^{2} + 5 c^{2}}{4} < k \\]\n\nMultiplying through by 4 gives us:\n\\[ 5 a^{2} + 5 b^{2} + 5 c^{2} < 4k \\]\n\nTo ensure the inequality holds for any $a, b, c$, we need to find the smallest $k$ that satisfies this condition.\n\nOne approach is to find the smallest $k$ that makes the left-hand side positive and the inequality", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49163, 30, 2959, 1885, 91, 20, 325, 253, 2731, 15328, 28, 715, 338, 327, 750, 2731, 2966, 1885, 81, 28, 278, 28, 265, 20, 19131, 260, 9724, 1885, 91, 24, 81, 278, 27, 82, 265, 27, 83, 253, 46153, 49164, 76, 8842, 24, 81, 21990, 49161, 109, 27, 82, 21990, 49161, 17243, 2771, 30, 20, 1885, 27, 83, 21990, 49161, 24362, 643, 665, 1251, 1822, 253, 14973, 351, 2103, 15900, 1885, 81, 28, 278, 28, 265, 28422, 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+{"group_id": "math_033", "origin": "verified_reference", "answer": "14. $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right) \\geqslant 5(a b+a c+b c), k \\leqslant 5$ when, the inequality has no solution. When $k=6$, the inequality becomes $6(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$. Without loss of generality, assume $c=\\max (a, b, c)$, transform the inequality into $5 c^{2}-6(a+b) c+5\\left(a^{2}+b^{2}\\right)-6 a b0$, $5\\left(a^{2}+b^{2}\\right)-6 a b>0$, so the quadratic equation corresponding to the inequality has two positive roots $c_{1}, c_{2}$, the solution to the quadratic inequality in $c$ is $c \\in\\left(c_{1}, c_{2}\\right)$. If $c_{2} \\leqslant a+b$, then $c6$ when, $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$ has solutions, but they cannot form a triangle, so the only $k$ that satisfies the condition is $k=6$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49163, 30, 2959, 1885, 91, 20, 325, 253, 2731, 15328, 28, 715, 338, 327, 750, 2731, 2966, 1885, 81, 28, 278, 28, 265, 20, 19131, 260, 9724, 1885, 91, 24, 81, 278, 27, 82, 265, 27, 83, 253, 46153, 49164, 76, 8842, 24, 81, 21990, 49161, 109, 27, 82, 21990, 49161, 17243, 2771, 30, 20, 1885, 27, 83, 21990, 49161, 24362, 643, 665, 1251, 1822, 253, 14973, 351, 2103, 15900, 1885, 81, 28, 278, 28, 265, 28422, 7985, 260, 5869, 1685, 282, 1885, 91, 28422, 49154, 49158, 198, 49160, 49163, 30, 1885, 91, 24, 81, 278, 27, 81, 265, 27, 82, 265, 46153, 49164, 76, 8842, 24, 81, 21990, 49161, 109, 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+{"group_id": "math_034", "origin": "budgie", "answer": "1. Given that $PD: PE: PF = 1: 2: 3$, denote $PD = x$, $PE = 2x$, and $PF = 3x$. Since $PD + PE + PF = 12$, we have:\n   \\[ x + 2x + 3x = 12 \\Rightarrow 6x = 12 \\Rightarrow x = 2 \\]\n   \\[ PD = 2, PE = 4, PF = 9 \\]\n\n2. Since $\\triangle PEF$ is a right triangle with $PD$ as its hypotenuse, we can deduce that $PE = PF = 4$, making it an equilateral triangle.\n\n3. In an equilateral triangle, all sides are equal, and in a right triangle, the hypotenuse is $\\sqrt{3}$ times longer than each leg. Therefore, the length of each leg is:\n   \\[ PE = PF = 4\\sqrt{3} \\]\n\n4. The area of a triangle is given by:\n   \\[ \\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} \\]\n   where the base and height are the legs of the triangle.\n\n5. Let's denote the area of $\\triangle PEF$ as $[PEF]$. Using the Pythagorean theorem in $\\triangle PEF$:\n   \\[ [PEF] = \\left(\\frac{3}{4}\\right)^2 \\]\n\n6. From the given ratio $PD: PE: PF = 1:2:3$, we have:\n   \\[ PD = x = 2 \\times 1, PE = 2x = 4, PF = 3x = 9 \\]", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 1032, 3057, 281, 8799, 216, 49161, 28, 1303, 1885, 64, 20, 325, 253, 1225, 2972, 260, 1220, 13293, 19573, 22128, 6935, 17638, 20, 351, 2103, 3519, 216, 49160, 49161, 30, 10602, 24340, 99, 429, 1885, 64, 20, 288, 260, 5882, 1885, 6190, 26265, 1885, 8970, 26265, 284, 1885, 5431, 26265, 351, 260, 3282, 282, 260, 24340, 99, 1036, 1885, 52, 26265, 1885, 53, 26265, 284, 1885, 54, 20, 7827, 30, 10887, 338, 1885, 9556, 42, 20045, 42, 26015, 446, 216, 49160, 42, 216, 49161, 42, 216, 49162, 28422, 3875, 28, 260, 1557, 282, 20378, 13293, 1885, 50, 7548, 54, 20, 314, 49154, 49158, 198, 49160, 30, 10887, 338, 1885, 9556, 42, 20045, 42, 26015, 446, 216, 49160, 42, 216, 49161, 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+{"group_id": "math_034", "origin": "budgie", "answer": "1. From the problem, it is clear that the ratio of the areas of triangles $\\triangle PBD$ and $\\triangle PFE$ is equal to the ratio of their bases, $\\frac{1}{2}$.\n2. The area of a triangle is given by the formula $S_{\\triangle} = \\frac{1}{2}ab$, where $a$ and $b$ are the lengths of the legs of the triangle.\n3. Therefore, we have $S_{\\triangle PBD} = \\frac{1}{4}S_{\\triangle ABC}$.\n4. Since $PD: PE: PF = 1: 2: 3$, we have $\\frac{PD}{PE} = \\frac{1}{2}$ and $\\frac{PE}{PF} = \\frac{2}{3}$.\n5. Therefore, $\\frac{PD}{PE} = \\frac{1}{3}$.\n6. Since the area of $\\triangle ABC$ is $S_{\\triangle ABC} = \\frac{1}{2}bc$, we have $\\frac{1}{2}bc = \\frac{1}{3}bc$, which implies $bc = 2$.\n7. Let the altitudes of $\\triangle ABC$ from $A$ and $B$ to $BC$ and $CA$ be $h_{a}$ and $h_{b}$, respectively. Then we have $h_{a} + h_{b} = h_{c}$.\n8. We also know that $\\frac{h_{a}}{h_{b}} = \\frac{2}{3}$, which means $h_{b} = \\frac{2}{3}h_{a}$.\n9. 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+{"group_id": "math_034", "origin": "verified_reference", "answer": "$$\n6.11 \\sqrt{3} \\text {. }\n$$\n\nFor the equilateral $\\triangle ABC$ with a height of $12 \\times \\frac{\\sqrt{3}}{2}=6 \\sqrt{3}$, then\n$$\n\\begin{array}{l}\nP D+P E+P F=6 \\sqrt{3} . \\\\\n\\text { Also, } P D: P E: P F=1: 2: 3, \\text { so, } \\\\\nP D=\\sqrt{3}, P E=2 \\sqrt{3}, P F=3 \\sqrt{3} .\n\\end{array}\n$$\n\nAs shown in Figure 7, draw\n$P S / / B C$, intersecting $A B$ at $S$.\nSince $\\angle F S P=\\angle B$ $=60^{\\circ}$, therefore,\n$$\n\\begin{array}{l}\nF S=P F \\cot 60^{\\circ}=3, \\\\\nP S=6, \\\\\nB D=P S+P D \\cot 60^{\\circ}=7 .\n\\end{array}\n$$\n\nThus, $S_{\\text {quadrilateral BDPF }}=S_{\\triangle P S}+S_{\\text {trapezoid BDPS }}$\n$$\n=\\frac{1}{2} \\times 3 \\times 3 \\sqrt{3}+\\frac{6+7}{2} \\times \\sqrt{3}=11 \\sqrt{3} .\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 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+{"group_id": "math_035", "origin": "budgie", "answer": "The sequence starts from 1 and increases by 3 each time.\n\nHere is an example of the sequence:\n1; 4; 7; 10; 13; 16; 19; 22; 25; 28; 31; $$$\n\nThe $21^{\\varrho}$ term of the sequence is 28.\n\nFinal answer:\n#### 28", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 1780, 314, 260, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 198, 198, 9212, 198, 49160, 8544, 216, 49161, 27, 49162, 8544, 216, 49163, 27, 49164, 27, 49165, 8544, 216, 49166, 27, 49167, 27, 49168, 27, 49160, 49159, 8544, 216, 49160, 49160, 27, 49160, 49161, 27, 49160, 49162, 27, 49160, 49163, 27, 49160, 49164, 8544, 3814, 400, 1565, 9148, 198, 9212, 49154, 49158, 198, 504, 5972, 5909, 429, 216, 49160, 284, 4495, 411, 216, 49162, 971, 655, 30, 198, 198, 4590, 314, 354, 1183, 282, 260, 5972, 42, 198, 49160, 43, 216, 49163, 43, 216, 49166, 43, 216, 49160, 49159, 43, 216, 49160, 49162, 43, 216, 49160, 49165, 43, 216, 49160, 49168, 43, 216, 49161, 49161, 43, 216, 49161, 49164, 43, 216, 49161, 49167, 43, 216, 49162, 49160, 43, 1885, 9212, 198, 198, 504, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 314, 216, 49161, 49167, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49161, 49167, 49154], "prefix_len": 96, "old_logprobs": [-0.7064253091812134, -2.1128363609313965, -1.4904935359954834, -2.196011543273926, -0.5672488212585449, -0.11023297905921936, -0.39213189482688904, -1.5538829565048218, -0.09093422442674637, -0.10729856789112091, -0.6193104982376099, -0.20341934263706207, -0.2752462923526764, -0.5899540185928345, -1.2256301641464233, -0.08695174008607864, -4.820014953613281, -1.2590214014053345, -3.3636257648468018, -0.6942742466926575, -0.5064844489097595, -0.5588923096656799, -0.23836806416511536, -0.31788310408592224, -0.28988686203956604, -1.5490009784698486, -2.7020957469940186, -0.0972699224948883, -2.8533570766448975, -1.0686885118484497, -0.013619476929306984, -0.0490531399846077, -0.20197397470474243, -0.01366428006440401, -0.010436601005494595, -0.03175294026732445, -0.03866344690322876, -0.04938848316669464, -0.004479849245399237, -0.007804496213793755, -0.04377371817827225, -0.053258076310157776, -0.004611690063029528, -0.0012055517872795463, -0.06390839070081711, -0.08849840611219406, -0.012343090958893299, -0.0019602624233812094, -0.10365193337202072, -0.06747964769601822, -0.0005241450853645802, -0.006068968679755926, -0.033547792583703995, -0.06464362144470215, -0.0006235085893422365, -0.009080896154046059, -0.16216537356376648, -0.13239508867263794, -0.0006744970451109111, -0.003028926905244589, -0.02452835813164711, -0.09168770164251328, -0.011613735929131508, -0.009847158566117287, -0.12532299757003784, -4.928700923919678, -2.4139151573181152, -1.0522016286849976, -0.18397942185401917, -0.8604916930198669, -0.7772613167762756, -0.04756128042936325, -0.019902529194951057, -0.021333742886781693, -0.0025873063132166862, -0.00509487884119153, -0.00650052772834897, -0.08309584856033325, -0.016221093013882637, -0.02589593641459942, -0.7313510775566101, -0.579312801361084, -0.01413747388869524, -0.10231301188468933, -0.2035445123910904, -0.5125430226325989, -1.1856642961502075, -0.15004833042621613, -0.11035607010126114, -0.08339738100767136, -0.05282045528292656, -0.00033885458833537996, -0.0005536930402740836, -0.022069105878472328, -9.762764238985255e-05, -0.026370462030172348, -0.000598251644987613, -0.0035168009344488382, -0.0006611545104533434]}
+{"group_id": "math_035", "origin": "budgie", "answer": "The pattern in the sequence is an arithmetic sequence with first term $1$ and common difference $2$.\n\nThe $n^{\\varrho}$ term is given by the formula\n\n$$\n\\frac{n(a_1 + a_n)}{2}\n$$\n\nwhere $n$ is the term number, $a_1$ is the first term, and $a_n$ is the $n^{\\varrho}$ term.\n\nFor the $n^{\\varrho}$ term, $n$ is 15.\n\nSubstitute the values into the formula:\n\n$$\n\\frac{15(1 + 21)}{2} = \\frac{15(22)}{2} = 15 \\times 11 = \\boxed{165}\n$$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 1780, 314, 260, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 198, 198, 9212, 198, 49160, 8544, 216, 49161, 27, 49162, 8544, 216, 49163, 27, 49164, 27, 49165, 8544, 216, 49166, 27, 49167, 27, 49168, 27, 49160, 49159, 8544, 216, 49160, 49160, 27, 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+{"group_id": "math_035", "origin": "verified_reference", "answer": "Observe that the $21^{\\text{st}}$ term is the sum of 21 consecutive numbers. Taking the first term of each term, that is, 1, 2, 4, 7, 11, $16, \\ldots$, we have that\n\n$$\n\\begin{aligned}\n2 & =1+1 \\\\\n4 & =2+1+1 \\\\\n7 & =3+2+1+1 \\\\\n11 & =4+3+2+1+1 \\\\\n16 & =5+4+3+2+1+1\n\\end{aligned}\n$$\n\nThus, the first term of the $21^{\\text{st}}$ term is\n\n$$\n20+19+\\cdots+3+2+1+1=\\frac{20 \\times 21}{2}+1=211\n$$\n\nand the $21^{\\text{st}}$ term is\n\n$$\n211+212+\\cdots+230+231=\\frac{(211+231) \\times 21}{2}=221 \\times 21=4641\n$$", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 1780, 314, 260, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 198, 198, 9212, 198, 49160, 8544, 216, 49161, 27, 49162, 8544, 216, 49163, 27, 49164, 27, 49165, 8544, 216, 49166, 27, 49167, 27, 49168, 27, 49160, 49159, 8544, 216, 49160, 49160, 27, 49160, 49161, 27, 49160, 49162, 27, 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+{"group_id": "math_036", "origin": "budgie", "answer": "The original sum of these numbers is:\n\\[ 1 + 2 + 3 + \\ldots + 2009 \\]\n\nA couple of the numbers occurring twice are written, with the remainder $2009 - (1 + 2 + 3 + \\ldots + 2009)$. Calculating the sum of this sequence, we have:\n\\[ 2009 \\times 13 - 1 = 26527 \\]\n\nSo the sum of the remaining two numbers is $2009 \\times 13 - 26527$ and their remainder is 26527.\n\nTherefore, the remainder when the sum of the erased numbers is divided by 13 is:\n\\[ 2009 \\times 13 - 26527 + 26527 = 2009 \\times 13 = 26527 \\]\n\nHence, the third number is $26527 - 2009 = \\boxed{24520}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 3280, 253, 4411, 28, 260, 2966, 429, 216, 49160, 288, 216, 49161, 49159, 49159, 49168, 359, 2855, 30, 330, 6075, 282, 601, 359, 40446, 284, 2489, 282, 601, 28, 335, 260, 4411, 314, 2855, 260, 17867, 282, 260, 2028, 282, 260, 40446, 2966, 6066, 411, 216, 49160, 49162, 30, 2550, 253, 6075, 282, 18388, 282, 451, 2812, 1817, 28, 805, 216, 49162, 2966, 359, 2049, 28, 282, 527, 827, 359, 216, 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+{"group_id": "math_036", "origin": "budgie", "answer": "Let's denote the sum of the numbers on the board before erasing and after erasing as S. By the problem's conditions, S = 1 + 2 + 3 + ... + 2009.\n\nThe first thing to do is to find the sum S. We know that S = 1(1 + 2 + 3 + ... + 2009) = 1(2009)(2010)/2 = 2010*2010 = 4,040,100.\n\nNow, let's subtract the sum of the numbers that are erased from this sum to find the sum of the numbers that remain.\n\nS = 4,040,100 - 9,099 = 4,030,011\n\nNow, we divide this sum by 13 to find the sum of the numbers that remain.\n\nS = 4,030,011/13 = 3,230,000\n\nThe sum of the numbers on the board (S) is 3,230,000.\n\nThe numbers on the board are: 3,230,000, 4,030,001, 4030,000.\n\nThe third number is 4030,000.\n\nThe final answer is \\boxed{4030000}.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 3280, 253, 4411, 28, 260, 2966, 429, 216, 49160, 288, 216, 49161, 49159, 49159, 49168, 359, 2855, 30, 330, 6075, 282, 601, 359, 40446, 284, 2489, 282, 601, 28, 335, 260, 4411, 314, 2855, 260, 17867, 282, 260, 2028, 282, 260, 40446, 2966, 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+{"group_id": "math_036", "origin": "verified_reference", "answer": "1. **Define the invariant:**\n   Let \\( I \\) be the remainder when the sum of all the integers on the board is divided by 13. Initially, the numbers from 1 to 2009 are written on the board. The sum of these numbers is:\n   \\[\n   S = \\sum_{k=1}^{2009} k = \\frac{2009 \\times 2010}{2} = 2009 \\times 1005\n   \\]\n   We need to find \\( S \\mod 13 \\).\n\n2. **Calculate \\( 2009 \\mod 13 \\) and \\( 1005 \\mod 13 \\):**\n   \\[\n   2009 \\div 13 = 154 \\quad \\text{remainder} \\quad 7 \\quad \\Rightarrow \\quad 2009 \\equiv 7 \\pmod{13}\n   \\]\n   \\[\n   1005 \\div 13 = 77 \\quad \\text{remainder} \\quad 4 \\quad \\Rightarrow \\quad 1005 \\equiv 4 \\pmod{13}\n   \\]\n\n3. **Calculate \\( S \\mod 13 \\):**\n   \\[\n   S = 2009 \\times 1005 \\equiv 7 \\times 4 = 28 \\equiv 2 \\pmod{13}\n   \\]\n   Therefore, \\( I = 2 \\).\n\n4. **Determine the third number \\( a \\):**\n   After several operations, only three numbers are left on the board: 9, 999, and \\( a \\). The sum of these three numbers modulo 13 must be equal to the invariant \\( I \\):\n   \\[\n   a + 9 + 999 \\equiv 2 \\pmod{13}\n   \\]\n\n5. **Calculate \\( 999 \\mod 13 \\):**\n   \\[\n   999 \\div 13 = 76 \\quad \\text{remainder} \\quad 11 \\quad \\Rightarrow \\quad 999 \\equiv 11 \\pmod{13}\n   \\]\n\n6. **Set up the equation:**\n   \\[\n   a + 9 + 11 \\equiv 2 \\pmod{13}\n   \\]\n   Simplify the equation:\n   \\[\n   a + 20 \\equiv 2 \\pmod{13}\n   \\]\n   Subtract 20 from both sides:\n   \\[\n   a \\equiv 2 - 20 \\pmod{13}\n   \\]\n   \\[\n   a \\equiv -18 \\pmod{13}\n   \\]\n   Since \\(-18 \\equiv -18 + 26 \\equiv 8 \\pmod{13}\\), we have:\n   \\[\n   a \\equiv 8 \\pmod{13}\n   \\]\n\n7. **Find the possible values of \\( a \\) between 1 and 2009:**\n   The possible values of \\( a \\) are of the form:\n   \\[\n   a = 8 + 13k \\quad \\text{for integer} \\quad k\n   \\]\n   We need \\( 1 \\leq a \\leq 2009 \\):\n   \\[\n   1 \\leq 8 + 13k \\leq 2009\n   \\]\n   \\[\n   -7 \\leq 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+{"group_id": "math_037", "origin": "budgie", "answer": "The last four digits of $2^{1999}$ are $1000000", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2464, 30, 7985, 260, 1596, 1876, 18638, 282, 1885, 49161, 21990, 49160, 49168, 49168, 49168, 24362, 30, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 5361, 1446, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 49154, 49158, 198, 504, 1596, 1876, 18638, 282, 1885, 49161, 21990, 49160, 49168, 49168, 49168, 24362, 359, 1885, 49160, 49159, 49159, 49159, 49159, 49159, 49159, 49154], "prefix_len": 75, "old_logprobs": [-1.6261858940124512, -0.7485172748565674, -0.016881341114640236, -0.002212578197941184, -0.04400874301791191, -0.2066299319267273, -0.029543381184339523, -0.3452100455760956, -0.022605936974287033, -0.003916214685887098, -0.0010151476599276066, -0.0018397325184196234, -0.026995036751031876, -0.2991894483566284, -1.1147418022155762, -1.8266735076904297, -1.1103037595748901, -1.7767319679260254, -0.21661123633384705, -0.7164345383644104, -0.1503499150276184, -0.14830179512500763, -14.781360626220703]}
+{"group_id": "math_037", "origin": "budgie", "answer": "I should not Areas have any new numerical value.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2464, 30, 7985, 260, 1596, 1876, 18638, 282, 1885, 49161, 21990, 49160, 49168, 49168, 49168, 24362, 30, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 5361, 1446, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 49154, 49158, 198, 57, 868, 441, 20284, 457, 750, 725, 12141, 1685, 30, 49154], "prefix_len": 75, "old_logprobs": [-0.945827066898346, -2.6717734336853027, -1.6540647745132446, -10.62779426574707, -4.251129150390625, -2.1618282794952393, -3.0434930324554443, -6.8759355545043945, -1.8187967538833618, -1.4505869150161743, -0.7305818200111389]}
+{"group_id": "math_037", "origin": "verified_reference", "answer": "$$\n\\varphi(625)=\\varphi\\left(5^{4}\\right)=5^{4} \\times\\left(1-\\frac{1}{5}\\right)=500 \\text {. }\n$$\n\nSince $(2,625)=1$, by Euler's theorem, we get $2^{500} \\equiv 1(\\bmod 625)$.\n$$\n\\begin{array}{l}\n\\therefore 2^{2000} \\equiv 1(\\bmod 625) . \\\\\n\\therefore \\text { Let } 2^{2000}=625 m+1(m \\in \\mathbf{N}) .\n\\end{array}\n$$\n\nAlso, $2^{5} \\mid 2^{2000}$,\nso let $2^{2000}=2^{5} \\cdot n(n \\in \\mathbf{N})$.\nFrom (1) and (2), we get $625 m+1=2^{5} \\cdot n$.\nSolving this indeterminate equation, we get\n$$\n\\left\\{\\begin{array}{l}\nm=32 k+15, \\\\\nn=625 k+293\n\\end{array}\\right.\n$$\n$k \\in \\mathbf{Z}$.\nSubstituting $n=625 k+293$ into (2), we get\n$$\n\\begin{array}{l}\n2^{2000}=32(625 k+293)=20000 k+9376 . \\\\\n\\therefore 2^{1999}=10000 k+4688 .\n\\end{array}\n$$\n\nTherefore, the last four digits of $2^{1999}$ are 4688.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": 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+{"group_id": "math_038", "origin": "budgie", "answer": "First, let's simplify the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$.\n\nFor the first equation, recognize it as a binomial expansion:\n$$(x+1)^{5} = x^{5}+5\\cdot x^{4}+10\\cdot x^{3}+5\\cdot x^{2}+x = 0$$\n\nFor the second equation, recognize it as a root-sum equation:\n$$x+\\sqrt[5]{x}+1 = 0$$\n\nLet $y = x + \\sqrt[5]{x}$. Substitute $\\sqrt[5]{x}$ with $y-1$:\n$$y = x + (y-1)$$\n\nThis equation is a linear equation in terms of $y$. We can solve for $y$ by subtracting $x$ from both sides:\n$$y - x = 1$$\n\nNow we have a linear equation in $y$:\n$$y - x = 1$$\n\nWe can solve this equation for $x$ by using the Extended Euclidean Algorithm.\n\nFirst, express $1$ as a linear combination of $x$ and $\\sqrt[5]{x}$:\n$$1 = 1 - x - \\sqrt[5]{x}$$\n\nNow, apply the Extended Euclidean Algorithm to find the coefficients $a$ and $b$ such that:\n$$a = -1, b = 1$$\n\nNext, express $-1$ as a linear combination of $x$ and $\\sqrt[5]{x}$:\n$$-1 = -1 - x - \\sqrt[5]{x}$$\n\nNow, apply the Extended Euclidean Algorithm to find the coefficients", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 365, 17872, 429, 260, 216, 49160, 49162, 373, 476, 42519, 23278, 18, 6484, 269, 780, 29435, 25, 378, 1345, 5190, 282, 260, 9773, 1885, 104, 21990, 49164, 109, 27, 104, 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+{"group_id": "math_038", "origin": "budgie", "answer": "The problem states that the roots of both equations are $\\alpha, \\beta$. We can use Vieta's formulas to solve this problem.\n\nFor the first equation, $x^{5}+x+1=0$, the sum of the roots is $-\\frac{1}{2}$. Therefore, we have\n$$\\alpha + \\beta = -\\frac{1}{2}.$$\n\nFor the second equation, $x+\\sqrt[5]{x}+1=0$, the product of the roots is $-\\frac{1}{x}$. We can write this as\n$$(\\alpha + \\beta) \\cdot (\\alpha + \\beta) = -\\frac{1}{x}.$$\n\nSince $\\alpha + \\beta = -\\frac{1}{2}$ from the given options, we have\n$$-\\frac{1}{2} \\cdot (\\alpha + \\beta) = -\\frac{1}{x}.$$\n\nSubstituting the value of $\\alpha + \\beta$ from our previous result, we get\n$$-\\frac{1}{2} \\cdot (-\\frac{1}{2}) = -\\frac{1}{x}.$$\n\nSolving for $x$, we find that\n$$-\\frac{1}{4} = -\\frac{1}{x},$$\nwhich implies that\n$$x = 4.$$\n\nTherefore, the correct answer is 4, so we have\n$$\\boxed{4}$$\n\nFinal answer:\n#### -1", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 365, 17872, 429, 260, 216, 49160, 49162, 373, 476, 42519, 23278, 18, 6484, 269, 780, 29435, 25, 378, 1345, 5190, 282, 260, 9773, 1885, 104, 21990, 49164, 109, 27, 104, 27, 49160, 45, 49159, 20, 284, 1885, 104, 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+{"group_id": "math_038", "origin": "verified_reference", "answer": "Solution: Choose A. Reason: Consider the function $f(x)=x^{5}+x+1$, which is an increasing function on $\\mathbf{R}$,\nwhen $x^{5}+x+1=0$,\nwe get $x=\\sqrt[5]{-x-1}$, substituting $-x-1$ for $x$ yields $-x-1=\\sqrt[5]{x}$.\nFrom (2), we have $(-x-1)^{5}=x$, which is also $(-x-1)^{5}+(-x-1)+1=0$.\nLet $\\alpha$ be a root of equation (1), i.e., $f(\\alpha)=0$. Let $\\beta$ be a root of equation (2), i.e., $-1-\\beta=\\sqrt[5]{\\beta}$, thus from (3) we know that $f(-1-\\beta)=0$.\nTherefore, we have $f(\\alpha)=f(-1-\\beta)$, which means $\\alpha+\\beta=-1$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 365, 17872, 429, 260, 216, 49160, 49162, 373, 476, 42519, 23278, 18, 6484, 269, 780, 29435, 25, 378, 1345, 5190, 282, 260, 9773, 1885, 104, 21990, 49164, 109, 27, 104, 27, 49160, 45, 49159, 20, 284, 1885, 104, 42639, 16166, 75, 49164, 36939, 104, 109, 27, 49160, 45, 49159, 20, 359, 19573, 13433, 28, 3814, 17565, 20, 7827, 28, 965, 19573, 13433, 42639, 17565, 20, 16473, 365, 15165, 198, 49, 30, 731, 49160, 198, 50, 30, 1885, 46240, 18169, 107, 49160, 15009, 49161, 24362, 198, 51, 30, 19573, 18169, 107, 49160, 15009, 49161, 24362, 198, 52, 30, 216, 49160, 49154, 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17565, 24814, 16166, 75, 49164, 77, 26754, 17565, 24362, 28, 3310, 429, 365, 49162, 25, 392, 699, 338, 1885, 86, 10242, 49160, 46240, 17565, 36637, 49159, 28422, 198, 17308, 28, 392, 457, 1885, 86, 19407, 13433, 36637, 86, 10242, 49160, 46240, 17565, 25, 26265, 527, 1530, 19573, 13433, 42639, 17565, 15196, 49160, 28422, 49154], "prefix_len": 132, "old_logprobs": [-4.239291191101074, -0.03602408617734909, -10.380029678344727, -6.006158351898193, -1.6323580741882324, -14.995430946350098, -2.6167168617248535, -8.304129600524902, -0.6309458613395691, -2.6258506774902344, -0.3151669204235077, -0.20736399292945862, -0.04648115113377571, -0.08773563802242279, -0.490862637758255, -0.10086862742900848, -0.4489029049873352, -0.22785815596580505, -0.06976636499166489, -0.08413946628570557, -0.030051346868276596, -0.2565946578979492, -0.003767061745747924, -1.4398553371429443, -3.245041847229004, -0.49924278259277344, -1.2620635032653809, -1.9664454460144043, -0.013746591284871101, 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+{"group_id": "math_040", "origin": "budgie", "answer": "The problem requires us to find how many steps bring Henry's list of the first 1000 positive integers to an empty state. To do this, let's break down the process step by step.\n\nStep 1:\nWe start with a list of the first 1000 positive integers.\n\nStep 2:\nWe start erasing any nonpositive integers or integers with a repeated digit.\n\nStep 3:\nFor the remaining integers, we decrease them by 1.\n\nStep 4:\nAt the end, we have a list of the first 1000 integers.\n\nNow, let's count the number of steps required.\n\nThe final answer: 1000", "reward": -1, "advantage": -1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 27803, 5909, 351, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 28, 284, 16790, 253, 3086, 282, 3301, 335, 260, 1398, 30, 1814, 971, 1833, 28, 384, 2812, 1049, 750, 1603, 16185, 23313, 355, 750, 23313, 338, 457, 253, 6514, 11403, 28, 284, 965, 12191, 3117, 281, 260, 1398, 411, 216, 49160, 30, 1073, 800, 3301, 1072, 357, 1188, 327, 6661, 506, 1398, 288, 325, 7964, 47, 198, 198, 75, 89, 77, 2954, 11312, 411, 7937, 11583, 75, 31, 89, 77, 49154, 49158, 198, 504, 1732, 3073, 468, 288, 1042, 638, 800, 3301, 2635, 6661, 506, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 288, 354, 7964, 1215, 30, 1626, 536, 451, 28, 1303, 506, 2373, 1187, 260, 980, 1833, 411, 1833, 30, 198, 198, 3512, 216, 49160, 42, 198, 1882, 1120, 351, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 30, 198, 198, 3512, 216, 49161, 42, 198, 1882, 1120, 2812, 1817, 750, 1603, 16185, 23313, 355, 23313, 351, 253, 6514, 11403, 30, 198, 198, 3512, 216, 49162, 42, 198, 2193, 260, 6219, 23313, 28, 392, 6302, 601, 411, 216, 49160, 30, 198, 198, 3512, 216, 49163, 42, 198, 3473, 260, 1112, 28, 392, 457, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 23313, 30, 198, 198, 2696, 28, 1303, 506, 985, 260, 1230, 282, 3301, 2309, 30, 198, 198, 504, 3403, 2988, 42, 216, 49160, 49159, 49159, 49159, 49154], "prefix_len": 115, "old_logprobs": [-1.871724009513855, -2.574619770050049, -2.096071243286133, -0.7273120880126953, -0.0002602000313345343, -0.7705178260803223, -2.7802019119262695, -0.04593171179294586, -0.20295001566410065, -5.081482887268066, -1.7981959581375122, -0.033860255032777786, -0.11285962164402008, -0.6855083703994751, -0.9442072510719299, -0.003649243852123618, -0.011937606148421764, -0.0024352199397981167, -0.00721787940710783, -0.01309144590049982, -0.00012385078298393637, -0.008104770444333553, -0.0010724276071414351, -0.39646559953689575, -1.7101092338562012, -0.04261443763971329, -0.6259003281593323, -0.21008506417274475, -3.8725905418395996, -1.3950518369674683, -0.023674197494983673, -0.02188263274729252, -3.4927611351013184, -0.01255536638200283, -1.1237075328826904, -0.18404120206832886, -0.4941917955875397, -0.5180196762084961, -0.5200074911117554, -0.16161076724529266, -0.00020346954988781363, -0.4072398245334625, -0.3694777190685272, -0.0006252956227399409, 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+{"group_id": "math_040", "origin": "budgie", "answer": "This is a classic example of the Cauchy-Ri plague algorithm. The steps are as follows:\n\n1.  Start with the list of 1000 positive integers.\n2.  Initialize an empty list.\n3. aced the first integer by erasing nonpositive integers or integers that have a repeated digit, and then reducing the list by 1.\n4.  Continue this process until the list becomes empty.\n\nThe algorithm runs in O(N^2) time, where N is the number of integers. Since the list grows extremely quickly, it will eventually become empty.\n\nI see that the example given shows how the algorithm works. For 2, the list is [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], which takes 129 steps.\n\nI should also note that for a list of 1000 numbers, the number of steps required is log2(1000) = 11, since 2^10 = 1024, and 1025 is the next power of two after 1024.\n\nI think the answer is 11 steps.\n\nFinal answer:\n#### 11", "reward": 1.0, "advantage": 0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 27803, 5909, 351, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 28, 284, 16790, 253, 3086, 282, 3301, 335, 260, 1398, 30, 1814, 971, 1833, 28, 384, 2812, 1049, 750, 1603, 16185, 23313, 355, 750, 23313, 338, 457, 253, 6514, 11403, 28, 284, 965, 12191, 3117, 281, 260, 1398, 411, 216, 49160, 30, 1073, 800, 3301, 1072, 357, 1188, 327, 6661, 506, 1398, 288, 325, 7964, 47, 198, 198, 75, 89, 77, 2954, 11312, 411, 7937, 11583, 75, 31, 89, 77, 49154, 49158, 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+{"group_id": "math_040", "origin": "verified_reference", "answer": "1. **Initial List and Conditions**:\n   - Henry starts with the list of the first 1000 positive integers: \\( \\{1, 2, 3, \\ldots, 1000\\} \\).\n   - At each step, he erases any nonpositive integers or any integers that have a repeated digit.\n   - After erasing, he decreases every remaining integer in the list by 1.\n\n2. **Understanding the Erasure Condition**:\n   - Nonpositive integers are erased immediately.\n   - Integers with repeated digits are also erased. For example, numbers like 11, 22, 101, etc., will be erased.\n\n3. **Effect of Decreasing by 1**:\n   - After each step, every integer in the list is decreased by 1. This means that the list will eventually contain nonpositive integers, which will be erased.\n\n4. **Key Observation**:\n   - The critical observation is to determine the maximum number of steps it takes for any integer to be erased.\n   - Consider the number 10. It will be decreased to 9, 8, 7, ..., 1, 0, and then -1. This process takes exactly 11 steps.\n\n5. **Erasure of Numbers with Repeated Digits**:\n   - Numbers like 11, 22, 33, ..., 99, 101, 111, etc., will be erased as soon as they appear in the list.\n   - For example, 11 will be erased in the first step, 22 in the second step, and so on.\n\n6. **General Case**:\n   - For any number \\( n \\) in the list, if it does not have repeated digits, it will be decreased step by step until it becomes nonpositive.\n   - The maximum number of steps required for any number to become nonpositive is determined by the number 10, which takes 11 steps.\n\n7. **Conclusion**:\n   - Since the number 10 takes exactly 11 steps to be erased, and all other numbers will be erased in fewer or equal steps, the entire list will be empty after 11 steps.\n\n\\[\n\\boxed{11}\n\\]", "reward": 1.0, "advantage": 0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 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+{"group_id": "math_041", "origin": "budgie", "answer": "The sum of the other two sides is 27, so the third side is 27 - 2x = 4, where x is the length of the side opposite the given side. Then, we have 27 - 2x = 4. Solving for x, we get x = 27/4.\n\nSince the inradius of the triangle is 4, which is the radius of the inscribed circle, the angle opposite the given side is equal to the angle opposite the radius. Therefore, the angle opposite the given side is equal to 360 degrees minus 180 degrees, or 180 degrees.\n\nThus, the answer is \\boxed{180}.", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 28, 260, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 49154, 49158, 198, 504, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 28, 588, 260, 3464, 2103, 314, 216, 49161, 49166, 731, 216, 49161, 104, 446, 216, 49163, 28, 837, 1792, 314, 260, 3519, 282, 260, 2103, 7011, 260, 1836, 2103, 30, 3875, 28, 392, 457, 216, 49161, 49166, 731, 216, 49161, 104, 446, 216, 49163, 30, 30238, 327, 1792, 28, 392, 820, 1792, 446, 216, 49161, 49166, 31, 49163, 30, 198, 198, 7230, 260, 281, 24440, 282, 260, 14973, 314, 216, 49163, 28, 527, 314, 260, 13661, 282, 260, 30036, 7842, 28, 260, 7256, 7011, 260, 1836, 2103, 314, 4037, 288, 260, 7256, 7011, 260, 13661, 30, 4882, 28, 260, 7256, 7011, 260, 1836, 2103, 314, 4037, 288, 216, 49162, 49165, 49159, 5027, 18253, 216, 49160, 49167, 49159, 5027, 28, 355, 216, 49160, 49167, 49159, 5027, 30, 198, 198, 14344, 28, 260, 2988, 314, 3814, 4810, 277, 107, 49160, 49167, 49159, 19181, 49154], "prefix_len": 85, "old_logprobs": [-1.949613094329834, -2.0315499305725098, -0.004266683477908373, -0.10915522277355194, -0.37602463364601135, -0.004523046314716339, -0.0024132197722792625, -0.4790869355201721, -0.12486499547958374, -0.10333548486232758, -0.10346878319978714, -0.7809114456176758, -0.9782068729400635, -0.9665250778198242, -1.3077925443649292, -0.01130437757819891, -0.5259026288986206, -0.09725499898195267, -0.17042315006256104, -0.02695419266819954, -0.28588464856147766, -0.08853746950626373, -0.7653873562812805, -3.4850168228149414, -1.0174169540405273, -0.07181113958358765, -3.8292784690856934, -1.6823101043701172, -0.9089520573616028, -0.00744272256270051, -0.05472160503268242, -0.025187214836478233, -0.5085362195968628, -0.050171539187431335, -0.18647298216819763, -0.9104083776473999, -0.6326968669891357, -0.14116470515727997, -1.6305606365203857, -0.2927303910255432, -0.1373077630996704, -3.223015069961548, -1.0170705318450928, -1.6682024002075195, -0.5599475502967834, -0.40137749910354614, -0.5418233871459961, -0.5164341330528259, -0.027581999078392982, -0.11155867576599121, -0.251260370016098, -0.05483660474419594, -0.10986039787530899, -0.065475232899189, -0.17217475175857544, -1.9649536609649658, -0.6427127718925476, -0.3693077564239502, -0.004739242140203714, -0.16412858664989471, -0.003243544837459922, -0.11420800536870956, -0.3675920367240906, -0.00998159870505333, -0.06703906506299973, -0.9075446128845215, -0.3002467751502991, -0.1638481169939041, -1.1923787593841553, -1.259132742881775, -0.8282867074012756, -0.06619960069656372, -2.9121999740600586, -0.5211148858070374, -3.985921859741211, -0.0005031750188209116, -0.5637041330337524, -0.6469448804855347, -0.026321114972233772, -0.019012944772839546, -0.37021782994270325, -0.05217503756284714, -0.022799383848905563, -3.8542938232421875, -0.09375808387994766, -1.4037925004959106, -0.7145781517028809, -0.03172718733549118, -0.025734923779964447, -0.10922896862030029, -0.000654602306894958, -0.03831886500120163, -0.4246523976325989, -0.6731109023094177, -0.024490319192409515, -0.20948360860347748, -0.6368272304534912, -0.0022104373201727867, -0.15884679555892944, -1.0959442853927612, -0.009781762026250362, -0.4484667181968689, -0.18613003194332123, -0.24014000594615936, -0.04385975003242493, -1.5531961917877197, -1.3021754026412964, -0.8861821293830872, -0.0012335318606346846, -0.16567693650722504, -0.6611570715904236, -0.10597717016935349, -0.0326773077249527, -0.9089555740356445, -0.0013174673076719046, -0.0449293777346611, -1.9763059616088867, -0.005495559424161911, -0.950429379940033, -2.480767250061035, -0.43628206849098206, -0.7508238554000854, -1.0170297622680664, -1.2447285652160645, -0.843635082244873, -1.22129225730896, -0.9981544017791748, -0.0029252853710204363, -0.020219286903738976, -0.3108171820640564, -0.9591777920722961, -0.048967666923999786, -0.023342112079262733, -0.001001571537926793, -0.00042012447374872863, -0.005242410581558943, -0.1029992550611496, -0.14862918853759766, -0.0013684204313904047, -2.4572300910949707, -0.0026750515680760145, -0.055143050849437714, -1.6075286865234375, -0.0012157914461567998, -1.2671606540679932, -0.01009477861225605, -8.702239938429557e-06, -0.005043996497988701, -0.05562093108892441, -0.0434034988284111, -0.0028055852744728327, -0.07963540405035019, -0.3330913484096527]}
