choices stringclasses 1
value | cont_tokens stringlengths 2.12k 163k | example stringlengths 444 1.27k | full_prompt stringlengths 594 1.42k | gold stringlengths 7 7 | gold_index stringclasses 1
value | input_tokens stringlengths 795 2.48k | instruction stringclasses 1
value | metrics stringclasses 2
values | num_asked_few_shots stringclasses 1
value | num_effective_few_shots stringclasses 1
value | padded stringclasses 1
value | pred_logits stringclasses 1
value | predictions stringlengths 9.65k 846k | specifics stringlengths 70 117 | truncated stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
[] | [[(1249, 11625, 279, 3491, 11, 582, 1191, 553, 26305, 429, 1124, 11520, 32696, 57758, 374, 264, 27594, 220, 16, 18, 339, 3704, 315, 30326, 11, 7290, 1124, 11520, 32696, 47822, 16, 18, 92, 284, 220, 16, 57758, 323, 1124, 11520, 32696, 61, 74, 1124, 811, 80, 220, 16, 57758, 369, 17767, 15, 366, 595, 366, 220, 16, 18, 59,... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['321'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we start by noting that \\(\\omega\\) is a primitive 13th root of unity, meaning \\(\\omega^{13} = 1\\) and \\(\\omega^k \\neq 1\\) for \\(0 < k < 13\\). The expression we need to evaluate is:\n\n\\[\n\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\n\\]\n\nFirst, we simplify the term inside the p... | {'extracted_predictions': ['147', '147'], 'extracted_golds': ['321', '321']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 12196, 1931, 1372, 429, 374, 2686, 1091, 17767, 39310, 61, 17, 1124, 8, 369, 678, 21669, 2855, 72, 17767, 1867, 6484, 57758, 1640, 17433, 304, 279, 17071, 65, 7924, 1124, 11520, 37018, 45340, 61, 17, 15170, 17, 15, 92, 481, 1124, 37018, 90, 88, 61, 17, 15170, 17... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['480'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To solve the problem, we need to find the greatest real number that is less than \\( BD^2 \\) for all rhombi \\(ABCD\\) inscribed in the hyperbola \\(\\frac{x^2}{20} - \\frac{y^2}{24} = 1\\) with diagonals intersecting at the origin.\n\n### Step 1: Understanding the Hyperbola and Rhombus Properties\nThe given hyper... | {'extracted_predictions': ['80', '80'], 'extracted_golds': ['480', '480']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 3084, 315, 17767, 26711, 1124, 8, 2661, 279, 4682, 315, 279, 76192, 17767, 19360, 35, 1124, 8, 323, 17767, 468, 12001, 39, 1124, 8, 323, 279, 4530, 482, 10748, 315, 3501, 17767, 422, 11, 468, 11, 356, 11, 434, 1124, 568, 1205, 1083, 1414, 429, 3501, 17767, 362, ... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['104'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to find the length of \\( CE \\) given the conditions of the rectangles \\( ABCD \\) and \\( EFGH \\) and the collinearity of points \\( D, E, C, F \\). We also know that points \\( A, D, H, G \\) lie on a circle.\n\nFirst, let's denote the coordinates of the points:\n- \\( A = (0, 0) ... | {'extracted_predictions': ['185', '185'], 'extracted_golds': ['104', '104']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 3535, 279, 5025, 1948, 279, 7289, 17767, 32, 57758, 323, 17767, 33, 59, 568, 44763, 11, 582, 1184, 311, 8253, 279, 1372, 315, 7289, 17767, 33, 57758, 429, 646, 387, 14122, 3118, 389, 279, 5424, 315, 17767, 32, 59, 3593, 10061, 594, 78064, 279, 5424, 315, 17767, 32, 57758, ... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['055'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to understand the relationship between the sets \\(A\\) and \\(B\\). Specifically, we need to determine the number of sets \\(B\\) that can be formed based on the elements of \\(A\\).