prompt
listlengths 2
2
| answer
stringlengths 3
48
| src
stringclasses 4
values | messages
listlengths 2
2
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|---|---|---|---|
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $f(x)$ be a third-degree polynomial with real coefficients satisfying \\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\\] Find $|f(0)|$ .",
"type": "text"
}
],
"role": "user"
}
] |
$72$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $f(x)$ be a third-degree polynomial with real coefficients satisfying \\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\\] Find $|f(0)|$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$ -th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$ . Find $10h$ .",
"type": "text"
}
],
"role": "user"
}
] |
$350$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$ -th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$ . Find $10h$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nIn triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$ , where $m\\,$ and $n\\,$ are relatively prime positive integers. Find $m+n.\\,$",
"type": "text"
}
],
"role": "user"
}
] |
$450$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nIn triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$ , where $m\\,$ and $n\\,$ are relatively prime positive integers. Find $m+n.\\,$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color. [asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]",
"type": "text"
}
],
"role": "user"
}
] |
$732$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color. [asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAlex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting those two points. Find the expected number of regions into which these $27$ line segments divide the disk.",
"type": "text"
}
],
"role": "user"
}
] |
$204$
|
MathArena/aime_2025_I
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAlex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting those two points. Find the expected number of regions into which these $27$ line segments divide the disk.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAssume that $x_1,x_2,\\ldots,x_7$ are real numbers such that \\begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\\\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\\\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \\end{align*} Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$ .",
"type": "text"
}
],
"role": "user"
}
] |
$334$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAssume that $x_1,x_2,\\ldots,x_7$ are real numbers such that \\begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\\\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\\\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \\end{align*} Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$3$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "An integer is called snakelike if its decimal representation $a_1a_2a_3\\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?",
"type": "text"
}
],
"role": "user"
}
] |
$882$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "An integer is called snakelike if its decimal representation $a_1a_2a_3\\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Compute the number of ways to arrange the numbers $1,2,3,4,5,6$, and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most 20. (Rotations and reflections count as different arrangements.)",
"type": "text"
}
],
"role": "user"
}
] |
$56$
|
MathArena/hmmt_feb_2025
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Compute the number of ways to arrange the numbers $1,2,3,4,5,6$, and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most 20. (Rotations and reflections count as different arrangements.)",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "A paper equilateral triangle $ABC$ has side length $12$ . The paper triangle is folded so that vertex $A$ touches a point on side $\\overline{BC}$ a distance $9$ from point $B$ . The length of the line segment along which the triangle is folded can be written as $\\frac{m\\sqrt{p}}{n}$ , where $m$ , $n$ , and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$ . [asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP(\"B\", (0,0), dir(200)); pair A = MP(\"A\", (9,0), dir(-80)); pair C = MP(\"C\", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP(\"B\", B+shift, dir(200)); pair A1 = MP(\"A\", K+shift, dir(90)); pair C1 = MP(\"C\", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]",
"type": "text"
}
],
"role": "user"
}
] |