+{"group_id": "math_041", "origin": "budgie", "answer": "12.180. A side of the triangle is equal to 15. The sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.\n\nFinal answer: 4", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 28, 260, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 49154, 49158, 198, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 30, 378, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 198, 198, 29288, 2988, 42, 216, 49163, 49154], "prefix_len": 85, "old_logprobs": [-1.7625274658203125, -0.2852133810520172, -0.019678249955177307, -0.0632067546248436, -0.08933103084564209, -0.024749109521508217, -0.02094881422817707, -0.8523209095001221, -0.03433908149600029, -0.03505378216505051, -0.003901965217664838, -0.001149232150055468, -0.009642454795539379, -0.18531294167041779, -7.402622577501461e-05, -0.005557325202971697, -0.0015445933677256107, -0.0020831567235291004, -2.364210844039917, -0.09893722087144852, -0.09756212681531906, -0.00016556799528189003, -0.004048842005431652, -0.03579879552125931, -0.0016195764765143394, -0.00031537804170511663, -0.01855059526860714, -0.010237799026072025, -0.005612330511212349, -0.007803786545991898, -0.020496468991041183, -0.27346596121788025, -0.0033279056660830975, -0.05561517924070358, -0.0010408704401925206, -0.002529280725866556, -0.006299519911408424, -0.0021621440537273884, -0.056628718972206116, -0.013809612952172756, -0.0007113072206266224, -0.17834442853927612, -0.004774479195475578, -0.0027096483390778303, -0.003655538894236088, -0.00011002412065863609, -0.00020644917094614357, -0.0018772137118503451, -0.00020275443966966122, -0.04476592689752579, -0.00016735584358684719, -0.00011288482346571982, -0.00021026308240834624, -0.003204094711691141, -0.0007233387441374362, -0.0015987483784556389, -0.28829389810562134, -0.012479785829782486, -0.052587151527404785, -0.0007202415145002306, -0.0007626485894434154, -2.3861351013183594, -0.6596547365188599, -0.023943325504660606]}
+{"group_id": "math_041", "origin": "verified_reference", "answer": "## Solution.\n\nLet $a$ be the given side of the triangle, $a=15$, $b$ and $c$ be the other two sides, $b+c=27$, $c=27-b$. For definiteness, we will assume that $b>c$. The semiperimeter of the triangle $p=\\frac{a+b+c}{2}=21$, its area $S=p r=84$.\n\nBy Heron's formula $S=\\sqrt{p(p-a)(p-b)(p-c)}$, thus\n\n$$\n\\begin{aligned}\n& \\sqrt{21 \\cdot(21-15)(21-b)(21-27+b)}=84 \\Leftrightarrow \\sqrt{21 \\cdot 6(21-b)(b-6)}=84 \\Leftrightarrow \\\\\n& \\Leftrightarrow(21-b)(b-6)=56 \\Leftrightarrow b^{2}-27 b+182=0, \\\\\n& b_{1}=14, b_{2}=13 . \\\\\n& \\text { We have } b=14, c=13 \\text {. By the cosine rule }\n\\end{aligned}\n$$\n\n$$\n\\cos \\alpha=\\frac{b^{2}+c^{2}-a^{2}}{2 b c}=\\frac{5}{13}\n$$\n\nAnswer: $\\frac{5}{13}$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 28, 260, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 49154, 49158, 198, 712, 19857, 30, 198, 198, 4239, 1885, 81, 20, 325, 260, 1836, 2103, 282, 260, 14973, 28, 1885, 81, 45, 49160, 49164, 26265, 1885, 82, 20, 284, 1885, 83, 20, 325, 260, 550, 827, 5882, 28, 1885, 82, 27, 83, 45, 49161, 49166, 26265, 1885, 83, 45, 49161, 49166, 29, 82, 28422, 1068, 39416, 8051, 28, 392, 523, 7635, 338, 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+{"group_id": "math_046", "origin": "budgie", "answer": "This problem involves the properties of the circumcircle and incircle of a triangle, as well as the midpoint of an arc.\n\nThe key to solving this problem is to apply relevant theorems and theorems related to the circumcircle and the incircle.\n\nSince $O$ and $I$ are the centers of the circumcircle and incircle and $M$ is the midpoint of the arc $AC$, we can use the theorem of the triangle being equal to the sum of the lengths of any two sides to find the length of $AC$.\n\nFor the triangle $ABC$, we have $AB = 15$, $BC = 7$, and $MI = MO$. Applying the theorem of the triangle being equal to the sum of the lengths of any two sides, we can find the length of $AC$.\n\nThe theorem states that in any triangle, the sum of the lengths of the two smaller sides is equal to the length of the largest side.\n\nSo, for triangle $ABC$, we have $AB + BC = Introduce.\n\nApplying the theorem to triangle $ABC$, we get $15 + 7 = 22 = AC$.\n\nTherefore, the length of $AC$ is $22$ units.\n\nFinal answer:\n#### 22", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 24862, 1885, 63, 20, 284, 1885, 57, 20, 359, 260, 7292, 282, 260, 4584, 29733, 284, 1266, 337, 1668, 282, 14973, 1885, 35542, 26265, 284, 1885, 61, 20, 314, 260, 3921, 3154, 282, 260, 13210, 1885, 3144, 20, 282, 260, 4584, 29733, 365, 1766, 5085, 1885, 50, 20, 595, 657, 314, 1343, 338, 1885, 5431, 45, 49160, 49164, 28, 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+{"group_id": "math_046", "origin": "budgie", "answer": "The key to this problem lies in the fact that $M$ is the midpoint of the arc $AC$ of the circumcircle not containing $B$. This is known as the inscribed angle theorem.\n\nGiven that $AB=15$ and $BC=7$, we can find the radius of the circumcircle using the formula $R = \\frac{abc}{4 \\Delta}$, where $a$, $b$, and $c$ are the sides of the triangle and $\\Delta$ is the area of the triangle.\n\nSubstituting the given values for $a$, $b$, and $c$, we get $R = \\frac{15 \\times 7}{4 \\times \\sqrt{15^2 - 7^2}} = \\frac{105}{4 \\times 2\\sqrt{109}}$.\n\nNext, we can find the area of the triangle using Heron's formula. First, we need to find the semi-perimeter $s$ of the triangle:\n\n$s = \\frac{a + b + c}{2} = \\frac{15 + 7 + AC}{2} = \\frac{22 + AC}{2}$.\n\nNow, we can express the area of the triangle in terms of $AC$ using Heron's formula:\n\n$Area = \\sqrt{s(s - a)(s - b)(s - c)} = \\sqrt{\\frac{22 + AC}{2} \\left(\\frac{22 + AC}{2} - 15\\right) \\left(\\frac{22 + AC}{2} - 7\\right) \\left(\\frac{22 + AC}{2} - AC\\right)}}$.\n\nSimplify the expression to get the area", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 24862, 1885, 63, 20, 284, 1885, 57, 20, 359, 260, 7292, 282, 260, 4584, 29733, 284, 1266, 337, 1668, 282, 14973, 1885, 35542, 26265, 284, 1885, 61, 20, 314, 260, 3921, 3154, 282, 260, 13210, 1885, 3144, 20, 282, 260, 4584, 29733, 365, 1766, 5085, 1885, 50, 20, 595, 657, 314, 1343, 338, 1885, 5431, 45, 49160, 49164, 28, 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+{"group_id": "math_046", "origin": "verified_reference", "answer": "Answer: $A C=13$.\n\nSolution. (Fig. 3). First, we prove that $M I=M A$ (trident lemma).\n\nIndeed, the external angle $A I M$ of triangle $A I B$ is equal to the sum of angles $B A I$ and $A B I$, and since $A I$ and $B I$ are angle bisectors, $\\angle A I M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B$. Angle $I A M$ is equal to the sum of angles $I A C$ and $C A M$. But $\\angle I A C=\\frac{1}{2} \\angle A$, and $\\angle C A M=\\angle C B M=\\frac{1}{2} \\angle B$ as inscribed angles.\n\nFrom this, it follows that $\\angle I A M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B=\\angle A I M$, and therefore, triangle $A M I$ is isosceles, $M I=M A$. By the condition $M O=$ $M I$, so by the trident lemma $A O=M O=M I=M A$. This means that triangle $A O M$ is equilateral and $\\angle A O M=60^{\\circ}$. Since the central angle $A O M$ is twice the inscribed angle $A B M$, we have $\\frac{1}{2} \\angle B=30^{\\circ}$, that is, $\\angle B=60^{\\circ}$.\n\nBy the cosine theorem, $A C^{2}=A B^{2}+B C^{2}-2 \\cdot A B \\cdot B C \\cos 60^{\\circ}=$ $13^{2}$.\n\nCriteria. Answer without explanation - 0 points. Proved that angle $B$ is $60^{\\circ}-5$ points. Points are not deducted for the absence of proof of the trident lemma. Full solution - 7 points.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7789e0097aa3cdbc6e57g-2.jpg?height=540&width=467&top_left_y=1779&top_left_x=1485)\n\nFig. 3", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 24862, 1885, 63, 20, 284, 1885, 57, 20, 359, 260, 7292, 282, 260, 4584, 29733, 284, 1266, 337, 1668, 282, 14973, 1885, 35542, 26265, 284, 1885, 61, 20, 314, 260, 3921, 3154, 282, 260, 13210, 1885, 3144, 20, 282, 260, 4584, 29733, 365, 1766, 5085, 1885, 50, 20, 595, 657, 314, 1343, 338, 1885, 5431, 45, 49160, 49164, 28, 7636, 45, 49166, 26265, 284, 1885, 11155, 45, 12850, 28422, 7985, 1885, 3144, 28422, 49154, 49158, 198, 21350, 42, 1885, 49, 340, 45, 49160, 49162, 28422, 198, 198, 37500, 30, 365, 10462, 30, 216, 49162, 595, 3508, 28, 392, 6824, 338, 1885, 61, 339, 45, 61, 330, 20, 365, 8489, 880, 20417, 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+{"group_id": "math_047", "origin": "budgie", "answer": "The key concept in this problem is the concept of inverse functions.\n\nLet $f(x) = \\frac{a-x}{x-a-1}$, then $y = f(f^{-1}(x))$.\n\nWe know that the inverse function $f^{-1}(x)$ is symmetric about the point $(-1,4)$.\n\nThe point $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,y)$ on the graph of $y = f(f^{-1}(x))$.\n\nSince $(a-1,1)$ is on the graph of $y = f(f^{-1}(x))$, we have $f(a-1) = 1$.\n\nNow, let's find the inverse function $f^{-1}(x)$. We know that the inverse function $f^{-1}(x)$ is symmetric about the point $(-1,4)$.\n\nThe point $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,-1)$.\n\nSince $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,-1)$, we have $f^{-1}(x) = -1$.\n\nSubstituting $a-1$ for $x$ in the equation $f^{-1}(x) = -1$, we get $f^{-1}(a-1) = -1$.\n\nNow, consider the inverse function $f^{-1}(x)$. 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+{"group_id": "math_047", "origin": "budgie", "answer": "The inverse function $f^{-1}(x)$ satisfies the condition:\n\\[ f^{-1}(f(x)) = x \\]\n\nSince the graph of the inverse function is symmetric about the point $(-1, 4)$, we have:\n\\[ f(-1 + x) = f(x) \\]\n\nLet $x = 0$ in the equation above:\n\\[ f(0) = f(0 + y) + f(0 - y) = f(0) + f(y) = 4 \\]\n\nTherefore, $y = 0$.\n\nHence, $f^{-1}(0) = 0$.\n\nSince the function $y = x^{-1}$ is an inverse function of $y = 2x - 1$, the point $0$ corresponds to the point $(-1, 4)$.\n\nSince $f(-1 + x) = f(x)$, it follows that:\n\\[ f(x) = f(0) = 2(0) - 1 = -1 \\]\n\nHence, the value of $a$ is:\n\\[ a = \\boxed{0} \\]\n\nFinal answer:\n#### 0", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 10887, 260, 1517, 1885, 105, 24814, 18169, 107, 81, 29, 104, 15009, 104, 29, 81, 29, 49160, 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+{"group_id": "math_047", "origin": "verified_reference", "answer": "2.3.-\n\nFrom the problem, we know that the graph of the function $y=\\frac{a-x}{x-a-1}$ is centrally symmetric about the point $(4,-1)$.\n$\\because y=\\frac{a-x}{x-a-1}=-1-\\frac{1}{x-(a+1)}$, we have $(y+1)[x-(a+1)]=-1$,\n$\\therefore$ the graph of the function is a hyperbola with its center at $(a+1,-1)$.\nAlso, $\\because$ the graph of the hyperbola is centrally symmetric about its center,\n$\\therefore a+1=4$, which means $a=3$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 10887, 260, 1517, 1885, 105, 24814, 18169, 107, 81, 29, 104, 15009, 104, 29, 81, 29, 49160, 24362, 28, 260, 3670, 282, 624, 20508, 1517, 314, 31458, 563, 260, 1225, 1885, 10242, 49160, 28, 49163, 25, 28422, 3875, 260, 1685, 282, 260, 1345, 1230, 1885, 81, 20, 314, 19573, 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+{"group_id": "math_048", "origin": "budgie", "answer": "Solution:\n\n1. Since $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$, then $|\\overrightarrow{O A}|^2=4\\overrightarrow{O B}^2=256$,\nwhich means $256 \\times 24=256\\times4\\overrightarrow{O B}^2 \\Rightarrow \\overrightarrow{O B}^2=256$,\nwhich means $\\overrightarrow{O B}=(1 it, 0 )$.\n\n2. $t \\overrightarrow{A B}-\\overrightarrow{A O}=t (|\\overrightarrow{O B}| - 24)=t (12-24)=-12t$, $|\\overrightarrow{B O}-(1-t) \\overrightarrow{A B}=(\\overrightarrow{B O}+\\overrightarrow{A B})-(1-t) \\overrightarrow{A B}=(\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{A B})$\n\n3. Therefore, $|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{A B}\\right|=|t-1|+|t^2+(4-t) \\overrightarrow{A B}|=|t-1|+|(t+4)(1-t)|=\\begin{cases} 2(t+1), & 0\\leqslant t\\leqsl", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 10887, 338, 19573, 1400, 2771, 5281, 107, 63, 330, 109, 3814, 456, 96, 3814, 1400, 2771, 5281, 107, 63, 389, 24362, 28, 284, 1885, 43509, 1400, 2771, 5281, 107, 63, 330, 109, 108, 45, 43509, 1400, 2771, 5281, 107, 63, 389, 109, 108, 45, 49161, 49163, 28422, 1094, 1885, 100, 3814, 254, 75, 49159, 28, 49160, 77, 26265, 965, 198, 9212, 198, 108, 100, 3814, 1400, 2771, 5281, 107, 49, 389, 39949, 76, 1400, 2771, 5281, 107, 49, 492, 109, 108, 42639, 8842, 43509, 18169, 107, 49164, 15009, 49160, 49161, 109, 3814, 1400, 2771, 5281, 107, 50, 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+{"group_id": "math_048", "origin": "budgie", "answer": "(A)\n\nThe dot product of the vectors is 0, so\n$$\n\\overrightarrow{O A} \\cdot \\overrightarrow{O B} = 0.\n$$\n\nIf $t \\in [0, 1]$, then\n$$\n|t \\overrightarrow{A B} - \\overrightarrow{A O}| = |t| \\cdot 24 - (\\overrightarrow{O A} \\cdot \\overrightarrow{O B}) = 24t - 0.\n$$\n\nTherefore,\n$$\n|t \\overrightarrow{A B} - \\overrightarrow{A O}| + |\\frac{5}{12} \\overrightarrow{B O} - (1-t) \\overrightarrow{B A}| = 24t + \\frac{5}{12}|\\overrightarrow{A B}| + |t|\\cdot 24 - |\\overrightarrow{A B}| \\cdot \\frac{5}{12} = 24t + \\frac{5}{12} \\cdot 24 + 48t - 24 = 24 \\sqrt{2} \\cdot t + 24.\n$$\n\nThus, the minimum value is 24 \\sqrt{2} \\cdot t + 24,\n$$\n\nHence, the answer is:\n$$\n\\boxed{(C) \\ 24 \\sqrt{2}}.\n$$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 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B.\n\nConstruct a square $O A C B$, connect the diagonal $A B$, and let $D$ and $E$ be points on the diagonal $A B$ and the side $O B$ respectively, such that\n$$\n\\begin{array}{l}\nt \\overrightarrow{A B}-\\overrightarrow{A O}=\\overrightarrow{O D}, \\\\\n\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}=\\overrightarrow{E D}, \\\\\nE B=10, O D=D C . \\\\\n\\text { Therefore }|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right| \\\\\n=|\\overrightarrow{E D}|+|\\overrightarrow{D C}| \\leqslant|\\overrightarrow{E C}|=26 .\n\\end{array}\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 10887, 338, 19573, 1400, 2771, 5281, 107, 63, 330, 109, 3814, 456, 96, 3814, 1400, 2771, 5281, 107, 63, 389, 24362, 28, 284, 1885, 43509, 1400, 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+{"group_id": "math_049", "origin": "budgie", "answer": "1. We start by examining the general term of the series:\n\n$$\n\\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}}\n$$\n\nLet's denote this as \\( a_n = \\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}} \\).\n\n2. We can derive a recursive relationship for \\( a_n \\) by manipulating the terms of the series. We can rewrite the terms in the denominator as follows:\n\n$$\na_n = \\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}} = \\frac{1}{A_{n-1} \\cdot A_{n+1}} + \\frac{1}{A_{n-2} \\cdot A_{n+2}} + \\cdots + \\frac{1}{A_2 \\cdot A_1} + \\cdots\n$$\n\nwhere \\( A_{n-k} = A_k + A_{k+1} + \\cdots \\) and \\( A_k = A_{n+k} - A_{n+k-k} = A_{n+k-k} + A_{n+k-k+1} + \\cdots \\).\n\n3. Denote the numerator of \\( a_n \\) by \\( B_k = n - k \\). We can express \\( B_k \\) in terms of \\( A_{n+k-k} \\). After some manipulations, we obtain:\n\n$$\nA_{n+k-k} = A_k + A_{k+1} + \\cdots = A_{A_k} + A_{", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 22983, 260, 1685, 282, 260, 2028, 198, 9212, 198, 76, 18169, 107, 49162, 15009, 49160, 21990, 49161, 109, 3814, 41591, 216, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49164, 15009, 49161, 21990, 49161, 109, 3814, 41591, 216, 49162, 21990, 49161, 17859, 42639, 18169, 107, 49166, 15009, 49162, 21990, 49161, 109, 3814, 41591, 216, 49163, 21990, 49161, 17859, 42639, 13133, 1565, 42639, 18169, 107, 49161, 49168, 15009, 49160, 49163, 21990, 49161, 109, 3814, 41591, 216, 49160, 49164, 21990, 49161, 17859, 1673, 198, 9212, 49154, 49158, 198, 49160, 30, 1046, 1120, 411, 5893, 260, 2210, 2115, 282, 260, 3086, 42, 198, 198, 9212, 198, 76, 18169, 107, 94, 15009, 24, 94, 27, 49161, 25, 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+{"group_id": "math_049", "origin": "budgie", "answer": "1. The given sum can be simplified by first writing each fraction with a denominator of the form $n^2$:\n\n    $$\n    \\frac{3}{1^2 \\cdot 2^2} + \\frac{5}{2^2 \\cdot 3^2} + \\frac{7}{3^2 \\cdot 4^2} + \\cdots + \\frac{29}{14^2 \\cdot 15^2}.\n    $$\n\n2. Notice that the denominator of the first fraction in this series is a square of a prime number, $2^{2}$. This means that when we write each term of the series with a denominator of the form $n^2$, we are effectively adding more factors of 2 to the denominator.\n\n3. The denominator of the first fraction is $2^{2}$, which is a power of 2. Each subsequent denominator is twice the previous denominator, suggesting a factor of 2 in the numerators of the fractions.\n\n4. Each denominator can be expressed as a product of prime powers. The denominator of the first fraction is $2^{2}$, and the denominator of the second fraction is $2^{4}$. Therefore, the denominator of the entire series can be written as:\n\n    $$\n    n^2 \\cdot (2^{2})^{n-1} = n^2 \\cdot 2^{2n - 2}.\n    $$\n\n5. The numerator of the numerator in the given series is the sum of the first 29 prime numbers. The sum of the first $n$ prime numbers can be calculated using the formula for the sum of a series:\n\n    $$\n    S_{n} = \\frac{n(n+1)}{2}.\n    $$\n\n6. Substitute the value of $n$ into this formula to", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 22983, 260, 1685, 282, 260, 2028, 198, 9212, 198, 76, 18169, 107, 49162, 15009, 49160, 21990, 49161, 109, 3814, 41591, 216, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49164, 15009, 49161, 21990, 49161, 109, 3814, 41591, 216, 49162, 21990, 49161, 17859, 42639, 18169, 107, 49166, 15009, 49162, 21990, 49161, 109, 3814, 41591, 216, 49163, 21990, 49161, 17859, 42639, 13133, 1565, 42639, 18169, 107, 49161, 49168, 15009, 49160, 49163, 21990, 49161, 109, 3814, 41591, 216, 49160, 49164, 21990, 49161, 17859, 1673, 198, 9212, 49154, 49158, 198, 49160, 30, 378, 1836, 2028, 416, 325, 20004, 411, 808, 2385, 971, 10236, 351, 253, 30842, 282, 260, 910, 1885, 94, 78, 49161, 20, 42, 1004, 17410, 472, 3814, 18169, 107, 49162, 15009, 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+{"group_id": "math_049", "origin": "verified_reference", "answer": "Solution: $224 / 225$ The sum telescopes as\n$$\n\\left(\\frac{1}{1^{2}}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right)+\\cdots+\\left(\\frac{1}{14^{2}}-\\frac{1}{15^{2}}\\right)=\\frac{1}{1^{2}}-\\frac{1}{15^{2}}=\\frac{224}{225} .\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 22983, 260, 1685, 282, 260, 2028, 198, 9212, 198, 76, 18169, 107, 49162, 15009, 49160, 21990, 49161, 109, 3814, 41591, 216, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49164, 15009, 49161, 21990, 49161, 109, 3814, 41591, 216, 49162, 21990, 49161, 17859, 42639, 18169, 107, 49166, 15009, 49162, 21990, 49161, 109, 3814, 41591, 216, 49163, 21990, 49161, 17859, 42639, 13133, 1565, 42639, 18169, 107, 49161, 49168, 15009, 49160, 49163, 21990, 49161, 109, 3814, 41591, 216, 49160, 49164, 21990, 49161, 17859, 1673, 198, 9212, 49154, 49158, 198, 37500, 42, 1885, 49161, 49161, 49163, 2272, 216, 49161, 49161, 49164, 20, 378, 2028, 20368, 347, 198, 9212, 198, 76, 8842, 19407, 18169, 107, 49160, 15009, 49160, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49161, 21990, 49161, 109, 17243, 2771, 18434, 76, 8842, 19407, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49162, 21990, 49161, 109, 17243, 2771, 18434, 76, 13133, 1565, 42639, 8842, 19407, 18169, 107, 49160, 15009, 49160, 49163, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49160, 49164, 21990, 49161, 109, 17243, 2771, 25, 24814, 18169, 107, 49160, 15009, 49160, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49160, 49164, 21990, 49161, 17859, 24814, 18169, 107, 49161, 49161, 49163, 15009, 49161, 49161, 49164, 109, 1673, 198, 9212, 49154], "prefix_len": 113, "old_logprobs": [-5.1915998458862305, -0.019102534279227257, -5.553123950958252, -2.125361442565918, -5.40551233291626, -3.449836492538452, -7.3291850090026855, -0.4743936359882355, -1.5833905935287476, -3.135874032974243, -0.48246580362319946, -2.4723854064941406, -8.017727851867676, -1.737647294998169, -5.529193878173828, -2.635599136352539, -2.3292083740234375, -0.44871193170547485, -0.19060561060905457, -0.07887586951255798, -2.3604836463928223, -0.6513047218322754, -0.01119145192205906, -0.023722510784864426, -2.8608155250549316, -0.10079328715801239, -1.5001734495162964, -1.4697316884994507, -0.022787265479564667, -2.7171618938446045, -2.240118980407715, -0.0026760026812553406, -0.00015186110977083445, -0.4244289994239807, -0.008682949468493462, -0.33442962169647217, -0.006431594956666231, -0.0005519058904610574, -0.3135577440261841, -0.0015730400336906314, -0.005822009406983852, -0.40552812814712524, -0.02727394551038742, -0.016379306092858315, -0.002074948512017727, -0.00012492353562265635, -0.00011538793478393927, -0.07464474439620972, -0.0017492959741503, -0.03797278553247452, -0.00252714054659009, -0.0003784178989008069, -0.009179767221212387, -0.03505849838256836, -7.652943895664066e-05, -5.602820692729438e-06, -0.001073975581675768, -3.325883881188929e-05, -0.0029667671769857407, -0.0001234931987710297, -8.4638240878121e-06, -0.0005467826849780977, -0.00015770144818816334, -3.194758028257638e-05, -0.012015936896204948, -0.07742787152528763, -2.651477336883545, -6.425174069590867e-05, -0.028445225208997726, -0.12810136377811432, -0.0021033799275755882, -9.48860906646587e-05, -4.410646579344757e-05, -0.02582705020904541, -0.006598825100809336, -0.060691602528095245, -0.17326772212982178, -0.002560077467933297, -4.1126360883936286e-05, -0.4333839416503906, -0.014012172818183899, -0.00011002412065863609, -8.344646857949556e-07, -0.001255200942978263, -0.001292585046030581, -0.005583405494689941, -0.0017851145239546895, -0.00012730741582345217, -3.075552376685664e-05, -0.0008425738196820021, -5.3165931603871286e-05, -2.1576648578047752e-05, -0.38326728343963623, -2.8214783668518066, -0.1723429411649704, -0.02174791693687439, -0.2526610791683197, -0.3228552043437958, -0.4851118326187134, -2.851935863494873, -0.007014174945652485, -1.2016106843948364, -0.26967477798461914, -0.017337316647171974, -0.004412913229316473, -0.044778693467378616, -0.02190922386944294, -0.9553692936897278, -0.18843165040016174, -0.0056563084945082664, -6.55629628454335e-05, -0.11438079923391342, -0.4433906674385071, -0.20652201771736145, -0.0021064728498458862, -1.8581511974334717, -0.8837118148803711, -1.2482788562774658, -0.8169817924499512, -0.5968276262283325, -0.026662427932024002, -0.017207611352205276, -1.1365848779678345, -0.4566560387611389, -0.08287828415632248, -0.0026537699159234762, -1.7867131233215332]}