\n\nLet's denote the elements of \\(A\\) as \\(a_1, a_2, \\ldots, a_n\\) where \\(a_1 < a_2 < \\cdots <... | {'extracted_predictions': ['sin(2*x - 1*pi/6)', '\\sin (2x - \\frac{\\pi}{6})'], 'extracted_golds': ['55', '55']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 264, 1140, 315, 6785, 25780, 429, 67901, 279, 2661, 4682, 323, 1221, 11047, 279, 2629, 315, 279, 31340, 315, 678, 279, 3589, 304, 279, 1140, 13, 6771, 594, 1438, 1495, 279, 3491, 3019, 553, 3019, 382, 16, 13, 3070, 9190, 315, 279, 3589, 304, 279, 1140, 374, 220, 18, ... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['236'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to find a list of positive integers that satisfies the given conditions and then calculate the sum of the squares of all the items in the list. Let's break down the problem step by step.\n\n1. **Sum of the items in the list is 30.**\n2. **The unique mode of the list is 9.**\n3. **The m... | {'extracted_predictions': ['242', '242'], 'extracted_golds': ['236', '236']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 10578, 17767, 435, 1124, 8, 315, 279, 24632, 25366, 429, 646, 6644, 1817, 315, 279, 51424, 14697, 304, 279, 738, 1124, 11520, 10374, 5416, 90, 33, 11035, 701, 1380, 1817, 3745, 702, 264, 7329, 3082, 315, 220, 20, 19, 323, 264, 8123, 315, 220, 17, 18, 13, 1205, 6... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['721'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To solve the problem, we need to find the radius \\( r \\) of the smallest sphere that can contain each of the rectangular boxes in the set \\(\\mathcal{B}\\), where each box has a surface area of 54 and a volume of 23. We will then determine \\( r^2 \\) and express it in the form \\(\\frac{p}{q}\\), where \\( p \\... | {'extracted_predictions': ['31', '31'], 'extracted_golds': ['721', '721']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 6010, 504, 279, 1459, 17767, 358, 1124, 8, 311, 1817, 315, 279, 12300, 315, 279, 27853, 32507, 291, 2248, 17767, 19360, 35, 1124, 568, 1096, 6010, 374, 279, 304, 26715, 315, 279, 27853, 32507, 291, 2248, 13, 576, 14806, 369, 279, 304, 26715, 17767, 435, 1124, 8,... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['104'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to find the distance from the point \\( I \\) to each of the faces of the tetrahedron \\( ABCD \\). This distance is the inradius of the tetrahedron. The formula for the inradius \\( r \\) of a tetrahedron with volume \\( V \\) and surface area \\( S \\) is given by:\n\n\\[ r = \\frac{... | {'extracted_predictions': ['17', '17'], 'extracted_golds': ['104', '104']} | [0] | |
[] | [[(1249, 1477, 279, 7772, 3204, 1931, 949, 315, 279, 7493, 1124, 1188, 22, 20, 10, 16, 16, 22, 72, 8, 89, 488, 1124, 37018, 90, 24, 21, 10, 16, 19, 19, 72, 15170, 89, 11035, 8, 1380, 17767, 89, 57758, 374, 264, 6351, 1372, 448, 17767, 91, 89, 87803, 19, 59, 701, 582, 1191, 553, 36710, 17767, 89, 57758, 304, 24660, 1352... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['540'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To find the largest possible real part of the expression \\((75+117i)z + \\frac{96+144i}{z}\\) where \\(z\\) is a complex number with \\(|z|=4\\), we start by expressing \\(z\\) in polar form. Let \\(z = 4e^{i\\theta}\\), where \\(\\theta\\) is a real number. Then, the expression becomes:\n\n\\[\n(75+117i)4e^{i\\th... | {'extracted_predictions': ['512', '512'], 'extracted_golds': ['540', '540']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 315, 9271, 279, 1372, 315, 12716, 315, 3084, 220, 16, 21, 389, 458, 17767, 23, 1124, 15136, 220, 23, 57758, 5827, 429, 2297, 5106, 6896, 3040, 3039, 11, 582, 1184, 311, 15516, 23643, 279, 5944, 315, 1741, 12716, 382, 14374, 14822, 220, 16, 25, 45451, 279, 7933, 28596, 198, 32, 1815, 315, 3084... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['294'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To solve the problem of finding the number of paths of length 16 on an \\(8 \\times 8\\) grid that change direction exactly four times, we need to carefully analyze the structure of such paths.\n\n### Step 1: Understanding the Path Structure\nA path of length 16 on an \\(8 \\times 8\\) grid from the lower left corn... | {'extracted_predictions': ['1365', '1365'], 'extracted_golds': ['294', '294']} | [0] | |
[] | [[(1249, 11625, 419, 3491, 11, 582, 686, 990, 279, 17508, 315, 27187, 10187, 8957, 13, 6771, 594, 6979, 279, 7289, 438, 11017, 510, 12, 17767, 422, 1124, 8, 374, 279, 738, 315, 10826, 879, 1828, 264, 22205, 10058, 624, 12, 17767, 479, 1124, 8, 374, 279, 738, 315, 10826, 879, 1828, 264, 738, 315, 19120, 18890, 624, 12, ... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['073'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve this problem, we will use the principle of inclusion-exclusion. Let's define the sets as follows:\n- \\( D \\) is the set of residents who own a diamond ring.\n- \\( G \\) is the set of residents who own a set of golf clubs.\n- \\( S \\) is the set of residents who own a garden spade.\n- \\( C \\) is the s... | {'extracted_predictions': ['437', '195 + 367 + 562 + 900 - 437 -'], 'extracted_golds': ['73', '73']} | [0] | |
[] | [[(1249, 11625, 419, 3491, 11, 582, 1184, 311, 11047, 279, 18927, 315, 24893, 10987, 279, 6662, 21882, 2661, 429, 1340, 2765, 264, 21882, 13, 6771, 594, 1438, 432, 1495, 3019, 553, 3019, 382, 5338, 11, 582, 11047, 279, 2790, 1372, 315, 5510, 311, 5157, 220, 19, 5109, 504, 279, 738, 17767, 328, 284, 1124, 90, 16, 11, 22... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['116'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | [(["To solve this problem, we need to calculate the probability of Jen winning the grand prize given that she won a prize. Let's break it down step by step.\n\nFirst, we calculate the total number of ways to choose 4 numbers from the set \\( S = \\{1, 2, 3, \\ldots, 10\\} \\). This is given by the combination formula \... | {'extracted_predictions': ['116', '116'], 'extracted_golds': ['116', '116']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 3535, 279, 52484, 6546, 315, 279, 12371, 6113, 323, 279, 21495, 14122, 553, 32359, 1181, 11067, 13, 6771, 594, 78064, 279, 3108, 3084, 315, 279, 12371, 6113, 438, 17767, 274, 1124, 3593, 5338, 11, 22986, 429, 279, 12371, 6113, 17767, 19360, 13649, 1124, 8, 374, 3232, 42939... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['080'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to understand the geometric configuration of the hexagon and the triangle formed by extending its sides. Let's denote the side length of the hexagon as \\( s \\).\n\nFirst, observe that the hexagon \\( ABCDEF \\) is equilateral and all pairs of opposite sides are parallel. This means t... | {'extracted_predictions': ['120', '120'], 'extracted_golds': ['80', '80']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 8253, 279, 1372, 315, 5510, 311, 1992, 1045, 315, 279, 220, 17, 20, 4158, 23595, 323, 220, 17, 20, 3691, 23595, 304, 264, 17767, 20, 1124, 15136, 220, 20, 57758, 5827, 1741, 429, 1817, 2779, 5610, 518, 1429, 825, 16392, 11, 678, 23595, 304, 279, 1852, 2802, 323, 678, 23595... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['902'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To solve the problem, we need to determine the number of ways to place some of the 25 white chips and 25 black chips in a \\(5 \\times 5\\) grid such that each cell contains at most one chip, all chips in the same row and all chips in the same column have the same color, and no additional chip can be placed without... | {'extracted_predictions': ['1024', '1024'], 'extracted_golds': ['902', '902']} | [0] | |