$113$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "A paper equilateral triangle $ABC$ has side length $12$ . The paper triangle is folded so that vertex $A$ touches a point on side $\\overline{BC}$ a distance $9$ from point $B$ . The length of the line segment along which the triangle is folded can be written as $\\frac{m\\sqrt{p}}{n}$ , where $m$ , $n$ , and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$ . [asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP(\"B\", (0,0), dir(200)); pair A = MP(\"A\", (9,0), dir(-80)); pair C = MP(\"C\", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP(\"B\", B+shift, dir(200)); pair A1 = MP(\"A\", K+shift, dir(90)); pair C1 = MP(\"C\", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nPositive real numbers $b \\not= 1$ and $n$ satisfy the equations \\[\\sqrt{\\log_b n} = \\log_b \\sqrt{n} \\qquad \\text{and} \\qquad b \\cdot \\log_b n = \\log_b (bn).\\] The value of $n$ is $\\frac{j}{k},$ where $j$ and $k$ are relatively prime positive integers. Find $j+k.$",
"type": "text"
}
],
"role": "user"
}
] |
$881$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nPositive real numbers $b \\not= 1$ and $n$ satisfy the equations \\[\\sqrt{\\log_b n} = \\log_b \\sqrt{n} \\qquad \\text{and} \\qquad b \\cdot \\log_b n = \\log_b (bn).\\] The value of $n$ is $\\frac{j}{k},$ where $j$ and $k$ are relatively prime positive integers. Find $j+k.$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nReal numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$ , and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$ . Find the sum of all possible values of $|b|$ .",
"type": "text"
}
],
"role": "user"
}
] |
$420$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nReal numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$ , and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$ . Find the sum of all possible values of $|b|$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3$ . What is the sum of the first $2001$ terms of this sequence if the sum of the first $1492$ terms is $1985$ , and the sum of the first $1985$ terms is $1492$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$986$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3$ . What is the sum of the first $2001$ terms of this sequence if the sum of the first $1492$ terms is $1985$ , and the sum of the first $1985$ terms is $1492$ ?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor positive integers $N$ and $k$ , define $N$ to be $k$ -nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$ -nice nor $8$ -nice.",
"type": "text"
}
],
"role": "user"
}
] |
$749$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor positive integers $N$ and $k$ , define $N$ to be $k$ -nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$ -nice nor $8$ -nice.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "The sequence $\\{a_n\\}$ is defined by \\[a_0 = 1,a_1 = 1, \\text{ and } a_n = a_{n - 1} + \\frac {a_{n - 1}^2}{a_{n - 2}}\\text{ for }n\\ge2.\\] The sequence $\\{b_n\\}$ is defined by \\[b_0 = 1,b_1 = 3, \\text{ and } b_n = b_{n - 1} + \\frac {b_{n - 1}^2}{b_{n - 2}}\\text{ for }n\\ge2.\\] Find $\\frac {b_{32}}{a_{32}}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$561$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "The sequence $\\{a_n\\}$ is defined by \\[a_0 = 1,a_1 = 1, \\text{ and } a_n = a_{n - 1} + \\frac {a_{n - 1}^2}{a_{n - 2}}\\text{ for }n\\ge2.\\] The sequence $\\{b_n\\}$ is defined by \\[b_0 = 1,b_1 = 3, \\text{ and } b_n = b_{n - 1} + \\frac {b_{n - 1}^2}{b_{n - 2}}\\text{ for }n\\ge2.\\] Find $\\frac {b_{32}}{a_{32}}$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nMark writes the expression $\\sqrt{\\underline{a b c d}}$ on the board, where $\\underline{a b c d}$ is a four-digit number and $a \\neq 0$. Derek, a toddler, decides to move the $a$, changing Mark's expression to $a \\sqrt{\\underline{b c d}}$. Surprisingly, these two expressions are equal. Compute the only possible four-digit number $\\underline{a b c d}$.",
"type": "text"
}
],
"role": "user"
}
] |
$3375$
|
MathArena/hmmt_feb_2025
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nMark writes the expression $\\sqrt{\\underline{a b c d}}$ on the board, where $\\underline{a b c d}$ is a four-digit number and $a \\neq 0$. Derek, a toddler, decides to move the $a$, changing Mark's expression to $a \\sqrt{\\underline{b c d}}$. Surprisingly, these two expressions are equal. Compute the only possible four-digit number $\\underline{a b c d}$.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its nonempty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract succesive numbers. For example, the alternating sum for $\\{1, 2, 3, 6,9\\}$ is $9-6+3-2+1=5$ and for $\\{5\\}$ it is simply $5$ . Find the sum of all such alternating sums for $n=7$ .",