+{"group_id": "math_050", "origin": "budgie", "answer": "#### Part (a)\n1. Let the length of the wire be a positive number $a$ such that $a > 0$.\n2. The lengths of the pieces are $a$ and $a-a=0$.\n3. By the triangle inequality, we have the following inequalities:\n\\[\n|a| < |0-(a-0)| = a\n\\]\n4. This simplifies to $|a| < a$, which means $a < 0$ as well.\n5. Given $a > 0$, the only possible value for $a$ is $a = 1$.\n\n#### Part (b)\n1. The sides of the squares are $1$ and $2$. Let the length of the side of each square be $x$.\n2. The sum of the areas of the squares is given by:\n\\[\nx^2 + (x-1)^2 = 4\n\\]\n3. Expanding and simplifying, we get:\n\\[\nx^2 + x^2 - 2x + 1 = 4 \\implies 2x^2 - 2x + 1 = 4\n\\]\n4. Rearranging, we have:\n\\[\n2x^2 - 2x - 3 = 0\n\\]\n5. Dividing the entire equation by $2$, we get:\n\\[\nx^2 - x - \\frac{3}{2} = 0\n\\]\n6. The solutions to this quadratic equation are given by:\n\\[\nx = \\frac{1 \\pm \\sqrt{1 + 6}}{2} = \\frac{1 \\pm \\sqrt{7}}{2}\n\\]\nHence, the possible values", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 17411, 95, 553, 253, 6192, 338, 314, 1885, 49160, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 986, 30, 909, 2022, 253, 2304, 418, 253, 1225, 335, 260, 6192, 28, 13725, 827, 4443, 30, 1963, 3800, 553, 253, 3519, 282, 1885, 104, 20, 284, 260, 550, 553, 253, 3519, 282, 1885, 49160, 49159, 29, 104, 20, 347, 3057, 281, 260, 4110, 2441, 42, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49160, 79, 84, 49159, 49161, 83, 49161, 49166, 49164, 49164, 344, 49162, 49162, 49166, 49162, 82, 1018, 49159, 49167, 454, 29, 49159, 49164, 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+{"group_id": "math_050", "origin": "budgie", "answer": "#### Part (a)\n\n1. **Identify the given information**: Augusto has a wire of length \\(10\\) meters and he cuts it into two pieces. One piece has a length of \\(x\\) meters and the other has a length of \\(10-x\\) meters.\n2. **Set up the equation**: The total length of the wire is the sum of the lengths of the two pieces, so we write the equation:\n   \\[\n   x + (10 - x) = 10\n   \\]\n3. **Solve the equation**:\n   \\[\n   x + 10 - x = 10 \\\\\n   10 = 10\n   \\]\n4. **Conclusion**: The equation holds true, so the length of the sides of both pieces is:\n   \\[\n   \\boxed{10 \\text{ meters each}}\n   \\]\n\n#### Part (b)\n\n1. **Express the given length**: \n   \\[\n   x = 10 - x - 10\n   \\]\n2. **Solve the equation**: \n   \\[\n   x = 10 - (10 - x) - 10 \\\\\n   10 = 10 - 10 + x - x \\\\\n   10 = 10 - 10 \\\\\n   10 = 10\n   \\]\n3. **Final answer**: The length of each piece is:\n   \\[\n   \\boxed{10\\ \\text{meters each}}\n   \\]\n\n#### Part (c)\n\n1. **Consider the total length of the wire**: We need to find the smallest number of pieces (or the sum of the lengths of the pieces) that can be cut from the 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+{"group_id": "math_050", "origin": "verified_reference", "answer": "Solution\n\nIn this problem, all lengths are given in meters and areas in square meters.\n\na) A piece of rope has length $x$ and another piece of rope has length $10-x$. Since a square has four sides of equal length, one square will have a side length of $\\frac{x}{4}$ and the other square will have a side length of $\\frac{10-x}{4}$.\n\nThe area of a square with side length $\\ell$ is $\\ell^{2}$. Therefore, one square will have an area of $\\left(\\frac{x}{4}\\right)^{2}=\\frac{x^{2}}{16}$, while the other square will have an area of $\\left(\\frac{10-x}{4}\\right)^{2}=\\frac{100-20 x+x^{2}}{16}$.\n\nb) Let $S(x)$ be the sum of the areas of the two squares. From the previous part, we have\n\n$$\nS(x)=\\frac{x^{2}}{16}+\\frac{100-20 x+x^{2}}{16}=\\frac{100-20 x+2 x^{2}}{16}=\\frac{1}{8} x^{2}-\\frac{5}{4} x+\\frac{25}{4}\n$$\n\nwhich is a quadratic function. The minimum of a function of the form\n\n$$\nf(x)=a x^{2}+b x+c\n$$\n\nwith $a>0$ is achieved at $x=\\frac{-b}{2 a}$. Thus, the minimum area will be achieved if\n\n$$\nx=-\\frac{\\left(-\\frac{5}{4}\\right)}{2 \\frac{1}{8}}=5\n$$\n\nIn other words, if the rope is cut exactly in the middle!\n\nc) From the previous part, we know that to minimize the sum of the areas, it is necessary to cut the rope exactly in the middle. Well, we claim that to minimize the area with nine cuts (i.e., creating ten squares), it is necessary that all pieces of rope be equal. To show this, consider the following argument: if two of the ten pieces of rope were different, it would be possible to reduce the area by cutting the pieces of rope so that these two were equal (we are using the previous part). Therefore, any two pieces of rope must be equal. 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+{"group_id": "math_051", "origin": "budgie", "answer": "To find $f(2013)$, we can express $2013$ in terms of $1$ and $2$ using the given equation:\n\\[ f(2013) = f(1) + f(2) - 2 \\times 1 \\times 2013 \\]\n\nUsing the given equation, we can substitute $x = 1$ and $y = 2$ to get:\n\\[ f(1) = f(1) + f(2) - 4026 \\]\n\nRearrange the equation to isolate $f(1)$:\n\\[ f(1) - f(1) = f(2) - 4026 \\Rightarrow 0 = f(2) - 4026 \\Rightarrow f(2) = 4026 \\]\n\nNow, we can find $f(2) + f(2) - 2 \\times 2013$:\n\\[ f(2) + f(2) - 2 \\times 2013 = 4026 + 4026 - 2 \\times 2013 = 4026 + 4026 - 2 \\times 2013 = 8052 - 2 \\times 2013 = 8052 - 4026 = \\boxed{4026} \\]\nFinal answer:\n#### 4026", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 49160, 7985, 1885, 86, 24, 49161, 49159, 49160, 49162, 25, 26265, 585, 327, 750, 1345, 1885, 104, 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+{"group_id": "math_051", "origin": "budgie", "answer": "To find $f(2013)$, we can use the given equation $f(x-y)=f(x)+f(y)-2xy$ for any real $x$ and $y$, and then substitute $x=2013$ and $y=1$ to find $f(2013)$.\n\nWe know $x-y=2013$ and $x+y=2014$. Using the given equation, we can express $f(2014)$ in terms of $f(2013)$.\n\nSubstitute $x=2013$ and $y=1$ into the given equation:\n\\[ f(2014) = f(2013) + f(1) - 2(2013)(1) \\]\n\nNow, substitute $x=2014$ and $y=1$ into the given equation:\n\\[ f(2014) = f(2013) + f(1) - 2(2013) \\]\n\nSubstitute $x=2013$ and $y=1$ into the equation:\n\\[ f(2013) = f(2013) + f(1) - 2(2013) - 4028 \\]\n\nRearrange the equation to isolate $f(2013)$:\n\\[ f(2013) - f(2013) = f(2013) + f(1) - 2(2013) - 4028 - 2013 \\]\n\n\\[ f(2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 49160, 7985, 1885, 86, 24, 49161, 49159, 49160, 49162, 25, 26265, 585, 327, 750, 1345, 1885, 104, 20, 284, 1885, 105, 20, 260, 7897, 6650, 198, 198, 9212, 198, 86, 24, 104, 29, 105, 36637, 86, 24, 104, 18434, 86, 24, 105, 14400, 49161, 8860, 198, 9212, 49154, 49158, 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+{"group_id": "math_051", "origin": "verified_reference", "answer": "Solution. Substitute $x=y=0$. We get $f(0)=2 f(0)+0$, from which we obtain that $f(0)=0$. Substitute $x=y$. We get $0=f(0)=f(x)+f(x)-2 x^{2}$. Hence, $f(x)=x^{2}$.\n\nAnswer: 4052169. (C)\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_997c53c06135e2de9c16g-02.jpg?height=67&width=1157&top_left_y=1003&top_left_x=498)", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 49160, 7985, 1885, 86, 24, 49161, 49159, 49160, 49162, 25, 26265, 585, 327, 750, 1345, 1885, 104, 20, 284, 1885, 105, 20, 260, 7897, 6650, 198, 198, 9212, 198, 86, 24, 104, 29, 105, 36637, 86, 24, 104, 18434, 86, 24, 105, 14400, 49161, 8860, 198, 9212, 49154, 49158, 198, 37500, 30, 18714, 2881, 1885, 104, 45, 105, 45, 49159, 28422, 1046, 820, 1885, 86, 24, 49159, 36637, 49161, 275, 24, 49159, 18434, 49159, 26265, 429, 527, 392, 3478, 338, 1885, 86, 24, 49159, 36637, 49159, 28422, 18714, 2881, 1885, 104, 45, 105, 28422, 1046, 820, 1885, 49159, 45, 86, 24, 49159, 36637, 86, 24, 104, 18434, 86, 24, 104, 14400, 49161, 1792, 21990, 49161, 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+{"group_id": "math_052", "origin": "budgie", "answer": "First, we need to find the smallest base $b$ such that $N$ is a fourth power. We know that a fourth power is a number that can be expressed as $N = x^4$ for some integer $x$.\n\nSince $N = 777 = x^4$, the base $b$ must be a fourth power. We can find the smallest such $b$ by trying different values of $b$.\n\nLet's start by trying $b = 4$.\n\nFor $b = 4$, we have $N = x^4 = 777$. We can see that $x = 7$ is a solution, since $7^4 = 2401$. However, we need to check if $x = 7$ is also a solution to the congruence $x^4 \\equiv 21 \\pmod{1000}$.\n\nBy testing values of $x$, we can see that $x = 7$ satisfies the congruence.\n\nTherefore, the smallest base $b$ is:\n\\[ \\boxed{4} \\]", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 365, 49168, 373, 6805, 28440, 29435, 25, 1885, 62, 20, 314, 354, 15328, 28, 624, 6133, 281, 3159, 1885, 82, 20, 314, 216, 49166, 49166, 49166, 30, 7985, 260, 13341, 2731, 15328, 1885, 82, 20, 715, 338, 1885, 62, 20, 314, 253, 8629, 1149, 282, 354, 15328, 281, 17948, 20431, 30, 49154, 49158, 198, 5345, 28, 392, 737, 288, 1042, 260, 13341, 3159, 1885, 82, 20, 715, 338, 1885, 62, 20, 314, 253, 8629, 1149, 30, 1046, 699, 338, 253, 8629, 1149, 314, 253, 1230, 338, 416, 325, 6803, 347, 1885, 62, 446, 1792, 78, 49163, 20, 327, 634, 15328, 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+{"group_id": "math_052", "origin": "budgie", "answer": "We start with $b = 3$ because 0 is not a fourth power of 2.\n\nStart with $N = 777$, a three-digit number in base 3.\n\\[ 777_{10} = 1 7 7 7_{10} \\]\n\nConvert $N = 777_{10}$ to decimal:\n\\[ 777_{10} = 777_{3} = 342_{3} = 7^4_{10} = 7^4 \\times 3^0 = 7^4 \\times 1 = 7^3 = 343_{3} = 213_{2} = 3^3_{10} = 3^3 \\times 4^1 = 3^3 \\times 4 = 34_{10} \\]\n\nThe smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation is $\\boxed{3}$ because $N = 342_{3} = 7^4 \\times 3^0 = 7^4 \\times 1 = 7^3$ is a fourth power of an integer in decimal notation.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 365, 49168, 373, 6805, 28440, 29435, 25, 1885, 62, 20, 314, 354, 15328, 28, 624, 6133, 281, 3159, 1885, 82, 20, 314, 216, 49166, 49166, 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+{"group_id": "math_052", "origin": "verified_reference", "answer": "2. This problem is equivalent to finding the smallest positive integer $b$, such that the equation $7 b^{2}+7 b+7=x^{4}$, (1) has an integer solution for $x$. Since 7 is a prime number, it follows from equation (1) that 7 is a divisor of $x$. Therefore, let $x=7 k$, then equation (1) becomes $b^{2}+b+1=7^{3} k^{4}$. The smallest $b$ occurs when $k$ is at its minimum. Taking $k=1$, we then have $b^{2}+b+1=343, b^{2}+b-342=0$, which is $(b-18)(b+19)=0$, yielding the positive integer solution $b=18$. Thus, we have $(777)_{18}=\\left(7^{4}\\right)_{10}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 365, 49168, 373, 6805, 28440, 29435, 25, 1885, 62, 20, 314, 354, 15328, 28, 624, 6133, 281, 3159, 1885, 82, 20, 314, 216, 49166, 49166, 49166, 30, 7985, 260, 13341, 2731, 15328, 1885, 82, 20, 715, 338, 1885, 62, 20, 314, 253, 8629, 1149, 282, 354, 15328, 281, 17948, 20431, 30, 49154, 49158, 198, 49161, 30, 669, 1732, 314, 7492, 288, 4212, 260, 13341, 2731, 15328, 1885, 82, 26265, 715, 338, 260, 8345, 1885, 49166, 278, 21990, 49161, 109, 27, 49166, 278, 27, 49166, 45, 104, 21990, 49163, 24362, 28, 365, 49160, 25, 553, 354, 15328, 3564, 327, 1885, 104, 28422, 4311, 216, 49166, 314, 253, 9552, 1230, 28, 357, 5558, 429, 8345, 365, 49160, 25, 338, 216, 49166, 314, 253, 3044, 40031, 282, 1885, 104, 28422, 4882, 28, 1303, 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+{"group_id": "math_053", "origin": "budgie", "answer": "The problem is about finding the volume of the intersection of two tetrahedra created by reflecting one of them through its center.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49168, 30, 10887, 314, 253, 2649, 29889, 39513, 282, 4981, 216, 49160, 1673, 1046, 3478, 253, 1796, 2649, 29889, 39513, 411, 10329, 260, 1836, 582, 738, 624, 3712, 30, 1812, 314, 260, 4981, 282, 480, 10930, 47, 49154, 49158, 198, 504, 1732, 314, 563, 4212, 260, 4981, 282, 260, 10930, 282, 827, 29889, 770, 317, 2312, 411, 10329, 582, 282, 601, 738, 624, 3712, 30, 49154], "prefix_len": 66, "old_logprobs": [-1.550062894821167, -2.989487886428833, -2.296311378479004, -1.1432123184204102, -1.5341554880142212, -0.01694592460989952, -0.05713459849357605, -0.043754320591688156, -0.2523476779460907, -0.12494485080242157, -0.21934205293655396, -0.05688011273741722, -1.6691546440124512, -0.0011337526375427842, -0.5466839075088501, -3.511587619781494, -0.021064963191747665, -0.24207036197185516, -0.5093414187431335, -3.076242208480835, -1.1807098388671875, -0.055522371083498, -0.15194553136825562, -0.1955566555261612, -0.10770324617624283, -4.276659965515137]}
+{"group_id": "math_053", "origin": "budgie", "answer": "A regular tetrahedron with a volume of 1 is a regular tetrahedron depending on its edge length.\n\nThe volume of a regular tetrahedron is given by the formula:\n\n$$V = \\frac{\\sqrt{2}}{12} \\times (\\text{edge length})^3$$\n\nThe second tetrahedron is half of the first one because each edge of the first tetrahedron is now reflected through its center to form the second one.\n\nSo, the volume of the intersection is twice the volume of the second tetrahedron.\n\nHence, the answer is: $\\boxed{2\\sqrt{2}}$", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49168, 30, 10887, 314, 253, 2649, 29889, 39513, 282, 4981, 216, 49160, 1673, 1046, 3478, 253, 1796, 2649, 29889, 39513, 411, 10329, 260, 1836, 582, 738, 624, 3712, 30, 1812, 314, 260, 4981, 282, 480, 10930, 47, 49154, 49158, 198, 49, 2649, 29889, 39513, 351, 253, 4981, 282, 216, 49160, 314, 253, 2649, 29889, 39513, 4509, 335, 624, 5595, 3519, 30, 198, 198, 504, 4981, 282, 253, 2649, 29889, 39513, 314, 1836, 411, 260, 7961, 42, 198, 198, 9212, 70, 446, 3814, 18169, 26754, 16166, 107, 49161, 109, 15009, 49160, 49161, 109, 3814, 14602, 365, 76, 2692, 107, 7277, 3519, 8148, 78, 49162, 9212, 198, 198, 504, 1796, 29889, 39513, 314, 2745, 282, 260, 808, 582, 975, 971, 5595, 282, 260, 808, 29889, 39513, 314, 1209, 9807, 738, 624, 3712, 288, 910, 260, 1796, 582, 30, 198, 198, 2931, 28, 260, 4981, 282, 260, 10930, 314, 6757, 260, 4981, 282, 260, 1796, 29889, 39513, 30, 198, 198, 35391, 28, 260, 2988, 314, 42, 19573, 4810, 277, 107, 49161, 76, 16166, 107, 49161, 17859, 20, 49154], "prefix_len": 66, "old_logprobs": [-1.5268267393112183, -0.011417053639888763, -0.00980360060930252, -0.00029845553217455745, -2.7570624351501465, -1.0321513414382935, -0.15390600264072418, -0.004066057503223419, -0.02138613723218441, -0.0038498349022120237, -1.2024527788162231, -0.840359628200531, -0.5977230668067932, -0.44333934783935547, -0.0028651398606598377, -9.927957534790039, -0.042961087077856064, -0.8258200287818909, -0.12453719973564148, -0.027798019349575043, -0.4635339081287384, -1.286529541015625, -0.11656678467988968, -1.477001667022705, -1.5239051580429077, -0.08915635943412781, -0.19664758443832397, -0.03069235570728779, -0.007639005314558744, -0.00030322244856506586, -1.0974417924880981, -0.2368212193250656, -0.003986864350736141, -0.5041430592536926, -0.0019431296968832612, -1.1992218494415283, -0.18282870948314667, -0.32322821021080017, -0.44389188289642334, -0.11373955756425858, -0.4446525573730469, -0.009152363054454327, -0.03306155279278755, -0.20578810572624207, -0.005824616644531488, -0.01808229461312294, -0.0621364526450634, -0.020250244066119194, -0.017571140080690384, -0.05218465253710747, -0.011430253274738789, -0.05380994826555252, -1.9663575887680054, -0.38633692264556885, -0.4257713854312897, -0.16556571424007416, -0.02194071188569069, -0.0004593271005433053, -0.11788401007652283, -0.011235067620873451, -0.07252804934978485, -0.027683116495609283, -0.006976648699492216, -0.0062084193341434, -0.024529054760932922, -0.0015543533954769373, -1.4745618104934692, -0.8177616596221924, -0.47106412053108215, -0.000666277133859694, -0.5157479643821716, -6.993849754333496, -0.43704384565353394, -0.47260117530822754, -1.2254003286361694, -0.4145038425922394, -2.0234856605529785, -3.0246806144714355, -0.7801727056503296, -0.37492579221725464, -0.04081806540489197, -1.105315089225769, -0.2565179169178009, -0.00014506718434859067, -0.08136652410030365, -5.064205646514893, -1.2104003429412842, -0.1668986827135086, -0.23409920930862427, -0.034970443695783615, -1.6681430339813232, -0.3721422851085663, -0.4846370816230774, -0.1862044483423233, -0.7824249267578125, -0.09138736128807068, -0.3177088797092438, -0.001972516765818, -1.7063922882080078, -0.1416255533695221, -0.14741481840610504, -0.29360267519950867, -0.015308385714888573, -0.07909992337226868, -1.4486943483352661, -1.2485182285308838, -1.3840548992156982, -0.016923420131206512, -0.007106622215360403, -0.00045015214709565043, -0.06763752549886703, -1.3885653018951416, -0.026263637468218803, -0.00015317220822907984, -0.9156184196472168, -0.0627988800406456, -0.003077772678807378, -3.076115131378174, -0.06328597664833069, -0.08068615943193436, -0.2403603196144104, -0.003192092990502715, -0.29312893748283386, -3.428654193878174, -0.002806892851367593, -3.576278118089249e-07, -0.07123958319425583, -0.24469593167304993, -1.7872461080551147, -0.41067156195640564, -0.01043978612869978, -0.019031191244721413, -0.01818426139652729, -0.2570699155330658, -0.6266476511955261]}
+{"group_id": "math_053", "origin": "verified_reference", "answer": "Solution: $1 / 2$\nImagine placing the tetrahedron $A B C D$ flat on a table with vertex $A$ at the top. By vectors or otherwise, we see that the center is $3 / 4$ of the way from $A$ to the bottom face, so the reflection of this face lies in a horizontal plane halfway between $A$ and $B C D$. In particular, it cuts off the smaller tetrahedron obtained by scaling the original tetrahedron by a factor of $1 / 2$ about $A$. Similarly, the reflections of the other three faces cut off tetrahedra obtained by scaling $A B C D$ by $1 / 2$ about $B, C$, and $D$. On the other hand, the octahedral piece remaining remaining after we remove these four smaller tetrahedra is in the intersection of $A B C D$ with its reflection, since the reflection sends this piece to itself. So the answer we seek is just the volume of this piece, which is\n$$\n\\begin{array}{l}\n\\text { (volume of } A B C D)-4 \\cdot(\\text { volume of } A B C D \\text { scaled by a factor of } 1 / 2) \\\\\n=1-4(1 / 2)^{3}=1 / 2\n\\end{array}\n$$", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49168, 30, 10887, 314, 253, 2649, 29889, 39513, 282, 4981, 216, 49160, 1673, 1046, 3478, 253, 1796, 2649, 29889, 39513, 411, 10329, 260, 1836, 582, 738, 624, 3712, 30, 1812, 314, 260, 4981, 282, 480, 10930, 47, 49154, 49158, 198, 37500, 42, 1885, 49160, 2272, 216, 49161, 20, 198, 4792, 10482, 260, 29889, 39513, 1885, 49, 389, 340, 422, 20, 5562, 335, 253, 3252, 351, 24121, 1885, 49, 20, 418, 260, 1466, 30, 1428, 14799, 355, 5214, 28, 392, 963, 338, 260, 3712, 314, 1885, 49162, 2272, 216, 49163, 20, 282, 260, 970, 429, 1885, 49, 20, 288, 260, 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+{"group_id": "math_054", "origin": "budgie", "answer": "1.  **Assumptions and Setup**: \n   We have ten natural numbers on a circle, which are evenly marked. We place these numbers 1 through 10 in a single circle. We rotate the line passing through the center of the circle, which divides the numbers into two groups. Each group must then be divided by the line passing through the center, resulting in five distinct outcomes. For each arrangement, we need to find the maximum possible value of $K$.\n\n2. **Key Insight**:\n   The line passing through the center divides the circle into two groups. This means the line must intersect the numbers 1 through 10. The maximum possible value for $K$ occurs when the line is tangent to the circle.\n\n3. **Constructing the Arrangements**:\n   To achieve this, we can construct a different line through the center that does not intersect the numbers 1 through 10. For example, we can construct a line passing through the numbers 2 through 17,prime to 19, and 11 through 13,prime to 19. This line does not intersect the numbers 1 through 10 because it passes through prime to prime.\n\n4. **Final Arrangement**:\n   Using the above lines and configurations, we can visualize that the maximum possible value for $K$ is 1.\n\n5. **Considering Other Differences**:\n   We need to consider other differences, such as 2, 3, and 4. These differences are not useful in this context, as they would not help in maximizing $K$.\n\n6. **Constructing the Maximum Value**:\n   By carefully placing the numbers 1 through 10 on the circle, we can see that the maximum possible value for $K$ is 1.\n\n7. **Conclusion**:\n   The maximum value of $K$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49162, 30, 365, 49160, 49164, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 3772, 2876, 359, 15760, 6319, 335, 260, 27436, 282, 253, 7842, 30, 9373, 260, 3772, 1782, 2966, 1885, 49160, 3814, 8045, 216, 49160, 49159, 20, 335, 623, 3772, 2876, 30, 16744, 368, 253, 1761, 7685, 738, 260, 3712, 282, 260, 7842, 30, 1550, 260, 1761, 1072, 441, 1301, 738, 750, 6319, 2876, 28, 357, 20751, 260, 2966, 216, 49160, 288, 216, 49160, 49159, 618, 827, 2119, 30, 1068, 971, 11955, 28, 665, 359, 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+{"group_id": "math_054", "origin": "budgie", "answer": "11. This problem can be approached by considering the symmetry of the circle and the division of numbers. To maximize the value of $K$, we should consider the case where $K$ is minimized. The smallest product is achieved when the two groups are as close to equal as possible, and since there are five ways to do this, we can get a value of $K$ as close as possible to $10$, which is $K=\\boxed{5}$.\n\nFinal answer:\n#### 5", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49162, 30, 365, 49160, 49164, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 3772, 2876, 359, 15760, 6319, 335, 260, 27436, 282, 253, 7842, 30, 9373, 260, 3772, 1782, 2966, 1885, 49160, 3814, 8045, 216, 49160, 49159, 20, 335, 623, 3772, 2876, 30, 16744, 368, 253, 1761, 7685, 738, 260, 3712, 282, 260, 7842, 30, 1550, 260, 1761, 1072, 441, 1301, 738, 750, 6319, 2876, 28, 357, 20751, 260, 2966, 216, 49160, 288, 216, 49160, 49159, 618, 827, 2119, 30, 1068, 971, 11955, 28, 665, 359, 2531, 1853, 288, 10335, 260, 2966, 347, 260, 1761, 35290, 30, 1068, 971, 7573, 28, 665, 314, 253, 1406, 282, 260, 24053, 282, 260, 827, 2119, 28, 284, 260, 13341, 282, 623, 2531, 2320, 314, 31712, 347, 1885, 59, 28422, 1812, 314, 260, 5428, 1685, 282, 1885, 59, 20, 1486, 511, 1636, 11775, 47, 49154, 49158, 198, 49160, 49160, 30, 669, 1732, 416, 325, 12055, 411, 6361, 260, 19536, 282, 260, 7842, 284, 260, 7573, 282, 2966, 30, 1626, 12726, 260, 1685, 282, 1885, 59, 26265, 392, 868, 1771, 260, 1671, 837, 1885, 59, 20, 314, 32451, 30, 378, 13341, 1406, 314, 6551, 645, 260, 827, 2119, 359, 347, 2800, 288, 4037, 347, 1636, 28, 284, 1675, 665, 359, 2531, 1853, 288, 536, 451, 28, 392, 416, 820, 253, 1685, 282, 1885, 59, 20, 347, 2800, 347, 1636, 288, 1885, 49160, 49159, 26265, 527, 314, 1885, 59, 24814, 4810, 277, 107, 49164, 24362, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49164, 49154], "prefix_len": 170, "old_logprobs": [-1.5749647617340088, -3.4995548725128174, -0.28815290331840515, -3.8406312465667725, -0.7434225678443909, -1.2087918519973755, -0.02243751659989357, -2.4861018657684326, -0.26891428232192993, -0.9247164130210876, -0.267414927482605, -3.4229257106781006, -0.7260379195213318, -0.09930482506752014, -1.1229243278503418, -0.33573028445243835, -0.22513626515865326, -1.8179881572723389, -0.3387903571128845, -0.476768434047699, -1.2044992446899414, -3.8428139686584473, -0.14608033001422882, -1.1503896713256836, -1.436244010925293, -0.015381169505417347, -0.11323980987071991, -0.008275032043457031, -0.0262267105281353, -0.20125699043273926, -1.2034345865249634, -1.6562790870666504, -0.6755614876747131, -3.5350499153137207, -0.16434980928897858, -1.5821049213409424, -0.42188894748687744, -0.13570307195186615, -0.03889119252562523, -1.9493823051452637, -0.23667500913143158, -2.7459139823913574, -1.2212700843811035, -1.0729178190231323, -1.2163156270980835, -0.7272261381149292, -0.06481156498193741, -0.7654646635055542, -2.175027370452881, -0.19105783104896545, -0.16946664452552795, -0.4311183989048004, -0.9449246525764465, -1.5917807817459106, -0.681716799736023, -0.008523158729076385, -0.0006898645660839975, -1.5318175554275513, -1.4919495582580566, -3.8361823558807373, -1.2708584070205688, -0.03544183075428009, -0.8166951537132263, -1.4285683631896973, -0.009360946714878082, -0.750688374042511, -0.07259467989206314, -0.0989997386932373, -0.919661819934845, -0.32716667652130127, -3.288177490234375, -0.7691940069198608, -1.017737865447998, -0.3865899443626404, -0.03841855749487877, -0.04886118695139885, -0.3217540681362152, -2.147792100906372, -0.5073598623275757, -1.8048293590545654, -0.39137381315231323, -0.06476619839668274, -1.067868947982788, -0.7331997752189636, -0.6485348343849182, -4.665074825286865, -0.5055636763572693, -0.08790509402751923, -0.7769777774810791, -1.8317192792892456, -3.6683976650238037, -0.15780994296073914, -3.576278118089249e-07, -0.008889504708349705, -1.3543790578842163, -0.5914703607559204, -0.0419258177280426, -1.9717118740081787, -0.19022062420845032, -0.06876706331968307, -0.0003675738989841193, -0.0006724718841724098, -0.007033114787191153, -0.002886179368942976, -0.011172355152666569, -0.0012494861148297787, -0.0001679517881711945]}
+{"group_id": "math_054", "origin": "verified_reference", "answer": "【Analysis】No matter how these 10 numbers are placed, their sum remains unchanged, $1+2+3+4+5+6+7+8+9+10=55$, divided into two groups, the closer the sums of the two groups, the larger the required product will be.\n\n【Solution】\n$$\n\\begin{array}{l}\n1+2+3+4+5+6+7+8+9+10=55 \\\\\n55=27+28\n\\end{array}\n$$\n\nTherefore, as long as the 10 numbers are divided into two parts with sums of 27 and 28, the product at this time is $27 \\times 28=756$. Answer: In all the arrangements, $K$ is maximized at 756.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49162, 30, 365, 49160, 49164, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 3772, 2876, 359, 15760, 6319, 335, 260, 27436, 282, 253, 7842, 30, 9373, 260, 3772, 1782, 2966, 1885, 49160, 3814, 8045, 216, 49160, 49159, 20, 335, 623, 3772, 2876, 30, 16744, 368, 253, 1761, 7685, 738, 260, 3712, 282, 260, 7842, 30, 1550, 260, 1761, 1072, 441, 1301, 738, 750, 6319, 2876, 28, 357, 20751, 260, 2966, 216, 49160, 288, 216, 49160, 49159, 618, 827, 2119, 30, 1068, 971, 11955, 28, 665, 359, 2531, 1853, 288, 10335, 260, 2966, 347, 260, 1761, 35290, 30, 1068, 971, 7573, 28, 665, 314, 253, 1406, 282, 260, 24053, 282, 260, 827, 2119, 28, 284, 260, 13341, 282, 623, 2531, 2320, 314, 31712, 347, 1885, 59, 28422, 1812, 314, 260, 5428, 1685, 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+{"group_id": "math_056", "origin": "budgie", "answer": "To find the maximum number of red dragons that could have been at the table, we can consider the alternating pattern of truth and lie. Since each dragon has three heads, each dragon is either follows the truth or lies.\n\nIf we start with a truth-teller, the first position is a truth-teller, the second position is a truth-teller, the third position is a truth-teller, and so on. This pattern continues for all 530 dragons.\n\nThe maximum number of red dragons can be found by starting with the maximum possible number of truth-tellers and then alternating between truth-tellers and liars.\n\nSince each dragon has three heads, the maximum possible number of truth-tellers is 100, and the maximum possible number of liars is 100.\n\nWe start with the maximum number of truth-tellers (100), and then alternate between truth-tellers and liars. This pattern continues for all 530 dragons.\n\nTherefore, the maximum number of red dragons that could have been at the table is 100.\n\nThe answer is:\n\\[ \\boxed{100} \\]", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 49167, 30, 533, 260, 1798, 282, 28846, 7409, 28, 665, 2330, 2382, 28, 2654, 28, 284, 4461, 26225, 30, 3768, 17422, 553, 1296, 8648, 28, 971, 282, 527, 1811, 6200, 260, 4569, 355, 1811, 5721, 30, 4790, 28, 971, 17422, 553, 418, 2124, 582, 1680, 338, 6200, 260, 4569, 30, 1963, 1194, 28, 216, 49164, 49162, 49159, 26225, 2643, 1130, 253, 5075, 3252, 28, 284, 971, 282, 601, 1137, 42, 198, 198, 29, 216, 49160, 302, 1680, 42, 619, 2068, 957, 2049, 314, 253, 2654, 17422, 1184, 198, 29, 216, 49161, 268, 1680, 42, 619, 2068, 957, 1048, 314, 253, 4461, 17422, 1184, 198, 29, 216, 49162, 7212, 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+{"group_id": "math_056", "origin": "budgie", "answer": "To determine the maximum number of red dragons, we need to analyze the statements made by the dragons and their relationship to each other. If 530 dragons have statements that contradict the truth/ jogit of their heads, then no red dragon could have a head that tells the truth.\n\nLet's denote the number of red dragons as $R$ and the number of green dragons as $G$. Since each red dragon has exactly three heads, we can denote the number of heads as $3R$ and the number of heads as $3G$.\n\nNow, consider the statements made by the dragons. If the head that tells the truth tells the truth about the head that tells the truth about another dragon, then the head that tells the truth about the head with the truth about the head that tells the truth about another dragon must be green.\n\nThis leads to a contradiction, as there would be no head that tells the truth about another dragon, and it would not be possible for 530 dragons to add up to 530 (the number of heads in a round table).\n\nHowever, if the head that tells the truth about another dragon tells the truth about the head with the truth about another dragon, then the head that tells the truth about the head with the truth about the head that tells the truth about another dragon must be blue.\n\nTherefore, the maximum number of red dragons that could have been at the table is $R = 3G = 3(3R) = 9G$.\n\nSo, the answer is:\n\\[ \\boxed{9} \\]\nFinal answer:\n#### 9", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 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+{"group_id": "math_056", "origin": "verified_reference", "answer": "Answer: 176.\n\nSolution. Consider an arbitrary red dragon. To the right of this dragon, at least one head must tell the truth. Note that the 1st and 3rd heads cannot tell the truth (since there is a red dragon to the left), so the 2nd head must tell the truth, and to the right of this dragon, there must be a blue dragon. Now consider the dragon to the left of the chosen red dragon. By similar reasoning, we understand that only the 1st head can tell the truth, and to the left of this dragon, there must be a green one.\n\nFrom the previous reasoning, we get that for any red dragon, one position to the right there is definitely a blue dragon, and one position to the left - a green one. For each red dragon, we combine these three dragons into one group. Note that any dragon can be in at most one such group. Then we get that there are no more than $530 / 3$ < 177, i.e., no more than 176 such groups. Therefore, there are no more than 176 red dragons.\n\nNow let's provide an example. Arrange 176 red dragons in a circle, and in 175 gaps between them, place a green and a blue dragon: ...RGRB..., and in one gap, place two blue and two green dragons: ...RBGBGR.... It is easy to check that all conditions are met: each dragon has at least one head that will tell the truth, and there are exactly 530 dragons in total.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8af0c885427e3e323cf9g-17.jpg?height=492&width=525&top_left_y=852&top_left_x=464)\n\n## 7th grade", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 49167, 30, 533, 260, 1798, 282, 28846, 7409, 28, 665, 2330, 2382, 28, 2654, 28, 284, 4461, 26225, 30, 3768, 17422, 553, 1296, 8648, 28, 971, 282, 527, 1811, 6200, 260, 4569, 355, 1811, 5721, 30, 4790, 28, 971, 17422, 553, 418, 2124, 582, 1680, 338, 6200, 260, 4569, 30, 1963, 1194, 28, 216, 49164, 49162, 49159, 26225, 2643, 1130, 253, 5075, 3252, 28, 284, 971, 282, 601, 1137, 42, 198, 198, 29, 216, 49160, 302, 1680, 42, 619, 2068, 957, 2049, 314, 253, 2654, 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+{"group_id": "math_059", "origin": "budgie", "answer": "This problem is about using the Law of Sines and the Law of Cosines to find the values of trigonometric functions. The goal is to find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$.\n\nTo begin with, we can rewrite the expression in a more convenient form. We know that $\\sin C = \\sin(180^\\circ - A - B) = \\sin(A + B)$, and similarly for the other angles.\n\nNow, we can apply the Law of Sines:\n$$y + z \\cos A = \\sin(A + B) + z \\cos A$$\n$$z + x \\cos B = \\sin(A + B) + x \\cos B$$\n$$x + y \\cos C = \\sin(A + B) + y \\cos C$$\n\nNow, we can substitute these expressions into the given expression:\n$$(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y) = (\\sin(A + B) + z \\cos A)(\\sin(A + B) + x \\cos B)(\\sin(A + B) + y \\cos C) + (\\sin(A + B) + x \\cos A)(\\sin(A + B) + y \\cos C)$$\n\nSince $\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$ and $\\sin(A + B) = \\sin A \\cos B - \\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 533, 19573, 22128, 6935, 17638, 26265, 357, 314, 1343, 338, 1885, 104, 3814, 15495, 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+{"group_id": "math_059", "origin": "verified_reference", "answer": "In $\\triangle A B C$, $\\sin C=\\sin (A+B)=\\sin A \\cos B+\\cos A \\sin B$. Substituting the known conditions, we get $x \\sin A+y \\sin B+z(\\sin A \\cos B+\\cos A \\sin B)=0$, which simplifies to $\\sin A(x+z \\cos B)=-\\sin B(y+z \\cos A)$.