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151643)]] | 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 answering.
Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when
\[\prod_{k=0}^{12}(2-2\omega^k+\omega^{2k})\]
is divided by 1000. | <|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' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before answering.
Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when
\[\prod_{k=0}^{12}(2-2\omega^k+\omega^{2k})\]
is divided by 1000.<|im_end|>
<|im_start|>assistant
| ['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, 79075, 90, 11692, 39351, 92, 12947, 358, 3900, 432, 374, 4396, 6, 320, 28996, 17194, 8, 1380, 96704, 640, 374, 1101, 279, 1590, 1372, 476, 7493, 429, 67477, 279, 3491, 13, 21149, 3019, 553, 3019, 1573, 35764, 382, 10061, 57960, 32696, 59, 811, 80, 220, 16, 3, 387, 264, 220, 16, 18, 339, 3704, 315, 30326, 13, 7379, 279, 26313, 979, 198, 59, 26056, 19748, 15159, 74, 28, 15, 92, 47822, 16, 17, 25547, 17, 12, 17, 59, 32696, 61, 74, 41715, 32696, 47822, 17, 74, 5410, 59, 921, 285, 17779, 553, 220, 16, 15, 15, 15, 13, 151645, 198, 151644, 77091, 198]] | {'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 product. For \\(k = 0\\), we have:\n\n\\[\n2 - 2\\omega^0 + \\omega^0 = 2 - 2 \\cdot 1 + 1 = 1\n\\]\n\nFor \\(k \\neq 0\\), we can rewrite the term as:\n\n\\[\n2 - 2\\omega^k + \\omega^{2k} = 2(1 - \\omega^k) + \\omega^{2k}\n\\]\n\nWe need to find the product of these terms for \\(k = 0, 1, 2, \\ldots, 12\\). Notice that the polynomial \\(x^{13} - 1\\) can be factored as:\n\n\\[\nx^{13} - 1 = (x-1)(x-\\omega)(x-\\omega^2) \\cdots (x-\\omega^{12})\n\\]\n\nWe are interested in the polynomial \\(P(x) = x^{13} - 1\\) evaluated at \\(x = 2\\), but with a modification. Specifically, we need to evaluate the product of the terms \\(2 - 2\\omega^k + \\omega^{2k}\\) for \\(k = 0, 1, 2, \\ldots, 12\\). This can be related to the polynomial \\(Q(x) = x^{13} - 1\\) by considering the transformation \\(x \\to 2x - x^2\\).\n\nHowever, a more straightforward approach is to use the fact that the product of the roots of the polynomial \\(x^{13} - 1\\) shifted by a certain transformation can be related to the polynomial itself. Specifically, we can use the fact that the product of the terms \\(2 - 2\\omega^k + \\omega^{2k}\\) is related to the constant term of the polynomial \\(P(x) = x^{13} - 1\\) evaluated at a specific transformation.\n\nBy properties of roots of unity and polynomial transformations, we can deduce that the product of the terms \\(2 - 2\\omega^k + \\omega^{2k}\\) for \\(k = 0, 1, 2, \\ldots, 12\\) is equal to \\(3^{13}\\). This is because each term \\(2 - 2\\omega^k + \\omega^{2k}\\) can be seen as a transformation of the roots of unity, and the product of these transformations is a power of 3.\n\nThus, we have:\n\n\\[\n\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k}) = 3^{13}\n\\]\n\nWe need to find the remainder when \\(3^{13}\\) is divided by 1000. We can use modular arithmetic to simplify this calculation. First, we find \\(3^{13} \\mod 8\\) and \\(3^{13} \\mod 125\\), and then use the Chinese Remainder Theorem to combine these results.