"type": "text"
}
],
"role": "user"
}
] |
$448$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its nonempty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract succesive numbers. For example, the alternating sum for $\\{1, 2, 3, 6,9\\}$ is $9-6+3-2+1=5$ and for $\\{5\\}$ it is simply $5$ . Find the sum of all such alternating sums for $n=7$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions: (a) $0\\le A<B<C\\le99$ , (b) there exist integers $a$ , $b$ , and $c$ , and prime $p$ where $0\\le b<a<c<p$ , (c) $p$ divides $A-a$ , $B-b$ , and $C-c$ , and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$ .",
"type": "text"
}
],
"role": "user"
}
] |
$272$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions: (a) $0\\le A<B<C\\le99$ , (b) there exist integers $a$ , $b$ , and $c$ , and prime $p$ where $0\\le b<a<c<p$ , (c) $p$ divides $A-a$ , $B-b$ , and $C-c$ , and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nDefine $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$401$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nDefine $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nEuler's formula states that for a convex polyhedron with $V\\,$ vertices, $E\\,$ edges, and $F\\,$ faces, $V-E+F=2\\,$ . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V\\,$ vertices, $T\\,$ triangular faces and $P^{}_{}$ pentagonal faces meet. What is the value of $100P+10T+V\\,$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$250$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nEuler's formula states that for a convex polyhedron with $V\\,$ vertices, $E\\,$ edges, and $F\\,$ faces, $V-E+F=2\\,$ . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V\\,$ vertices, $T\\,$ triangular faces and $P^{}_{}$ pentagonal faces meet. What is the value of $100P+10T+V\\,$ ?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor distinct complex numbers $z_1,z_2,\\dots,z_{673}$ , the polynomial \\[(x-z_1)^3(x-z_2)^3 \\cdots (x-z_{673})^3\\] can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$ , where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$ . The value of \\[\\left| \\sum_{1 \\le j <k \\le 673} z_jz_k \\right|\\] can be expressed in the form $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$352$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor distinct complex numbers $z_1,z_2,\\dots,z_{673}$ , the polynomial \\[(x-z_1)^3(x-z_2)^3 \\cdots (x-z_{673})^3\\] can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$ , where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$ . The value of \\[\\left| \\sum_{1 \\le j <k \\le 673} z_jz_k \\right|\\] can be expressed in the form $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTriangle $ABC$ has side lengths $AB=120,BC=220$ , and $AC=180$ . Lines $\\ell_A,\\ell_B$ , and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$ , and $\\overline{AB}$ , respectively, such that the intersections of $\\ell_A,\\ell_B$ , and $\\ell_C$ with the interior of $\\triangle ABC$ are segments of lengths $55,45$ , and $15$ , respectively. Find the perimeter of the triangle whose sides lie on lines $\\ell_A,\\ell_B$ , and $\\ell_C$ .",
"type": "text"
}
],
"role": "user"
}
] |
$715$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTriangle $ABC$ has side lengths $AB=120,BC=220$ , and $AC=180$ . Lines $\\ell_A,\\ell_B$ , and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$ , and $\\overline{AB}$ , respectively, such that the intersections of $\\ell_A,\\ell_B$ , and $\\ell_C$ with the interior of $\\triangle ABC$ are segments of lengths $55,45$ , and $15$ , respectively. Find the perimeter of the triangle whose sides lie on lines $\\ell_A,\\ell_B$ , and $\\ell_C$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "If $\\tan x+\\tan y=25$ and $\\cot x + \\cot y=30$ , what is $\\tan(x+y)$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$150$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "If $\\tan x+\\tan y=25$ and $\\cot x + \\cot y=30$ , what is $\\tan(x+y)$ ?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nJane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible?",
"type": "text"
}
],
"role": "user"
}
] |
$25$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nJane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$ ? ",
"type": "text"
}
],
"role": "user"
}
] |
$816$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$ ? ",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $a = \\pi/2008$ . Find the smallest positive integer $n$ such that \\[2[\\cos(a)\\sin(a) + \\cos(4a)\\sin(2a) + \\cos(9a)\\sin(3a) + \\cdots + \\cos(n^2a)\\sin(na)]\\] is an integer.",