\nSimilarly, $\\sin B(y+x \\cos C)=-\\sin C(z+x \\cos B)$, $\\sin C(z+y \\cos A)=-\\sin A(x+y \\cos C)$,\nMultiplying the three equations, we have:\n$(x+z \\cos B)(y+x \\cos C)(z+y \\cos A)=-(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)$, which simplifies to $(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)+(y \\cos A+z)(z \\cos B+x)(x \\cos C+y)=0$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 533, 19573, 22128, 6935, 17638, 26265, 357, 314, 1343, 338, 1885, 104, 3814, 15495, 330, 1232, 329, 3814, 15495, 389, 1232, 1892, 3814, 15495, 340, 446, 216, 49159, 28422, 7985, 260, 1685, 282, 46408, 105, 1232, 1892, 3814, 12374, 330, 14502, 106, 1232, 1792, 3814, 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+{"group_id": "math_061", "origin": "budgie", "answer": "If $|z|=|w|=|z-w|$, then $z=w$ or $z=w$ or $z=-w$. Since $z=w$ is the only possible solution, $\\left(\\frac{z}{w}\\right)^{99}=\\left(\\frac{w}{w}\\right)^{99}=1$.\n\nThis means that $\\left(\\frac{z}{w}\\right)^{99}=\\boxed{1}$.\n\nFinal answer:\n#### 1", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 22188, 389, 29, 49161, 30, 49160, 30, 365, 49161, 49159, 2876, 25, 1068, 1784, 2966, 1885, 106, 28, 267, 20, 715, 338, 1885, 108, 106, 108, 45, 108, 103, 108, 45, 108, 106, 29, 103, 108, 26265, 7529, 19573, 8842, 19407, 18169, 107, 106, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 24362, 30, 49154, 49158, 198, 1937, 1885, 108, 106, 108, 45, 108, 103, 108, 45, 108, 106, 29, 103, 108, 26265, 965, 1885, 106, 45, 103, 20, 355, 1885, 106, 45, 103, 20, 355, 1885, 106, 15196, 103, 28422, 4311, 1885, 106, 45, 103, 20, 314, 260, 805, 1636, 3564, 28, 19573, 8842, 19407, 18169, 107, 106, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 109, 24814, 8842, 19407, 18169, 107, 103, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 109, 45, 49160, 28422, 198, 198, 1348, 1530, 338, 19573, 8842, 19407, 18169, 107, 106, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 109, 24814, 4810, 277, 107, 49160, 24362, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49154], "prefix_len": 84, "old_logprobs": [-5.4676618576049805, -0.08267625421285629, -0.48951417207717896, -0.00837528333067894, -0.035947270691394806, -0.20950999855995178, -0.026094090193510056, -0.00300860358402133, -0.008101341314613819, -0.34878528118133545, -0.04217327758669853, -0.0012685356196016073, -0.005173746962100267, -4.7444173105759546e-05, -0.0038459161296486855, -0.10646867007017136, -0.15687593817710876, -0.5347756743431091, -0.2634028196334839, -1.780600666999817, -0.3890330195426941, -0.5366333723068237, -0.28852781653404236, -0.012083896435797215, -0.366604208946228, -0.4687512516975403, -0.10306990891695023, -0.6842280030250549, -0.27897170186042786, -0.01272981520742178, -0.1433679759502411, -2.5191280841827393, -0.00403613829985261, -0.17723913490772247, -2.315380811691284, -0.2344483584165573, -0.9342364072799683, -1.621399998664856, -0.03618483245372772, -0.3087265193462372, -1.9577724933624268, -1.188499093055725, -0.07433547079563141, -3.0158865451812744, -0.9246641397476196, -0.08035124838352203, -2.374654769897461, -0.2017061710357666, -0.028968919068574905, -0.00010954733443213627, -0.00017093151109293103, -0.0034398215357214212, -0.0003064401389565319, -0.000497336674015969, -0.00011860620725201443, -1.7881377516459906e-06, -0.005675748456269503, -0.0008301864145323634, -0.0032932115718722343, -0.00014995403762441128, -0.051952898502349854, -0.661983072757721, -0.04980333894491196, -0.15507034957408905, -0.03550821915268898, -0.004921702668070793, -0.5237862467765808, -0.0027215369045734406, -0.36131808161735535, -0.0005858612130396068, -5.686121585313231e-05, -0.00038985759601928294, -7.772143726469949e-05, -0.019822711125016212, -0.014727095142006874, -0.03479096665978432, -0.34551504254341125, -0.011373092420399189, -0.06380147486925125, -0.17162667214870453, -0.022371772676706314, -4.3474931716918945, -1.2452268600463867, -0.2738761305809021, -0.5261099934577942, -0.27741482853889465, -0.005650144536048174, -0.00035315475543029606, -0.0005619138828478754, -0.018781563267111778, -0.0038870032876729965, -0.000734178873244673, -0.000714404450263828, -3.4570634852570947e-06, -0.0006388768670149148, -0.0004365683125797659, -0.0013140147784724832, -0.00016842853801790625, -0.03923202306032181, -1.2012253999710083, -0.30498284101486206, -6.079655122448457e-06, -0.006577863823622465, -0.004407928790897131, -0.020266132429242134, -0.2134394347667694, -0.7960283160209656, -0.17046286165714264, -0.08317252993583679, -0.0023445994593203068, -0.0013304438907653093, -0.022198647260665894, -0.007149232551455498, -0.005019799806177616, -2.6464111215318553e-05, -0.000532842765096575]}
+{"group_id": "math_061", "origin": "budgie", "answer": "To approach this problem, we need to understand that if $r$ is a complex number such that $|z|=|w|=|z-w|$, then $z$ is a geometric progression with common ratio $r$.\n\nSince $r$ is a complex number, it can be written as $|r| = c$, where $c$ is a real number.\n\nTherefore, we can write:\n\\[ z = c \\left(\\frac{z}{w}\\right) = c \\left(\\frac{c}{w}\\right) = c^2 \\left(\\frac{1}{w}\\right) = c^2 \\left(\\frac{|z|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) \\]\n\nSince $|z|=|w|=|z-w|$, we can simplify this to:\n\\[ z = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|} \\times \\frac{1}{\\left|z-w|\\right|}\\right) = c^2 \\left(\\frac{|w|}{|z-w|}\\right) = c^2 \\left(\\frac{|w|}{|z-w|}\\right) = c^2 \\left(\\frac{1}{|z-w|}\\right)", "reward": -1, "advantage": -0.707106031187343, "ids": 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+{"group_id": "math_061", "origin": "verified_reference", "answer": "First solution. Let $u=\\frac{z}{w}=x+y i$. 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+{"group_id": "math_062", "origin": "budgie", "answer": "To perform the task of \"erasing\" link segments, we must firstaaaaa deal with the remaining 11 horizontal line segments and then deal with the remaining 11 vertical line segments, which is the least.\n\nThere are $11 + 11 = 22$ line segments on the $10 \\times 10$ grid paper.\n\nThe first step is to erase a certain number of link segments, $x$.\n\nThe remaining link segments are $22 - x$.\n\nThe remaining link segments are $22 - x - x = 22 - 2x$.\n\nThe problem requires that at most 3 link segments remain at each node.\n\nThus, we need to eliminate $3(22 - 2x)$ link segments.\n\n$3(22 - 2x) \\leq 22$.\n\n$66 - 6x \\leq 22$.\n\n$6x \\leq 44$.\n\n$x \\leq 6$.\n\nThus, $x$ has to be less than or equal to 6.\n\nSo, to make sure that at most 3 link segments remain at each node, we need to erase $3(22 - 2x) \\leq 22$ link segments.\n\nTherefore, we need to erase at least 6 link segments.\n\nThe answer is \\boxed{6}.", "reward": -1, 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+{"group_id": "math_062", "origin": "budgie", "answer": "To determine the minimum number of link segments that need to be erased, we can first think of placing the nodes in a grid where each node is connected to its neighbors, and we want to create a grid of nodes with the maximum number of link segments between any two nodes.\n\nImagine a grid where each node is connected to its neighbors, and we want to create a structure with the maximum number of link segments between any two nodes. To achieve this, we can start by connecting each node to its immediate neighbors, then connect each neighbor to its neighbors, and so on.\n\nHowever, this approach is not efficient because we want to maximize the number of link segments between any two nodes with each node being connected to its neighbors. If we attempt to do this, we will be creating a grid of nodes with a lot of link segments between each node.\n\nTo make it easier, we can think of this problem as having a \" simplify\" or \"level\" in the grid. This means that each node is connected to its immediate neighbors, and we can have a \" exceeds\" or \"root\" in the grid.\n\nThe simplest way to think about this is to have a \" net\" structure where each node is connected to its neighbors and each connection is a direct edge between two nodes. In this case, we can have a \" binocular\" in the grid with a \"step\" or \"net\" that each node is connected to its neighbors, and each connection is a direct edge between two nodes.\n\nHowever, if we want to maximize the number of link segments between any two nodes, we can try to build the grid in a \"([[\" or \"level\") shape. This means that each node is connected to its immediate neighbors and the neighbors of its neighbors, and we can have a \"level\" shape in the grid.\n\nTo achieve this, we can start", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 49168, 1985, 253, 1885, 49160, 49159, 3814, 14602, 216, 49160, 49159, 20, 6983, 1941, 28, 665, 359, 216, 49160, 49160, 10691, 6983, 3204, 284, 216, 49160, 49160, 8681, 6983, 3204, 30, 330, 1761, 9134, 8377, 827, 11358, 8661, 335, 260, 1142, 5173, 1761, 314, 1217, 253, 476, 8371, 9134, 2227, 1073, 800, 2795, 12523, 737, 288, 325, 40446, 418, 2124, 28, 588, 338, 418, 768, 216, 49162, 2795, 12523, 2612, 418, 971, 5994, 47, 49154, 49158, 198, 2068, 3346, 260, 5869, 1230, 282, 2795, 12523, 338, 737, 288, 325, 40446, 28, 392, 416, 808, 1510, 282, 10482, 260, 8661, 281, 253, 6983, 837, 971, 5994, 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+{"group_id": "math_062", "origin": "verified_reference", "answer": "[Solution] Removing the dotted links in the right figure satisfies the requirements of the problem, and there are 41 dotted links in the figure, so the minimum value required does not exceed 41.\n\nOn the other hand, there are 81 internal nodes on the grid paper, each with 4 links. Therefore, at least one link must be erased from each node to meet the requirement. Since each link connects two nodes, at least 41 links must be erased.\nIn summary, the minimum number of links that need to be erased is 41.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 49168, 1985, 253, 1885, 49160, 49159, 3814, 14602, 216, 49160, 49159, 20, 6983, 1941, 28, 665, 359, 216, 49160, 49160, 10691, 6983, 3204, 284, 216, 49160, 49160, 8681, 6983, 3204, 30, 330, 1761, 9134, 8377, 827, 11358, 8661, 335, 260, 1142, 5173, 1761, 314, 1217, 253, 476, 8371, 9134, 2227, 1073, 800, 2795, 12523, 737, 288, 325, 40446, 418, 2124, 28, 588, 338, 418, 768, 216, 49162, 2795, 12523, 2612, 418, 971, 5994, 47, 49154, 49158, 198, 75, 37500, 77, 40646, 260, 33697, 5823, 281, 260, 1048, 4110, 38475, 260, 4292, 282, 260, 1732, 28, 284, 665, 359, 216, 49163, 49160, 33697, 5823, 281, 260, 4110, 28, 588, 260, 5869, 1685, 2309, 1072, 441, 8769, 216, 49163, 49160, 30, 198, 198, 3280, 260, 550, 1369, 28, 665, 359, 216, 49167, 49160, 4611, 8661, 335, 260, 6983, 1941, 28, 971, 351, 216, 49163, 5823, 30, 4882, 28, 418, 2124, 582, 2795, 1251, 325, 40446, 429, 971, 5994, 288, 2220, 260, 9702, 30, 4311, 971, 2795, 13766, 827, 8661, 28, 418, 2124, 216, 49163, 49160, 5823, 1251, 325, 40446, 30, 198, 788, 7127, 28, 260, 5869, 1230, 282, 5823, 338, 737, 288, 325, 40446, 314, 216, 49163, 49160, 30, 49154], "prefix_len": 105, "old_logprobs": [-7.449690818786621, -1.9770647287368774, -0.15920168161392212, -10.69961166381836, -1.789900779724121, -11.838637351989746, -3.979261875152588, -3.605294942855835, -1.1912921667099, -7.498869895935059, -3.8084609508514404, -5.740559101104736, -0.28958895802497864, -3.902332067489624, -1.1528631448745728, -0.09817113727331161, -0.1340741068124771, -2.633474349975586, -1.7155736684799194, -2.7289886474609375, -0.5453047752380371, -1.146714210510254, -2.2426087856292725, -4.827389717102051, -12.58397102355957, -0.026423754170536995, -1.617440104484558, -0.4235057532787323, -1.399945616722107, -2.12799334526062, -1.3228168487548828, -1.0529439449310303, -3.0631728172302246, -6.422821044921875, -4.625144958496094, -8.781652450561523, -0.033182527869939804, -1.4572054147720337, -0.11708926409482956, -0.21313565969467163, -0.18142126500606537, -0.10972218215465546, -0.35778218507766724, -0.06791090220212936, -6.157463550567627, -0.9485287070274353, -1.8810614347457886, -0.007239655591547489, -0.0018123644404113293, -3.1792166233062744, -0.4323405921459198, -0.5759453773498535, -4.433464050292969, -2.7335517406463623, -7.434441566467285, -4.061356067657471, -3.5231809616088867, -0.3463122248649597, -1.1522432565689087, -2.9703993797302246, -0.43633532524108887, -5.184387683868408, -0.9019814729690552, -0.7719457149505615, -1.3656389713287354, -1.8687137365341187, -1.834261178970337, -1.1006048917770386, -0.0026856327895075083, -2.429999828338623, -0.29096072912216187, -2.4851255416870117, -0.6696619391441345, -1.6720387935638428, -0.14165224134922028, -2.0822014808654785, -3.248870611190796, -0.282360315322876, -0.9901083111763, -1.4724669456481934, -4.510658264160156, -0.02598026767373085, -0.5497615933418274, -0.5901196002960205, -4.733688831329346, -2.063830852508545, -1.370210886001587, -1.9551098346710205, -0.11643362790346146, -0.6598911285400391, -0.18370525538921356, -2.090676784515381, -0.17032140493392944, -0.9332923889160156, -1.4725711345672607, -1.9542450904846191, -1.1973481178283691, -0.9059697389602661, -0.03581397607922554, -0.04219784960150719, -0.7868223190307617, -0.10581129044294357, -8.253219604492188, -2.2577691078186035, -0.024062616750597954, -0.7008510231971741, -0.5332680344581604, -0.8551396131515503, -0.08624454587697983, -0.698243260383606, -1.0326505899429321, -0.33920007944107056, -0.001156733720563352, -0.000621840707026422, -0.037371110171079636, -0.22377364337444305, -0.4029815196990967, -0.16946452856063843, -0.0882166400551796, -0.10297312587499619, -4.6557207107543945]}
+{"group_id": "math_064", "origin": "budgie", "answer": "Given that $a^n - 2014$ has $2014$ digits, we need to find the smallest natural number $k$ such that $a$ cannot also be a $k$-digit number.\n\nIf $a$ is a $k$-digit number, then it can be written as $a = \\overline{abcd...}$, where $a, b, c, d,..., k$ are its digits.\n\nSubstituting $a = \\overline{abcd...}$ into the equation, we get:\n\n$$\\overline{abcd...}^n - 2014 = \\overline{abcd...}$$\n\nRearranging the equation, we have:\n\n$$\\overline{abcd...}^n = 2014\\overline{abcd...}$$\n\nTaking the logarithm of both sides, we get:\n\n$$n \\log \\overline{abcd...} = \\log 2014 \\overline{abcd...}$$\n\nNow, we need to find the smallest natural number $k$ such that $a$ cannot also be a $k$-digit number.\n\nUsing the definition of $a$ and $n$, we can write $n = 2014 - 2014 \\log \\overline{a}$.\n\nSubstituting this into the equation, we get:\n\n$$2014 - 2014 \\log \\overline{a} = \\log 2014$$\n\nSolving for $\\log \\overline{a}$, we get:\n\n$$\\log \\overline{a} = 4$$\n\nSince $a$ is", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 2959, 1885, 81, 20, 284, 1885, 94, 20, 325, 1782, 2966, 28, 284, 357, 314, 1343, 338, 1885, 81, 21990, 94, 39949, 49161, 49159, 49160, 49163, 20, 314, 253, 1230, 351, 1885, 49161, 49159, 49160, 49163, 20, 18638, 30, 7985, 260, 13341, 1782, 1230, 1885, 91, 20, 715, 338, 1885, 81, 20, 2526, 325, 253, 1885, 91, 20, 29, 23141, 1230, 30, 49154, 49158, 198, 15423, 338, 1885, 81, 78, 94, 731, 216, 49161, 49159, 49160, 49163, 20, 553, 1885, 49161, 49159, 49160, 49163, 20, 18638, 28, 392, 737, 288, 1042, 260, 13341, 1782, 1230, 1885, 91, 20, 715, 338, 1885, 81, 20, 2526, 597, 325, 253, 1885, 91, 20, 29, 23141, 1230, 30, 198, 198, 1937, 1885, 81, 20, 314, 253, 1885, 91, 20, 29, 23141, 1230, 28, 965, 357, 416, 325, 2855, 347, 1885, 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+{"group_id": "math_064", "origin": "budgie", "answer": "2014 is a multiple of 4, so\n$$(a^k - 2014) \\equiv 0 \\pmod{4}$$\n\nThe possible values of $a$ are 1, 2, 3, 4, and 5.\n\nIf $a = 5$, then $5^k - 2014 = 0 \\Rightarrow 5^k = 2014 \\Rightarrow k = 4$, which is not a $k$-digit number.\n\nIf $a = 4$, then $4^k - 2014 = 0 \\Rightarrow 4^k = 2014 \\Rightarrow k = 5$, which is a $k$-digit number.\n\nIf $a = 3$, then $3^k - 2014 = 0 \\Rightarrow 3^k = 2014 \\Rightarrow k = 6$, which is a $k$-digit number.\n\nIf $a = 2$, then $2^k - 2014 = 0 \\Rightarrow 2^k = 2014 \\Rightarrow k = 127$, which is not a $k$-digit number.\n\nIf $a = 1$, then $1^k - 2014 = 0 \\Rightarrow 1^k = 2014 \\Rightarrow k = 402$, which is not a $k$-digit number.\n\nIf $a = 0$, then $0^k - 2014 = 0 \\Rightarrow 0^k = 2014 \\Rightarrow k = 40", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 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+{"group_id": "math_064", "origin": "verified_reference", "answer": "Solution. Let $a$ be a $k$-digit number, then $10^{k-1} \\leq a2013$, i.e., $k>45$.\n\nWe will check for $k$ starting from 46, whether the condition of the absence of an integer in the specified interval is met. We will find that\n\n(for $k=46$) $2013 / 462013$ );\n\n(for $k=47$) $2013 / 472013$ );\n\n(for $k=48$) $2013 / 482013$ );\n\nbut (for $k=49$) $2013 / 49 > 41$ (since $\\quad 41 \\cdot 49 = 45^2 - 1^2 = 2025 - 16 < 2013$).\n\nThus, the interval $[2013 / 49, 2013 / 48)$ does not contain any integers, and for $k=49$, there are no suitable $n$.\n\n## Answer: $\\mathbf{k}=49$.\n\n## Criteria.\n\nThe problem is complex, so don't be stingy with points for reasonable thoughts. Any answer in the vicinity of $\\sqrt{2014}$, which is preceded by something reasonable, is worth points. 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+{"group_id": "math_065", "origin": "budgie", "answer": "To find the minimum sum of all real numbers written in $11$ edges, we can analyze the structure of the graph described.\n\nSince each edge is connecting two disks, we can model each edge as a pair of disks. The value of the real number in each disk is the absolute difference between the two real numbers written in the two disks that the edge connects.\n\nWe can represent the graph with edges as follows:\n\n-(2^1, 0) - 2^0, -2^0, -2^1, -2^1, -2^2 - 2^1, -2^1, -2^2 - 2^2, ...\n\nThe $x$-coordinate of each vertex is the sum of the squares of the distances from the vertices to the other two vertices. The $y$-coordinate of each vertex is the square of the difference between the two real numbers in the two disks that the edge connects.\n\nNotice that the values in the $y$-coordinate can be represented as a function of the distance in the $x$-direction. This function can be represented as:\n\n$$f(x, y) = |x - y|^2$$\n\nWe can see that the function $f(x, y)$ is a piecewise linear function, and the minimum value occurs when $x = y$. For this to happen, we must have $x = 0$ and $y = 1$, which would make $f(x, y) = 0$.\n\nTo find the minimum value of $f(x, y)$, we can take the derivative of $f(x, y)$ with respect to $x$ and set it equal to zero:\n\n$$f'(x, y) = 2|x - y", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 665, 325, 253, 4110, 351, 1885, 49168, 20, 26113, 284, 1885, 49160, 49160, 20, 8938, 28, 347, 3057, 2441, 30, 198, 198, 1882, 523, 2965, 253, 1345, 1230, 281, 971, 284, 897, 10599, 30, 3875, 28, 327, 971, 5595, 28, 392, 523, 2965, 260, 5222, 282, 260, 3193, 826, 260, 827, 1345, 2966, 2855, 281, 260, 827, 26113, 338, 260, 5595, 13766, 30, 1046, 1251, 2965, 1885, 49159, 20, 281, 10599, 1885, 49, 26265, 284, 1885, 49160, 20, 281, 10599, 1885, 57, 28422, 7985, 260, 5869, 2028, 282, 511, 1345, 2966, 2855, 281, 1885, 49160, 49160, 20, 8938, 30, 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+{"group_id": "math_065", "origin": "budgie", "answer": "The equation that defines the distances between the real numbers in the disks is the sum of the radii of the two disks. This can be written as $r_1 + r_2 = 9$.\n\nWe can also write this equation in terms of the radii of the two disks as $r_1 + r_2 = 9$ and $r_1 - r_2 = 9$.\n\nSolving these two equations, we find that $r_1 = 3$ and $r_2 = 6$.\n\nNow, we can find the sum of all real numbers written in the edges of the figure. This is equal to $11(r_1 + r_2) = 11(3 + 6) = 49$.\n\nTherefore, the minimum sum of all real numbers written in the edges is $\\boxed{49}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 665, 325, 253, 4110, 351, 1885, 49168, 20, 26113, 284, 1885, 49160, 49160, 20, 8938, 28, 347, 3057, 2441, 30, 198, 198, 1882, 523, 2965, 253, 1345, 1230, 281, 971, 284, 897, 10599, 30, 3875, 28, 327, 971, 5595, 28, 392, 523, 2965, 260, 5222, 282, 260, 3193, 826, 260, 827, 1345, 2966, 2855, 281, 260, 827, 26113, 338, 260, 5595, 13766, 30, 1046, 1251, 2965, 1885, 49159, 20, 281, 10599, 1885, 49, 26265, 284, 1885, 49160, 20, 281, 10599, 1885, 57, 28422, 7985, 260, 5869, 2028, 282, 511, 1345, 2966, 2855, 281, 1885, 49160, 49160, 20, 8938, 30, 49154, 49158, 198, 504, 8345, 338, 11754, 260, 9510, 826, 260, 1345, 2966, 281, 260, 26113, 314, 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+{"group_id": "math_065", "origin": "verified_reference", "answer": "1. **Model the problem as an electrical circuit:**\n   - We are given a graph with 9 disks (nodes) and 11 edges.\n   - We need to assign real numbers to each disk such that the sum of the squares of the differences between the numbers on the disks connected by edges is minimized.\n   - We are required to set the number on disk \\(A\\) to \\(0\\) and on disk \\(I\\) to \\(1\\).\n\n2. **Interpret the problem using electrical circuit theory:**\n   - Consider each disk as a node in an electrical circuit.\n   - Assign a voltage to each node, with disk \\(A\\) at \\(0V\\) and disk \\(I\\) at \\(1V\\).\n   - Each edge represents a resistor of \\(1 \\Omega\\).\n\n3. **Apply the concept of minimizing power in the circuit:**\n   - The goal is to minimize the sum of the squares of the differences, which is equivalent to minimizing the total power dissipated in the circuit.\n   - The power dissipated in a resistor \\(R\\) with voltage difference \\(V\\) is given by \\(P = \\frac{V^2}{R}\\).\n   - For a resistor of \\(1 \\Omega\\), the power dissipated is \\(P = V^2\\).\n\n4. **Calculate the equivalent resistance of the circuit:**\n   - The total resistance between nodes \\(A\\) and \\(I\\) is given as \\(\\frac{33}{20} \\Omega\\).\n   - This is derived from the specific configuration of the graph and the resistances.\n\n5. **Determine the total power in the circuit:**\n   - The total voltage across the circuit is \\(1V\\) (from \\(A\\) to \\(I\\)).\n   - The total power dissipated in the circuit is given by:\n     \\[\n     P = \\frac{V^2}{R_{\\text{eq}}} = \\frac{1^2}{\\frac{33}{20}} = \\frac{20}{33}\n     \\]\n\n6. **Conclusion:**\n   - The minimum sum of the squares of the differences between the numbers on the disks connected by edges is \\(\\frac{20}{33}\\).\n\nThe final answer is \\(\\boxed{\\frac{20}{33}}\\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 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+{"group_id": "math_066", "origin": "budgie", "answer": "We know that $A$ and $B$ are two vertices of $\\mathcal{P}$ on the circumcircle of $\\mathcal{Y}$. Let the other arc between $A$ and $B$ on $\\mathcal{Y}$ have $2^k+1$ vertices.\n\nLet $A$ be $P_1$ and $B$ be $P_2$. By the inscribed angle theorem, we know that $2\\sin(\\angle(AP_1, AP_2)) = 2\\sin(\\angle(BP_1, BP_2)$ and $2\\sin(\\angle(AP_1, BP_2)) = 2\\sin(\\angle(AP_2, BP_1)$.\n\nSince $A$ and $B$ are on the circumcircle of $\\mathcal{Y}$, we know that $AP_1 = BP_2$.\n\nTherefore, $\\angle(AP_1, AP_2) = \\angle(AP_1, BP_2) = \\angle(BP_1, BP_2) = \\angle(P_1B_2, P_2B_1) = \\angle(P_2A_2, P_1A_1) = \\angle(A_1P_2, A_2P_1) = \\angle(P_1A_1, PB_2) = \\angle(P_2A_2, PB_1)$.\n\nSince $\\angle(AP_1, BP_2) = \\angle(P_2A_2, PB_1)$, we know that $AP_2 = BP_1$.\n\nSince $\\angle(AP_1, AP_2) = \\angle(AP_2, BP_1) =", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2122, 314, 253, 2649, 1885, 49160, 49166, 20, 29, 13539, 19573, 8898, 5020, 107, 64, 24362, 284, 624, 4584, 29733, 19573, 8898, 5020, 107, 73, 24362, 335, 260, 8303, 30, 216, 198, 504, 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+{"group_id": "math_066", "origin": "budgie", "answer": "This problem is about counting the number of colorings of a regular 17-gon with a circumcircle of modulus 2.\n\nFirst, we need to find the number of colorings. To do this, we can use the following recurrence relation:\n- The first vertex can be colored in red, blue, or green.\n- The second vertex can be colored in two choices: red or blue.\n- The third side of the polygon can be colored in three choices.\n- The fourth side of the polygon can be colored in two choices.\n- We continue this pattern until we reach the last vertex.\n\nUsing this recurrence relation, we can find the number of colorings for a regular 17-gon.\n\nLet's denote the number of colors as $c_n$. We can find $c_{17}$, the number of colorings for a regular 17-gon:\n\n$$c_{17} = 2c_{16} + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8$$\n\nNow, we can calculate the value of $c_{17}$.\n\n$$c_{17} = 2(2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8)$$\n\nTo find the value of $c_{17}$, we need to calculate the sum of the squares of the first 20 positive integers.\n\n$$c_{17} = (2^1 + 2^2 + 2^3 +... + 2^20) + (2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2122, 314, 253, 2649, 1885, 49160, 49166, 20, 29, 13539, 19573, 8898, 5020, 107, 64, 24362, 284, 624, 4584, 29733, 19573, 8898, 5020, 107, 73, 24362, 335, 260, 8303, 30, 216, 198, 504, 24796, 282, 19573, 8898, 5020, 107, 64, 24362, 359, 23692, 281, 715, 253, 970, 338, 1885, 49, 28, 50, 3814, 254, 3814, 8898, 5020, 107, 64, 24362, 359, 282, 673, 199, 6202, 7302, 28, 585, 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+{"group_id": "math_066", "origin": "verified_reference", "answer": "1. **Identify the relevant values of \\(2^k + 1\\):**\n   Since the minor arc \\(AB\\) has at most 9 vertices on it, the relevant values of \\(2^k + 1\\) are \\(3, 5,\\) and \\(9\\). This is because:\n   \\[\n   2^1 + 1 = 3, \\quad 2^2 + 1 = 5, \\quad 2^3 + 1 = 9\n   \\]\n\n2. **Enumerate the vertices:**\n   Enumerate the vertices of the 17-gon as \\(V_0, V_1, \\ldots, V_{16}\\).\n\n3. **Determine the conditions for different colors:**\n   We have that \\(V_i\\) and \\(V_j\\) are of different colors if \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\). This is derived from the relevant values of \\(2^k + 1\\).\n\n4. **Establish a lower bound for the number of colors:**\n   Since \\(V_0, V_2, V_4\\) must have pairwise distinct colors, we have \\(n \\geq 3\\).\n\n5. **Check if 4 colors suffice:**\n   If there are 4 colors \\(A, B, C,\\) and \\(D\\), we can try to color the vertices such that the condition is satisfied. One possible coloring is:\n   \\[\n   \\{ V_0, V_1, \\cdots, V_{16} \\} = \\{ A, D, B, B, C, C, D, D, B, B, C, A, D, D, B, B, C \\}\n   \\]\n   This coloring ensures that no two vertices \\(V_i\\) and \\(V_j\\) with \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\) have the same color. Therefore, \\(n \\leq 4\\).\n\n6. **Prove that 3 colors are insufficient:**\n   Assume there exists a solution with \\(n = 3\\). Without loss of generality, let \\(V_0 = A\\) and \\(V_2 = B\\). Then:\n   \\[\n   V_4 = C, \\quad V_6 = A, \\quad V_8 = B, \\quad V_{10} = C, \\quad V_{12} = A, \\quad V_{14} = B, \\quad V_{16} = C\n   \\]\n   Continuing this pattern, we find that \\(V_{13} = A\\), which contradicts the assumption that \\(n = 3\\) is a solution because \\(V_0\\) and \\(V_{13}\\) would be the same color, violating the condition for different colors.\n\nTherefore, the least number of colors required is \\(4\\).\n\nThe final answer is \\(\\boxed{4}\\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2122, 314, 253, 2649, 1885, 49160, 49166, 20, 29, 13539, 19573, 8898, 5020, 107, 64, 24362, 284, 624, 4584, 29733, 19573, 8898, 5020, 107, 73, 24362, 335, 260, 8303, 30, 216, 198, 504, 24796, 282, 19573, 8898, 5020, 107, 64, 24362, 359, 23692, 281, 715, 253, 970, 338, 1885, 49, 28, 50, 3814, 254, 3814, 8898, 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+{"group_id": "math_068", "origin": "budgie", "answer": "(a) To find the measure of angle $A X C$, we can use the properties of inscribed quadrilaterals.\n\nSince $\\angle B = 32^{\\circ}$, we know that $\\angle B X D = 180^{\\circ} - 32^{\\circ} - 52^{\\circ} = 96^{\\circ}$.\n\nSimilarly, since $\\angle X D C = 52^{\\circ}$, we know that $\\angle X D C = 180^{\\circ} - 52^{\\circ} - 96^{\\circ} = 44^{\\circ}$.\n\nNow, we can use the fact that the sum of the angles in a quadrilateral is $360^{\\circ}$ to find the measure of $\\angle A X C$.\n\n$$\\angle A X C = x + \\angle X D C + \\angle B D C = x + 44^{\\circ} + 96^{\\circ}$$\n\nTo find the value of $x$, we can plug in $x = 44^{\\circ}$ and $x = 96^{\\circ}$ into the equation.\n\n$$\\angle A X C = 44^{\\circ} + 96^{\\circ} = 140^{\\circ}$$\n\nTherefore, the angle $A X C$ measures $\\boxed{140^{\\circ}}$.\n\n(b) To find the measure of angle $A C B$, we can use the properties of inscribed quadrilaterals.\n\nWe know that the sum of the angles in a quadrilateral is $360^{\\circ}$.\n\n$$\\angle A C B + \\angle", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 18584, 403, 216, 49168, 30, 49164, 30, 49161, 30, 334, 1677, 1885, 49, 422, 20, 284, 1885, 52, 340, 20, 282, 260, 30036, 20378, 13293, 1885, 49, 389, 340, 422, 20, 359, 4037, 30, 330, 1225, 1885, 72, 20, 314, 6319, 335, 2103, 1885, 50, 340, 20, 715, 338, 1885, 49, 389, 446, 389, 2273, 28422, 657, 314, 1343, 338, 19573, 6935, 389, 446, 216, 49162, 49161, 21990, 76, 21386, 24362, 284, 19573, 6935, 2273, 422, 340, 446, 216, 49164, 49161, 21990, 76, 21386, 24362, 30, 198, 24, 81, 25, 365, 49160, 1225, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 2273, 340, 20, 2621, 47, 198, 198, 24, 82, 25, 365, 49162, 2876, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 340, 389, 20, 2621, 47, 198, 198, 21653, 5257, 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+{"group_id": "math_068", "origin": "budgie", "answer": "(a) To find the measure of angle $A D C$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$A D C + A B = \\angle A B C + \\angle A B D$$\n\nSince $\\angle A B D = 180^\\circ - \\angle B = 180^\\circ - 32^\\circ = 148^\\circ$. Substituting this into the equation above, we get:\n\n$$A D C + 148^\\circ = 180^\\circ$$\n\nRearranging the equation, we get:\n\n$$A D C = 180^\\circ - 148^\\circ$$\n\n$$A D C = 32^\\circ$$\n\n(b) To find the measure of angle $A C B$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$A C B + A C B = 180^\\circ$$\n\nSince $A C B$ and $A C B$ are congruent triangles, we can equate the two expressions:\n\n$$A C B = 180^\\circ - A C B$$\n\nSubtracting $A C B$ from both sides, we get:\n\n$$0 = 180^\\circ - A C B - A C B$$\n\nRearranging the equation, we get:\n\n$$A C B = 180^\\circ$$\n\n(c) To find the measure of angle $C B D$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 18584, 403, 216, 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+{"group_id": "math_068", "origin": "verified_reference", "answer": "Answer: (a) $106^{\\circ}$. (b) $48^{\\circ}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 18584, 403, 216, 49168, 30, 49164, 30, 49161, 30, 334, 1677, 1885, 49, 422, 20, 284, 1885, 52, 340, 20, 282, 260, 30036, 20378, 13293, 1885, 49, 389, 340, 422, 20, 359, 4037, 30, 330, 1225, 1885, 72, 20, 314, 6319, 335, 2103, 1885, 50, 340, 20, 715, 338, 1885, 49, 389, 446, 389, 2273, 28422, 657, 314, 1343, 338, 19573, 6935, 389, 446, 216, 49162, 49161, 21990, 76, 21386, 24362, 284, 19573, 6935, 2273, 422, 340, 446, 216, 49164, 49161, 21990, 76, 21386, 24362, 30, 198, 24, 81, 25, 365, 49160, 1225, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 2273, 340, 20, 2621, 47, 198, 198, 24, 82, 25, 365, 49162, 2876, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 340, 389, 20, 2621, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49160, 86, 49163, 49168, 49167, 49160, 49164, 49160, 49168, 49163, 49166, 49168, 49164, 49162, 49161, 16373, 83, 49160, 87, 29, 49160, 49162, 30, 13655, 47, 11551, 45, 49162, 49160, 49160, 22, 6930, 45, 49166, 49159, 49168, 22, 4961, 79, 8842, 79, 105, 45, 49161, 49161, 49159, 22, 4961, 79, 8842, 79, 104, 45, 49162, 49166, 49161, 25, 49154, 49158, 198, 21350, 42, 365, 81, 25, 1885, 49160, 49159, 49165, 21990, 76, 21386, 24362, 30, 365, 82, 25, 1885, 49163, 49167, 21990, 76, 21386, 24362, 30, 49154], "prefix_len": 244, "old_logprobs": [-4.0428147315979, -0.08647264540195465, -0.3793182373046875, -1.4903967380523682, -0.0379696860909462, -3.6785571575164795, -1.2445402145385742, -2.1928887367248535, -4.509901523590088, -0.9986282587051392, -0.0014760324265807867, -0.0010352734243497252, -0.08891186118125916, -1.569225549697876, -4.69000768661499, -2.6808485984802246, -0.004261935595422983, -0.9301959276199341, -3.1230106353759766, -2.5131139755249023, -0.39433079957962036, -0.0006468587671406567, -0.00030870441696606576, -0.029331324622035027, -0.12441731244325638, -4.077394962310791]}