[] | [[(1249, 1477, 279, 1372, 315, 76192, 429, 646, 387, 14122, 4766, 264, 8356, 5792, 294, 534, 66, 6113, 320, 16, 17, 2371, 263, 8, 1380, 1817, 3108, 315, 279, 22756, 15448, 389, 2987, 264, 3108, 476, 264, 37236, 315, 279, 294, 534, 66, 6113, 11, 582, 1184, 311, 2908, 279, 5888, 315, 279, 294, 534, 66, 6113, 323, 279, 32... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['315'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To find the number of rectangles that can be formed inside a fixed regular dodecagon (12-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon, we need to consider the properties of the dodecagon and the possible ways to form rectangles.\n\nA regular dodecagon has 12 vertices an... | {'extracted_predictions': ['15', '15'], 'extracted_golds': ['315', '315']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 990, 279, 2661, 4682, 323, 5888, 315, 279, 21495, 11, 10198, 3057, 11, 323, 304, 3057, 13, 6771, 594, 10354, 3019, 553, 3019, 382, 16, 13, 3070, 22043, 31834, 25, 1019, 256, 481, 576, 10198, 26715, 17767, 431, 284, 220, 16, 18, 1124, 4292, 256, 481, 576, 304, 26715, 17767,... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['468'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to use the given conditions and properties of the triangle, circumcenter, and incenter. Let's proceed step by step.\n\n1. **Given Conditions:**\n - The circumradius \\( R = 13 \\).\n - The inradius \\( r = 6 \\).\n - The line segment \\( \\overline{IA} \\) is perpendicular to \\(... | {'extracted_predictions': ['384', '384'], 'extracted_golds': ['468', '468']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 12196, 3040, 48342, 6785, 7546, 17767, 451, 1124, 8, 1741, 429, 10018, 894, 825, 315, 1181, 18509, 311, 220, 16, 3059, 304, 264, 1372, 74916, 553, 220, 22, 13, 6771, 594, 78064, 17767, 451, 1124, 8, 438, 17767, 1124, 1975, 1056, 90, 68644, 92, 1124, 701, 1380, 1... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['699'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to find the greatest four-digit positive integer \\( N \\) such that changing any one of its digits to 1 results in a number divisible by 7. Let's denote \\( N \\) as \\( \\overline{abcd} \\), where \\( a, b, c, \\) and \\( d \\) are its digits. This means \\( N = 1000a + 100b + 10c + ... | {'extracted_predictions': ['3', '3'], 'extracted_golds': ['699', '699']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 686, 990, 279, 5888, 315, 279, 21878, 805, 323, 279, 2355, 315, 264, 1459, 57817, 13, 6771, 594, 1191, 553, 26305, 429, 2474, 17767, 35, 57758, 374, 279, 19246, 315, 279, 21878, 805, 311, 1124, 11520, 32696, 57758, 518, 17767, 33, 57758, 323, 17767, 34, 59, 701, 17767, 35, 57758, 374... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['113'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we will use the properties of the tangents and the power of a point theorem. Let's start by noting that since \\(D\\) is the intersection of the tangents to \\(\\omega\\) at \\(B\\) and \\(C\\), \\(D\\) is the excenter of \\(\\triangle ABC\\) opposite to \\(A\\). This means that \\(D\\) is the... | {'extracted_predictions': ['17', '17'], 'extracted_golds': ['113', '113']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 8253, 279, 304, 26715, 315, 21495, 17767, 25411, 57758, 2661, 279, 26112, 315, 25362, 13, 6771, 594, 1438, 1495, 279, 3491, 3019, 553, 3019, 382, 16, 13, 3070, 70151, 279, 26112, 315, 25362, 25, 1019, 256, 481, 1205, 614, 8063, 25362, 315, 10578, 17767, 18, 19, 57758, 429,... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['197'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to determine the inradius of triangle \\(ABC\\) given the arrangement of circles. Let's break down the problem step by step.\n\n1. **Understanding the arrangement of circles:**\n - We have eight circles of radius \\(34\\) that are sequentially tangent to each other.\n - Two of thes... | {'extracted_predictions': ['r', 'r'], 'extracted_golds': ['197', '197']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 8253, 279, 4911, 1459, 17767, 34, 57758, 389, 279, 1555, 10238, 1124, 11520, 1975, 1056, 90, 1867, 11035, 8, 429, 1558, 537, 9173, 311, 894, 10238, 504, 279, 2997, 1124, 11520, 10374, 5416, 90, 37, 11035, 8, 1008, 1091, 1124, 11520, 1975, 1056, 90, 1867, 11035, 568, 576, 3... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['023'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To solve the problem, we need to determine the unique point \\(C\\) on the line segment \\(\\overline{AB}\\) that does not belong to any segment from the family \\(\\mathcal{F}\\) other than \\(\\overline{AB}\\). The points \\(A\\) and \\(B\\) are given as \\(A = \\left(\\frac{1}{2}, 0\\right)\\) and \\(B = \\left(... | {'extracted_predictions': ['1', '1'], 'extracted_golds': ['23', '23']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 8253, 279, 18927, 429, 264, 5792, 18491, 6113, 646, 387, 45620, 773, 429, 678, 6303, 17228, 835, 705, 518, 9892, 1380, 1052, 1033, 13214, 2518, 17228, 13, 1096, 3363, 429, 1283, 12695, 11, 1817, 6303, 11936, 1969, 387, 304, 264, 2309, 429, 572, 13214, 24367, 553, 264, 2518... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['371'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to determine the probability that a regular octagon can be rotated so that all blue vertices end up at positions where there were originally red vertices. This means that after rotation, each blue vertex must be in a position that was originally occupied by a red vertex.\n\nFirst, let'... | {'extracted_predictions': ['3', '3'], 'extracted_golds': ['371', '371']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1191, 553, 41018, 279, 2661, 37906, 510, 78045, 264, 488, 293, 488, 272, 284, 220, 18, 15, 15, 1124, 921, 78045, 264, 61, 17, 65, 488, 264, 61, 17, 66, 488, 293, 61, 17, 64, 488, 293, 61, 17, 66, 488, 272, 61, 17, 64, 488, 272, 61, 17, 65, 284, 220, 21, 11, 15, 15, 15, 11, 15, 15, 15... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['601'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we start by analyzing the given equations:\n\\[ a + b + c = 300 \\]\n\\[ a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000. \\]\n\nFirst, we rewrite the second equation in a more convenient form:\n\\[ a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = ab(a + b) + ac(a + c) + bc(b + c). \\]\n\nWe know... | {'extracted_predictions': ['3', '3'], 'extracted_golds': ['601', '601']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 8253, 279, 11900, 72, 17767, 435, 5318, 1124, 8, 323, 17767, 435, 14179, 1124, 8, 315, 279, 25362, 315, 21878, 2251, 1948, 279, 5683, 355, 17767, 350, 1124, 8, 323, 279, 25366, 17767, 328, 1124, 8, 979, 279, 5683, 355, 374, 68342, 68660, 311, 279, 25366, 304, 1378, 2155, 3... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['127'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to determine the radii \\( r_i \\) and \\( r_o \\) of the circles of tangency between the torus \\( T \\) and the sphere \\( S \\) when the torus is externally tangent to the sphere in two different configurations. Let's proceed step by step.