"type": "text"
}
],
"role": "user"
}
] |
$251$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $a = \\pi/2008$ . Find the smallest positive integer $n$ such that \\[2[\\cos(a)\\sin(a) + \\cos(4a)\\sin(2a) + \\cos(9a)\\sin(3a) + \\cdots + \\cos(n^2a)\\sin(na)]\\] is an integer.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $ABCD$ be an isosceles trapezoid with $\\overline{AD}||\\overline{BC}$ whose angle at the longer base $\\overline{AD}$ is $\\dfrac{\\pi}{3}$ . The diagonals have length $10\\sqrt {21}$ , and point $E$ is at distances $10\\sqrt {7}$ and $30\\sqrt {7}$ from vertices $A$ and $D$ , respectively. Let $F$ be the foot of the altitude from $C$ to $\\overline{AD}$ . The distance $EF$ can be expressed in the form $m\\sqrt {n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$32$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $ABCD$ be an isosceles trapezoid with $\\overline{AD}||\\overline{BC}$ whose angle at the longer base $\\overline{AD}$ is $\\dfrac{\\pi}{3}$ . The diagonals have length $10\\sqrt {21}$ , and point $E$ is at distances $10\\sqrt {7}$ and $30\\sqrt {7}$ from vertices $A$ and $D$ , respectively. Let $F$ be the foot of the altitude from $C$ to $\\overline{AD}$ . The distance $EF$ can be expressed in the form $m\\sqrt {n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$ , and $\\left(P(2)\\right)^2 = P(3)$ . Then $P(\\tfrac72)=\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$109$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$ , and $\\left(P(2)\\right)^2 = P(3)$ . Then $P(\\tfrac72)=\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA bug walks all day and sleeps all night. On the first day, it starts at point $O,$ faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point $P.$ Then $OP^2=\\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$",
"type": "text"
}
],
"role": "user"
}
] |
$103$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA bug walks all day and sleeps all night. On the first day, it starts at point $O,$ faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point $P.$ Then $OP^2=\\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2 - 2x + 2 \\le P(x) \\le 2x^2 - 4x + 3$ for all real numbers $x$ , and suppose $P(11) = 181$ . Find $P(16)$ .",
"type": "text"
}
],
"role": "user"
}
] |
$406$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2 - 2x + 2 \\le P(x) \\le 2x^2 - 4x + 3$ for all real numbers $x$ , and suppose $P(11) = 181$ . Find $P(16)$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.",
"type": "text"
}
],
"role": "user"
}
] |
$945$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nThe following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 1; i < 13; ++i) { label(\"\\small\" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy] Initially, both hands point to the number $12$ . The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move. Let $N$ be the number of sequences of $144$ hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the $144$ movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$608$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nThe following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 1; i < 13; ++i) { label(\"\\small\" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy] Initially, both hands point to the number $12$ . The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move. Let $N$ be the number of sequences of $144$ hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the $144$ movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by $1000$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor a positive integer $p$ , define the positive integer $n$ to be $p$ -safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$ . For example, the set of $10$ -safe numbers is $\\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \\ldots\\}$ . Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$ -safe, $11$ -safe, and $13$ -safe.",
"type": "text"
}
],
"role": "user"
}
] |
$958$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor a positive integer $p$ , define the positive integer $n$ to be $p$ -safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$ . For example, the set of $10$ -safe numbers is $\\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \\ldots\\}$ . Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$ -safe, $11$ -safe, and $13$ -safe.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\\le x\\le2,$ $0\\le y\\le3,$ and $0\\le z\\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$",
"type": "text"
}
],
"role": "user"
}
] |