\n\nSince \\(3^2 = 9 \\equiv 1 \\mod 8\\), we have:\n\n\\[\n3^{13} = 3 \\cdot (3^2)^6 \\equiv 3 \\cdot 1^6 \\equiv 3 \\mod 8\n\\]\n\nNext, we find \\(3^{13} \\mod 125\\). Using Euler's theorem, since \\(\\phi(125) = 100\\), we have \\(3^{100} \\equiv 1 \\mod 125\\). Therefore:\n\n\\[\n3^{13} \\equiv 3^{13} \\mod 125\n\\]\n\nWe can calculate \\(3^{13} \\mod 125\\) directly:\n\n\\[\n3^2 = 9, \\quad 3^4 = 81, \\quad 3^8 = 6561 \\equiv 61 \\mod 125, \\quad 3^{12} = 3^{12} \\cdot 3^2 = 531441 \\cdot 9 \\equiv 61 \\cdot 9 = 549 \\equiv 49 \\mod 125\n\\]\n\nThus:\n\n\\[\n3^{13} = 3^{12} \\cdot 3 \\equiv 49 \\cdot 3 = 147 \\equiv 22 \\mod 125\n\\]\n\nNow we use the Chinese Remainder Theorem to combine the results \\(3^{13} \\equiv 3 \\mod 8\\) and \\(3^{13} \\equiv 22 \\mod 125\\). We need to find a number \\(x\\) such that:\n\n\\[\nx \\equiv 3 \\mod 8 \\quad \\text{and} \\quad x \\equiv 22 \\mod 125\n\\]\n\nWe can write \\(x = 125k + 22\\) for some integer \\(k\\), and we need:\n\n\\[\n125k + 22 \\equiv 3 \\mod 8 \\implies 5k + 6 \\equiv 3 \\mod 8 \\implies 5k \\equiv -3 \\equiv 5 \\mod 8 \\implies k \\equiv 1 \\mod 8\n\\]\n\nSo \\(k = 8m + 1\\) for some integer \\(m\\), and:\n\n\\[\nx = 125(8m + 1) + 22 = 1000m + 147\n\\]\n\nThus, \\(x \\equiv 147 \\mod 1000\\). Therefore, the remainder when \\(3^{13}\\) is divided by 1000 is:\n\n\\[\n\\boxed{147}\n\\]"], [-0.16153813898563385, -0.226798877120018, -0.060627762228250504, -0.25245317816734314, -0.0535094179213047, -0.07014262676239014, -0.6477265954017639, -0.002493845531716943, -0.48017948865890503, -0.0002857038634829223, -0.021843327209353447, -1.3708974620385561e-05, -1.2755313036905136e-05, -0.005066057667136192, -1.5497195136049413e-06, -8.987976616481319e-05, -0.5768731236457825, -0.0001399419124936685, -8.4638240878121e-06, -2.825220326485578e-05, -2.1576648578047752e-05, -7.510157047363464e-06, 0.0, -4.768370445162873e-07, -0.05490950122475624, -0.7593214511871338, -0.001919809728860855, -8.344646857949556e-07, -3.3378546504536644e-06, -0.014757518656551838, -7.152555099310121e-07, -1.0251946150674485e-05, -0.00010954733443213627, -2.3841855067985307e-07, -4.768370445162873e-07, 0.0, -9.7508447652217e-05, -8.344646857949556e-07, -0.14374035596847534, -9.65590606938349e-06, 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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 answering.
Let $A$, $B$, $C$, and $D$ be point on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi. | <|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' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before answering.
Let $A$, $B$, $C$, and $D$ be point on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi.<|im_end|>
<|im_start|>assistant
| ['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, 79075, 90, 11692, 39351, 92, 12947, 358, 3900, 432, 374, 4396, 6, 320, 28996, 17194, 8, 1380, 96704, 640, 374, 1101, 279, 1590, 1372, 476, 7493, 429, 67477, 279, 3491, 13, 21149, 3019, 553, 3019, 1573, 35764, 382, 10061, 400, 32, 54876, 400, 33, 54876, 400, 34, 54876, 323, 400, 35, 3, 387, 1459, 389, 279, 17071, 65, 7924, 57960, 37018, 45340, 61, 17, 15170, 17, 15, 19732, 1124, 37018, 90, 88, 61, 17, 15170, 17, 19, 92, 284, 220, 16, 3, 1741, 429, 400, 1867, 6484, 3, 374, 264, 21669, 2855, 355, 6693, 84018, 1127, 31796, 518, 279, 6238, 13, 7379, 279, 12196, 1931, 1372, 429, 374, 2686, 1091, 400, 9548, 61, 17, 3, 369, 678, 1741, 21669, 2855, 72, 13, 151645, 198, 151644, 77091, 198]] | {'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 hyperbola is \\(\\frac{x^2}{20} - \\frac{y^2}{24} = 1\\). The diagonals of the rhombus intersect at the origin, which means the rhombus is symmetric with respect to both the x-axis and y-axis.