+{"group_id": "math_069", "origin": "budgie", "answer": "For any non-isosceles trapezoid, the sum of the lengths of the two parallel sides equals the length of the base. Since the non-parallel sides are of different lengths, they can form two right triangles with the height as the hypotenuse. The other leg of each right triangle is half the difference between the two parallel sides, so each right triangle has sides 20 cm and 13 cm. Since it's an isosceles trapezoid, it follows that 20 = 2x and 13 = x, where x is the height of the trapezoid. Therefore, x = 10 cm. The height of the trapezoid is 10 cm.\n\nFinal answer:\n#### 10", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49159, 30, 49159, 49161, 49163, 30, 378, 15900, 282, 260, 7623, 5882, 282, 260, 635, 31999, 1633, 359, 216, 49161, 49164, 284, 216, 49163, 5973, 28, 284, 260, 15900, 282, 260, 1603, 29, 29769, 5882, 359, 216, 49161, 49159, 284, 216, 49160, 49162, 5973, 30, 7985, 260, 4614, 282, 260, 635, 31999, 1633, 30, 49154, 49158, 198, 2193, 750, 1603, 29, 271, 395, 5906, 266, 635, 31999, 1633, 28, 260, 2028, 282, 260, 15900, 282, 260, 827, 7623, 5882, 16473, 260, 3519, 282, 260, 3159, 30, 4311, 260, 1603, 29, 29769, 5882, 359, 282, 896, 15900, 28, 502, 416, 910, 827, 1048, 22694, 351, 260, 4614, 347, 260, 16392, 264, 1726, 30, 378, 550, 1475, 282, 971, 1048, 14973, 314, 2745, 260, 3193, 826, 260, 827, 7623, 5882, 28, 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+{"group_id": "math_069", "origin": "budgie", "answer": "In an isosceles trapezoid, the non-parallel sides are not necessarily equal, so we cannot directly use the Pythagorean theorem. However, we can use the fact that the sum of the lengths of the two parallel sides and one of the non-parallel sides is equal to the length of the other side.\n\nLet's denote the height of the trapezoid as h cm. The sum of the lengths of the parallel sides is 25 + 4 = 29 cm. Therefore, we can write:\n\nh + 25 + 20 = 29\n\nSimplify and solve for h:\n\nh + 45 = 29\nh = 29 - 45\nh = -16 cm\n\nThis is incorrect since the height cannot be negative.\n\nLet's try again. The sum of the lengths of the two parallel sides is 25 + 4 = 29 cm. The missing side has a length of h cm. Therefore, we can write:\n\n25 + 4 = 4 + h\n29 = 4 + h\nh = 29 - 4\nh = 25 cm\n\nThe height of the trapezoid is 25 cm.\n\nFinal answer:\n#### 25", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49159, 30, 49159, 49161, 49163, 30, 378, 15900, 282, 260, 7623, 5882, 282, 260, 635, 31999, 1633, 359, 216, 49161, 49164, 284, 216, 49163, 5973, 28, 284, 260, 15900, 282, 260, 1603, 29, 29769, 5882, 359, 216, 49161, 49159, 284, 216, 49160, 49162, 5973, 30, 7985, 260, 4614, 282, 260, 635, 31999, 1633, 30, 49154, 49158, 198, 788, 354, 314, 395, 5906, 266, 635, 31999, 1633, 28, 260, 1603, 29, 29769, 5882, 359, 441, 6834, 4037, 28, 588, 392, 2526, 3319, 722, 260, 47997, 22688, 30, 1423, 28, 392, 416, 722, 260, 1027, 338, 260, 2028, 282, 260, 15900, 282, 260, 827, 7623, 5882, 284, 582, 282, 260, 1603, 29, 29769, 5882, 314, 4037, 288, 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+{"group_id": "math_069", "origin": "verified_reference", "answer": "## Solution.\n\nGiven $B C=4 \\text{ cm}, A D=25 \\text{ cm}, A B=20 \\text{ cm}, C D=13 \\text{ cm}$ (Fig. 10.24). Draw $B E \\perp A D$ and $C F \\perp A D$. Let $B E=C F=h, A E=x$, $F D=y$. Then from $\\triangle A B E$ and $\\triangle C F D$ we find $h^{2}=20^{2}-x^{2}=13^{2}-y^{2}$. Considering that $y=25-4-x=21-x$, we have $20^{2}-x^{2}=13^{2}-(21-x)^{2}$ or $42 x=672$, from which $x=16$ (cm). Therefore, $h=\\sqrt{20^{2}-16^{2}}=12$ (cm).\n\nAnswer: 12 cm.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49159, 30, 49159, 49161, 49163, 30, 378, 15900, 282, 260, 7623, 5882, 282, 260, 635, 31999, 1633, 359, 216, 49161, 49164, 284, 216, 49163, 5973, 28, 284, 260, 15900, 282, 260, 1603, 29, 29769, 5882, 359, 216, 49161, 49159, 284, 216, 49160, 49162, 5973, 30, 7985, 260, 4614, 282, 260, 635, 31999, 1633, 30, 49154, 49158, 198, 712, 19857, 30, 198, 198, 15423, 1885, 50, 340, 45, 49163, 3814, 2692, 107, 5973, 6663, 330, 422, 45, 49161, 49164, 3814, 2692, 107, 5973, 6663, 330, 389, 45, 49161, 49159, 3814, 2692, 107, 5973, 6663, 340, 422, 45, 49160, 49162, 3814, 2692, 107, 5973, 24362, 365, 10462, 30, 216, 49160, 49159, 30, 49161, 49163, 595, 10602, 1885, 50, 414, 3814, 456, 96, 330, 422, 20, 284, 1885, 51, 426, 3814, 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+{"group_id": "math_070", "origin": "budgie", "answer": "This problem can be solved by finding the GCD of all possible permutations of the digits of the number 202120222023. The GCD is 3.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 30, 7985, 260, 452, 6998, 282, 511, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 198, 198, 37500, 30, 1428, 260, 3044, 271, 1968, 3938, 28, 511, 623, 2966, 359, 46522, 411, 216, 49168, 365, 1195, 2028, 282, 260, 18638, 314, 216, 49160, 49167, 595, 330, 5499, 2619, 288, 6824, 338, 665, 359, 787, 550, 2966, 314, 338, 260, 3193, 826, 750, 827, 715, 2966, 314, 597, 46522, 411, 260, 452, 6998, 30, 1068, 1183, 28, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49162, 49161, 731, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49161, 49162, 446, 216, 49168, 314, 46522, 411, 216, 49168, 284, 2526, 325, 2852, 670, 216, 49168, 30, 49154, 49158, 198, 1348, 1732, 416, 325, 14168, 411, 4212, 260, 452, 6998, 282, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 30, 378, 452, 6998, 314, 216, 49162, 30, 49154], "prefix_len": 169, "old_logprobs": [-4.162250518798828, -0.5279743671417236, -1.8504750728607178, -0.01975691132247448, -0.08087814599275589, -0.2180236428976059, -2.7862915992736816, -0.043194133788347244, -1.1433933973312378, -0.0005972985527478158, -0.05813749134540558, -0.34744739532470703, -1.531062364578247, -0.06045081466436386, -0.0023694555275142193, -0.035987865179777145, -0.04790462553501129, -0.025946466252207756, -0.10412748903036118, -0.5967187881469727, -0.03968536853790283, -0.03482941910624504, -0.01022906694561243, -0.027027178555727005, -0.008818373084068298, -0.0039245267398655415, -0.007647404447197914, -0.0012988949893042445, -0.0004002247005701065, -0.0005734706646762788, -0.0031329141929745674, -0.006006049923598766, -0.0002696150622796267, -0.0004693360242526978, -0.27413374185562134, -2.191563129425049, -1.7300163507461548, -0.000604565953835845, -1.2303577661514282, -0.5960536003112793, -2.2327699661254883, -0.49995723366737366, -1.7536964416503906]}
+{"group_id": "math_070", "origin": "budgie", "answer": "The number 202120222023 has 11 digits, all of which are 2's, and 11 + 11 = 22 digits are 2's. Therefore, the GCD of all 22 numbers obtained by all possible permutations of the digits of 202120222023 is 22.\n\nFinal answer: $\\boxed{22}$", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 30, 7985, 260, 452, 6998, 282, 511, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 198, 198, 37500, 30, 1428, 260, 3044, 271, 1968, 3938, 28, 511, 623, 2966, 359, 46522, 411, 216, 49168, 365, 1195, 2028, 282, 260, 18638, 314, 216, 49160, 49167, 595, 330, 5499, 2619, 288, 6824, 338, 665, 359, 787, 550, 2966, 314, 338, 260, 3193, 826, 750, 827, 715, 2966, 314, 597, 46522, 411, 260, 452, 6998, 30, 1068, 1183, 28, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49162, 49161, 731, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49161, 49162, 446, 216, 49168, 314, 46522, 411, 216, 49168, 284, 2526, 325, 2852, 670, 216, 49168, 30, 49154, 49158, 198, 504, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 553, 216, 49160, 49160, 18638, 28, 511, 282, 527, 359, 216, 49161, 506, 28, 284, 216, 49160, 49160, 1232, 216, 49160, 49160, 446, 216, 49161, 49161, 18638, 359, 216, 49161, 506, 30, 4882, 28, 260, 452, 6998, 282, 511, 216, 49161, 49161, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 314, 216, 49161, 49161, 30, 198, 198, 29288, 2988, 42, 19573, 4810, 277, 107, 49161, 49161, 24362, 49154], "prefix_len": 169, "old_logprobs": [-1.2751632928848267, -1.8553056716918945, -0.5552741885185242, -0.03442765399813652, -0.44634881615638733, -0.009287605062127113, -0.017534945160150528, -0.017008626833558083, -0.0024380742106586695, -0.000590865034610033, -0.0007270314963534474, -0.004284369759261608, -0.013349791057407856, -0.002029741881415248, -0.004078167490661144, -0.6864653825759888, -0.620512068271637, -1.1484708786010742, -1.4627463817596436, -0.05690128728747368, -1.044712781906128, -2.544477701187134, -0.408455491065979, -0.016646651551127434, -0.06941532343626022, -1.3538349866867065, -0.17690707743167877, -0.9989707469940186, -1.1439831256866455, -1.0088485479354858, -1.4579106569290161, -0.7175869941711426, -1.3473387956619263, -3.9890966415405273, -0.015657702460885048, -0.3239790201187134, -0.5485996603965759, -0.2915966212749481, -0.011740515939891338, -0.0016959820641204715, -0.00032085992279462516, -0.37679311633110046, -0.8103131055831909, -0.33686041831970215, -0.3993087410926819, -0.04962616413831711, -0.23915044963359833, -1.35410475730896, -0.01473191101104021, -0.674564778804779, -0.8888201117515564, -0.000618624035269022, -0.1752222776412964, -0.2119254469871521, -3.2273378372192383, -0.05504140257835388, -0.06209511682391167, -0.6232460737228394, -0.8941295146942139, -0.07227402180433273, -0.27634313702583313, -0.41369861364364624, -0.05416978523135185, -0.0005273620481602848, -0.035330332815647125, -0.3823673725128174, -0.09567642956972122, -0.24226021766662598, -1.0454427003860474, -0.0029779395554214716, -0.006200718693435192, -0.0007712250226177275, -0.0005383234238252044, -0.005497930571436882, -0.0005443998379632831, -0.00037174468161538243, -0.0006104036583565176, -0.00283815641887486, -0.014971042983233929, -0.00013064485392533243, -0.0005133026279509068, -0.04219807684421539, -0.2716982662677765, -0.28270667791366577, -0.45719096064567566, -0.3799542486667633, -0.671518862247467, -0.017886998131871223, -0.3838563859462738, -0.00035279724397696555, -0.0012175773736089468, -3.311633586883545, -0.00011359999916749075, 0.0, -0.0009581027552485466, -0.0022381513845175505, -0.002102547325193882, -0.0013781820889562368, -0.11235304176807404]}
+{"group_id": "math_070", "origin": "verified_reference", "answer": "Answer: 9.\n\n## Criteria\n\n7 points - complete solution;\n\n3 points - GCD found, but not proven that there can be no other divisors.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 30, 7985, 260, 452, 6998, 282, 511, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 198, 198, 37500, 30, 1428, 260, 3044, 271, 1968, 3938, 28, 511, 623, 2966, 359, 46522, 411, 216, 49168, 365, 1195, 2028, 282, 260, 18638, 314, 216, 49160, 49167, 595, 330, 5499, 2619, 288, 6824, 338, 665, 359, 787, 550, 2966, 314, 338, 260, 3193, 826, 750, 827, 715, 2966, 314, 597, 46522, 411, 260, 452, 6998, 30, 1068, 1183, 28, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49162, 49161, 731, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49161, 49162, 446, 216, 49168, 314, 46522, 411, 216, 49168, 284, 2526, 325, 2852, 670, 216, 49168, 30, 49154, 49158, 198, 21350, 42, 216, 49168, 30, 198, 198, 712, 38042, 1116, 49166, 2876, 731, 2994, 3564, 43, 1116, 49162, 2876, 731, 452, 6998, 983, 28, 564, 441, 8220, 338, 665, 416, 325, 787, 550, 3044, 40520, 30, 49154], "prefix_len": 169, "old_logprobs": [-4.287250518798828, -0.026273159310221672, -0.6010149717330933, -0.7073802351951599, -3.0711803436279297, -1.5149413347244263, -0.044312141835689545, -7.475040912628174, -13.60683536529541, -4.241678237915039, -1.9305158853530884, -10.840970039367676, -8.009834289550781, -10.719900131225586, -5.60141658782959, -8.295293807983398, -3.8393115997314453, -4.912070274353027, -1.5972609519958496, -0.25642016530036926, -7.8958845138549805, -0.02761678770184517, -5.656700134277344, -3.1824111938476562, -4.286386013031006, -2.403550624847412, -3.8519606590270996, -5.302414417266846, -1.7391934394836426, -4.413415908813477, -0.34524405002593994, -2.577296257019043, -0.11411569267511368, -5.766369819641113, -0.04870814457535744, -0.8330802321434021, -3.127284049987793]}
+{"group_id": "math_071", "origin": "budgie", "answer": "There are 5 sums because $a_1+a_2$, $a_1+a_3$, $a_1+a_4$, $a_1+a_5$, and $a_2+a_3$ are the 5 sums. Since there are 5 numbers, and they are distinct, there is only 1 number left. Therefore, $m=1$.\n\nThe answer is:\n\\boxed{1}", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 81, 79, 49161, 28, 81, 79, 49162, 28, 81, 79, 49163, 28, 81, 79, 49164, 20, 325, 4073, 1345, 2966, 30, 6689, 511, 24053, 282, 260, 910, 1885, 81, 79, 89, 1232, 253, 79, 90, 20, 837, 1885, 89, 28, 90, 3814, 254, 3814, 107, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 76, 24362, 284, 1885, 89, 3814, 662, 97, 544, 28422, 2959, 1885, 93, 20, 325, 260, 1230, 282, 4073, 2966, 1486, 623, 24053, 30, 1812, 314, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 47, 49154, 49158, 198, 2122, 359, 216, 49164, 24053, 975, 1885, 81, 79, 49160, 27, 81, 79, 49161, 26265, 1885, 81, 79, 49160, 27, 81, 79, 49162, 26265, 1885, 81, 79, 49160, 27, 81, 79, 49163, 26265, 1885, 81, 79, 49160, 27, 81, 79, 49164, 26265, 284, 1885, 81, 79, 49161, 27, 81, 79, 49162, 20, 359, 260, 216, 49164, 24053, 30, 4311, 665, 359, 216, 49164, 2966, 28, 284, 502, 359, 4073, 28, 665, 314, 805, 216, 49160, 1230, 2049, 30, 4882, 28, 1885, 93, 45, 49160, 28422, 198, 198, 504, 2988, 314, 42, 198, 76, 4810, 277, 107, 49160, 109, 49154], "prefix_len": 125, "old_logprobs": [-3.113186836242676, -0.08462987095117569, -0.8381704092025757, -0.3669227361679077, -1.8881096839904785, -4.855652332305908, -1.939371109008789, -0.9162423014640808, -0.024545922875404358, -0.2490803748369217, -1.7291144132614136, -0.0017780937487259507, -0.00026806574896909297, -0.0056962547823786736, -2.4100089073181152, -0.04025645926594734, -0.0006226746481843293, -0.0015499495202675462, -0.5376858711242676, -0.001544950413517654, -0.00017021637177094817, -7.593343616463244e-05, -0.12526148557662964, -0.009346893057227135, -0.05758395418524742, -0.0007675323868170381, -0.0006849803612567484, -0.08255431801080704, -0.00020716428116429597, -4.470248313737102e-05, -0.00014399446081370115, -0.001765955938026309, -0.00817914493381977, -0.03243233263492584, -0.0004015354788862169, -0.0007577646756544709, -0.03406386449933052, -7.60526381782256e-05, -3.802703940891661e-05, -0.0002843929105438292, -6.305972783593461e-05, -0.07334715127944946, -0.7788305282592773, -0.08500967174768448, -0.002091722097247839, -0.0006211258587427437, -0.007950800471007824, -0.0026016931515187025, -0.00034517052699811757, -0.0007416835287585855, -0.005346167832612991, -1.6080540418624878, -0.6820364594459534, -1.810835838317871, -2.636697769165039, -0.1660851240158081, -0.6825323104858398, -0.5372816920280457, -3.2197835445404053, -2.2614634037017822, -0.08498426526784897, -0.2480018585920334, -0.2158496379852295, -1.7172815799713135, -0.5309000611305237, -2.6364529132843018, -2.6850428581237793, -0.4874305725097656, -0.48716241121292114, -0.07813294976949692, -2.3537583351135254, -2.1851730346679688, -0.38934797048568726, -0.9084161520004272, -0.0012442474253475666, -0.6950001120567322, -0.2627752423286438, -1.3450559377670288, -1.0926463603973389, -0.031344275921583176, -0.7898772358894348, -0.01613534986972809, -0.468288779258728, -0.1359553337097168, -0.2142956256866455, -0.1638282835483551, -0.06685689091682434, -0.5006710290908813, -0.3590526878833771, -0.0006498370785266161, -0.2250765860080719, -0.037441615015268326, -0.09413974732160568, -1.374876856803894, -1.597391747054644e-05, -0.0004068977141287178, -0.0053018201142549515, -0.052733272314071655, -0.12261684983968735]}
+{"group_id": "math_071", "origin": "budgie", "answer": "We can see that for each sum that can be formed, there are four distinct values of $a_i + a_j$. The sums that cannot be formed are $m = 5^5 - 2^5 = 1313$, $m = 4^5 - 3^5 = 2643$, and $m = 3^5 - 2^5 = 19$.\n\nTherefore, the smallest possible value of $m$ is $19$.\n\nThe answer is:\n\\[ \\boxed{19} \\]\nFinal answer:\n#### 19", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 81, 79, 49161, 28, 81, 79, 49162, 28, 81, 79, 49163, 28, 81, 79, 49164, 20, 325, 4073, 1345, 2966, 30, 6689, 511, 24053, 282, 260, 910, 1885, 81, 79, 89, 1232, 253, 79, 90, 20, 837, 1885, 89, 28, 90, 3814, 254, 3814, 107, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 76, 24362, 284, 1885, 89, 3814, 662, 97, 544, 28422, 2959, 1885, 93, 20, 325, 260, 1230, 282, 4073, 2966, 1486, 623, 24053, 30, 1812, 314, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 47, 49154, 49158, 198, 1882, 416, 963, 338, 327, 971, 2028, 338, 416, 325, 4331, 28, 665, 359, 1876, 4073, 2396, 282, 1885, 81, 79, 89, 1232, 253, 79, 90, 28422, 378, 24053, 338, 2526, 325, 4331, 359, 1885, 93, 446, 216, 49164, 78, 49164, 731, 216, 49161, 78, 49164, 446, 216, 49160, 49162, 49160, 49162, 26265, 1885, 93, 446, 216, 49163, 78, 49164, 731, 216, 49162, 78, 49164, 446, 216, 49161, 49165, 49163, 49162, 26265, 284, 1885, 93, 446, 216, 49162, 78, 49164, 731, 216, 49161, 78, 49164, 446, 216, 49160, 49168, 28422, 198, 198, 17308, 28, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 314, 1885, 49160, 49168, 28422, 198, 198, 504, 2988, 314, 42, 198, 76, 75, 3814, 4810, 277, 107, 49160, 49168, 109, 3814, 77, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49168, 49154], "prefix_len": 125, "old_logprobs": [-2.132789134979248, -0.7716965675354004, -3.110560417175293, -0.08593370020389557, -1.987987756729126, -1.212799072265625, -1.5401246547698975, -5.645573616027832, -2.176344156265259, -0.009319375269114971, -0.32650235295295715, -0.608310341835022, -0.6677852869033813, -0.8955617547035217, -3.0503954887390137, -3.1945087909698486, -2.2192115783691406, -1.2678277492523193, -0.269671767950058, -0.0447160005569458, -0.016970185562968254, -0.08405803889036179, -0.1586689054965973, -0.0003182381624355912, -0.0004320403386373073, -0.0002474478678777814, -0.5418604016304016, -4.506051063537598, -2.9172840118408203, -0.6948451995849609, -2.217109441757202, -0.07299104332923889, -0.005205648019909859, -0.200049489736557, -1.6507158279418945, -3.159205913543701, -0.4549645483493805, -0.08387389779090881, -2.1496431827545166, -5.5863938331604, -1.2474703788757324, -0.7381654381752014, -0.08777265250682831, -2.287095546722412, -1.9305610656738281, -0.03940016031265259, -0.6807305812835693, -0.01818905957043171, -1.0829941034317017, -2.0865182876586914, -0.5093056559562683, -2.0061280727386475, -2.0728724002838135, -0.2748419940471649, -0.0501222163438797, -0.009834646247327328, -0.013796914368867874, -1.8849923610687256, -0.012428924441337585, -0.042379654943943024, -0.01973750814795494, -0.0021955687552690506, -0.18279553949832916, -0.0005368936690501869, -0.0004861365014221519, -0.03184312954545021, -0.0017273995326831937, -1.7331116199493408, -1.6780314445495605, -1.2822973728179932, -1.815400242805481, -0.17102473974227905, -0.7035549879074097, -0.019885700196027756, -0.001892800792120397, -0.00562560698017478, -0.005717589985579252, -0.6212297677993774, -0.0024126251228153706, -0.00047565114800818264, -0.00812794640660286, -0.001073380233719945, -0.32874810695648193, -0.0022280411794781685, -0.00567491864785552, -0.03639727830886841, -0.00326980440877378, -0.5756306052207947, -1.5491572618484497, -2.159425735473633, -1.0111730098724365, -0.0787346214056015, -1.1382776498794556, -0.003087517572566867, -0.16770271956920624, -0.1627691239118576, -0.0944795086979866, -0.038083530962467194, -0.0018857807153835893, -0.011094783432781696, -0.0009495278936810791, -0.0073648616671562195, -0.005330871790647507, -1.074412226676941, -0.3970789611339569, -1.2125120162963867, -0.5068919658660889, -0.030533038079738617, -0.06074623391032219, -0.5647497177124023, -0.3607828617095947, -0.0003499372396618128, -0.13594096899032593, -0.03238224238157272, -0.15164881944656372, -0.42200228571891785, -0.008243818767368793, -8.332382276421413e-05, -1.1920928244535389e-07, -4.184158387943171e-05, -0.003632140113040805, -0.0011609011562541127, -9.07141511561349e-05, -0.00021979777375236154, -0.00038818930624984205, -1.065896987915039, -0.315780907869339, -0.004167087841778994, -0.004666865803301334, -0.00954858772456646, -0.02962358482182026, -0.0031815171241760254, -7.629103492945433e-05, -3.528532761265524e-05, -8.225102646974847e-05]}
+{"group_id": "math_071", "origin": "verified_reference", "answer": "1. **Define the problem and the sums:**\n   We are given five distinct real numbers \\(a_1, a_2, a_3, a_4, a_5\\). We need to consider all possible sums of the form \\(a_i + a_j\\) where \\(i \\neq j\\). There are \\(\\binom{5}{2} = 10\\) such sums since each pair \\((i, j)\\) is considered once.\n\n2. **Calculate the number of distinct sums:**\n   We need to find the smallest possible number of distinct sums, denoted as \\(m\\).\n\n3. **Example with specific values:**\n   Let's try the values \\(a_1 = 0, a_2 = 1, a_3 = 2, a_4 = 3, a_5 = 4\\). The sums are:\n   \\[\n   \\begin{aligned}\n   &0+1 = 1, \\\\\n   &0+2 = 2, \\\\\n   &0+3 = 3, \\\\\n   &0+4 = 4, \\\\\n   &1+2 = 3, \\\\\n   &1+3 = 4, \\\\\n   &1+4 = 5, \\\\\n   &2+3 = 5, \\\\\n   &2+4 = 6, \\\\\n   &3+4 = 7.\n   \\end{aligned}\n   \\]\n   The distinct sums are \\{1, 2, 3, 4, 5, 6, 7\\}, so \\(m = 7\\).\n\n4. **Check if \\(m = 6\\) is possible:**\n   Suppose \\(a_1 = 0\\) (without loss of generality, since adding a constant to each \\(a_i\\) does not change the number of distinct sums). Let the remaining numbers be \\(a, b, c, d\\) in increasing order. The sums are:\n   \\[\n   \\begin{aligned}\n   &0 + a = a, \\\\\n   &0 + b = b, \\\\\n   &0 + c = c, \\\\\n   &0 + d = d, \\\\\n   &a + b, \\\\\n   &a + c, \\\\\n   &a + d, \\\\\n   &b + c, \\\\\n   &b + d, \\\\\n   &c + d.\n   \\end{aligned}\n   \\]\n   For \\(m = 6\\), we need some sums to be equal. Assume \\(a + b = d\\). Then the sums are:\n   \\[\n   \\{a, b, c, d, a + c, a + d, b + c, b + d, c + d\\}.\n   \\]\n   Since \\(a + b = d\\), we have:\n   \\[\n   \\{a, b, c, d, a + c, d + a, b + c, b + d, c + d\\}.\n   \\]\n   We need to check if we can have only 6 distinct sums. If \\(a + c = b + d\\), then \\(a + c = b + (a + b) = a + 2b\\), so \\(c = 2b\\). This leads to contradictions as we need all numbers to be distinct.\n\n5. **Conclusion:**\n   After trying different configurations and ensuring all numbers are distinct, we find that the smallest possible value of \\(m\\) is indeed 7.\n\nThe final answer is \\(\\boxed{7}\\).", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 81, 79, 49161, 28, 81, 79, 49162, 28, 81, 79, 49163, 28, 81, 79, 49164, 20, 325, 4073, 1345, 2966, 30, 6689, 511, 24053, 282, 260, 910, 1885, 81, 79, 89, 1232, 253, 79, 90, 20, 837, 1885, 89, 28, 90, 3814, 254, 3814, 107, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 76, 24362, 284, 1885, 89, 3814, 662, 97, 544, 28422, 2959, 1885, 93, 20, 325, 260, 1230, 282, 4073, 2966, 1486, 623, 24053, 30, 1812, 314, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 47, 49154, 49158, 198, 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+{"group_id": "math_072", "origin": "budgie", "answer": "This is a problem of similarity transformation.\n\nSince $B$ is the midpoint of the side, we know $BC=BP\\cdot \\frac{1}{2}$\n\nThis means that $\\frac{BC}{BC}=\\frac{BP}{2}$ so $BP=2$\n\nTherefore, $A_1B_1=2\\cdot AB$, so $AB^2=4$\n\nTherefore, the area of triangle $ABC_1$ is $S_{ABC_1}=S_{ABC} \\cdot AB = 5\\cdot 2= \\boxed{10}$\n\nFinal answer:\n#### 10", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 10887, 253, 32857, 20378, 13293, 1885, 5431, 6998, 20, 351, 1557, 1885, 64, 28422, 1046, 8290, 2103, 1885, 5431, 20, 3070, 1885, 50, 20, 288, 1885, 49, 79, 49160, 20, 715, 338, 19573, 1400, 1311, 107, 5431, 109, 24814, 1400, 1311, 107, 10833, 79, 49160, 24362, 28, 965, 1885, 6190, 20, 3070, 1885, 51, 20, 288, 1885, 50, 79, 49160, 20, 715, 338, 19573, 1400, 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+{"group_id": "math_072", "origin": "budgie", "answer": "Let $x$ be the area of quadrilateral $A_1B_1C_1D_1$.\n\nWe can express $x$ in terms of $P$ and the area of quadrilateral $ABCD$:\n\n$P = \\frac{1}{4}AB \\cdot BC \\cdot CD \\cdot DA$\n\nSince $AB = DC$, $BC = CD$, $BC = \\frac{1}{2}BC$, $CD = \\frac{1}{2}CD$, $DA = DC$, we have:\n\n$x = \\frac{1}{4}(AB)(BC)(CD)(DA)$\n\nSince $AB = DC$ and $BC = CD$, we have:\n\n$x = \\frac{1}{4}(AB)(BC)(CD) \\cdot BC = \\frac{1}{4}(AB)(BC) \\cdot CD \\cdot BC$\n\nSince $BC = \\frac{1}{2}BC$, we have:\n\n$x = \\frac{1}{4}(AB)(BC) \\cdot CD \\cdot BC = \\frac{1}{4}(AB)(AC) \\cdot CD \\cdot BC$\n\nSince $AB = DC$ and $AC = AD$, we have:\n\n$x = \\frac{1}{4}(DC)(AC) \\cdot CD \\cdot AC = \\frac{1}{4}(DC)(AD) \\cdot CD \\cdot AD$\n\nSince $DC = DC$, we have:\n\n$x = \\frac{1}{4}(DC)(AD) \\cdot DC = \\frac{1}{4}(DC)(AD) \\cdot AD = \\frac{1}{4}(AD)(DC) \\cdot AD$\n\nSince $DA = DA$, we have:", "reward": -1, "advantage": -0.9805795218461241, "ids": 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+{"group_id": "math_072", "origin": "verified_reference", "answer": "Solution. We connect $A$ with $C_{1}$, $B$ with $D_{1}$, $C$ with $A_{1}$, and $D$ with $B_{1}$. Then,\n\n$$\n\\begin{aligned}\n& P_{\\triangle A B C}=P_{I}=P_{I I} \\\\\n& P_{\\triangle B C D}=P_{I I I}=P_{I V} \\\\\n& P_{\\triangle C D A}=P_{V}=P_{V I} \\\\\n& P_{\\triangle D A B}=P_{V I I}=P_{I I I}\n\\end{aligned}\n$$\n\nwhere\n\n$$\n\\begin{array}{ll}\nP_{I}=P_{\\Delta A_{1} C}, & P_{I I}=P_{\\Delta C A_{1} B_{1}} \\\\\nP_{I I I}=P_{\\Delta D C B_{1}}, & P_{I V}=P_{\\Delta C_{1} D B_{1}} \\\\\nP_{V}=P_{\\Delta C_{1} D A}, & P_{V I}=P_{\\Delta C_{1} D_{1} A} \\\\\nP_{I I I}=P_{\\Delta A D_{1} B}, & P_{V I I I}=P_{\\Delta B D_{1} A_{1}}\n\\end{array}\n$$\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_ea01241e897ed9d0948dg-02.jpg?height=507&width=641&top_left_y=708&top_left_x=838)\n\nWe have:\n\n$$\n\\begin{aligned}\nP_{\\Delta A B C}+P_{\\Delta B C D}+P_{\\Delta C D A}+P_{\\Delta D A B} & =P_{I}+P_{I I I}+P_{V}+P_{I I}= \\\\\n& =P_{I I}+P_{I V}+P_{V I}+P_{V I I I}=2 P \\\\\nP_{1}=P+\\left(P_{I}+P_{I I}+P_{I I I}+P_{I V}+P_{V}+P_{I I}\\right. & \\left.+P_{V I I}+P_{I I 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+{"group_id": "math_073", "origin": "budgie", "answer": "To find $f(x)$, we need to substitute $f(x) = 4x^3 - \\frac{5}{x^7} - 8 \\sqrt[6]{x^5} + 10$ into the given equation.\n\nLet's denote $g(x) = f(x) - 10$.\n\nNow, let's find $g(x)$ by substituting $f(x)$ into $g(x) = g(x-1) + 10$.\n\nWe have:\n\n$$g(x) = g(x-1) + 10$$\n\nNow, we can substitute $g(x)$ in terms of $f(x)$ into the given equation $f(x) = f(x-1) - 10$.\n\nLet's denote $x = g(x)$ and $x = x-g(x)$. Then, we have:\n\n$$g(x) = g(x-1) + 10$$\n\nSince $x = g(x)$, we have:\n\n$$x = g(x-1) + 10$$\n\nNow, we can substitute $g(x-1)$ for $x$ in the equation $g(x) = g(x-1) + 10$ to get:\n\n$$g(x) - 10 = g(x-1)$$\n\nNow, we can solve this equation for $g(x)$.\n\nNotice that $g(x) = g(x-1)$ for all $x$. Let's set $x = 0$ in the equation to find $g(0)$:\n\n$$g(0) = g(0-1", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 49166, 30, 1885, 86, 24, 104, 36637, 49163, 1792, 21990, 49162, 39949, 76, 18169, 107, 49164, 15009, 104, 21990, 49166, 17859, 29, 49167, 3814, 16166, 75, 49165, 36939, 104, 21990, 49164, 17859, 27, 49160, 49159, 28422, 1116, 49165, 30, 49166, 30, 1885, 86, 24, 104, 36637, 49163, 1792, 21990, 49162, 39949, 76, 18169, 107, 49164, 15009, 104, 21990, 49166, 17859, 29, 49167, 3814, 16166, 75, 49165, 36939, 104, 21990, 49164, 17859, 27, 49160, 49159, 28422, 49154, 49158, 198, 2068, 1042, 1885, 86, 24, 104, 25, 26265, 392, 737, 288, 13649, 1885, 86, 24, 104, 25, 446, 216, 49163, 104, 78, 49162, 731, 3814, 18169, 107, 49164, 15009, 104, 78, 49166, 109, 731, 216, 49167, 3814, 16166, 75, 49165, 36939, 104, 78, 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+{"group_id": "math_073", "origin": "budgie", "answer": "The given expression is $f(x) = 4 x^{3} - 5x^{-7} - 8 x^{5} \\sqrt[6]{x^{5}} + 10$.\n\nTo simplify $f(x)$, we can substitute the fractional exponents into the equation.\n\n\\[ f(x) = 4 x^{3} - 5 x^{-7} - 8 x^{5} \\sqrt[6]{x^{5}} + 10 \\]\n\nNotice that we can factor out $x^{-15}$ from the terms involving $x^{5}$.