$200$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\\le x\\le2,$ $0\\le y\\le3,$ and $0\\le z\\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nThe wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$ . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path $AJABCHCHIJA$ , which has $10$ steps. Let $n$ be the number of paths with $15$ steps that begin and end at point $A.$ Find the remainder when $n$ is divided by $1000$ . [asy] size(6cm); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } dot(1 * dir( 90), linewidth(5)); dot(1 * dir(162), linewidth(5)); dot(1 * dir(234), linewidth(5)); dot(1 * dir(306), linewidth(5)); dot(1 * dir(378), linewidth(5)); dot(2 * dir(378), linewidth(5)); dot(2 * dir(306), linewidth(5)); dot(2 * dir(234), linewidth(5)); dot(2 * dir(162), linewidth(5)); dot(2 * dir( 90), linewidth(5)); label(\"$A$\", 1 * dir( 90), -dir( 90)); label(\"$B$\", 1 * dir(162), -dir(162)); label(\"$C$\", 1 * dir(234), -dir(234)); label(\"$D$\", 1 * dir(306), -dir(306)); label(\"$E$\", 1 * dir(378), -dir(378)); label(\"$F$\", 2 * dir(378), dir(378)); label(\"$G$\", 2 * dir(306), dir(306)); label(\"$H$\", 2 * dir(234), dir(234)); label(\"$I$\", 2 * dir(162), dir(162)); label(\"$J$\", 2 * dir( 90), dir( 90)); [/asy]",
"type": "text"
}
],
"role": "user"
}
] |
$4$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nThe wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$ . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path $AJABCHCHIJA$ , which has $10$ steps. Let $n$ be the number of paths with $15$ steps that begin and end at point $A.$ Find the remainder when $n$ is divided by $1000$ . [asy] size(6cm); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } dot(1 * dir( 90), linewidth(5)); dot(1 * dir(162), linewidth(5)); dot(1 * dir(234), linewidth(5)); dot(1 * dir(306), linewidth(5)); dot(1 * dir(378), linewidth(5)); dot(2 * dir(378), linewidth(5)); dot(2 * dir(306), linewidth(5)); dot(2 * dir(234), linewidth(5)); dot(2 * dir(162), linewidth(5)); dot(2 * dir( 90), linewidth(5)); label(\"$A$\", 1 * dir( 90), -dir( 90)); label(\"$B$\", 1 * dir(162), -dir(162)); label(\"$C$\", 1 * dir(234), -dir(234)); label(\"$D$\", 1 * dir(306), -dir(306)); label(\"$E$\", 1 * dir(378), -dir(378)); label(\"$F$\", 2 * dir(378), dir(378)); label(\"$G$\", 2 * dir(306), dir(306)); label(\"$H$\", 2 * dir(234), dir(234)); label(\"$I$\", 2 * dir(162), dir(162)); label(\"$J$\", 2 * dir( 90), dir( 90)); [/asy]",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $x_1\\leq x_2\\leq \\cdots\\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \\cdots + |x_{100}| = 1$ and $x_1 + x_2 + \\cdots + x_{100} = 0$ . Among all such $100$ -tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$841$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $x_1\\leq x_2\\leq \\cdots\\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \\cdots + |x_{100}| = 1$ and $x_1 + x_2 + \\cdots + x_{100} = 0$ . Among all such $100$ -tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number? ",
"type": "text"
}
],
"role": "user"
}
] |
$126$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number? ",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$ . Find the greatest positive integer $n$ such that $2^n$ divides $K$ .",
"type": "text"
}
],
"role": "user"
}
] |
$150$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$ . Find the greatest positive integer $n$ such that $2^n$ divides $K$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $x_1, x_2, x_3, \\ldots$ be a sequence of rational numbers defined by $x_1 = \\frac{25}{11}$ and\n$$x_{k+1} = \\frac{1}{3}\\left(x_k + \\frac{1}{x_k} - 1\\right)$$\nfor all $k \\ge 1$. Then $x_{2025}$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by $1000$.",
"type": "text"
}
],
"role": "user"
}
] |
$248$
|
MathArena/aime_2025_II
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $x_1, x_2, x_3, \\ldots$ be a sequence of rational numbers defined by $x_1 = \\frac{25}{11}$ and\n$$x_{k+1} = \\frac{1}{3}\\left(x_k + \\frac{1}{x_k} - 1\\right)$$\nfor all $k \\ge 1$. Then $x_{2025}$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by $1000$.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Circle $C$ with radius 2 has diameter $\\overline{AB}$ . Circle $D$ is internally tangent to circle $C$ at $A$ . Circle $E$ is internally tangent to circle $C$ , externally tangent to circle $D$ , and tangent to $\\overline{AB}$ . The radius of circle $D$ is three times the radius of circle $E$ , and can be written in the form $\\sqrt{m}-n$ , where $m$ and $n$ are positive integers. Find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$254$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Circle $C$ with radius 2 has diameter $\\overline{AB}$ . Circle $D$ is internally tangent to circle $C$ at $A$ . Circle $E$ is internally tangent to circle $C$ , externally tangent to circle $D$ , and tangent to $\\overline{AB}$ . The radius of circle $D$ is three times the radius of circle $E$ , and can be written in the form $\\sqrt{m}-n$ , where $m$ and $n$ are positive integers. Find $m+n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFind the smallest integer $k$ for which the conditions (1) $a_1, a_2, a_3, \\ldots$ is a nondecreasing sequence of positive integers (2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$ (3) $a_9=k$ are satisfied by more than one sequence.",
