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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
|
['321']
|
[]
|
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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 the problem, we need to find the remainder when the product \\(\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\) is divided by 1000, where \\(\\omega\\) is a 13th root of unity. Let's break this down step by step.\n\n1. **Understanding the 13th roots of unity:**\n The 13th roots of unity are the solutions to the equation \\(x^{13} = 1\\). These roots can be written as \\(\\omega^k\\) for \\(k = 0, 1, 2, \\ldots, 12\\), where \\(\\omega = e^{2\\pi i / 13}\\).\n\n2. **Expression simplification:**\n We need to evaluate the product \\(\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\). Notice that \\(\\omega^{13} = 1\\), so \\(\\omega^{2k} = \\omega^{2k \\mod 13}\\).\n\n3. **Symmetry and properties of roots of unity:**\n The roots of unity have symmetric properties. For example, \\(\\omega^{13-k} = \\omega^{-k} = \\overline{\\omega^k}\\) (the complex conjugate of \\(\\omega^k\\)). This symmetry can help simplify our product.\n\n4. **Rewriting the product:**\n Let's rewrite the product in a more manageable form:\n \\[\n \\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\n \\]\n Notice that \\(\\omega^{2k} = \\omega^{2k \\mod 13}\\), so we can consider the product over the first 13 terms.\n\n5. **Using polynomial properties:**\n Consider the polynomial \\(P(x) = x^{13} - 1\\). The roots of this polynomial are the 13th roots of unity. We can factorize \\(P(x)\\) as:\n \\[\n P(x) = (x - 1)(x - \\omega)(x - \\omega^2) \\cdots (x - \\omega^{12})\n \\]\n We are interested in the product of the form \\(2 - 2\\omega^k + \\omega^{2k}\\). This can be seen as a transformation of the roots of unity.\n\n6. **Evaluating the product:**\n To evaluate the product, we can use the fact that the product of the roots of a polynomial \\(P(x)\\) is given by \\((-1)^n a_0 / a_n\\), where \\(a_0\\) is the constant term and \\(a_n\\) is the leading coefficient. For our polynomial \\(P(x) = x^{13} - 1\\), the product of the roots is \\((-1)^{13} \\cdot (-1) = 1\\).\n\n7. **Final calculation:**\n The product \\(\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\) can be evaluated by considering the transformation of the roots. Since the roots are symmetric and the product of the roots of unity is 1, the product of the transformed roots will also be 1.\n\n8. **Remainder calculation:**\n Finally, we need to find the remainder when this product is divided by 1000. Since the product is 1, the remainder when 1 is divided by 1000 is 1.\n\nTherefore, the remainder when \\(\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\) is divided by 1000 is \\(\\boxed{1}\\)."], [-0.8440272212028503, -0.36763298511505127, -0.4548359513282776, -0.5803064107894897, -0.1406562179327011, -0.29492780566215515, -0.6434818506240845, -0.00046361665590666234, -0.2791346609592438, -0.0009129646932706237, -0.0717688649892807, -0.05507141351699829, -0.5737462639808655, -0.09044037014245987, -0.4130168855190277, -0.016555199399590492, -0.00011514954530866817, -8.380061626667157e-05, -1.2397689715726301e-05, -1.537788011773955e-05, -0.000263894529780373, -1.2159273865108844e-05, -0.00016282663273159415, -1.2755313036905136e-05, -6.770858453819528e-05, -0.026629462838172913, -8.4638240878121e-06, -0.1428954303264618, -1.5020257706055418e-05, -0.0002623452164698392, -0.00012838016846217215, -0.000328367663314566, -2.3841830625315197e-06, -0.018220549449324608, 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{'extracted_predictions': ['1', '1'], '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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0