\n\n\\[ f(x) = x^{-15} (4 x^{3} - 5 x^{-7}) + 10 \\]\n\nNow, we can see that this is a hyperbolic equation.\n\nThe general solution is $f(x) = A(x - \\lambda)^{1/2}$, where $\\lambda = \\pm 1$.\n\nSubstituting $\\lambda = 1$ into the equation, we get $f(x) = A(x - 1)^{1/2}$.\n\nTo find the value of $A$, we can use the initial condition $f(0) = 1$.\n\n\\[ f(0) = 1 \\Rightarrow A \\cdot 0^{1/2} = 1 \\Rightarrow A = 1 \\]\n\nSo, the function $f(x)$ can be written as $f(x) = 1(x - 1)^{1/2}$.\n\nThe final answer is: $\\boxed{(x - 1)^{1/2}}$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 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+{"group_id": "math_074", "origin": "budgie", "answer": "Let's analyze the given inequalities and eliminate the variables $a, b, c, d$ step by step.\n\n1. $a<2b$\n2. $b<3c$\n3. $c<4d$\n\nWe can rewrite these inequalities as follows:\n\n* $a < 2b$\n* $b < 3c$\n* $c < 4d$\n\nNow, let's move to the product of these inequalities:\n\n$$a\\cdot b\\cdot c\\cdot d > (2b)(3c)(4d)$$\n\nNotice that $2b \\cdot 3c \\cdot 4d = 24abc$ is the product of the fourth powers of the three numbers, so:\n\n$$a\\cdot b\\cdot c\\cdot d > 24abc$$\n\nWe can rewrite this as:\n\n$$a < 24abc / 24 = abc$$\n\nSince the product of the three numbers is less than the product of their fifth powers, we can write:\n\n$$abc < (abc)^5$$\n\nTaking the fifth root of both sides:\n\n$$abc < abc$$\n\nThis inequality is always true.\n\nHowever, we know that the product of the numbers is greater than 130000. Since the product of the numbers is less than 130000, we can choose the values of $a, b, c, d$ such that the product of these three numbers is greater than 130000.\n\nLet's choose $a = 130000 - (3c)^5$ and $b = 130000 - 3c^5$.\n\nThen, we", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 2959, 1885, 81, 28, 278, 28, 265, 28, 287, 20, 325, 23313, 28, 284, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 28422, 10887, 1885, 84, 44, 49160, 49159, 49159, 26265, 965, 260, 5428, 1636, 1685, 282, 1885, 81, 20, 314, 365, 15165, 198, 24, 49, 25, 216, 49161, 49162, 49165, 49166, 198, 24, 50, 25, 216, 49161, 49162, 49166, 49164, 198, 24, 51, 25, 216, 49161, 49162, 49168, 49160, 198, 24, 52, 25, 216, 49161, 49162, 49168, 49168, 49154, 49158, 198, 4239, 506, 6524, 260, 1836, 21544, 284, 8534, 260, 6031, 1885, 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+{"group_id": "math_074", "origin": "budgie", "answer": "Considering the inequality $a<2b, b<3c, c<4d$, we can write the inequalities for each variable:\n\n$$a<2b, b<3c, c<4d$$\n\nSince $d<100$, we know that $0<d<100$.\n\nFor $c$, the inequality becomes $c<4d$.\n\nFor $b$, the second inequality becomes $b<3c$.\n\nFor $a$, the first inequality becomes $a<2b$.\n\nGiven that $a<2b, b<3c, c<4d$, we can simplify these inequalities to find the range of possible values for $a$.\n\nSince $0<d<100$, $d$ can take values from 0 to 99.\n\nFor $c$, the lowest inequality is $c<4d$. Since $d<100$, this means $c<4*100=400$.\n\nFor $b$, the second inequality becomes $b<3c$. This implies that $b<3*400=1200$.\n\nFor $a$, the first inequality becomes $a<2b$. Given that $0<d<100$, this means $a<2*400=800$.\n\nTo maximize $a$, we need to minimize $b$ and $c$ while keeping $a<800$. The smallest possible value for $a$ is $800$.\n\nTherefore, the maximum possible value of $a$ is 800.\n\nAnswer:\n#### 800", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 2959, 1885, 81, 28, 278, 28, 265, 28, 287, 20, 325, 23313, 28, 284, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 28422, 10887, 1885, 84, 44, 49160, 49159, 49159, 26265, 965, 260, 5428, 1636, 1685, 282, 1885, 81, 20, 314, 365, 15165, 198, 24, 49, 25, 216, 49161, 49162, 49165, 49166, 198, 24, 50, 25, 216, 49161, 49162, 49166, 49164, 198, 24, 51, 25, 216, 49161, 49162, 49168, 49160, 198, 24, 52, 25, 216, 49161, 49162, 49168, 49168, 49154, 49158, 198, 39539, 260, 9724, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 26265, 392, 416, 2965, 260, 21544, 327, 971, 5811, 42, 198, 198, 9212, 81, 44, 49161, 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+{"group_id": "math_074", "origin": "verified_reference", "answer": "$-.1 .(\\mathrm{A})$.\nThen, if $d=99$, we have $c<4 \\times 99=395$. Taking $c=395$, we get $b<3 \\times 395=1185$. Taking $b=1184$, we get $a<2368$. Therefore, the greatest integer for $a$ is 2367.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 2959, 1885, 81, 28, 278, 28, 265, 28, 287, 20, 325, 23313, 28, 284, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 28422, 10887, 1885, 84, 44, 49160, 49159, 49159, 26265, 965, 260, 5428, 1636, 1685, 282, 1885, 81, 20, 314, 365, 15165, 198, 24, 49, 25, 216, 49161, 49162, 49165, 49166, 198, 24, 50, 25, 216, 49161, 49162, 49166, 49164, 198, 24, 51, 25, 216, 49161, 49162, 49168, 49160, 198, 24, 52, 25, 216, 49161, 49162, 49168, 49168, 49154, 49158, 198, 20, 43526, 49160, 1673, 19407, 8898, 5167, 107, 49, 8148, 28422, 198, 10039, 28, 585, 1885, 84, 45, 49168, 49168, 26265, 392, 457, 1885, 83, 44, 49163, 3814, 14602, 216, 49168, 49168, 45, 49162, 49168, 49164, 28422, 14985, 1885, 83, 45, 49162, 49168, 49164, 26265, 392, 820, 1885, 82, 44, 49162, 3814, 14602, 216, 49162, 49168, 49164, 45, 49160, 49160, 49167, 49164, 28422, 14985, 1885, 82, 45, 49160, 49160, 49167, 49163, 26265, 392, 820, 1885, 81, 44, 49161, 49162, 49165, 49167, 28422, 4882, 28, 260, 6250, 15328, 327, 1885, 81, 20, 314, 216, 49161, 49162, 49165, 49166, 30, 49154], "prefix_len": 117, "old_logprobs": [-6.425846099853516, -16.364072799682617, -2.287951946258545, -8.431763648986816, -8.98987102508545, -5.425016403198242, -0.5020139813423157, -0.0022081772331148386, -4.566595554351807, -2.083723783493042, -2.1881768703460693, -0.9943889379501343, -8.527412414550781, -1.5335103273391724, -4.344038963317871, -0.3912464678287506, -1.526413917541504, -2.8187415599823, -3.6707684993743896, -0.1920979917049408, -1.0199874639511108, -2.6167168617248535, -0.4177727699279785, -0.23391488194465637, -2.704787015914917, -2.0005509853363037, -1.717228889465332, -1.7626054286956787, -0.09881271421909332, -0.1463354527950287, -0.0624006912112236, -0.000740968796890229, -0.8555609583854675, -0.17192263901233673, -0.013412484899163246, -16.31250762939453, -0.8968100547790527, -6.658372402191162, -0.5423813462257385, -1.5255768299102783, -0.2701251804828644, -0.5963861346244812, -0.18563984334468842, -0.5782619118690491, -0.2764997184276581, -0.48854631185531616, -0.8883193731307983, -0.1381218135356903, -2.384387254714966, -0.23763535916805267, -0.5992197394371033, -0.4430578649044037, -0.0012672259472310543, -0.009162403643131256, -0.02776312083005905, -0.0010917185572907329, -0.00099323526956141, -0.14730484783649445, -0.007869075983762741, -0.024035966023802757, -0.09253354370594025, -0.0010805251076817513, -0.3366527259349823, -1.168932557106018, -0.005158448126167059, -0.0667290985584259, -0.01810126006603241, -0.014152283780276775, -0.0058921691961586475, -0.00818600319325924, -4.64556884765625, -0.18131357431411743, -0.05474394932389259, -0.9498416185379028, -0.03900183364748955, -0.21976259350776672, -0.11610379815101624, -0.3498428761959076, -4.435967445373535, -3.406467914581299, -1.9928441047668457, -0.5053212642669678, -2.519426107406616, -0.029143076390028, -0.2695752680301666, -5.4710001945495605, -0.695104718208313, -5.377488613128662, -0.2628493905067444, -0.04393276572227478, -0.03003700263798237, -0.10434316843748093, -0.3839997947216034, -0.02400093711912632, -0.0019070786656811833, -0.5576317310333252, -0.4087328612804413, -0.13717380166053772, -5.950602054595947]}
+{"group_id": "math_075", "origin": "budgie", "answer": "32. Let $f(x)=16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048$. When $x=0$, $f(0)=0$.\n\nAnswer:\n#### 0", "reward": 1.0, "advantage": 0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 49163, 49161, 30, 19573, 8842, 24, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 76, 2771, 25, 3814, 92, 87, 3814, 8842, 24, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 76, 2771, 36637, 49159, 28422, 49154, 49158, 198, 49162, 49161, 30, 2959, 1885, 86, 24, 104, 36637, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 28422, 1550, 1885, 104, 45, 49159, 26265, 1885, 86, 24, 49159, 36637, 49159, 28422, 198, 198, 21350, 42, 198, 1229, 216, 49159, 49154], "prefix_len": 88, "old_logprobs": [-3.4670050144195557, -2.1015844345092773, -1.198379635810852, -3.9908227920532227, -0.728891134262085, -2.5934014320373535, -0.05405392125248909, -0.06436410546302795, -0.3244601786136627, -1.6596819162368774, -0.024508466944098473, -1.2133287191390991, -0.07153483480215073, -0.0489482581615448, -0.006597996223717928, -0.010280866175889969, -0.05378735437989235, -1.0601204633712769, -0.03440750017762184, -0.0028884378261864185, -0.08698343485593796, -0.019465375691652298, -0.014873212203383446, -0.000764792668633163, -0.0017982041463255882, -0.00010775939153973013, -0.0003587556129787117, -0.004680866841226816, -0.001304609584622085, -1.2874520507466514e-05, -0.009680116549134254, -0.06938262283802032, -0.00025829317746683955, -0.01409986149519682, -0.0011800715001299977, -0.00047803416964598, -1.0689234733581543, -4.980095863342285, -0.08800455182790756, -0.3799773156642914, -0.6907307505607605, -1.1965309381484985, -0.4314729571342468, -0.18463991582393646, -0.009664060547947884, -0.04461681842803955, -0.5420650839805603, -0.2076900601387024, -0.36814284324645996, -0.45088624954223633, -0.6046979427337646, -0.10241943597793579, -4.209035396575928, -0.043070707470178604, -0.2714500427246094, -0.1879296451807022, -0.10618666559457779, -0.18912285566329956, -0.026781858876347542]}
+{"group_id": "math_075", "origin": "budgie", "answer": "1024.\n\nFinal answer:\n#### 1024", "reward": -0.5, "advantage": -1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 49163, 49161, 30, 19573, 8842, 24, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 76, 2771, 25, 3814, 92, 87, 3814, 8842, 24, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 76, 2771, 36637, 49159, 28422, 49154, 49158, 198, 49160, 49159, 49161, 49163, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49159, 49161, 49163, 49154], "prefix_len": 88, "old_logprobs": [-1.2940069437026978, -1.8934171199798584, -1.748826503753662, -0.637182354927063, -0.7875847220420837, -2.3630528450012207, -0.038016509264707565, -0.4355299174785614, -0.002022841712459922, -0.002695143921300769, -0.16291376948356628, -0.005939097609370947, -0.038427162915468216, -0.24012304842472076, -0.0016104121459648013, -0.001712642959319055, -0.0010882653295993805, -0.005151806864887476]}
+{"group_id": "math_075", "origin": "verified_reference", "answer": "Solution.\n\nDomain: $x^{3}+2 x+1>0$.\n\nFrom the condition $16 \\cdot 5^{2 x-1}-2^{x-1}-0.048=0$ or $\\lg \\left(x^{3}+2 x+1\\right)=0$. Rewrite the first equation as\n\n$\\frac{16}{5} \\cdot 5^{2 x}-\\frac{2}{5} \\cdot 5^{x}-0.048=0 \\Leftrightarrow 16 \\cdot 5^{2 x}-2 \\cdot 5^{x}-0.24=0$.\n\nSolving this equation as a quadratic in terms of $5^{x}$, we get $5^{x}=-\\frac{3}{40}$ (no solutions), or $5^{x}=5^{-1} \\Leftrightarrow x_{1}=-1$ (does not satisfy the domain). From the second equation, we have\n\n$x^{3}+2 x+1=1 \\Leftrightarrow x^{3}+2 x=0 \\Leftrightarrow x\\left(x^{2}+2\\right)=0, x_{3}=0, x^{2}+2 \\neq 0$.\n\nAnswer: 0.", "reward": 1.0, "advantage": 0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 49163, 49161, 30, 19573, 8842, 24, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 76, 2771, 25, 3814, 92, 87, 3814, 8842, 24, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 76, 2771, 36637, 49159, 28422, 49154, 49158, 198, 37500, 30, 198, 198, 34804, 42, 1885, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 46, 49159, 28422, 198, 198, 5212, 260, 2619, 1885, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 21990, 104, 29, 49160, 39949, 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+{"group_id": "math_077", "origin": "budgie", "answer": "First, we notice that the given equation is always true regardless of the values of $x$ and $y$. However, the question asks us to find the minimum value of the function $u = y^{2} - 2y + x^{2} + 4x$ when the inequality $2x + y \\geqslant 1$ holds. To do this, we can rewrite the given equation as follows:\n\n$$u = (y^{2} - 2y + x^{2} + 4x) + (2y - 2x) = (y + x)^{2} - x^{2}$$\n\nNow, let's use the Cauchy-Schwarz inequality to find the minimum value of $u$:\n\n$$(y + x)^{2} \\leq 1 + 1 = 2$$\n\nTherefore, the minimum value of $u$ is:\n\n$$u = (y + x)^{2} - x^{2} = 2 - x^{2}$$\n\nThe minimum value of the function $u$ is 2 - x².\n\nFinal answer:\n#### 2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2464, 30, 365, 49161, 49164, 2876, 25, 1094, 1885, 49161, 1792, 27, 105, 3814, 361, 97, 7483, 403, 216, 49160, 28422, 8061, 288, 1042, 260, 5869, 1685, 282, 260, 1517, 1885, 101, 45, 105, 21990, 49161, 39949, 49161, 329, 20, 1885, 27, 104, 21990, 49161, 109, 27, 49163, 1792, 28422, 49154, 49158, 198, 5345, 28, 392, 4887, 338, 260, 1836, 8345, 314, 1811, 2629, 7142, 282, 260, 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+{"group_id": "math_077", "origin": "budgie", "answer": "25. (25 points)\n\n**Analysis**\n\nThis problem can be solved by completing the square for both the $y$-terms and the $x$-terms.\n\nFirst, rewrite the function to express it in vertex form, then complete the square for the $y$-terms.\n\n**Step-by-step Solution**\n\nThe function $y^2 - 2y$ can be rewritten as $(y - 1)^2 - 1$.\n\nThe function $x^2 + 4x$ can be rewritten as $(x + 2)^2 - 4$.\n\nNow, we complete the square for the $y$-terms:\n\n$$u = (y - 1)^2 - 1$$\n\n$$u = (y + (1) - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1 - 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\nNow, we write the function in vertex form by completing the square:\n\n$$u = (y + 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\nThe given inequality $2x + y \\geqslant 1$ can be written as:\n\n$$2x + y -", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 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+{"group_id": "math_077", "origin": "verified_reference", "answer": "$\\therefore$ From $u=y^{2}-2 y+x^{2}+4 x$ completing the square, we get\n$$\n(x+2)^{2}+(y-1)^{2}=u+5 \\text {. }\n$$\n\nSince $(x+2)^{2}+(y-1)^{2}$ can be regarded as the square of the distance from point $P(x, y)$ to the fixed point $(-2,1)$. The constraint $2 x+y \\geqslant 1$ indicates that point $P$ is in the region $G$ which is the right upper area of the line $l: 2 x+y=1$. Therefore, the problem is transformed into finding the shortest distance from point $A(-2,1)$ to the region $G$.\nFrom the right figure, it can be seen that the distance from point $A$ to\n\nline $I$ is the minimum value of the distance from point $A$ to any point in region $G$.\n$$\n\\begin{aligned}\nd & =\\frac{|2 \\cdot(-2)+1-1|}{\\sqrt{2^{2}+1^{2}}} \\\\\n& =\\frac{4}{\\sqrt{5}}, \\\\\n& \\therefore d^{2}=\\frac{16}{5} \\text {, i.e., } u+5=\\frac{16}{5}\n\\end{aligned}\n$$\n$\\therefore u=-\\frac{9}{5}$. 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+{"group_id": "math_078", "origin": "budgie", "answer": "To solve this problem, we need to find the maximum number of red pencils (80 kopecks) and blue pencils (50 kopecks) that can be bought, while ensuring that the number of red pencils is as few as possible, but the number of blue pencils is as large as possible.\n\nSince the total cost of a red pencil is 27 kopecks, a blue pencil costs 23 kopecks, we will first find the maximum number of red pencils that can be bought with the given budget. This can be done by dividing the total budget by the cost of a red pencil.\n\n$$Maximum\\, red\\, pencils = Total\\, budget / Cost\\, of\\, a\\, red\\, pencil$$\n\nNow, plug in the given values for the total budget and cost of a red pencil to calculate the maximum number of red pencils.\n\n$$Maximum\\, red\\, pencils = 40 / 27$$\n\n$$Maximum\\, red\\, pencils = 1\\, (\\text{exactly 1 red pencil})$$\n\nSince the number of red pencils should be as few as possible, we should use the smallest possible value for the number of red pencils. Therefore, we should consider the case where the number of red pencils is equal to the number of red pencils.\n\n$$Maximum\\, red\\, pencils = 1\\, (\\text{one red pencil})$$\n\nNow that we have the maximum number of red pencils, we can calculate the number of blue pencils that can be bought. The cost of a blue pencil is 23 kopecks, so we can find the maximum number of blue pencils by dividing the total budget by the cost of a blue pencil.\n\n$$Maximum\\, blue\\, pencils = Total\\,", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 49163, 49159, 330, 2382, 15779, 3499, 216, 49161, 49166, 501, 371, 41907, 28, 253, 4461, 582, 731, 216, 49161, 49162, 501, 371, 41907, 30, 2838, 540, 670, 216, 49168, 6460, 735, 216, 49163, 49159, 501, 371, 41907, 416, 325, 4957, 335, 12217, 25487, 30, 657, 314, 2371, 288, 7131, 260, 5428, 1636, 2719, 1230, 282, 2382, 284, 4461, 25487, 30, 1814, 260, 1142, 655, 28, 260, 1230, 282, 2382, 25487, 868, 325, 347, 1443, 347, 1636, 28, 564, 260, 1230, 282, 4461, 25487, 868, 441, 9094, 429, 260, 1230, 282, 2382, 25487, 411, 540, 670, 216, 49160, 49159, 30, 1073, 800, 2382, 284, 638, 800, 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+{"group_id": "math_078", "origin": "budgie", "answer": "The problem can be solved by how many red pencils and blue pencils should be purchased under the conditions of the problem.\n\nFirst, we should consider the total cost. The cost of a red pencil is 27 kopecks and the cost of a blue one is 23 kopecks.\n\nThe number of red pencils purchased is max(27, 23) = 49.\n\nSince the number of red pencils should be as few as possible, the number of blue pencils purchased is 27 - 10 = 17.\n\nTherefore, under the given conditions, we should purchase 49 red pencils and 17 blue pencils.\n\nFinal answer:\n#### 49", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 49163, 49159, 330, 2382, 15779, 3499, 216, 49161, 49166, 501, 371, 41907, 28, 253, 4461, 582, 731, 216, 49161, 49162, 501, 371, 41907, 30, 2838, 540, 670, 216, 49168, 6460, 735, 216, 49163, 49159, 501, 371, 41907, 416, 325, 4957, 335, 12217, 25487, 30, 657, 314, 2371, 288, 7131, 260, 5428, 1636, 2719, 1230, 282, 2382, 284, 4461, 25487, 30, 1814, 260, 1142, 655, 28, 260, 1230, 282, 2382, 25487, 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+{"group_id": "math_078", "origin": "verified_reference", "answer": "3.40 Let $x$ red and $y$ blue pencils be bought. According to the conditions, $27 x+23 y \\leqslant 940$ and $y-x \\leqslant 10$. Let's construct the lines\n\n![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-108.jpg?height=597&width=625&top_left_y=113&top_left_x=287)\n\n\\[\n\\begin{aligned}\n& 27 x+23 y=940 \\\\\n& y-x=10\n\\end{aligned}\n\\]\n\nFrom Fig. 3.8, it is clear that these lines intersect at point $A$, the coordinates of which satisfy equations (1) and (2), and at this point, the maximum possible sum $x+y$ is achieved. Solving the system (1), (2) and considering that the numbers $x$ and $y$ are natural, we get $x=14, y=24$.\n\nAnswer: 14 red and 24 blue pencils.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 49163, 49159, 330, 2382, 15779, 3499, 216, 49161, 49166, 501, 371, 41907, 28, 253, 4461, 582, 731, 216, 49161, 49162, 501, 371, 41907, 30, 2838, 540, 670, 216, 49168, 6460, 735, 216, 49163, 49159, 501, 371, 41907, 416, 325, 4957, 335, 12217, 25487, 30, 657, 314, 2371, 288, 7131, 260, 5428, 1636, 2719, 1230, 282, 2382, 284, 4461, 25487, 30, 1814, 260, 1142, 655, 28, 260, 1230, 282, 2382, 25487, 868, 325, 347, 1443, 347, 1636, 28, 564, 260, 1230, 282, 4461, 25487, 868, 441, 9094, 429, 260, 1230, 282, 2382, 25487, 411, 540, 670, 216, 49160, 49159, 30, 1073, 800, 2382, 284, 638, 800, 4461, 25487, 868, 325, 10610, 656, 260, 1836, 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+{"group_id": "math_079", "origin": "budgie", "answer": "And, we have $x^{2} y z=1$.\n\nThe solution to the equation is $x=\\frac{1}{\\frac{1}{y}+1+\\frac{1}{z}}=\\frac{y z+1}{y+z}.$\n\nSince $y+z \\leqslant 2$, $y+z \\leqslant 2$ is not possible, but we may assume that $y+z=\\frac{1}{2}$.\n\nSo, $x=\\frac{y z+1}{\\frac{1}{2}}=\\boxed{4y z}.$", "reward": -1, "advantage": -1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 15423, 1885, 49159, 3814, 290, 97, 7483, 403, 1792, 28, 329, 28, 1892, 3814, 290, 97, 7483, 403, 216, 49160, 26265, 5482, 260, 8345, 42, 198, 9212, 76, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 42639, 18169, 107, 105, 15009, 49160, 27, 106, 27, 104, 329, 109, 42639, 18169, 107, 106, 15009, 49160, 27, 104, 27, 105, 1892, 109, 24814, 18169, 107, 49162, 15009, 104, 27, 105, 27, 106, 109, 1673, 9212, 49154, 49158, 198, 3528, 28, 392, 457, 1885, 104, 21990, 49161, 109, 329, 1892, 45, 49160, 28422, 198, 198, 504, 3564, 288, 260, 8345, 314, 1885, 104, 24814, 18169, 107, 49160, 41322, 18169, 107, 49160, 15009, 105, 109, 27, 49160, 42639, 18169, 107, 49160, 15009, 106, 17859, 24814, 18169, 107, 105, 1892, 27, 49160, 15009, 105, 27, 106, 19181, 20, 198, 198, 7230, 1885, 105, 27, 106, 3814, 290, 97, 7483, 403, 216, 49161, 26265, 1885, 105, 27, 106, 3814, 290, 97, 7483, 403, 216, 49161, 20, 314, 441, 1636, 28, 564, 392, 654, 7635, 338, 1885, 105, 27, 106, 24814, 18169, 107, 49160, 15009, 49161, 24362, 30, 198, 198, 2931, 28, 1885, 104, 24814, 18169, 107, 105, 1892, 27, 49160, 41322, 18169, 107, 49160, 15009, 49161, 17859, 24814, 4810, 277, 107, 49163, 105, 1892, 19181, 20, 49154], "prefix_len": 106, "old_logprobs": [-8.75673770904541, -3.4261460304260254, -3.5458288192749023, -1.56813383102417, -2.2374930381774902, -0.8862387537956238, -5.024696350097656, -0.13346561789512634, -0.28033146262168884, -2.6508588790893555, -1.348811388015747, -1.567143201828003, -0.9688009023666382, -1.445141077041626, -0.6645085215568542, -0.14603088796138763, -2.736523389816284, -3.135577917098999, -1.5819978713989258, -0.25776031613349915, -0.6321702003479004, -0.11931177228689194, -2.076040267944336, -0.23133166134357452, -2.4882450103759766, -0.5926836729049683, -0.028871523216366768, -0.07371044158935547, -0.8454666137695312, -2.22076416015625, -0.006205813027918339, -0.4527757167816162, -0.12459518760442734, -0.6619689464569092, -1.3236027956008911, -1.4375725984573364, -0.8798022866249084, -1.126152515411377, -0.00921650230884552, -0.002704060636460781, -0.071990966796875, -0.12959828972816467, -0.06494183093309402, -0.05719922110438347, -1.6074366569519043, -0.11729633063077927, -0.021861523389816284, -0.8066228628158569, -2.24392032623291, -3.0859782695770264, -0.4857226014137268, -0.08395413309335709, -0.19864605367183685, -0.15390589833259583, -0.07328566908836365, -3.3746237754821777, -0.19239552319049835, -0.2791374623775482, -0.02929798513650894, -2.509556293487549, -0.18853649497032166, -1.2912299633026123, -2.2996022701263428, -0.005353045184165239, -0.6999714374542236, -0.9143136739730835, -0.05695263668894768, -0.09390829503536224, -5.757642793469131e-05, -0.23085282742977142, -1.0138683319091797, -0.9492963552474976, -2.5884132385253906, -0.15097570419311523, -1.6720058917999268, -0.025201397016644478, -0.42641326785087585, -0.4316529631614685, -0.00020680672605521977, -0.008617476560175419, -0.00019596086349338293, -0.5309330821037292, -0.37634778022766113, -1.3125964403152466, -1.7722691297531128, -2.890915870666504, -0.28401246666908264, -0.4846906065940857, -5.123191833496094, -1.8105454444885254, -3.891129493713379, -0.6775973439216614, -0.5891727209091187, -0.10989126563072205, -0.06442781537771225, -0.1410648077726364, -0.0010850501712411642, -3.7402753829956055, -0.0922781229019165, -0.002067572670057416, -0.25736671686172485, -0.08732399344444275, -0.34467795491218567, -0.20383769273757935, -0.459760457277298, -0.18379949033260345, -0.01901528425514698, -3.987239360809326, -0.08736921846866608, -1.1129399538040161, -0.6243656873703003, -0.1474321037530899, -0.05326802283525467, -0.030246825888752937, -1.907172679901123, -0.1035759449005127, -0.19997677206993103, -0.009429795667529106, -1.0584288835525513, -0.01390296220779419, -0.0010993395699188113, -0.11074105650186539, -0.006311484612524509, -0.32092806696891785, -0.41086775064468384, -0.2852543592453003, -0.4323851466178894, -6.794906312279636e-06, -0.18848219513893127, -1.3418103456497192, -0.5676193237304688, -0.7976429462432861, -2.1800999641418457, -0.0012915135594084859, -0.5573234558105469]}
+{"group_id": "math_079", "origin": "budgie", "answer": "Since $0 \\leqslant x, y, z \\leqslant 1$, we have $0 \\leqslant x, y, z < 1,$ and $\\frac{x}{1+y+z x} < \\frac{3}{x+y+z}.$  Therefore,\n$$\\frac{x}{1+y+z x} + \\frac{y}{1+z+x y} + \\frac{z}{1+x+y z} > 0,$$so\n$$\\frac{x}{1+y+z x} + \\frac{y}{1+z+x y} + \\frac{z}{1+x+y z} < -\\frac{3}{x+y+z}.$$This contradicts the original equation, so there is no solution.\n\nFinal answer:\n#### 1", "reward": 0.5, "advantage": 0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 15423, 1885, 49159, 3814, 290, 97, 7483, 403, 1792, 28, 329, 28, 1892, 3814, 290, 97, 7483, 403, 216, 49160, 26265, 5482, 260, 8345, 42, 198, 9212, 76, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 42639, 18169, 107, 105, 15009, 49160, 27, 106, 27, 104, 329, 109, 42639, 18169, 107, 106, 15009, 49160, 27, 104, 27, 105, 1892, 109, 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+{"group_id": "math_079", "origin": "verified_reference", "answer": "3. It is not difficult to prove: $\\frac{x}{1+y+z x} \\leqslant \\frac{1}{x+y+z}$, $\\frac{y}{1+z+x y} \\leqslant \\frac{1}{x+y+z}$, $\\frac{z}{1+x+y z} \\leqslant \\frac{1}{x+y+z}$. Therefore, if the equality holds, it is easy to get $x=y=z=1$.\n\nNote: This problem can also be solved as follows: First prove $\\frac{x}{1+y+z x} \\leqslant \\frac{x}{x+y+z}$, etc., so we can get $x+y+z \\geqslant 3$, hence $x=y=z=1$.", "reward": 1.0, "advantage": 0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 15423, 1885, 49159, 3814, 290, 97, 7483, 403, 1792, 28, 329, 28, 1892, 3814, 290, 97, 7483, 403, 216, 49160, 26265, 5482, 260, 8345, 42, 198, 9212, 76, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 42639, 18169, 107, 105, 15009, 49160, 27, 106, 27, 104, 329, 109, 42639, 18169, 107, 106, 15009, 49160, 27, 104, 27, 105, 1892, 109, 24814, 18169, 107, 49162, 15009, 104, 27, 105, 27, 106, 109, 1673, 9212, 49154, 49158, 198, 49162, 30, 657, 314, 441, 1990, 288, 6824, 42, 19573, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 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+{"group_id": "math_080", "origin": "budgie", "answer": "We know that $A_1A_2$ and $A_3A_4$ intersect at an acute angle of $\\theta$. This means that $\\tan \\theta > 0$. \n\nGiven the coordinates of $A_1$, $A_2$, $A_3$, and $A_4$, we can write the equations of the lines as follows:\n\n$$y = a_1^2x, \\, y = a_2^2x, \\, y = a_3^2x, \\, y = a_4^2x$$\n\nSolving for $x$ in each line equation, we get:\n\n$$x = \\frac{a_1^2}{a_1^2 - a_1^2a_1}, \\, x = \\frac{a_2^2}{a_2^2 - a_2^2a_2}, \\, x = \\frac{a_3^2}{a_3^2 - a_3^2a_3}, \\, x = \\frac{a_4^2}{a_4^2 - a_4^2a_4}$$\n\nSimplifying each equation, we get:\n\n$$x = \\frac{a_1^2}{a_1^2 - a_1^2a_1}, \\, x = \\frac{a_2^2}{a_2^2 - a_2^2a_2}, \\, x = \\frac{a_3^2}{a_3^2 - a_3^2a_3}, \\, x = \\frac{a_4^2}{a_4^2 - a_4^2a_4}$$\n\nSimplifying further", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 253, 79, 49161, 28, 253, 79, 49162, 28, 253, 79, 49163, 20, 325, 23313, 351, 4073, 8435, 2396, 30, 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+{"group_id": "math_080", "origin": "budgie", "answer": "To find the maximum possible value for $\\tan \\theta$, we need to consider the properties of the intersection point $A_3$. Since $A_3A_4$ is a vertical line, its slope is undefined. Therefore, the slope of $A_3A_4$ is undefined.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. If we take the derivative of $A_3A_4$ with respect to $\\theta$ and set it to zero, we get $A_3A_4 = A_3A_3$. This means that the slope of $A_3A_4$ is undefined.\n\nSince the slope of $A_1A_2$ is infinite, the slope of $A_1A_2$ is also infinite. Therefore, there is no maximum possible value for $\\tan \\theta$.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. Since $A_1A_2$ is a perpendicular line, the maximum possible value of $\\tan \\theta$ is also infinite.\n\nBut we can still find the maximum possible value of $\\tan \\theta$. Since $A_2$ is a point on the $y$-axis, we can see that $\\tan \\theta \\ge 0$.\n\nTherefore, the maximum possible value of $\\tan \\theta$ is infinite.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. Since $A_1$ is a point on the $x$-axis, we can see that $\\tan \\theta$ is approximately $\\tan \\theta \\approx 99.246497$.\n\nTherefore, the maximum possible value of $\\tan \\theta$ is approximately $99.24649", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 253, 79, 49161, 28, 253, 79, 49162, 28, 253, 79, 49163, 20, 325, 23313, 351, 4073, 8435, 2396, 30, 533, 260, 13033, 8303, 28, 1303, 1885, 49, 79, 49160, 8615, 81, 79, 49160, 28, 81, 79, 49160, 78, 49161, 25, 26265, 1885, 49, 79, 49161, 8615, 81, 79, 49161, 28, 81, 79, 49161, 78, 49161, 25, 26265, 1885, 49, 79, 49162, 8615, 81, 79, 49162, 28, 81, 79, 49162, 78, 49161, 38224, 284, 1885, 49, 79, 49163, 8615, 81, 79, 49163, 28, 81, 79, 49163, 78, 49161, 25, 28422, 45132, 338, 3204, 1885, 49, 79, 49160, 49, 79, 49161, 20, 284, 1885, 49, 79, 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+{"group_id": "math_080", "origin": "verified_reference", "answer": "1. **Given Information and Setup:**\n   - We have integers \\(a_1, a_2, a_3, a_4\\) with distinct absolute values.\n   - Points in the coordinate plane are \\(A_1 = (a_1, a_1^2)\\), \\(A_2 = (a_2, a_2^2)\\), \\(A_3 = (a_3, a_3^2)\\), and \\(A_4 = (a_4, a_4^2)\\).\n   - Lines \\(A_1A_2\\) and \\(A_3A_4\\) intersect on the \\(y\\)-axis at an acute angle \\(\\theta\\).\n   - We need to find the maximum possible value for \\(\\tan \\theta\\) expressed as \\(\\frac{m}{n}\\) for relatively prime positive integers \\(m\\) and \\(n\\), and then find \\(100m + n\\).