\n\n### Step 2: Coordinates of Points on the Hyperbola\nLet the coordinates of points \\(A\\), \\(B\\), \\(C\\), and \\(D\\) be \\((x_1, y_1)\\), \\((-x_1, -y_1)\\), \\((x_2, y_2)\\), and \\((-x_2, -y_2)\\) respectively. Since these points lie on the hyperbola, they satisfy the equation:\n\\[\n\\frac{x_1^2}{20} - \\frac{y_1^2}{24} = 1 \\quad \\text{and} \\quad \\frac{x_2^2}{20} - \\frac{y_2^2}{24} = 1\n\\]\n\n### Step 3: Length of Diagonals\nThe diagonals of the rhombus are \\(AC\\) and \\(BD\\). The length of diagonal \\(AC\\) is:\n\\[\nAC = \\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\n\\]\nThe length of diagonal \\(BD\\) is:\n\\[\nBD = \\sqrt{(x_1 + x_2)^2 + (y_1 + y_2)^2}\n\\]\n\n### Step 4: Relationship Between Diagonals\nSince \\(ABCD\\) is a rhombus, the diagonals are perpendicular bisectors of each other. Therefore, the product of the lengths of the diagonals is related to the area of the rhombus. However, we are interested in \\(BD^2\\).\n\n### Step 5: Simplifying \\(BD^2\\)\nUsing the properties of the hyperbola and the symmetry of the rhombus, we can express \\(BD^2\\) in terms of the coordinates of the points. Since the diagonals intersect at the origin, we can use the fact that the product of the slopes of the diagonals is \\(-1\\).\n\n### Step 6: Using the Hyperbola Equation\nFrom the hyperbola equation, we have:\n\\[\n\\frac{x_1^2}{20} - \\frac{y_1^2}{24} = 1 \\quad \\text{and} \\quad \\frac{x_2^2}{20} - \\frac{y_2^2}{24} = 1\n\\]\nMultiplying these two equations:\n\\[\n\\left(\\frac{x_1^2}{20} - \\frac{y_1^2}{24}\\right) \\left(\\frac{x_2^2}{20} - \\frac{y_2^2}{24}\\right) = 1 \\cdot 1\n\\]\n\\[\n\\frac{x_1^2 x_2^2}{400} - \\frac{x_1^2 y_2^2}{480} - \\frac{y_1^2 x_2^2}{480} + \\frac{y_1^2 y_2^2}{576} = 1\n\\]\n\n### Step 7: Finding \\(BD^2\\)\nTo find \\(BD^2\\), we use the fact that the diagonals of a rhombus are perpendicular and bisect each other. The length of \\(BD\\) is:\n\\[\nBD = \\sqrt{(x_1 + x_2)^2 + (y_1 + y_2)^2}\n\\]\nSquaring both sides:\n\\[\nBD^2 = (x_1 + x_2)^2 + (y_1 + y_2)^2\n\\]\n\n### Step 8: Maximizing \\(BD^2\\)\nTo maximize \\(BD^2\\), we need to consider the maximum possible values of \\(x_1, x_2, y_1,\\) and \\(y_2\\) that satisfy the hyperbola equation. The maximum value of \\(x_1^2\\) and \\(x_2^2\\) is 20, and the maximum value of \\(y_1^2\\) and \\(y_2^2\\) is 24.\n\n### Step 9: Calculating the Maximum \\(BD^2\\)\nThe maximum value of \\(BD^2\\) occurs when \\(x_1 = x_2\\) and \\(y_1 = -y_2\\) (or vice versa). 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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 answering.
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$,$AB=107$,$FG=17$, and $EF=184$, what is the length of $CE$? | <|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' (without quotes) where ANSWER is just the final number or expression that solves the problem. Think step by step before answering.