\n\n### Step 1: Understanding the Geometry o... | {'extracted_predictions': ['4', '4'], 'extracted_golds': ['127', '127']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 3245, 10250, 1372, 17767, 281, 1124, 8, 1741, 429, 1052, 6724, 264, 6785, 7546, 17767, 308, 1124, 8, 369, 892, 17767, 308, 61, 19, 488, 220, 16, 1124, 8, 374, 74916, 553, 17767, 281, 61, 17, 1124, 568, 5005, 11, 582, 1184, 311, 1477, 279, 3245, 6785, 7546, 17767... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['110'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to find the least prime number \\( p \\) such that there exists a positive integer \\( n \\) for which \\( n^4 + 1 \\) is divisible by \\( p^2 \\). Then, we need to find the least positive integer \\( m \\) such that \\( m^4 + 1 \\) is divisible by \\( p^2 \\).\n\nFirst, let's consider... | {'extracted_predictions': ['7', '7'], 'extracted_golds': ['110', '110']} | [0] | |
[] | [[(1249, 11625, 279, 2661, 1849, 315, 37906, 11, 582, 1191, 553, 36710, 279, 89936, 21220, 37906, 304, 3793, 315, 505, 2700, 13, 576, 2661, 37906, 525, 1447, 59, 9640, 59, 839, 62, 17, 59, 2359, 11520, 37018, 45340, 15170, 18348, 11035, 1291, 8, 284, 1124, 37018, 90, 16, 15170, 17, 532, 59, 921, 59, 9640, 59, 839, 62, ... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['033'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(['To solve the given system of equations, we start by expressing the logarithmic equations in terms of exponents. The given equations are:\n\n\\[\n\\log_2\\left(\\frac{x}{yz}\\right) = \\frac{1}{2}\n\\]\n\\[\n\\log_2\\left(\\frac{y}{xz}\\right) = \\frac{1}{3}\n\\]\n\\[\n\\log_2\\left(\\frac{z}{xy}\\right) = \\frac{1}... | {'extracted_predictions': ['8', '8'], 'extracted_golds': ['33', '33']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1191, 448, 279, 2661, 37906, 510, 59, 9640, 59, 839, 3212, 59, 2359, 7021, 61, 87, 59, 1291, 8, 284, 220, 16, 15, 1124, 31610, 1124, 1318, 90, 437, 92, 1124, 31610, 1124, 839, 4178, 59, 2359, 2075, 47822, 19, 88, 11035, 1291, 8, 284, 220, 16, 15, 624, 59, 2533, 5338, 11, 582, 39721, ... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['025'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | [(['To solve the problem, we start with the given equations:\n\\[\n\\log_x\\left(y^x\\right) = 10 \\quad \\text{and} \\quad \\log_y\\left(x^{4y}\\right) = 10.\n\\]\n\nFirst, we simplify each logarithmic expression using the properties of logarithms. Recall that \\(\\log_b(a^c) = c \\log_b(a)\\).\n\nFor the first equati... | {'extracted_predictions': ['25', '25'], 'extracted_golds': ['25', '25']} | [0] | |
[] | [[(1249, 11625, 419, 3491, 11, 582, 1184, 311, 23643, 279, 1809, 323, 8253, 279, 10987, 323, 13188, 9892, 13, 362, 2309, 374, 264, 13188, 2309, 421, 894, 3271, 504, 429, 2309, 11508, 311, 264, 10987, 2309, 11, 323, 264, 10987, 2309, 374, 825, 504, 892, 1052, 374, 518, 3245, 825, 3271, 311, 264, 13188, 2309, 382, 10061,... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['809'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve this problem, we need to analyze the game and determine the winning and losing positions. A position is a losing position if any move from that position leads to a winning position, and a winning position is one from which there is at least one move to a losing position.\n\nLet's denote the number of token... | {'extracted_predictions': ['202', '202'], 'extracted_golds': ['809', '809']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1992, 18509, 304, 264, 220, 17, 87, 18, 5827, 1741, 429, 279, 2629, 315, 279, 1378, 5109, 14122, 553, 5290, 2115, 311, 1290, 374, 220, 24, 24, 24, 11, 323, 279, 2629, 315, 279, 2326, 5109, 14122, 553, 5290, 1909, 311, 5622, 374, 220, 24, 24, 13, 6771, 594, 78064, 279, 1850... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['045'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to place digits in a 2x3 grid such that the sum of the two numbers formed by reading left to right is 999, and the sum of the three numbers formed by reading top to bottom is 99. Let's denote the digits in the grid as follows:\n\n\\[\n\\begin{array}{|c|c|c|}\n\\hline\na & b & c \\\\\n\... | {'extracted_predictions': ['{Eq(a, 9), Eq(b, 0), Eq(c, 9)}', 'a = 9, b = 0, c = 9'], 'extracted_golds': ['45', '45']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 8253, 279, 2750, 315, 17767, 274, 1124, 8, 323, 17767, 259, 1124, 8, 1156, 11, 323, 1221, 990, 1493, 2750, 311, 1477, 279, 882, 4429, 979, 362, 7755, 22479, 518, 17767, 274, 488, 1124, 37018, 90, 16, 15170, 17, 92, 1124, 8, 40568, 817, 6460, 382, 14374, 14822, 220, 16, 25,... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['204'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to determine the values of \\( s \\) and \\( t \\) first, and then use these values to find the time taken when Aya walks at \\( s + \\frac{1}{2} \\) kilometers per hour.\n\n### Step 1: Set up the equations for the given conditions\n\n1. **Condition 1: Walking at \\( s \\) km/h for 4 h... | {'extracted_predictions': ['3', '3'], 'extracted_golds': ['204', '204']} | [0] | |
[] | [[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 1372, 315, 64257, 315, 279, 38999, 315, 279, 5746, 17767, 379, 284, 220, 19, 342, 955, 11520, 15940, 320, 17, 1124, 2493, 856, 7705, 1124, 8, 323, 17767, 856, 284, 220, 19, 342, 955, 11520, 9407, 320, 18, 1124, 2493, 379, 7705, 1124, 3593, 5338, 11, 1077, 594, 2... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['385'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To solve the problem, we need to find the number of intersections of the graphs of the functions \\( y = 4 g(f(\\sin (2 \\pi x))) \\) and \\( x = 4 g(f(\\cos (3 \\pi y))) \\).\n\nFirst, let's analyze the functions \\( f(x) \\) and \\( g(x) \\).\n\nThe function \\( f(x) = ||x| - \\frac{1}{2}| \\) can be simplified a... | {'extracted_predictions': ['4', '4'], 'extracted_golds': ['385', '385']} | [0] | |
[] | [[(1249, 8253, 279, 3245, 7546, 17767, 293, 1124, 709, 80, 220, 17, 1124, 8, 369, 892, 1052, 525, 803, 1091, 5779, 17767, 293, 1124, 7287, 86944, 25780, 11, 582, 1184, 311, 1795, 1493, 7354, 1447, 16, 13, 3070, 16250, 2685, 279, 7271, 315, 17767, 293, 1124, 7287, 86944, 25780, 25, 1019, 256, 481, 362, 6785, 7546, 17767... | Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before an... | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\boxed{ANSWER}$. I hope it is correct' (with... | ['211'] | [] | [[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151645, 198, 151644, 872, 198, 50, 3948, 279, 2701, 6888, 3491, 29720, 323, 9355, 13, 220, 576, 1537, 1555, 315, 697, 2033, 1265, 387, 315, 279, 2701, 3561, 25, 364, 54815, 11, 279, 1590, 4226, 374, 25, 57960, 79... | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | [(["To determine the least integer \\( b \\geq 2 \\) for which there are more than ten \\( b \\)-beautiful integers, we need to follow these steps:\n\n1. **Understand the definition of \\( b \\)-beautiful integers:**\n - A positive integer \\( n \\) is \\( b \\)-beautiful if it has exactly two digits when expressed i... | {'extracted_predictions': ['7', '7'], 'extracted_golds': ['211', '211']} | [0] |
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