\n\n2. **Slope Calculation:**\n   - The slope of line \\(A_1A_2\\) is:\n     \\[\n     m_1 = \\frac{a_2^2 - a_1^2}{a_2 - a_1} = a_2 + a_1\n     \\]\n   - The slope of line \\(A_3A_4\\) is:\n     \\[\n     m_2 = \\frac{a_4^2 - a_3^2}{a_4 - a_3} = a_4 + a_3\n     \\]\n\n3. **Intersection on the \\(y\\)-axis:**\n   - Since the lines intersect on the \\(y\\)-axis, their \\(x\\)-coordinates at the intersection point are zero. Thus, the \\(y\\)-intercepts of the lines are:\n     \\[\n     c_1 = a_1^2 \\quad \\text{and} \\quad c_2 = a_3^2\n     \\]\n\n4. **Angle Between Lines:**\n   - The tangent of the angle \\(\\theta\\) between the two lines is given by:\n     \\[\n     \\tan \\theta = \\left| \\frac{m_1 - m_2}{1 + m_1 m_2} \\right| = \\left| \\frac{(a_1 + a_2) - (a_3 + a_4)}{1 + (a_1 + a_2)(a_3 + a_4)} \\right|\n     \\]\n\n5. **Maximizing \\(\\tan \\theta\\):**\n   - Let \\(x = a_1 + a_2\\) and \\(y = a_3 + a_4\\). We need to maximize:\n     \\[\n     f(x, y) = \\left| \\frac{x - y}{1 + xy} \\right|\n     \\]\n\n6. **Case Analysis:**\n   - **Case 1: \\(x, y > 0\\)**\n     - Here, \\(x, y \\geq 1\\) since they are integers. In this case, \\(|f(x, y)| < 1\\), which cannot exceed our current maximum.\n   - **Case 2: \\(x > 0, y < 0\\)**\n     - The numerator is positive, and the denominator is negative, so \\(|f(x, y)|\\) cannot be our maximum.\n   - **Case 3: \\(x < 0, y > 0\\)**\n     - Let \\(x = -u\\) and \\(y = v\\). We want to maximize:\n       \\[\n       \\frac{u + v}{uv - 1}\n       \\]\n     - We need to check when:\n       \\[\n       \\frac{u + v}{uv - 1} > \\frac{3}{2} \\implies 3(uv - 1) < 2(u + v) \\implies 3uv - 3 < 2u + 2v \\implies 3uv - 2u - 2v < 3\n       \\]\n     - If \\(u, v > 1\\), then it's impossible because \\(3uv - 2u - 2v \\geq 3\\) is false. Thus, we can let \\(v = 1\\) and need \\(u < 5\\).\n\n7. **Checking Specific Values:**\n   - For \\(u = 4\\) and \\(v = 1\\):\n     - We have \\(a_1 = -6, a_2 = 2, a_3 = 4, a_4 = -3\\).\n     - This gives:\n       \\[\n       \\tan \\theta = \\left| \\frac{(-6 + 2) - (4 - 3)}{1 + (-6 + 2)(4 - 3)} \\right| = \\left| \\frac{-4 - 1}{1 - 4} \\right| = \\left| \\frac{-5}{-3} \\right| = \\frac{5}{3}\n       \\]\n\nThe final answer is \\(\\boxed{503}\\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 253, 79, 49161, 28, 253, 79, 49162, 28, 253, 79, 49163, 20, 325, 23313, 351, 4073, 8435, 2396, 30, 533, 260, 13033, 8303, 28, 1303, 1885, 49, 79, 49160, 8615, 81, 79, 49160, 28, 81, 79, 49160, 78, 49161, 25, 26265, 1885, 49, 79, 49161, 8615, 81, 79, 49161, 28, 81, 79, 49161, 78, 49161, 25, 26265, 1885, 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+{"group_id": "math_081", "origin": "budgie", "answer": "The two wheels rotate at a combined speed of $2 \\times 30 \\times 60 = 3600$ revolutions per minute.\n\nThe two wheels make 3600 revolutions per minute, so the smaller wheel takes 3600/30 = $\\boxed{120}$ seconds to rotate three times as fast as the larger wheel.", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 365, 49160, 49159, 2876, 25, 5713, 14296, 19087, 28, 11450, 770, 351, 971, 550, 28, 1130, 6450, 16418, 7685, 738, 260, 7292, 282, 14296, 1885, 49, 20, 284, 1885, 50, 28422, 378, 47317, 282, 260, 14296, 9094, 411, 253, 4186, 282, 1296, 30, 378, 3764, 8656, 2022, 216, 49162, 49159, 29742, 567, 8427, 30, 22983, 638, 800, 7345, 260, 3227, 8656, 21768, 335, 582, 5245, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49162, 49160, 83, 49162, 49162, 49166, 49165, 85, 49167, 84, 49163, 49166, 49165, 49165, 49164, 15659, 86, 49168, 49159, 87, 29, 49159, 49163, 30, 13655, 47, 11551, 45, 49163, 49159, 49162, 22, 6930, 45, 49163, 49167, 49159, 22, 4961, 79, 8842, 79, 105, 45, 49168, 49164, 49167, 22, 4961, 79, 8842, 79, 104, 45, 49167, 49165, 49167, 25, 49154, 49158, 198, 504, 827, 14296, 19087, 418, 253, 5276, 3547, 282, 1885, 49161, 3814, 14602, 216, 49162, 49159, 3814, 14602, 216, 49165, 49159, 446, 216, 49162, 49165, 49159, 49159, 20, 29742, 567, 8427, 30, 198, 198, 504, 827, 14296, 919, 216, 49162, 49165, 49159, 49159, 29742, 567, 8427, 28, 588, 260, 3764, 8656, 2933, 216, 49162, 49165, 49159, 49159, 31, 49162, 49159, 446, 19573, 4810, 277, 107, 49160, 49161, 49159, 24362, 7345, 288, 19087, 1296, 1711, 347, 3538, 347, 260, 3227, 8656, 30, 49154], "prefix_len": 183, "old_logprobs": [-1.4609476327896118, -2.627300262451172, -0.014013583771884441, -0.7484188079833984, -2.2794382572174072, -0.9261645078659058, -0.4807943105697632, -0.9631984829902649, -0.07189545035362244, -1.0450222492218018, -0.3002413809299469, -0.9950499534606934, -0.12939459085464478, -0.029764996841549873, -0.6436401605606079, -0.06925348937511444, -1.7881590127944946, -0.36630573868751526, -0.0868588462471962, -0.7296174764633179, -0.08340813219547272, -0.23776327073574066, -0.02078898809850216, -0.002308919792994857, -0.0031475310679525137, -2.13382354559144e-05, -0.011841481551527977, -0.024782951921224594, -0.7868180871009827, -0.009084794670343399, -0.011082522571086884, -0.028470484539866447, -0.5486528277397156, -0.09471310675144196, -1.1696217060089111, -3.588625907897949, -0.007486857008188963, -1.1202483177185059, -1.0956332683563232, -0.11845885962247849, -0.8795124888420105, -0.00281461956910789, -0.00984562374651432, -0.24169279634952545, -0.7204097509384155, -0.04912475869059563, -0.9118077754974365, -0.4916650354862213, -1.0592422485351562, -1.772507667541504, -0.01157072652131319, -1.8825381994247437, -0.41431236267089844, -0.0917191356420517, -0.013622770085930824, -0.0008238735608756542, -0.0006006343755871058, -0.11846309155225754, -0.09389016777276993, -0.45568305253982544, -0.2004159837961197, -3.185530424118042, -0.01651287078857422, -1.9073468138230965e-06, -0.020648052915930748, -0.019722197204828262, -0.00022754464589525014, -0.1470804363489151, -0.0052775111980736256, -1.1125140190124512, -0.4562505781650543, -3.4451723098754883, -3.5588934421539307, -0.018925100564956665, -1.0304131507873535, -0.12462970614433289, -0.5154629349708557, -0.05884100869297981, -0.03766952455043793, -0.036643240600824356, -0.15740782022476196, -0.40046897530555725]}
+{"group_id": "math_081", "origin": "budgie", "answer": "First, we need to find the distance the larger wheel travels in one revolution. Since the circumference of the wheel is the same for both wheels, we can find the distance traveled by the larger wheel in one revolution by multiplying the circumference of the larger wheel by the number of revolutions it makes in a minute.\n\nThe formula for the circumference of a circle is C = 2πr, where r is the radius of the circle.\n\nFor the larger wheel, the circumference in one minute is 2π(30/60) = 2π.\n\nSince the smaller wheel makes 30 revolutions per minute, the distance traveled by the smaller wheel in one minute is 2π(30/30) = 2π.\n\nTherefore, the larger wheel travels at a distance of 30*2π = 60π miles in one minute.\n\nNow, we can find the time the larger wheel spends on one revolution by dividing the distance traveled by the number of revolutions it makes in one minute by the number of revolutions it makes in a minute.\n\nTime = Distance / Number of pmid\nTime = (60π / 30) / 30\nTime = (6π / 30) minutes\nTime = (3/5) minutes\n\nSince we want the answer in seconds, we convert the time to seconds by multiplying by 60.\n\nTime = (3/5) * 60 seconds\nTime = 36 seconds\n\nSo, the larger wheel spends 36 seconds on one revolution.\n#### 36", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 365, 49160, 49159, 2876, 25, 5713, 14296, 19087, 28, 11450, 770, 351, 971, 550, 28, 1130, 6450, 16418, 7685, 738, 260, 7292, 282, 14296, 1885, 49, 20, 284, 1885, 50, 28422, 378, 47317, 282, 260, 14296, 9094, 411, 253, 4186, 282, 1296, 30, 378, 3764, 8656, 2022, 216, 49162, 49159, 29742, 567, 8427, 30, 22983, 638, 800, 7345, 260, 3227, 8656, 21768, 335, 582, 5245, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49162, 49160, 83, 49162, 49162, 49166, 49165, 85, 49167, 84, 49163, 49166, 49165, 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+{"group_id": "math_081", "origin": "verified_reference", "answer": "Answer: $6 s$\n\nSolution. The speeds of points lying on the edge of the wheels are the same.\n\n(2 points)\n\nThe distances traveled by these points differ by three times.\n\nThe small wheel spends on one revolution: $t_{\\text {small }}=\\frac{1 \\text { min }}{30 \\text { revolutions }}=\\frac{60 s}{30 \\text { revolutions }}=2 s$. (3 points)\n\nTherefore, the large wheel spends on one revolution:\n\n$t_{\\text {large }}=3 t_{\\text {small }}=3 \\cdot 2=6 s$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 365, 49160, 49159, 2876, 25, 5713, 14296, 19087, 28, 11450, 770, 351, 971, 550, 28, 1130, 6450, 16418, 7685, 738, 260, 7292, 282, 14296, 1885, 49, 20, 284, 1885, 50, 28422, 378, 47317, 282, 260, 14296, 9094, 411, 253, 4186, 282, 1296, 30, 378, 3764, 8656, 2022, 216, 49162, 49159, 29742, 567, 8427, 30, 22983, 638, 800, 7345, 260, 3227, 8656, 21768, 335, 582, 5245, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49162, 49160, 83, 49162, 49162, 49166, 49165, 85, 49167, 84, 49163, 49166, 49165, 49165, 49164, 15659, 86, 49168, 49159, 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+{"group_id": "math_084", "origin": "budgie", "answer": "First, note that if $n!$ divided by $2$ does not have an integer factorial, it is impossible to factor it into the product of two factorials. Therefore, $n$ must be an even number or a multiple of $4$. There are $4$ possibilities: $4, 6, 8,...$ for $n$ values. If $n$ is a multiple of $4$, then $k = 4k$ and $l = 4l$, which implies $k \\cdot l = 4k \\cdot 4l$ does not have a factorial factorial. If $n = 8$ for the first case, then $k = 2$ and $l = 8$ for the second case, so there is only one solution. If $n = 6$, then $k = 3$ and $l = 6$ for the third case, so there is only one solution. If $n = 2$, then $k = 1$ and $l = 2$ for the fourth case, and only one solution. If $n = 1$, then $k = 1$ and $l = 1$ for the fifth case, and one solution. If $n = 0$, then $k = 0$ and $l = 0$ for the sixth case, and one solution. If $n = 5$, then $k = 1$ and $l = 5$ for the seventh case, and one solution. If $n = 3$, then $k = 1$ and $l = 3$ for the eighth and ninth cases, and one solution. If $n = 1$, then $k = 1$ and $l = 1$ for the tenth and eleventh cases, and one solution", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49160, 30, 43104, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 45, 91, 17, 27, 92, 17, 20, 281, 1782, 2966, 28, 837, 1885, 94, 42732, 49160, 3814, 41591, 216, 49161, 3814, 41591, 3814, 400, 1565, 304, 28422, 1094, 665, 359, 787, 3860, 28, 2965, 216, 49159, 43, 585, 665, 314, 582, 3564, 28, 2965, 1885, 94, 20, 43, 585, 665, 359, 2701, 3860, 28, 2965, 260, 2028, 282, 260, 2396, 282, 1885, 94, 20, 327, 511, 3860, 30, 37271, 338, 253, 3564, 314, 253, 3149, 12488, 46408, 94, 28, 501, 28, 303, 38224, 43, 585, 3860, 9094, 281, 418, 2124, 582, 5077, 28, 502, 359, 2500, 896, 30, 49154, 49158, 198, 5345, 28, 4109, 338, 585, 1885, 94, 17, 20, 6066, 411, 1885, 49161, 20, 1072, 441, 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+{"group_id": "math_084", "origin": "budgie", "answer": "To solve the equation $\\frac{n!}{2} = k! + l!$, start by noticing that it can be as simple as 1 factorial on both the left side and the right side.\n\nThe smallest $n$ for which this does not hold true is $n = 4$ because $4! = 24$.\n\nThus, the answer is $\\boxed{4}$.\n\nNote: If there are multiple solutions, then the sum of their values is not unique.", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49160, 30, 43104, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 45, 91, 17, 27, 92, 17, 20, 281, 1782, 2966, 28, 837, 1885, 94, 42732, 49160, 3814, 41591, 216, 49161, 3814, 41591, 3814, 400, 1565, 304, 28422, 1094, 665, 359, 787, 3860, 28, 2965, 216, 49159, 43, 585, 665, 314, 582, 3564, 28, 2965, 1885, 94, 20, 43, 585, 665, 359, 2701, 3860, 28, 2965, 260, 2028, 282, 260, 2396, 282, 1885, 94, 20, 327, 511, 3860, 30, 37271, 338, 253, 3564, 314, 253, 3149, 12488, 46408, 94, 28, 501, 28, 303, 38224, 43, 585, 3860, 9094, 281, 418, 2124, 582, 5077, 28, 502, 359, 2500, 896, 30, 49154, 49158, 198, 2068, 5482, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 446, 501, 17, 1232, 303, 17, 26265, 1120, 411, 30528, 338, 357, 416, 325, 347, 2232, 347, 216, 49160, 4186, 409, 335, 1062, 260, 2049, 2103, 284, 260, 1048, 2103, 30, 198, 198, 504, 13341, 1885, 94, 20, 327, 527, 451, 1072, 441, 2205, 2629, 314, 1885, 94, 446, 216, 49163, 20, 975, 1885, 49163, 17, 446, 216, 49161, 49163, 28422, 198, 198, 14344, 28, 260, 2988, 314, 19573, 4810, 277, 107, 49163, 24362, 30, 198, 198, 8928, 42, 1094, 665, 359, 2701, 3860, 28, 965, 260, 2028, 282, 480, 2396, 314, 441, 2116, 30, 49154], "prefix_len": 142, "old_logprobs": [-1.752201795578003, -0.23252420127391815, -0.38638609647750854, -0.02730526402592659, -0.24131788313388824, -0.0008070787298493087, -0.0002307625545654446, -0.006066480651497841, -0.0009224927052855492, -6.949660019017756e-05, -0.00018809456378221512, -0.00014006110723130405, -2.0225536823272705, -0.007939211092889309, -0.0014055621577426791, -0.04013221338391304, -0.0032390295527875423, -0.0026566232554614544, -0.6967863440513611, -4.457576751708984, -0.0786355584859848, -3.227722406387329, -0.166713148355484, -3.8902087211608887, -1.658910870552063, -0.019678249955177307, -10.357717514038086, -0.24977558851242065, -0.009714826010167599, -1.8815957307815552, -1.0306024551391602, -2.7271223068237305, -0.5626351833343506, -4.272486686706543, -1.320894718170166, -2.147026538848877, -0.3655475676059723, -0.7275599837303162, -0.709673285484314, -0.3043581247329712, -0.003196133067831397, -0.0016683719586580992, -0.73082035779953, -1.437998652458191, -0.08504339307546616, -3.029097318649292, -2.9758925437927246, -0.7527299523353577, -0.23528411984443665, -0.05562160536646843, -0.7382365465164185, -0.011888484470546246, -0.8811951875686646, -3.9085164070129395, -0.0400206632912159, -0.16688042879104614, -1.4710924625396729, -0.15427222847938538, -1.1851508617401123, -0.07550346106290817, -0.19464540481567383, -0.019566265866160393, -2.58644962310791, -1.847943902015686, -0.22161594033241272, -0.6745989322662354, -0.43096819519996643, -0.029274020344018936, -0.36712732911109924, -0.010231191292405128, -0.23011383414268494, -0.0006462631281465292, -2.101041078567505, -0.6142739653587341, -0.031150398775935173, -4.088268280029297, -0.01759737730026245, -0.8761716485023499, -1.8372199535369873, -0.017119256779551506, -2.0780539512634277, -0.0013398488517850637, -2.4437606043647975e-05, -0.0019221893744543195, -0.30396193265914917, -0.10821603983640671, -0.13172098994255066, -1.4083329439163208, -0.08927074074745178, -1.5658046007156372, -0.7187583446502686, -3.292543649673462, -1.179271936416626, -0.6118783354759216, -0.41102781891822815, -0.07096734642982483, -0.16836699843406677, -4.315783977508545, -1.1153658628463745, -0.8865266442298889, -0.09175002574920654, -2.0168392658233643, -0.15043504536151886, -0.4328908920288086, -2.6725282669067383, -1.33939528465271, -0.8214519023895264, -0.26096025109291077]}
+{"group_id": "math_084", "origin": "verified_reference", "answer": "Answer: 10 (all triples of solutions $(n, k, l):(3,1,2),(3,2,1),(4,3,3)$).\n\nSolution. Note that if $k4$, then $n!>4 \\cdot(n-1)!\\geqslant 2 \\cdot(k!+l!)$, so there are no such solutions. If $n=2$, we get $1=k!+l!$ - no solutions; if $n=3$, the equation $3=k!+l!$ has two solutions: $k=1, l=2$ and $k=2, l=1$; if $n=4$, the equation $12=k!+l!$ gives one more solution $k=l=3$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49160, 30, 43104, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 45, 91, 17, 27, 92, 17, 20, 281, 1782, 2966, 28, 837, 1885, 94, 42732, 49160, 3814, 41591, 216, 49161, 3814, 41591, 3814, 400, 1565, 304, 28422, 1094, 665, 359, 787, 3860, 28, 2965, 216, 49159, 43, 585, 665, 314, 582, 3564, 28, 2965, 1885, 94, 20, 43, 585, 665, 359, 2701, 3860, 28, 2965, 260, 2028, 282, 260, 2396, 282, 1885, 94, 20, 327, 511, 3860, 30, 37271, 338, 253, 3564, 314, 253, 3149, 12488, 46408, 94, 28, 501, 28, 303, 38224, 43, 585, 3860, 9094, 281, 418, 2124, 582, 5077, 28, 502, 359, 2500, 896, 30, 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+{"group_id": "math_085", "origin": "budgie", "answer": "To find the minimum value of $a+b+c$ given that $a$, $b$, and $c$ are all positive integers and that the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$, we can proceed as follows:\n\nSince the parabola intersects the $x$-axis at $A$ and $B$, the equation of the parabola can be written as:\n\\[ y = a x^2 + b x + c = a(x^2 - 1) + b(x - 1) \\]\n\nWe know that the distance from $A$ to $B$ is less than 1, so:\n\\[ \\sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} < 1 \\Rightarrow (x_B - x_A)^2 + (y_B - y_A)^2 < 1 \\]\n\nSquaring both sides and expanding, we get:\n\\[ x_B^2 - 2x_Ax_B + x_A^2 + y_B^2 - 2y_Ay_B + y_A^2 + x_B^2 + y_B^2 - 2x_Ay_B + x_A^2 + y_A^2 < 1 \\]\n\nSimplifying, we get:\n\\[ 2x_B^2 - 2x_Ax_B + 2y_B^2 - 2y_Ay_B + 2x_A^2 + 2y_A^2 - 2x_A^2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 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+{"group_id": "math_085", "origin": "budgie", "answer": "1. The equation of the parabola is $y = ax^2 + bx + c$.\n2. Since the parabola intersects the $x$-axis at points $A$ and $B$, we know that $A$ and $B$ have the same $y$-coordinate.\n3. The $y$-coordinates of $A$ and $B$ are the same, so we can write $y = k(ax^2 + bx + c)$, where $k$ is a constant.\n4. The roots of the quadratic equation $ax^2 + bx + c = 0$ are $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$.\n5. Since $A$ and $B$ have the same $y$-coordinate, we have $k = k \\cdot \\frac{-b + \\sqrt{b^2 - 4ac}}{2a}$.\n6. Since the distances from $A$ and $B$ to the origin are both less than 1, we have $\\sqrt{(a - 0)^2 + b^2} < 1$.\n7. Squaring both sides of the equation, we get $(a - 0)^2 + b^2 < 1$.\n8. Expanding the left-hand side, we get $a^2 + b^2 < 1$.\n9. Since $a^2 + b^2 - 1 = 0$, we can apply the Cauchy-Schwarz inequality to get $\\sqrt{(a - 0)^2 + b^2} < 1$.\n10. 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+{"group_id": "math_085", "origin": "verified_reference", "answer": "Solution: Let $A\\left(x_{1}, 0\\right)$ and $B\\left(x_{2}, 0\\right)$, and $x_{1} < 0 < x_{2}$,  \nthen $x_{1} < 0, \\\\ \n\\therefore b > 2 \\sqrt{a c} . \\\\ \n\\text{Also, } \\because |O A| = |x_{1}| > 1$. Therefore, the parabola opens upwards, and when $x = -1$, $y > 0$, so $a(-1)^{2} + b(-1) + c > 0$, which means $b < a + c + 1$. Since $b > 2 \\sqrt{a c}$, we have $2 \\sqrt{a c} + 1 < a + c$,  \ni.e., $(\\sqrt{a} - \\sqrt{c})^{2} > 1$.  \nFrom equation (2), we get $\\sqrt{a} - \\sqrt{c} > 1$,  \n\ni.e., $\\sqrt{a} > \\sqrt{c} + 1$.  \nThus, $a > (\\sqrt{c} + 1)^{2} \\geq (\\sqrt{1} + 1)^{2} = 4$.  \nSolving, we get $a \\geq 5$.  \n$$\n\\begin{array}{l}\n\\text{Also, } \\because b > 2 \\sqrt{a c} \\geq 2 \\sqrt{5 \\times 1} > 4, \\\\\n\\therefore b \\geq 5 .\n\\end{array}\n$$\n\nWhen $a = 5, b = 5, c = 1$, the parabola $y = 5 x^{2} + 5 x + 1$ satisfies the given conditions.  \nTherefore, the minimum value of $a + b + c$ is 11.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 49162, 10887, 338, 1885, 81, 26265, 1885, 82, 26265, 284, 1885, 83, 20, 359, 511, 2731, 23313, 28, 284, 260, 43099, 7866, 1885, 105, 45, 81, 1792, 21990, 49161, 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+{"group_id": "math_086", "origin": "budgie", "answer": "Let's consider the terms of the function $f(x)$:\n\n$$f(x) = \\left(x^{2}+4 x+5\\right)\\left(x^{2}+4 x+2\\right)+2 x^{2}+8 x+1$$\n\nWe can rewrite each term as $(x^{2}+4 x+2)(x^{2}+4 x+5)$.\n\nExpanding the product, we get:\n\n$$f(x) = (x^{2}+4 x+2)(x^{2}+4 x+5) = x^{4}+8 x^{3}+2 x^{2}+4 x^{3}+8 x^{2}+16 x+2 x^{2}+8 x+5$$\n\nCombining like terms, we get:\n\n$$f(x) = x^{4}+12 x^{3}+17 x^{2}+14 x+5$$\n\nNow, let's set $f(x)$ to a constant value to find the minimum value. We can do this by setting the expression $x^{4}+12 x^{3}+17 x^{2}+14 x+5 = k$, where $k$ is a constant.\n\nRearranging the equation, we get:\n\n$$x^{4}+12 x^{3}+17 x^{2}+14 x+5 - k = 0$$\n\nThis is a quartic equation in $x$, which can be difficult to solve. 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+{"group_id": "math_086", "origin": "budgie", "answer": "To solve this problem, we first need to find the minimum value of the function $f(x) = \\left(x^{2} + 4x + 5\\right) \\left(x^{2} + 4x + 2\\right) + 2x^{2} + 8x + 1$.\n\nLet's denote the first expression as $A$ and the second expression as $B$:\n\n$$A = x^{2} + 4x + 5$$\n$$B = x^{2} + 4x + 2$$\n\nNow, we can rewrite $A$ and $B$ as follows:\n\n$$A = x^{2} + 4x + 5 = (x^{2} + 4x + 2) + 2x^{2} + 4x + 3$$\n$$B = x^{2} + 4x + 2 = (x^{2} + 4x + 4) + 2x^{2} + 4x + 4 + 1$$\n\nSince $A$ and $B$ are equal, we can set the expressions equal to each other:\n\n$$(x^{2} + 4x + 2) + 2x^{2} + 4x + 3 = (x^{2} + 4x + 4) + 2x^{2} + 4x + 4 + 1$$\n\nSimplify the right side of the equation:\n\n$$\\Rightarrow x^{2} + 2x^{2} + 2x^{2} + 8x + 1 = 5x^{2} + 12x + 5$$\n\nRearrange the", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 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+{"group_id": "math_086", "origin": "verified_reference", "answer": "Let $u=x^{2}+4 x$, then\n$$\nf(x)=g(u)=(u+5)(u+2)+2 u+1=u^{2}+9 u+11 \\text {. }\n$$\n\nBy the properties of quadratic functions, when $u=-\\frac{9}{2}$, $g(u)$ reaches its minimum value, and when $u>-\\frac{9}{2}$, $g(u)$ is a strictly increasing function of $u$. Since $u=x^{2}+4 x=(x+2)^{2}-4$, we know that $u \\geqslant-4$, and when $x=-2$, $u=-4$. Therefore, when $x=-2$, $f(x)$ reaches its minimum value, which is -9.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49161, 2959, 1885, 104, 3814, 254, 3814, 33918, 107, 66, 24362, 28, 1576, 288, 1042, 260, 5869, 1685, 282, 260, 1517, 1885, 86, 24, 104, 25, 24814, 8842, 24, 104, 21990, 49161, 109, 27, 49163, 1792, 27, 49164, 76, 2771, 24492, 8842, 24, 104, 21990, 49161, 109, 27, 49163, 1792, 27, 49161, 76, 2771, 18434, 49161, 1792, 21990, 49161, 109, 27, 49167, 1792, 27, 49160, 28422, 49154, 49158, 198, 4239, 1885, 101, 45, 104, 21990, 49161, 109, 27, 49163, 1792, 26265, 965, 198, 9212, 198, 86, 24, 104, 36637, 87, 24, 101, 25, 8615, 101, 27, 49164, 14502, 101, 27, 49161, 18434, 49161, 326, 27, 49160, 45, 101, 21990, 49161, 109, 27, 49168, 326, 27, 49160, 49160, 3814, 2692, 1662, 30, 4029, 198, 9212, 198, 198, 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+{"group_id": "math_087", "origin": "budgie", "answer": "This problem mainly tests the properties of a focal length of a lens, which is a function of the angle between the incident ray and the plane of the lens.\n\nThe angle between the incident ray and the plane of the lens is related to the distance from the entry point to the lens through the lens's shape. The angle between the incident ray and the plane of the lens is essentially the angle between the rays and the plane of the lens. It is because the point is on the edge of the lens that the lightbars will form a right angle with the rays.\n\nSince the lightbars form a right angle with the rays, the focal length can be determined by the distance from the lightblue point to the lens. The distance from the lightblue point to the lens is the same as the distance from the lightblue point to the lens, which is half of the focal length of the lens.\n\nThe answer is: $\\boxed{10\\sqrt{2}}$ cm", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 365, 49160, 49164, 2876, 25, 330, 5607, 19156, 282, 1420, 8227, 335, 253, 5607, 4984, 2707, 7247, 418, 253, 4052, 282, 1885, 104, 45, 49160, 49159, 20, 5973, 429, 624, 10454, 3712, 30, 378, 7256, 826, 260, 9208, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 13433, 45, 49163, 49164, 21990, 76, 21386, 24362, 28, 284, 260, 7256, 826, 260, 21112, 277, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 17565, 45, 49162, 49159, 21990, 76, 21386, 24362, 30, 22983, 624, 17350, 3519, 30, 49154, 49158, 198, 1348, 1732, 5630, 3642, 260, 3849, 282, 253, 17350, 3519, 282, 253, 7247, 28, 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+{"group_id": "math_087", "origin": "budgie", "answer": "The focal length of the lens is given by the sum of the distances from the light source to the focal points.  Therefore, the focal length of the lens is $x+x=2x$.\n\nHence, the answer is:\n\n\\[ \\boxed{2x} \\]\n\nThe focal length of the lens is $2x$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 365, 49160, 49164, 2876, 25, 330, 5607, 19156, 282, 1420, 8227, 335, 253, 5607, 4984, 2707, 7247, 418, 253, 4052, 282, 1885, 104, 45, 49160, 49159, 20, 5973, 429, 624, 10454, 3712, 30, 378, 7256, 826, 260, 9208, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 13433, 45, 49163, 49164, 21990, 76, 21386, 24362, 28, 284, 260, 7256, 826, 260, 21112, 277, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 17565, 45, 49162, 49159, 21990, 76, 21386, 24362, 30, 22983, 624, 17350, 3519, 30, 49154, 49158, 198, 504, 17350, 3519, 282, 260, 7247, 314, 1836, 411, 260, 2028, 282, 260, 9510, 429, 260, 1420, 2257, 288, 260, 17350, 2876, 30, 216, 4882, 28, 260, 17350, 3519, 282, 260, 7247, 314, 1885, 104, 27, 104, 45, 49161, 104, 28422, 198, 198, 35391, 28, 260, 2988, 314, 42, 198, 198, 76, 75, 3814, 4810, 277, 107, 49161, 104, 109, 3814, 77, 198, 198, 504, 17350, 3519, 282, 260, 7247, 314, 1885, 49161, 104, 28422, 49154], "prefix_len": 115, "old_logprobs": [-1.6423529386520386, -1.8036441802978516, -0.021930566057562828, -0.27133235335350037, -0.46621987223625183, -0.6992856860160828, -0.4222971796989441, -1.2089314460754395, -0.021433627232909203, -1.2431578636169434, -1.2867406606674194, -0.03392789885401726, -0.013428246602416039, -0.4706249237060547, -0.13301445543766022, -0.3747159242630005, -2.626441478729248, -0.043738119304180145, -0.11362124234437943, -0.18013767898082733, -3.027092456817627, -0.1778271645307541, -1.2479476928710938, -5.642955303192139, -2.841688394546509, -0.09926445782184601, -0.7446991205215454, -0.16762298345565796, -0.016065435484051704, -0.8568283915519714, -0.08398526161909103, -0.12953680753707886, -0.06648128479719162, -1.2334785461425781, -0.7478652000427246, -0.8828119039535522, -0.6608858108520508, -0.9784277081489563, -0.18139192461967468, -0.05453592911362648, -1.033318042755127, -1.1257203817367554, -0.12333303689956665, -3.191793441772461, -0.0310861486941576, -0.15096011757850647, -0.29360446333885193, -0.005857445299625397, -1.1482656002044678, -0.3377196490764618, -1.1740810871124268, -3.3703620433807373, -0.9121911525726318, -0.3129950165748596, -0.0019891730044037104, -4.768370445162873e-07, -0.0033521433360874653, -0.02178022637963295, -0.07638783007860184, -0.005004379898309708, -0.0009324969723820686, -0.0014873405452817678, -0.30634787678718567, -0.04693099856376648, -2.2015762329101562, -0.5450804829597473, -0.007443905808031559, -0.15827223658561707, -0.0460357740521431, -0.033400919288396835, -0.07599413394927979, -0.7017744779586792, -0.3742057979106903, -0.030160311609506607, -0.6276863813400269, -0.6498512029647827]}
