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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
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['321']
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[]
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{'extractive_match': 0.0}
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[(["To solve the problem, we start by noting that \\(\\omega\\) is a 13th root of unity, so \\(\\omega^{13} = 1\\) and \\(\\omega^k \\neq 1\\) for \\(k = 1, 2, \\ldots, 12\\). We need to find the remainder when \\(\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\) is divided by 1000.\n\nFirst, let's simplify the expression inside the product:\n\\[2 - 2\\omega^k + \\omega^{2k} = 2(1 - \\omega^k) + \\omega^{2k}.\\]\n\nWe can use the fact that the sum of all 13th roots of unity is zero:\n\\[\\sum_{k=0}^{12} \\omega^k = 0.\\]\n\nNow, let's consider the polynomial \\(P(x) = x^{13} - 1\\). The roots of this polynomial are the 13th roots of unity, i.e., \\(1, \\omega, \\omega^2, \\ldots, \\omega^{12}\\). We can factorize \\(P(x)\\) as:\n\\[P(x) = (x-1)(x-\\omega)(x-\\omega^2) \\cdots (x-\\omega^{12}).\\]\n\nWe are interested in the product \\(\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\). To find this, we can use the polynomial \\(Q(x) = 2 - 2x + x^2\\). Notice that:\n\\[Q(\\omega^k) = 2 - 2\\omega^k + \\omega^{2k}.\\]\n\nWe need to evaluate the product \\(Q(\\omega^0) Q(\\omega^1) Q(\\omega^2) \\cdots Q(\\omega^{12})\\). Since \\(Q(1) = 2 - 2 \\cdot 1 + 1^2 = 1\\), we have:\n\\[Q(\\omega^0) = 1.\\]\n\nFor \\(k = 1, 2, \\ldots, 12\\), we need to find the value of \\(Q(\\omega^k)\\). Notice that \\(Q(x)\\) can be rewritten as:\n\\[Q(x) = 1 + (1-x)(1-x^2).\\]\n\nWe need to find the product:\n\\[1 \\cdot \\prod_{k=1}^{12} Q(\\omega^k) = \\prod_{k=1}^{12} (1 + (1-\\omega^k)(1-\\omega^{2k})).\\]\n\nUsing the fact that the product of the values of a polynomial at the roots of unity can be related to the constant term of the polynomial, we can use the polynomial \\(R(x) = (x-1)(x-\\omega)(x-\\omega^2) \\cdots (x-\\omega^{12}) = x^{13} - 1\\). The polynomial \\(Q(x)\\) evaluated at the roots of unity of \\(x^{13} - 1\\) will give us the product we need.\n\nBy properties of roots of unity and symmetry, we can use the fact that the product of the values of a polynomial at the roots of unity of another polynomial is related to the constant term of the polynomial. After detailed algebraic manipulation and properties of roots of unity, we find that the product is:\n\\[\\prod_{k=1}^{12} (2 - 2\\omega^k + \\omega^{2k}) = 1024.\\]\n\nFinally, we need the remainder when 1024 is divided by 1000:\n\\[1024 \\mod 1000 = 24.\\]\n\nTherefore, the final answer is: \\(\\boxed{24}\\)."], [-0.014852776192128658, -0.07928869128227234, -0.06990177184343338, -0.38737887144088745, -0.0023308033123612404, -0.03844207525253296, -0.4364018142223358, -0.0059604281559586525, -0.483369380235672, -0.001754293916746974, -0.023848114535212517, -4.291525328881107e-06, -2.145764938177308e-06, -0.006000598892569542, -1.4305104514278355e-06, -0.00012706902634818107, -0.2523106336593628, 0.0, -1.1920928244535389e-07, -8.940656698541716e-06, -0.00028713393840007484, 0.0, -5.960462772236497e-07, -0.035702865570783615, -0.6772614121437073, -0.0018156962469220161, -4.768370445162873e-07, -8.344646857949556e-07, -0.003322321455925703, -1.1920928244535389e-07, 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{'extracted_predictions': ['24', '24'], 'extracted_golds': ['321', '321']}
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[0]
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[]
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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.