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$,$AB=107$,$FG=17$, and $EF=184$, what is the length of $CE$?<|im_end|>
<|im_start|>assistant
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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) \\)\n- \\( B = (107, 0) \\)\n- \\( C = (107, 16) \\)\n- \\( D = (0, 16) \\)\n- \\( E = (x, y) \\)\n- \\( F = (x + 17, y) \\)\n- \\( G = (x + 17, y + 184) \\)\n- \\( H = (x, y + 184) \\)\n\nSince \\( D, E, C, F \\) are collinear, the slope of \\( DE \\) must be equal to the slope of \\( CF \\).\n\nThe slope of \\( DE \\) is:\n\\[\n\\text{slope of } DE = \\frac{y - 16}{x - 0} = \\frac{y - 16}{x}\n\\]\n\nThe slope of \\( CF \\) is:\n\\[\n\\text{slope of } CF = \\frac{y + 184 - 16}{x + 17 - 107} = \\frac{y + 168}{x - 90}\n\\]\n\nSetting the slopes equal to each other:\n\\[\n\\frac{y - 16}{x} = \\frac{y + 168}{x - 90}\n\\]\n\nCross-multiplying to solve for \\( y \\):\n\\[\n(y - 16)(x - 90) = (y + 168)x\n\\]\n\\[\nxy - 90y - 16x + 1440 = xy + 168x\n\\]\n\\[\n-90y - 16x + 1440 = 168x\n\\]\n\\[\n-90y = 184x - 1440\n\\]\n\\[\ny = \\frac{1440 - 184x}{90}\n\\]\n\\[\ny = \\frac{1440}{90} - \\frac{184x}{90}\n\\]\n\\[\ny = 16 - \\frac{184x}{90}\n\\]\n\\[\ny = 16 - \\frac{92x}{45}\n\\]\n\nNext, we use the fact that \\( A, D, H, G \\) lie on a circle. The coordinates of \\( A, D, H, G \\) are:\n- \\( A = (0, 0) \\)\n- \\( D = (0, 16) \\)\n- \\( H = (x, y + 184) \\)\n- \\( G = (x + 17, y + 184) \\)\n\nThe distance from \\( A \\) to \\( D \\) is:\n\\[\nAD = 16\n\\]\n\nThe distance from \\( A \\) to \\( H \\) is:\n\\[\nAH = \\sqrt{x^2 + (y + 184)^2}\n\\]\n\nThe distance from \\( A \\) to \\( G \\) is:\n\\[\nAG = \\sqrt{(x + 17)^2 + (y + 184)^2}\n\\]\n\nSince \\( A, D, H, G \\) lie on a circle, the distances \\( AD, AH, \\) and \\( AG \\) must be equal. However, we can use the fact that \\( D, E, C, F \\) are collinear and the slopes are equal to find \\( x \\).\n\nGiven the collinearity and the slopes, we can solve for \\( x \\) and \\( y \\):\n\\[\ny = 16 - \\frac{92x}{45}\n\\]\n\nSubstitute \\( y \\) back into the collinearity condition:\n\\[\n\\frac{16 - \\frac{92x}{45} - 16}{x} = \\frac{16 - \\frac{92x}{45} + 168}{x - 90}\n\\]\n\\[\n\\frac{-\\frac{92x}{45}}{x} = \\frac{184 - \\frac{92x}{45}}{x - 90}\n\\]\n\\[\n-\\frac{92}{45} = \\frac{184 - \\frac{92x}{45}}{x - 90}\n\\]\n\nSolving this equation, we find:\n\\[\nx = 90\n\\]\n\nSubstitute \\( x = 90 \\) back into the equation for \\( y \\):\n\\[\ny = 16 - \\frac{92 \\cdot 90}{45}\n\\]\n\\[\ny = 16 - \\frac{8280}{45}\n\\]\n\\[\ny = 16 - 184\n\\]\n\\[\ny = -168\n\\]\n\nFinally, the length of \\( CE \\) is:\n\\[\nCE = \\sqrt{(107 - 90)^2 + (16 - (-168))^2}\n\\]\n\\[\nCE = \\sqrt{17^2 + 184^2}\n\\]\n\\[\nCE = \\sqrt{289 + 33856}\n\\]\n\\[\nCE = \\sqrt{34145}\n\\]\n\\[\nCE = 185\n\\]\n\nThus, the length of \\( CE \\) is:\n\\[\n\\boxed{185}\n\\]"], [-0.18171542882919312, -0.12759929895401, -0.7387080192565918, -0.020594218745827675, -0.029647424817085266, -0.14310972392559052, -0.3151920437812805, -4.708655978902243e-05, -1.057112693786621, -3.814689989667386e-06, -4.6967357775429264e-05, -0.001505433232523501, -0.009581885300576687, -0.3875367343425751, -1.3232143828645349e-05, -0.5327608585357666, -0.013570202514529228, -0.0018749530427157879, -0.2722523808479309, -0.6012789607048035, -0.0017489389283582568, -0.20264595746994019, -0.25193458795547485, -0.08378532528877258, -5.960462772236497e-07, -2.3841855067985307e-07, -0.000337305391440168, -1.1920928244535389e-07, -8.344646857949556e-07, -0.00021550717065110803, -2.622600959512056e-06, -4.768360213347478e-06, -1.5497195136049413e-06, -0.8925571441650391, -0.031093312427401543, -0.1030079647898674, -0.883829653263092, -0.20171289145946503, -8.070142939686775e-05, -0.20363563299179077, 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-0.03319544345140457, -0.01607857272028923, -0.6648520827293396, -6.305972783593461e-05, -3.2543604902457446e-05, -2.753696753643453e-05, -0.6931726932525635, -2.7418097943154862e-06, -1.1920928244535389e-07, -0.0028054662980139256, 0.0, 0.0, -1.2040065485052764e-05, -2.50339189733495e-06, -1.4305104514278355e-06, 0.0, -0.14272521436214447, -1.4305104514278355e-06, -0.001764765940606594, -3.3854863431770355e-05, 0.0, -6.556489552167477e-06, -0.0024922997690737247, -0.0006574614671990275, -0.0001833270798670128, -3.349725011503324e-05, -0.0006554362480528653, -0.10295871645212173, -0.00038747431244701147])] | {'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, (...TRUNCATED) | "Solve the following math problem efficiently and clearly. The last line of your response should be(...TRUNCATED) | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ['055'] | [] | "[[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075(...TRUNCATED) | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | "[([\"To solve the problem, we need to understand the relationship between the sets \\\\(A\\\\) and (...TRUNCATED) | "{'extracted_predictions': ['sin(2*x - 1*pi/6)', '\\\\sin (2x - \\\\frac{\\\\pi}{6})'], 'extracted_g(...TRUNCATED) | [0] |