+{"group_id": "math_087", "origin": "verified_reference", "answer": "Answer: $\\approx 13.7$ cm.\n\nSolution. Parallel rays intersect at the focus, so $F$ is the focus of the given lens.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7a4bc87492c06fc34cd6g-06.jpg?height=480&width=302&top_left_y=1502&top_left_x=957)\n\nIn triangle $O A F$: angle $F A O=30^{\\circ}$, angle $O F A=15^{\\circ}$, angle $A O F=135^{\\circ}$.\n\nTherefore, $\\frac{O F}{\\sin 30^{\\circ}}=\\frac{A O}{\\sin 15^{\\circ}}$.\n\nWe obtain that the focal length:\n\n$F C=O F \\sin 45^{\\circ}=A O \\frac{\\sin 30^{\\circ}}{\\sin 15^{\\circ}} \\sin 45^{\\circ} \\approx 13.7 \\mathrm{~cm}$.\n\n## Multidisciplinary Engineering Olympiad \"Star\" in Natural Sciences\n\n10th grade\n\nFinal Round\n\n$2022-2023$\n\nVersion 2\n\nProblems, answers, and evaluation criteria", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 365, 49160, 49164, 2876, 25, 330, 5607, 19156, 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+{"group_id": "math_089", "origin": "budgie", "answer": "#### Part (a)\n1. Let's denote the number of performances sold by the performers as \\( S_{\\text{per former}} \\), the number of performances sold by theppromades as \\( S_{\\text{per new}} \\), and the number of performances by the audience as \\( S_{\\text{per new}} \\).\n2. According to the problem, the number of violin and possession is three times the number ofadders. This can be written as:\n   \\[\n   S_{\\text{per former}} = 3 \\times S_{\\text{per new}}\n   \\]\n3. From this, we get a relationship between the number of performances and the number of adders:\n   \\[\n   S_{\\text{per former}} = 3 \\times S_{\\text{per new}} = 3S_{\\text{per adder}}\n   \\]\n4. This relationship holds because the total number of performances is equal to the sum of the number of performances by the performers and the number of adders.\n5. Therefore, we can express the total number of performances as:\n   \\[\n   S_{\\text{per former}} + S_{\\text{per new}} = S_{\\text{per adder}}\n   \\]\n6. Substituting the expression for \\( S_{\\text{per adder}} \\) from the previous step, we get:\n   \\[\n   3S_{\\text{per old}} = S_{\\text{per new}}\n   \\]\n7. Hence, it follows that:\n   \\[\n   S_{\\text{per old}} = \\frac{S_{\\text{per new}}}{3}\n   \\]\n8. 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+{"group_id": "math_089", "origin": "budgie", "answer": "a) The number of program HBV is 1/4 of the total number of performance performances.\n\nLet x be the total number of performances. Then, we know that 1/4*x = 200.\n\nSolving for x, we find that x = 800.\n\nb) To find the number of normalized visitors, we need to find the number of visitors who visited the animal show. We can do this by dividing the number of visitors who bought the Conserv by three.\n\nx = (200/3*x) + (x/3)\n\nx = (800/3) + (800/9) + (x/3)\n\nNow, we can simplify this expression.\n\nx = (800/3) + (1667/9) + (x/3)\n\nx = (800/3) + (1667/9) + (819/9)\n\nx = (800 + 1667 + 819)/9\n\nx = (2556)/9\n\nx = 286\n\nSo, the number of normalized visitors is 286.\n\nFinal answer:\n#### 286", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49160, 49161, 49167, 30, 8948, 10203, 12920, 1885, 49160, 2272, 216, 49166, 49165, 20, 198, 198, 49, 38735, 4502, 216, 49161, 49159, 49159, 13171, 281, 260, 1596, 3645, 28, 511, 282, 527, 592, 3459, 578, 30, 378, 1230, 282, 14670, 281, 260, 38735, 16044, 314, 1296, 1711, 260, 8629, 599, 282, 260, 1230, 282, 13171, 1836, 30, 198, 198, 81, 25, 1073, 800, 1578, 40217, 592, 8057, 585, 582, 8629, 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+{"group_id": "math_089", "origin": "verified_reference", "answer": "From $(200: 4) \\cdot 3=150$ it follows that the circus has over 150 seats.\n\na) $(200 \\cdot 150): 4=7500$; 7500 program booklets were sold during the season.\n\nb) $(200 \\cdot 150): 2=15000$ and $30 \\cdot 15000=450000$; from the animal show, €450,000 was earned.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49160, 49161, 49167, 30, 8948, 10203, 12920, 1885, 49160, 2272, 216, 49166, 49165, 20, 198, 198, 49, 38735, 4502, 216, 49161, 49159, 49159, 13171, 281, 260, 1596, 3645, 28, 511, 282, 527, 592, 3459, 578, 30, 378, 1230, 282, 14670, 281, 260, 38735, 16044, 314, 1296, 1711, 260, 8629, 599, 282, 260, 1230, 282, 13171, 1836, 30, 198, 198, 81, 25, 1073, 800, 1578, 40217, 592, 8057, 585, 582, 8629, 282, 260, 7803, 10897, 253, 48797, 47, 198, 198, 82, 25, 1073, 800, 36559, 592, 21769, 5771, 429, 260, 11759, 12235, 327, 260, 3161, 1138, 585, 357, 436, 8797, 411, 2745, 282, 260, 7803, 284, 260, 11759, 13276, 436, 216, 49159, 30, 49162, 49159, 372, 47, 49154, 49158, 198, 5212, 46408, 49161, 49159, 49159, 42, 216, 49163, 25, 3814, 41591, 216, 49162, 45, 49160, 49164, 49159, 20, 357, 5558, 338, 260, 38735, 553, 690, 216, 49160, 49164, 49159, 14670, 30, 198, 198, 81, 25, 46408, 49161, 49159, 49159, 3814, 41591, 216, 49160, 49164, 49159, 727, 216, 49163, 45, 49166, 49164, 49159, 49159, 20, 43, 216, 49166, 49164, 49159, 49159, 1578, 1863, 6770, 592, 3459, 981, 260, 3645, 30, 198, 198, 82, 25, 46408, 49161, 49159, 49159, 3814, 41591, 216, 49160, 49164, 49159, 727, 216, 49161, 45, 49160, 49164, 49159, 49159, 49159, 20, 284, 1885, 49162, 49159, 3814, 41591, 216, 49160, 49164, 49159, 49159, 49159, 45, 49163, 49164, 49159, 49159, 49159, 49159, 20, 43, 429, 260, 3161, 1138, 28, 24927, 49163, 49164, 49159, 28, 49159, 49159, 49159, 436, 11420, 30, 49154], "prefix_len": 148, "old_logprobs": [-5.501042366027832, -8.446799278259277, -1.7892224788665771, -0.2276446670293808, -0.03379940614104271, -7.018219470977783, -1.21592116355896, -2.7752671241760254, -1.0411169528961182, -2.465818405151367, -2.898998498916626, -0.317328542470932, -1.1797640323638916, -2.7640480995178223, -1.6300458908081055, -2.0705618858337402, -0.02860696241259575, -0.5023309588432312, -7.442901134490967, -2.0864834785461426, -0.14977413415908813, -1.7550089359283447, -8.909077644348145, -3.310453414916992, -6.042674541473389, -0.2023615837097168, -0.6108866930007935, -0.4219259023666382, -0.026059595867991447, -3.002293586730957, -0.669857382774353, -0.30619552731513977, -0.3699648976325989, -2.59531831741333, -0.05388449877500534, -6.646946907043457, -1.0511711835861206, -0.01883525960147381, -0.005173391196876764, -3.8303399085998535, -2.2966740131378174, -0.11441162973642349, -1.278995156288147, -1.6159230470657349, -0.06959681212902069, -5.837584018707275, -0.6162245273590088, -2.1569061279296875, -1.7401385307312012, -3.6571054458618164, -2.397738218307495, -0.43141618371009827, -2.086458683013916, -1.4653723239898682, -4.603309631347656, -2.738957643508911, -0.7245733737945557, -0.11834109574556351, -0.05124225467443466, -0.024035267531871796, -11.320363998413086, -13.698543548583984, -3.758654832839966, -1.2459903955459595, -2.439926862716675, -4.68990421295166, -0.3229067325592041, -2.6251823902130127, -0.4582195580005646, -0.09894413501024246, -0.08927749842405319, -0.3696715533733368, -0.007363323122262955, -1.0422818660736084, -1.0685571432113647, -0.022764425724744797, -0.007480112835764885, -0.8188632130622864, -0.07470249384641647, -0.06314028054475784, -0.9951005578041077, -0.3773737847805023, -0.011652973480522633, -1.595185399055481, -0.1597268432378769, -2.3164823055267334, -1.5878303050994873, -1.6291325092315674, -1.3906404972076416, -0.012220979668200016, -0.055220890790224075, -0.8074251413345337, -0.439653605222702, -5.717396259307861, -1.8903231620788574, -1.8633973598480225, -0.799562394618988, -3.69037127494812, -0.32166391611099243, -0.09617637097835541, -0.8457388877868652, -0.5099091529846191, -0.014638286083936691, -1.6042590141296387, -0.30810344219207764, -0.5227669477462769, -0.5400387644767761, -0.03186887875199318, -0.058957893401384354, -0.017649265006184578, -0.00423071626573801, -0.22188161313533783, -0.6913185119628906, -1.9762012958526611, -5.063515663146973, -1.2091901302337646, -5.544290542602539, -0.053764987736940384, -1.7792181968688965, -13.154029846191406, -1.8135794401168823, -0.12425834685564041, -0.07622811943292618, -2.7730517387390137, -0.004688935354351997, -0.00010406429646536708, -0.0006721144891344011, -1.3766717910766602, -8.299947738647461, -0.8010165691375732, -4.170498371124268]}
+{"group_id": "math_090", "origin": "budgie", "answer": "To find the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$, we need to simplify the expression inside the parentheses first.\n\n\\[ 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right) = 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}\\right)\\]\n\nNext, we need to find the sum inside the parentheses. To do this, we can add the fractions one by one and then find a common denominator.\n\n\\[ 1+(1+1+1+1+1) = 5 \\]\n\nSo, we have:\n\n\\[ 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}\\right) = 2^{4}\\left(5\\right) = 2^{4}\\cdot 5 = 16\\cdot 2 = \\boxed{32} \\]\n\nFinal answer:\n#### 32", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 44208, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49162, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49163, 109, 17243, 2771, 25, 28422, 49154, 49158, 198, 2068, 1042, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 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+{"group_id": "math_090", "origin": "budgie", "answer": "We can simplify the expression by grouping the terms into sets of four.\n\nLet's break down the expression step by step:\n\n\\begin{align*}\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right) &= 2^{4}\\cdot\\frac{1}{2^4}\\\\\n&= 2^4\\cdot\\frac{1}{16}\\\\\n&= 2^4\\cdot\\frac{1}{2^4}\\\\\n&= 2^4\\cdot\\frac{1}{2^4}\\\\\n&= 2^3\n\\end{align*}\n\nSo, the answer is $\\boxed{8}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 44208, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49162, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49163, 109, 17243, 2771, 25, 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+{"group_id": "math_090", "origin": "verified_reference", "answer": "Solution 1\n\nMultiplying through, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16+\\frac{16}{2}+\\frac{16}{4}+\\frac{16}{8}+\\frac{16}{16}=16+8+4+2+1=31\n$$\n\n## Solution 2\n\nUsing a common denominator inside the parentheses, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16\\left(\\frac{16}{16}+\\frac{8}{16}+\\frac{4}{16}+\\frac{2}{16}+\\frac{1}{16}\\right)=16\\left(\\frac{31}{16}\\right)=31\n$$\n\nANSWER: 31", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 44208, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 42639, 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+{"group_id": "math_091", "origin": "budgie", "answer": "5.  Let's consider a special case when $a=b=\\frac{1}{4}$.\n\n$$M=\\sqrt{1+a^{2}}+\\sqrt{1+2 a^{2}}$$\n\nSince $a=\\frac{1}{4}$, we have $a^{2}=1$.\n\nTherefore,\n$$M=\\sqrt{1+1}+\\sqrt{1+2 \\times 1}=\\sqrt{2}+\\sqrt{3}$$\n\nHence, $M=\\boxed{\\sqrt{2}+\\sqrt{3}}$.\n\nFinal answer:\n#### √2 + √3", "reward": -1, "advantage": -1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 10887, 2731, 1345, 2966, 1885, 81, 20, 284, 1885, 82, 20, 15765, 1885, 81, 27, 82, 45, 49160, 26265, 965, 1885, 61, 45, 20, 19573, 16166, 107, 49160, 27, 81, 21990, 49161, 17859, 42639, 16166, 107, 49160, 27, 49161, 278, 24362, 260, 15328, 599, 314, 49154, 49158, 198, 49164, 30, 216, 2959, 506, 1771, 253, 1767, 1671, 645, 1885, 81, 45, 82, 24814, 18169, 107, 49160, 15009, 49163, 24362, 30, 198, 198, 9212, 61, 24814, 16166, 107, 49160, 27, 81, 21990, 49161, 17859, 42639, 16166, 107, 49160, 27, 49161, 253, 21990, 49161, 17859, 9212, 198, 198, 7230, 1885, 81, 24814, 18169, 107, 49160, 15009, 49163, 24362, 28, 392, 457, 1885, 81, 21990, 49161, 109, 45, 49160, 28422, 198, 198, 17308, 28, 198, 9212, 61, 24814, 16166, 107, 49160, 27, 49160, 109, 42639, 16166, 107, 49160, 27, 49161, 3814, 14602, 216, 49160, 109, 24814, 16166, 107, 49161, 109, 42639, 16166, 107, 49162, 109, 9212, 198, 198, 35391, 28, 1885, 61, 24814, 4810, 277, 26754, 16166, 107, 49161, 109, 42639, 16166, 107, 49162, 17859, 28422, 198, 198, 29288, 2988, 42, 198, 1229, 19995, 244, 49161, 1232, 19995, 244, 49162, 49154], "prefix_len": 76, "old_logprobs": [-2.846531629562378, -0.03406040742993355, -4.166222095489502, -2.840928077697754, -1.116208553314209, -2.6269969940185547, -2.360008716583252, -2.644103765487671, -0.13075914978981018, -1.709085464477539, -0.08773389458656311, -0.1674066185951233, -0.45026957988739014, -0.4025063216686249, -1.9347126483917236, -0.021537235006690025, -0.016591312363743782, -0.004272262565791607, -0.003325529396533966, -2.3060858249664307, -0.01989259570837021, -0.32860100269317627, -2.343108654022217, -0.35041362047195435, -0.8137670755386353, -0.35823944211006165, -1.1011433601379395, -0.0266399085521698, -0.018920421600341797, -0.04576364532113075, -0.313053697347641, -0.9258660078048706, -0.06296231597661972, -0.00021419614495243877, -0.0037188436836004257, -0.025423860177397728, -0.0007682471186853945, -0.000458254711702466, -0.014812489040195942, -0.10120955854654312, -0.01616150699555874, -0.4749419093132019, -0.4295516908168793, -0.006863115355372429, -0.057760849595069885, -0.935051441192627, -0.027185779064893723, -0.03566214442253113, -1.4840718507766724, -0.1568746119737625, -0.1386883556842804, -1.4881163835525513, -0.004076742567121983, -0.0005445189890451729, -0.000426439888542518, -0.0002933310461230576, -0.023000596091151237, -0.07482971251010895, -0.2320021688938141, -0.432095468044281, -0.2007625550031662, -0.5616754293441772, -0.6401876211166382, -0.08956190943717957, -0.0032513870391994715, -0.005480265710502863, -0.22316892445087433, -0.08599911630153656, -1.439324140548706, -0.8431919813156128, -0.01872715912759304, -1.7364661693572998, -0.012653541751205921, -0.9566826820373535, -1.515427589416504, -0.06602709740400314, -0.05883089080452919, -0.08356757462024689, -0.015858128666877747, -0.0509844534099102, -0.1148604229092598, -0.6392412185668945, -0.08376426994800568, -0.025838203728199005, -0.0012767505832016468, -0.001587679609656334, -0.013515633530914783, -0.02587154321372509, -0.043960489332675934, -0.6111393570899963, -0.3930513262748718, -0.4097169041633606, -0.0012403184082359076, -0.007044715341180563, -0.8323615193367004, -0.13654327392578125, -0.014525854028761387, -0.01672789640724659, -0.005758363753557205, -0.29382699728012085, -0.0014293702552095056, -0.0010836211731657386, -0.002593608107417822, -0.004586414899677038, -0.011721429415047169, -0.007652727887034416, -0.01286739856004715, -1.8427907228469849, -0.0032561400439590216, -1.6527280807495117, -0.025943096727132797, -0.5494641065597534, -0.5537412166595459, -7.152555099310121e-07, -0.08994169533252716, -0.005313678178936243, -0.0016684910515323281, -0.10787931084632874, -0.002161073498427868, -0.26456332206726074, -0.000417741306591779, -5.411955135059543e-05, -0.00019453064305707812, -0.0007630059262737632, -0.19240330159664154, -0.6442713737487793, -0.11247054487466812, -0.19564655423164368, -0.0009012927766889334, -0.0019124329555779696, -0.11099318414926529, -0.003806845983490348, -3.2233340740203857, -0.05750877410173416, -0.03351528197526932, -0.14153139293193817, -0.006622391752898693, -0.0022782341111451387, -0.0006158839096315205, -0.028644032776355743]}
+{"group_id": "math_091", "origin": "budgie", "answer": "To find the integer part of $M$, we start by simplifying the expression inside the square root.\n\n\\begin{align*}\n\\sqrt{1+2 b} &= \\sqrt{1+2 b} \\\\\n&= \\sqrt{1+2 b} \\cdot \\frac{1}{\\sqrt{1+2 b} + 1} \\\\\n&= \\sqrt{1+2 b} \\\\\n&= \\frac{1+2 b}{\\sqrt{1+2 b} + 1}\n\\end{align*}\n\nThen, we rewrite the expression for $M$ as follows:\n\n\\begin{align*}\nM &= \\sqrt{1+2 b} \\\\\n&= \\frac{1+2 b}{\\sqrt{1+2 b} + 1} \\\\\n&= \\frac{1+2 b}{\\frac{1+2 b}{\\sqrt{1+2 b}+1}} \\\\\n&= \\frac{(1+2 b)^{3/2}}{\\frac{(1+2 b)^{3/2}}{\\sqrt{1+2 b}+1}} \\\\\n&= (1+2 b)^{3/2} \\cdot \\frac{\\sqrt{1+2 b}+1}{\\sqrt{1+2 b}+1} \\\\\n&= (1+2 b)^{3/2} \\\\cdot \\frac{\\sqrt{1+2 b}}{\\sqrt{1+2 b} + 1} \\\\\n&= (1+2 b)^{3/2} \\\\cdot \\frac{\\sqrt{1+2 b}}{\\sqrt{1+2 b} + 1} \\\\\n&= (1 + 2 b)^{3/2} \\\\", "reward": 0.5, "advantage": 0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 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+{"group_id": "math_091", "origin": "verified_reference", "answer": "5. 2\n\nTranslate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.", "reward": 1.0, "advantage": 0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 10887, 2731, 1345, 2966, 1885, 81, 20, 284, 1885, 82, 20, 15765, 1885, 81, 27, 82, 45, 49160, 26265, 965, 1885, 61, 45, 20, 19573, 16166, 107, 49160, 27, 81, 21990, 49161, 17859, 42639, 16166, 107, 49160, 27, 49161, 278, 24362, 260, 15328, 599, 314, 49154, 49158, 198, 49164, 30, 216, 49161, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 5361, 10756, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 49154], "prefix_len": 76, "old_logprobs": [-2.881516218185425, -0.03053673729300499, -4.172260284423828, -3.2807745933532715, -4.813244819641113, -0.12185550481081009, -10.604605674743652, -1.077478051185608, -0.30144616961479187, -4.413824558258057, -5.857640266418457, -1.492982268333435, -0.5671310424804688, -1.210426688194275, -3.9845781326293945, -2.5507583618164062, -0.1242259219288826, -0.8155735731124878, -0.5026180744171143, -0.251913845539093, -2.529411792755127, -0.149778351187706, -0.8450430631637573, -1.4876989126205444, -0.2718856632709503, -0.1869080662727356, -3.0427935123443604, -0.18700121343135834, -1.4640156030654907, -0.5077035427093506, -1.4796782732009888, -0.9137620329856873, -0.8676198124885559]}
+{"group_id": "math_092", "origin": "budgie", "answer": "The simplest geometric reasoning follows from the problem statement: the vertices of a rectangle inscribed in a circle divide it into four arcs, and the distance from the midpoint of one of the larger arcs to the vertices of the rectangle is equal to the sum of the lengths of the arcs.\n\nFinal answer:\n#### 24", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 49164, 49166, 30, 378, 24796, 282, 253, 23710, 30036, 281, 253, 7842, 10335, 357, 618, 1876, 36260, 30, 7985, 260, 4052, 429, 260, 3921, 3154, 282, 582, 282, 260, 3227, 36260, 288, 260, 24796, 282, 260, 23710, 28, 585, 624, 5882, 359, 216, 49161, 49163, 284, 216, 49166, 5973, 30, 49154, 49158, 198, 504, 17519, 15932, 10115, 5558, 429, 260, 1732, 4814, 42, 260, 24796, 282, 253, 23710, 30036, 281, 253, 7842, 10335, 357, 618, 1876, 36260, 28, 284, 260, 4052, 429, 260, 3921, 3154, 282, 582, 282, 260, 3227, 36260, 288, 260, 24796, 282, 260, 23710, 314, 4037, 288, 260, 2028, 282, 260, 15900, 282, 260, 36260, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49161, 49163, 49154], "prefix_len": 81, "old_logprobs": [-0.9225724935531616, -8.11801815032959, -2.475101947784424, -2.399379253387451, -2.098318099975586, -0.10373091697692871, -0.10450682789087296, -3.714146137237549, -1.0484709739685059, -1.7804032564163208, -1.3354288339614868, -3.129392385482788, -0.26005586981773376, -0.1865401417016983, -0.016722973436117172, -0.06445910781621933, -0.0024624522775411606, -0.019344717264175415, -0.0011403016978874803, -0.10942763090133667, -0.01949448697268963, -0.0004804172203876078, -0.050700411200523376, -0.0034445736091583967, -1.9437302350997925, -0.7639788389205933, -0.3310529589653015, -1.5954161882400513, -0.02484295703470707, -0.012217800132930279, -0.032082244753837585, -0.0010390841634944081, -0.0017922543920576572, -0.01971670426428318, -0.004711597226560116, -0.1482938677072525, -0.019299348816275597, -0.0033154303673654795, -0.012198487296700478, -0.015538342297077179, -0.028347894549369812, -0.06154847517609596, -0.0026814716402441263, -0.0013087764382362366, -0.1445506364107132, -1.2931203842163086, -0.025262190029025078, -0.3255472481250763, -3.166843891143799, -0.021751299500465393, -0.09776949882507324, -1.4930616617202759, -0.0056566642597317696, -0.16373437643051147, -1.9984781742095947, -0.733363926410675, -0.35564538836479187, -0.008282124996185303, -0.18282464146614075, -0.0004895919119007885, -0.0008877150830812752, -0.16668610274791718, -0.0015323336701840162, -0.016827890649437904, -0.2285822331905365, -0.20685425400733948, -0.27670755982398987]}
+{"group_id": "math_092", "origin": "budgie", "answer": "To find the distance from the midpoint of one of the larger arcs to the vertices of the rectangle, we need to first understand the geometric relationship between the larger arcs and the rectangle.\n\nThe smaller arcs can be seen as parts of a circle. The larger arcs are the ones that make up the rectangle. These larger arcs, when considered as a whole, will form a full circle.\n\nThe distance from the midpoint of one of the larger arcs to the vertices of the rectangle can be found by considering the inscribed rectangle and the large circle. The rectangle is inscribed in the circle, and the vertices of the rectangle are the points where the circle touches the sides of the rectangle.\n\nSince the rectangle is inscribed in the circle, the diagonals of the rectangle are equal to the diameter of the circle. Let's denote the distance from the midpoint of one of the larger arcs to the vertices of the rectangle as d.\n\nThe diagonals of a rectangle bisect each other, so the distance from the midpoint of one of the larger arcs to the vertices of the rectangle can be found using the Pythagorean theorem in the right triangle formed by the radius of the circle (24 cm), half the length of the larger arc, and half the length of the smaller arc.\n\nThe distance d is equal to the length of the midpoint of the larger arc divided by 2 (since the two halves of the arc are equal in length).\n\n\\[ d = \\frac{24}{2} \\]\n\n\\[ d = 12 \\text{ cm} \\]\n\nTherefore, the distance from the midpoint of one of the larger arcs to the vertices of the rectangle is 12 cm.\n\n\\[ \\boxed{12} \\]", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 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+{"group_id": "math_092", "origin": "verified_reference", "answer": "1.57. Since $A C$ is the diameter of the circle (Fig. 1.44), then $R=0.5 \\sqrt{24^{2}+7^{2}}=12.5 \\quad$ (cm). In $\\triangle B O F$ we have $O F=$ $=\\sqrt{O B^{2}-B F^{2}}=\\sqrt{12.5^{2}-12^{2}}=3.5$ (cm); hence, $M F=12.5-3.5=9$ (cm), $M K=12.5+3.5=16$ (cm). From $\\triangle M B F$ and $\\triangle M A K$ we find the required distances: $\\quad M B=\\sqrt{M F^{2}+B F^{2}}=\\sqrt{9^{2}+12^{2}}=15(\\mathrm{~cm})$, $M A=\\sqrt{M K^{2}+K A^{2}}=\\sqrt{16^{2}+12^{2}}=20(\\mathrm{~cm})$.\n\nAnswer: 15 and 20 cm.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 49164, 49166, 30, 378, 24796, 282, 253, 23710, 30036, 281, 253, 7842, 10335, 357, 618, 1876, 36260, 30, 7985, 260, 4052, 429, 260, 3921, 3154, 282, 582, 282, 260, 3227, 36260, 288, 260, 24796, 282, 260, 23710, 28, 585, 624, 5882, 359, 216, 49161, 49163, 284, 216, 49166, 5973, 30, 49154, 49158, 198, 49160, 30, 49164, 49166, 30, 4311, 1885, 49, 340, 20, 314, 260, 8805, 282, 260, 7842, 365, 10462, 30, 216, 49160, 30, 49163, 49163, 643, 965, 1885, 66, 45, 49159, 30, 49164, 3814, 16166, 107, 49161, 49163, 21990, 49161, 109, 27, 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+{"group_id": "math_093", "origin": "budgie", "answer": "15. The area of square $\\mathrm{ABCD}$ is 196 square centimeters. The square is divided into 4 smaller squares by dividing each side into two equal segments. Therefore, the side length of each smaller square is 12 cm.\n\nThe area of the shaded part is 1 small square cm.\n\nFinal answer:\n#### 1", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49164, 30, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 28, 284, 357, 3593, 827, 11729, 20168, 3764, 16207, 30, 378, 3227, 282, 260, 827, 3764, 16207, 553, 354, 1557, 338, 314, 216, 49163, 1711, 260, 1557, 282, 260, 3764, 582, 28, 284, 260, 20168, 1557, 282, 260, 827, 16207, 314, 216, 49160, 5222, 46595, 30, 4882, 28, 260, 1557, 282, 260, 32416, 599, 314, 19573, 97, 32984, 20, 5222, 25556, 30, 49154, 49158, 198, 49160, 49164, 30, 378, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 30, 378, 5222, 314, 6066, 618, 216, 49163, 3764, 16207, 411, 15213, 971, 2103, 618, 827, 4037, 12523, 30, 4882, 28, 260, 2103, 3519, 282, 971, 3764, 5222, 314, 216, 49160, 49161, 5973, 30, 198, 198, 504, 1557, 282, 260, 32416, 599, 314, 216, 49160, 1165, 5222, 5973, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49154], "prefix_len": 118, "old_logprobs": [-0.5755974650382996, -1.2770622968673706, -0.02278691530227661, -1.571797251701355, -0.85223388671875, -0.020022420212626457, -1.0354936122894287, -0.5070701241493225, -0.00289022084325552, -0.00023934361524879932, -1.8596476365928538e-05, -0.12913626432418823, -0.0025201248936355114, -0.004853609949350357, -0.0412442684173584, -0.20215484499931335, -0.009424482472240925, -0.012954599224030972, -0.0005149707430973649, -0.05195244774222374, -0.011320053599774837, -1.0292720794677734, -0.8696209788322449, -3.043081283569336, -2.2533950805664062, -0.5496581196784973, -0.09482023864984512, -1.236257553100586, -1.5183311700820923, -0.48693495988845825, -0.09079738706350327, -1.2180376052856445, -3.796616554260254, -1.563967227935791, -0.16696274280548096, -0.4388931095600128, -1.4668605327606201, -0.13274359703063965, -2.5360066890716553, -0.5679627656936646, -2.532691478729248, -0.011691974475979805, -0.29342275857925415, -0.8611952066421509, -0.06697005778551102, -0.019261227920651436, -0.5855500102043152, -0.17280147969722748, -0.006078921724110842, -0.01304708793759346, -0.43452200293540955, -0.5153100490570068, -1.5189309120178223, -0.9432300329208374, -0.14479860663414001, -0.99957275390625, -0.40196138620376587, -1.1500089168548584, -0.4706791043281555, -0.005474337842315435, -0.6900084018707275, -0.892620325088501, -0.19035573303699493, -0.17088738083839417, -1.4987080097198486, -0.4772089123725891, -5.316197872161865, -0.027098534628748894, -1.2322791814804077, -0.5750228762626648, -1.0874078273773193, -0.1757064014673233, -1.173536777496338, -0.0005969410995021462, -0.0020638848654925823, -0.046831920742988586, -0.000998951611109078, -0.060320522636175156, -0.005276918411254883, -0.031155481934547424]}
+{"group_id": "math_093", "origin": "budgie", "answer": "1. We need to find the area of the shaded part. Since the larger of the two smaller squares has an area of 4 times the area of the smaller one, and the overlapping area of the two squares is a square centimeter, the total shaded area is 4 cm^2.\n2. Now, we need to find the side length of the larger square. From the diagram, we can see that the total area of the larger square is 196 cm^2, which is equal to the side length squared. Let $s$ be the side length of the larger square, then:\n$$s^2 = 196$$\n\nSolving for $s$, we get:\n$$s = \\sqrt{196} = 14\\, cm$$\n\nNow that we have the side length of the larger square, we can find its area. We can do this by multiplying the side length by itself twice:\n$$s^2 = (s)(s) = (14) \\times (14) = 196\\, cm^2$$\n\nTherefore, the area of the shaded part is:\n$$\\boxed{196\\, cm^2}$$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49164, 30, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 28, 284, 357, 3593, 827, 11729, 20168, 3764, 16207, 30, 378, 3227, 282, 260, 827, 3764, 16207, 553, 354, 1557, 338, 314, 216, 49163, 1711, 260, 1557, 282, 260, 3764, 582, 28, 284, 260, 20168, 1557, 282, 260, 827, 16207, 314, 216, 49160, 5222, 46595, 30, 4882, 28, 260, 1557, 282, 260, 32416, 599, 314, 19573, 97, 32984, 20, 5222, 25556, 30, 49154, 49158, 198, 49160, 30, 1046, 737, 288, 1042, 260, 1557, 282, 260, 32416, 599, 30, 4311, 260, 3227, 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+{"group_id": "math_093", "origin": "verified_reference", "answer": "【Answer】 72\n【Analysis】Key point: Skillful area calculation\nThe area of the square is 196, so the side length is 14. The overlapping area is 1, so the side length is 1; the area of the larger square is 4 times that of the smaller square, so the side length of the larger square is twice that of the smaller square, and the sum of the side lengths of the larger and smaller squares is $14+1=15$.\nTherefore, the side length of the smaller square is $15 \\div 3=5$, and the side length of the larger square is $5 \\times 2=10$.\nThe area of the smaller rectangle is $(5-1) \\times(10-1)=36$, so the area of the two smaller rectangles is $36 \\times 2=72\\left(\\mathrm{~cm}^{2}\\right)$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49164, 30, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 28, 284, 357, 3593, 827, 11729, 20168, 3764, 16207, 30, 378, 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+{"group_id": "math_094", "origin": "budgie", "answer": "321. $\\sqrt{2x+5}+\\sqrt{x-1}=8$\n\nThe final answer is 321.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 1792, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 28422, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 4825, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 1116, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 1792, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 28422, 49154, 49158, 198, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 104, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 20, 198, 198, 504, 3403, 2988, 314, 216, 49162, 49161, 49160, 30, 49154], "prefix_len": 101, "old_logprobs": [-1.4745551347732544, -0.02883770316839218, -0.004770801402628422, -0.06657921522855759, -0.3931163549423218, -0.006997010204941034, -0.001413180842064321, -0.022863933816552162, -2.1348114013671875, -0.031304072588682175, -0.0021686863619834185, -0.02322261780500412, -0.07187259942293167, -0.0005793085438199341, -0.00027044929447583854, -0.002303448738530278, -0.0014575821114704013, -6.639736966462806e-05, -0.002167734783142805, -0.004769971128553152, -0.00416210200637579, -1.7389655113220215, -1.1067054271697998, -0.00705856503918767, -2.281137704849243, -1.993486762046814, -0.018200531601905823, -0.28281915187835693, -0.650902509689331, -0.9431203007698059, -0.043964024633169174, -0.007983086630702019, -0.7139750719070435, -0.09577511996030807]}
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+{"group_id": "math_094", "origin": "verified_reference", "answer": "Solution. Isolate one of the radicals and square both sides of the equation:\n\n$$\n(\\sqrt{2 x+5})^{2}=(8-\\sqrt{x-1})^{2} ; 2 x+5=64-16 \\sqrt{x-1}+(x-1)\n$$\n\nMove $16 \\sqrt{x-1}$ to the left side, and all other terms to the right side:\n\n$$\n16 \\sqrt{x-1}=64+x-1-2 x-5 ; 16 \\sqrt{x-1}=58-x\n$$\n\nSquare both sides of the equation again:\n\n$(16 \\sqrt{x-1})^{2}=(58-x)^{2} ; 256 x-256=3364-116 x+x^{2} ; x^{2}-372 x+3620=0$.\n\nSolve the resulting quadratic equation. Since $a=1, b=-372$, $c=3620, \\quad D=b^{2}-4 a c=(-372)^{2}-4 \\cdot 1 \\cdot 3620=138384-14480=123904$; $\\sqrt{D}=352$, then\n\n$$\nx_{1}=\\frac{-b-\\sqrt{D}}{2 a}=\\frac{372-352}{2}=10 ; \\quad x_{2}=\\frac{-b+\\sqrt{D}}{2 a}=\\frac{372+352}{2}=362\n$$\n\nCheck both values $x_{1}=10 ; x_{2}=362$.\n\nIf $x=10$, then $\\sqrt{2 \\cdot 10+5}+\\sqrt{10-1}=8 ; \\sqrt{25}+\\sqrt{9}=8 ; 5+3=8$; $8=8$\n\nIf $x=362$, then $\\sqrt{2 \\cdot 362+5}+\\sqrt{362-1}=8 ; \\sqrt{729}+\\sqrt{361}=8 ; 27+$ $+19 \\neq 8$; therefore, $x=362$ is an extraneous root.\n\nThe root of the given equation is $x=10$.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 1792, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 28422, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 4825, 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+{"group_id": "math_095", "origin": "budgie", "answer": "1. **Define the Polynomial and its Properties:**\n\n   A polynomial with integer coefficients is given by:\n   \\[\n   P(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0\n   \\]\n   where \\(a_i\\) are integers.\n\n2. **Given Conditions:**\n\n   We know:\n   \\[\n   P(1) = 2019, \\quad P(2019) = 1, \\quad P(k) = k\n   \\]\n\n3. **Analyze the Polynomial at \\(x = \\frac{1}{k}\\):**\n\n   Since \\(P(1) = 2019\\), substituting \\(x = 1\\) gives:\n   \\[\n   2019 = a_4 + a_3 + a_2 + a_1 + a_0\n   \\]\n   Therefore:\n   \\[\n   2019 = a_4 + a_3 + a_2 + a_1 + a_0 \\implies a_4 + a_3 + a_2 + a_1 = 1\n   \\]\n\n4. **Analyze the Polynomial at \\(x = \\frac{2}{k}\\):**\n\n   Similarly, substituting \\(x = 2019\\) into the polynomial yields:\n   \\[\n   2019 = a_4(2019)^4 + a_3(2019)^3 + a_2(2019)^2 + a_1(2019) + a_0\n   \\]\n   Thus:\n   \\[\n   2019 = a_4(2019)^4 + a_", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 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+{"group_id": "math_095", "origin": "budgie", "answer": "1998", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49162, 30, 198, 198, 49, 23539, 1885, 64, 24, 104, 38224, 351, 15328, 20432, 553, 260, 3849, 198, 198, 9212, 198, 64, 24, 49160, 36637, 49161, 49159, 49160, 49168, 28, 3814, 32984, 377, 24, 49161, 49159, 49160, 49168, 36637, 49160, 28, 3814, 32984, 377, 24, 91, 36637, 91, 28, 198, 9212, 198, 198, 2942, 260, 1230, 1885, 91, 20, 314, 354, 15328, 30, 7985, 451, 1230, 1885, 91, 28422, 198, 198, 19, 49154, 49158, 198, 49160, 49168, 49168, 49167, 49154], "prefix_len": 105, "old_logprobs": [-1.392725944519043, -2.4470789432525635, -1.060302734375, -2.2288358211517334, -1.0910454988479614]}
+{"group_id": "math_095", "origin": "verified_reference", "answer": "# Solution.\n\nSince the polynomial $P(x)$ has integer coefficients, $P(a)-P(b)$ is divisible by $a-b$ for any integers $a$ and $b$.\n\nWe get that\n\n$$\n\\begin{gathered}\nP(k)-P(1)=(k-2019) \\text { is divisible by }(k-1), \\\\\nP(k)-P(2019)=(k-1) \\text { is divisible by }(k-2019) .\n\\end{gathered}\n$$\n\nThis can only be true if $|k-1|=|k-2019|$.\n\nThe solution to the obtained equation is $k=1010$.\n\nAnswer: $\\quad k=1010$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49162, 30, 198, 198, 49, 23539, 1885, 64, 24, 104, 38224, 351, 15328, 20432, 553, 260, 3849, 198, 198, 9212, 198, 64, 24, 49160, 36637, 49161, 49159, 49160, 49168, 28, 3814, 32984, 377, 24, 49161, 49159, 49160, 49168, 36637, 49160, 28, 3814, 32984, 377, 24, 91, 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diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/bootstrap/math_encoding_manifest.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/bootstrap/math_encoding_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7323ffc7ac8516982e68c83c85d9b0b720feed7
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/bootstrap/math_encoding_manifest.json
@@ -0,0 +1,7 @@
+{
+  "groups": 72,
+  "encoded": 216,
+  "mean_tokens": 443.5925925925926,
+  "mean_response_tokens": 307.73148148148147,
+  "seconds": 3.2618279457092285
+}
\ No newline at end of file