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<|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
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['480']
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[]
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{'extractive_match': 0.0}
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[(['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 hyperbola is given by:\n\\[\n\\frac{x^2}{20} - \\frac{y^2}{24} = 1\n\\]\nThe diagonals of the rhombus intersect at the origin, which is the center of the hyperbola. Let the diagonals be \\(2a\\) and \\(2b\\), where \\(a\\) and \\(b\\) are the semi-lengths of the diagonals.\n\n### Step 2: Coordinates of the Vertices\nSince the diagonals intersect at the origin and are perpendicular, the vertices of the rhombus can be represented as:\n\\[\nA = (a, b), \\quad B = (-a, b), \\quad C = (-a, -b), \\quad D = (a, -b)\n\\]\n\n### Step 3: Verifying the Points on the Hyperbola\nWe need to check if these points satisfy the hyperbola equation. For point \\(A = (a, b)\\):\n\\[\n\\frac{a^2}{20} - \\frac{b^2}{24} = 1\n\\]\n\n### Step 4: Length of the Diagonals\nThe length of the diagonals are:\n\\[\nBD = 2\\sqrt{a^2 + b^2}\n\\]\n\n### Step 5: Squaring the Diagonal Length\nWe need to find \\(BD^2\\):\n\\[\nBD^2 = (2\\sqrt{a^2 + b^2})^2 = 4(a^2 + b^2)\n\\]\n\n### Step 6: Using the Hyperbola Equation\nFrom the hyperbola equation:\n\\[\n\\frac{a^2}{20} - \\frac{b^2}{24} = 1\n\\]\nWe can express \\(b^2\\) in terms of \\(a^2\\):\n\\[\n\\frac{b^2}{24} = \\frac{a^2}{20} - 1\n\\]\n\\[\nb^2 = 24 \\left( \\frac{a^2}{20} - 1 \\right)\n\\]\n\\[\nb^2 = \\frac{24a^2}{20} - 24\n\\]\n\\[\nb^2 = \\frac{6a^2}{5} - 24\n\\]\n\n### Step 7: Substituting \\(b^2\\) into \\(BD^2\\)\nSubstitute \\(b^2\\) into the expression for \\(BD^2\\):\n\\[\nBD^2 = 4(a^2 + b^2)\n\\]\n\\[\nBD^2 = 4 \\left( a^2 + \\frac{6a^2}{5} - 24 \\right)\n\\]\n\\[\nBD^2 = 4 \\left( \\frac{5a^2 + 6a^2}{5} - 24 \\right)\n\\]\n\\[\nBD^2 = 4 \\left( \\frac{11a^2}{5} - 24 \\right)\n\\]\n\\[\nBD^2 = \\frac{44a^2}{5} - 96\n\\]\n\n### Step 8: Finding the Maximum Value of \\(BD^2\\)\nTo find the maximum value of \\(BD^2\\), we need to consider the range of \\(a^2\\). Since \\(a^2\\) can be any positive value, we need to find the minimum value of the expression \\(\\frac{44a^2}{5} - 96\\).\n\nThe minimum value of \\(\\frac{44a^2}{5} - 96\\) occurs when \\(a^2\\) is minimized. The smallest positive value for \\(a^2\\) is approaching 0, but not zero. As \\(a^2 \\to 0\\):\n\\[\n\\frac{44a^2}{5} \\to 0\n\\]\n\\[\nBD^2 \\to -96\n\\]\n\nHowever, since \\(a^2\\) cannot be zero, the expression \\(\\frac{44a^2}{5} - 96\\) is always greater than \\(-96\\). 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|
{'extracted_predictions': ['96', '96'], 'extracted_golds': ['480', '480']}
|
[0]
|
|
[]
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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 $CE$?
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<|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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['104']
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[]
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{'extractive_match': 0.0}
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0
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0
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[0]
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[]
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[(["To solve for the length of \\(CE\\), we will use the given information and properties of the geometric figures involved.\n\n1. **Identify the given lengths and properties:**\n - \\(AB = 107\\)\n - \\(BC = 16\\)\n - \\(FG = 17\\)\n - \\(EF = 184\\)\n - \\(D, E, C, F\\) are collinear.\n - \\(A, D, H, G\\) lie on a circle.\n\n2. **Determine the coordinates of the points:**\n - Place \\(A\\) at \\((0, 0)\\).\n - Place \\(B\\) at \\((107, 0)\\).\n - Place \\(C\\) at \\((107, 16)\\).\n - Place \\(D\\) at \\((107, 0)\\) (since \\(D\\) is on the same vertical line as \\(B\\)).\n\n3. **Determine the coordinates of \\(E\\) and \\(F\\):**\n - Since \\(D, E, C, F\\) are collinear, \\(E\\) and \\(F\\) must lie on the line passing through \\(D\\) and \\(C\\).\n - The slope of the line \\(DC\\) is \\(\\frac{16 - 0}{107 - 107} = \\text{undefined}\\), indicating a vertical line.\n - Therefore, \\(E\\) and \\(F\\) must have the same x-coordinate as \\(D\\) and \\(C\\), which is 107.\n\n4. **Determine the coordinates of \\(E\\) and \\(F\\):**\n - Let \\(E\\) be at \\((107, y_E)\\).\n - Let \\(F\\) be at \\((107, y_F)\\).\n\n5. **Use the given lengths \\(EF = 17\\) and \\(FG = 184\\):**\n - The distance between \\(E\\) and \\(F\\) is \\(|y_F - y_E| = 17\\).\n - The distance between \\(F\\) and \\(G\\) is \\(|y_G - y_F| = 184\\).\n\n6. **Determine the coordinates of \\(G\\):**\n - Since \\(A, D, H, G\\) lie on a circle, and \\(D\\) and \\(A\\) are fixed, \\(H\\) and \\(G\\) must be such that the circle passes through these points.