|
[] | "[[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 264, 1140, 315, 6785, 25780, 429, 67901, 279, (...TRUNCATED) | "Solve the following math problem efficiently and clearly. The last line of your response should be(...TRUNCATED) | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ['236'] | [] | "[[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075(...TRUNCATED) | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | "[([\"To solve the problem, we need to find a list of positive integers that satisfies the given con(...TRUNCATED) | {'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, 246(...TRUNCATED) | "Solve the following math problem efficiently and clearly. The last line of your response should be(...TRUNCATED) | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ['721'] | [] | "[[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075(...TRUNCATED) | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | "[(['To solve the problem, we need to find the radius \\\\( r \\\\) of the smallest sphere that can (...TRUNCATED) | {'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(...TRUNCATED) | "Solve the following math problem efficiently and clearly. The last line of your response should be(...TRUNCATED) | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ['104'] | [] | "[[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075(...TRUNCATED) | {'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(...TRUNCATED) | {'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, (...TRUNCATED) | "Solve the following math problem efficiently and clearly. The last line of your response should be(...TRUNCATED) | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ['540'] | [] | "[[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075(...TRUNCATED) | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | "[(['To find the largest possible real part of the expression \\\\((75+117i)z + \\\\frac{96+144i}{z}(...TRUNCATED) | {'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, 1776(...TRUNCATED) | "Solve the following math problem efficiently and clearly. The last line of your response should be(...TRUNCATED) | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ['294'] | [] | "[[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075(...TRUNCATED) | {'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\\\\) (...TRUNCATED) | {'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, 6(...TRUNCATED) | "Solve the following math problem efficiently and clearly. The last line of your response should be(...TRUNCATED) | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ['073'] | [] | "[[151644, 8948, 198, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075(...TRUNCATED) | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | "[([\"To solve this problem, we will use the principle of inclusion-exclusion. Let's define the sets(...TRUNCATED) | "{'extracted_predictions': ['437', '195 + 367 + 562 + 900 - 437 -'], 'extracted_golds': ['73', '73'](...TRUNCATED) | [0] |
End of preview. Expand
in Data Studio
Dataset Card for Evaluation run of Lansechen/Qwen2.5-7B-Open-R1-GRPO-math-lighteval
Dataset automatically created during the evaluation run of model Lansechen/Qwen2.5-7B-Open-R1-GRPO-math-lighteval.
The dataset is composed of 5 configuration, each one coresponding to one of the evaluated task.
The dataset has been created from 23 run(s). Each run can be found as a specific split in each configuration, the split being named using the timestamp of the run.The "train" split is always pointing to the latest results.
An additional configuration "results" store all the aggregated results of the run.
To load the details from a run, you can for instance do the following:
from datasets import load_dataset
data = load_dataset("Lansechen/details_Lansechen__Qwen2.5-7B-Open-R1-GRPO-math-lighteval",
"results",
split="train")
Latest results
These are the latest results from run 2025-04-10T14:54:30.856264(note that their might be results for other tasks in the repos if successive evals didn't cover the same tasks. You find each in the results and the "latest" split for each eval):
{
"all": {
"math_pass@1:32_samples": 0.08958333333333333,
"math_pass@1:32_samples_stderr": 0.04723243353647764,
"extractive_match": 0.06666666666666667,
"extractive_match_stderr": 0.046320555585310084
},
"lighteval|aime24|0": {
"math_pass@1:32_samples": 0.08958333333333333,
"math_pass@1:32_samples_stderr": 0.04723243353647764,
"extractive_match": 0.06666666666666667,
"extractive_match_stderr": 0.046320555585310084
}
}
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