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0
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[0]
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[(['To solve this 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-by-Step Solution:\n\n1. **Identify the properties of the rhombus:**\n - Since the diagonals of the rhombus intersect at the origin, they are perpendicular bisectors of each other.\n - Let the vertices of the rhombus be \\( A(x_1, y_1) \\), \\( B(x_2, y_2) \\), \\( C(-x_1, -y_1) \\), and \\( D(-x_2, -y_2) \\).\n\n2. **Use the equation of the hyperbola:**\n - The points \\( A \\) and \\( C \\) lie on the hyperbola, so they satisfy the equation \\( \\frac{x^2}{20} - \\frac{y^2}{24} = 1 \\).\n - Similarly, \\( B \\) and \\( D \\) also lie on the hyperbola.\n\n3. **Express the coordinates of the vertices:**\n - Let \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\) be such that \\( x_1^2/20 - y_1^2/24 = 1 \\) and \\( x_2^2/20 - y_2^2/24 = 1 \\).\n\n4. **Calculate the length of the diagonals:**\n - The length of diagonal \\( AC \\) is \\( 2\\sqrt{x_1^2 + y_1^2} \\).\n - The length of diagonal \\( BD \\) is \\( 2\\sqrt{x_2^2 + y_2^2} \\).\n\n5. **Relate the diagonals to the hyperbola:**\n - Since \\( A \\) and \\( C \\) are symmetric about the origin, \\( x_1 = -x_2 \\) and \\( y_1 = -y_2 \\).\n - Therefore, \\( x_1^2 = x_2^2 \\) and \\( y_1^2 = y_2^2 \\).\n\n6. **Calculate \\( BD^2 \\):**\n - \\( BD^2 = (x_2 - (-x_2))^2 + (y_2 - (-y_2))^2 = (2x_2)^2 + (2y_2)^2 = 4(x_2^2 + y_2^2) \\).\n\n7. **Express \\( x_2^2 + y_2^2 \\) in terms of the hyperbola equation:**\n - From the hyperbola equation, \\( x_2^2/20 - y_2^2/24 = 1 \\).\n - Multiply through by 240 to clear the denominators: \\( 12x_2^2 - 10y_2^2 = 240 \\).\n - Rearrange to solve for \\( x_2^2 \\): \\( 12x_2^2 = 240 + 10y_2^2 \\).\n - Divide by 12: \\( x_2^2 = 20 + \\frac{5}{6}y_2^2 \\).\n\n8. **Substitute \\( x_2^2 \\) back into \\( BD^2 \\):**\n - \\( BD^2 = 4(x_2^2 + y_2^2) = 4\\left(20 + \\frac{5}{6}y_2^2 + y_2^2\\right) = 4\\left(20 + \\frac{11}{6}y_2^2\\right) \\).\n\n9. **Find the maximum value of \\( BD^2 \\):**\n - To maximize \\( BD^2 \\), we need to maximize \\( y_2^2 \\).\n - Since \\( y_2^2 \\) can be arbitrarily large, \\( BD^2 \\) can also be arbitrarily large.\n\n10. **Determine the greatest real number less than \\( BD^2 \\):**\n - The greatest real number less than \\( BD^2 \\) is the limit as \\( y_2^2 \\) approaches infinity, which is \\( \\infty \\).\n\nTherefore, the greatest real number that is less than \\( BD^2 \\) for all such rhombi is \\( \\boxed{\\infty} \\).'], [-0.8213381767272949, -0.2314036339521408, -0.6600733995437622, -0.008595375344157219, -0.06429580599069595, -0.35841840505599976, -0.43001583218574524, -0.0016784878680482507, -0.8461112976074219, -0.014774550683796406, -0.36339613795280457, -0.04098936915397644, -8.856858039507642e-05, -0.17856542766094208, -0.001998928841203451, -0.0009233264136128128, 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|
{'extracted_predictions': ['oo', '\\infty'], 'extracted_golds': ['480', '480']}
|
[0]
|
|
[]
| "[[(1249, 11625, 419, 3491, 11, 582, 686, 990, 279, 5888, 315, 4428, 42446, 323, 279, 5355, 95362, 4(...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 this problem, we will use the properties of similar triangles and the Pythagorean theo(...TRUNCATED) |
{'extracted_predictions': ['176.500000000000', '176.5'], 'extracted_golds': ['104', '104']}
|
[0]
|
|
[]
| "[[(1249, 11625, 419, 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 this problem, we need to understand the relationship between the sets \\\\(A\\\\) and(...TRUNCATED) |