\n - The y-coordinate of \\(G\\) can be found by considering the circle's properties and the given distances.\n\n7. **Calculate the length of \\(CE\\):**\n - Since \\(C\\) is at \\((107, 16)\\) and \\(E\\) is at \\((107, y_E)\\), the distance \\(CE\\) is \\(|16 - y_E|\\).\n\n8. **Use the collinearity and circle properties:**\n - Since \\(D, E, C, F\\) are collinear and \\(A, D, H, G\\) lie on a circle, the y-coordinate of \\(E\\) must be such that the distances and circle properties are satisfied.\n - Given the distances and the circle's properties, we can deduce that \\(y_E = 16 - 17 = -1\\) (since \\(|y_F - y_E| = 17\\) and \\(y_F = 16 + 184 = 192\\)).\n\n9. **Final calculation:**\n - The length of \\(CE\\) is \\(|16 - (-1)| = 17\\).\n\nTherefore, the final answer is: \\(\\boxed{17}\\)."], [-0.0236908458173275, -0.0645395815372467, -0.4648008942604065, -0.0046577295288443565, -2.145764938177308e-06, -0.005968485958874226, -0.03007378801703453, -0.42876556515693665, -0.07889734953641891, -1.0847986231965479e-05, -0.07936213910579681, -1.0948537588119507, -0.5981098413467407, -0.2560431659221649, -0.14103436470031738, -0.44758614897727966, -0.13283149898052216, -0.16161665320396423, -0.00015293381875380874, -0.7528697848320007, -0.4742632210254669, -0.5361359715461731, -0.7858397960662842, -0.4085957109928131, -0.22429512441158295, -1.1920928244535389e-07, -0.011346926912665367, -0.011555642820894718, -0.008637921884655952, 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|
{'extracted_predictions': ['17', '17'], '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': ['2046 - 1*1', '2046 - 1'], 'extracted_golds': ['55', '55']}
|
[0]
|
|
[]
| "[[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 9245, 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 construct a list of positive integers that satisfies the give(...TRUNCATED) |
{'extracted_predictions': ['285', '285'], 'extracted_golds': ['236', '236']}
|
[0]
|
|
[]
| "[[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 1477, 279, 10578, 315, 279, 24632, 25366, 429, 646, (...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 of the smallest sphere that can contain each (...TRUNCATED) |
{'extracted_predictions': ['50', '50'], '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': ['1707', '1707'], 'extracted_golds': ['104', '104']}
|
[0]
|
|
[]
| "[[(1249, 11625, 279, 3491, 11, 582, 1191, 553, 36710, 17767, 1147, 1124, 8, 304, 1181, 24660, 1352,(...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': 1.0}
|
0
|
0
|
[0]
|
[]
| "[(['To solve the problem, we start by expressing \\\\( z \\\\) in its polar form. Since \\\\( |z| =(...TRUNCATED) |
{'extracted_predictions': ['540', '540'], 'extracted_golds': ['540', '540']}
|
[0]
|
|
[]
| "[[(1249, 11625, 279, 3491, 11, 582, 1184, 311, 8253, 279, 1372, 315, 12716, 315, 3084, 220, 16, 21,(...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, we need to determine the number of paths of length 16 on an \\\\(8 \\\\t(...TRUNCATED) |
{'extracted_predictions': ['4000', '4000'], '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': ['21', '21'], 'extracted_golds': ['73', '73']}
|
[0]
|
End of preview. Expand
in Data Studio
Dataset Card for Evaluation run of Lansechen/Qwen2.5-7B-Instruct-Distill-om220k-1k-simplified-batch32-epoch1-8192
Dataset automatically created during the evaluation run of model Lansechen/Qwen2.5-7B-Instruct-Distill-om220k-1k-simplified-batch32-epoch1-8192.
The dataset is composed of 3 configuration, each one coresponding to one of the evaluated task.
The dataset has been created from 9 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-Instruct-Distill-om220k-1k-simplified-batch32-epoch1-8192",
"results",
split="train")
Latest results
These are the latest results from run 2025-03-19T12:04:10.987530(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": {
"extractive_match": 0.3888888888888889,
"extractive_match_stderr": 0.0347327959083696
},
"custom|gpqa:diamond|0": {
"extractive_match": 0.3888888888888889,
"extractive_match_stderr": 0.0347327959083696
}
}
Dataset Details
Dataset Description
- Curated by: [More Information Needed]
- Funded by [optional]: [More Information Needed]
- Shared by [optional]: [More Information Needed]
- Language(s) (NLP): [More Information Needed]
- License: [More Information Needed]
Dataset Sources [optional]
- Repository: [More Information Needed]
- Paper [optional]: [More Information Needed]
- Demo [optional]: [More Information Needed]
Uses
Direct Use
[More Information Needed]
Out-of-Scope Use
[More Information Needed]
Dataset Structure
[More Information Needed]
Dataset Creation
Curation Rationale
[More Information Needed]
Source Data
Data Collection and Processing
[More Information Needed]
Who are the source data producers?
[More Information Needed]
Annotations [optional]
Annotation process
[More Information Needed]
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