{'extracted_predictions': ['66', '66'], 'extracted_golds': ['55', '55']}
|
[0]
|
|
[]
| "[[(1249, 11625, 419, 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 this problem, we need to find a list of positive integers that satisfies the given co(...TRUNCATED) |
{'extracted_predictions': ['85', '85'], 'extracted_golds': ['236', '236']}
|
[0]
|
|
[]
| "[[(1249, 11625, 419, 3491, 11, 582, 1184, 311, 1477, 279, 24632, 25366, 429, 646, 6644, 678, 279, 5(...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 this problem, we need to find the smallest sphere that can contain all the rectangula(...TRUNCATED) |
{'extracted_predictions': ['29', '29'], 'extracted_golds': ['721', '721']}
|
[0]
|
|
[]
| "[[(1249, 11625, 419, 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 this problem, we need to find the distance from the point \\\\( I \\\\) to each of th(...TRUNCATED) |
{'extracted_predictions': ['23', '23'], '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': ['384*sqrt(2)', '384\\sqrt{2}'], 'extracted_golds': ['540', '540']}
|
[0]
|
|
[]
| "[[(1249, 11625, 419, 3491, 11, 582, 1184, 311, 2908, 279, 12716, 315, 3084, 220, 16, 21, 429, 1795,(...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 this problem, we need to consider the paths of length 16 that follow the lines from th(...TRUNCATED) |
{'extracted_predictions': ['5760', '5760'], 'extracted_golds': ['294', '294']}
|
[0]
|
|
[]
| "[[(1249, 11625, 419, 3491, 11, 582, 646, 990, 279, 17508, 315, 27187, 10187, 8957, 13, 6771, 594, 7(...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 can use the principle of inclusion-exclusion. Let's denote the numbe(...TRUNCATED) |
{'extracted_predictions': ['687', '687'], 'extracted_golds': ['73', '73']}
|
[0]
|
End of preview. Expand
in Data Studio
Dataset Card for Evaluation run of Qwen/Qwen2.5-7B
Dataset automatically created during the evaluation run of model Qwen/Qwen2.5-7B.
The dataset is composed of 3 configuration, each one coresponding to one of the evaluated task.
The dataset has been created from 3 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_Qwen__Qwen2.5-7B",
"results",
split="train")
Latest results
These are the latest results from run 2025-03-28T11:31:03.198625(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": 0.1,
"extractive_match_stderr": 0.055708601453115555
},
"custom|aime24|0": {
"extractive_match": 0.1,
"extractive_match_stderr": 0.055708601453115555
}
}
Dataset Details
Dataset Description
- Curated by: [More Information Needed]
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- Shared by [optional]: [More Information Needed]
- Language(s) (NLP): [More Information Needed]
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- Repository: [More Information Needed]
- Paper [optional]: [More Information Needed]
- Demo [optional]: [More Information Needed]
Uses
Direct Use
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Out-of-Scope Use
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Dataset Structure
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Dataset Creation
Curation Rationale
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Source Data
Data Collection and Processing
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Who are the source data producers?
[More Information Needed]
Annotations [optional]
Annotation process
[More Information Needed]
Who are the annotators?
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Personal and Sensitive Information
[More Information Needed]
Bias, Risks, and Limitations
[More Information Needed]
Recommendations
Users should be made aware of the risks, biases and limitations of the dataset. More information needed for further recommendations.
Citation [optional]
BibTeX:
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