"type": "text"
}
],
"role": "user"
}
] |
$748$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFind the smallest integer $k$ for which the conditions (1) $a_1, a_2, a_3, \\ldots$ is a nondecreasing sequence of positive integers (2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$ (3) $a_9=k$ are satisfied by more than one sequence.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFind the value of $10\\cot(\\cot^{-1}3+\\cot^{-1}7+\\cot^{-1}13+\\cot^{-1}21).$",
"type": "text"
}
],
"role": "user"
}
] |
$15$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFind the value of $10\\cot(\\cot^{-1}3+\\cot^{-1}7+\\cot^{-1}13+\\cot^{-1}21).$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTriangle $ABC$ has side lengths $AB = 9$ , $BC =$ $5\\sqrt{3}$ , and $AC = 12$ . Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$ , and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$ . Furthermore, each segment $\\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2449$ , is parallel to $\\overline{BC}$ . The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2450$ , that have rational length.",
"type": "text"
}
],
"role": "user"
}
] |
$20$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTriangle $ABC$ has side lengths $AB = 9$ , $BC =$ $5\\sqrt{3}$ , and $AC = 12$ . Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$ , and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$ . Furthermore, each segment $\\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2449$ , is parallel to $\\overline{BC}$ . The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2450$ , that have rational length.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $A$ and $B$ be the endpoints of a semicircular arc of radius $2$ . The arc is divided into seven congruent arcs by six equally spaced points $C_1,C_2,\\dots,C_6$ . All chords of the form $\\overline{AC_i}$ or $\\overline{BC_i}$ are drawn. Let $n$ be the product of the lengths of these twelve chords. Find the remainder when $n$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$672$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $A$ and $B$ be the endpoints of a semicircular arc of radius $2$ . The arc is divided into seven congruent arcs by six equally spaced points $C_1,C_2,\\dots,C_6$ . All chords of the form $\\overline{AC_i}$ or $\\overline{BC_i}$ are drawn. Let $n$ be the product of the lengths of these twelve chords. Find the remainder when $n$ is divided by $1000$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\\sqrt{21}$ and $\\sqrt{31}$ . The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . A parallelepiped is a solid with six parallelogram faces such as the one shown below. [asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]",
"type": "text"
}
],
"role": "user"
}
] |
$125$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\\sqrt{21}$ and $\\sqrt{31}$ . The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . A parallelepiped is a solid with six parallelogram faces such as the one shown below. [asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$ . ",
"type": "text"
}
],
"role": "user"
}
] |
$376$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$ . ",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nCompute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.",
"type": "text"
}
],
"role": "user"
}
] |
$6300$
|
MathArena/hmmt_feb_2025
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nCompute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Hexagon $ABCDEF$ is divided into five rhombuses, $P, Q, R, S,$ and $T$ , as shown. Rhombuses $P, Q, R,$ and $S$ are congruent, and each has area $\\sqrt{2006}.$ Let $K$ be the area of rhombus $T$ . Given that $K$ is a positive integer, find the number of possible values for $K.$ [asy] // TheMathGuyd size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label(\"$\\mathcal{T}$\",(2.1,-1.6)); label(\"$\\mathcal{P}$\",(0,-1),NE); label(\"$\\mathcal{Q}$\",(4.2,-1),NW); label(\"$\\mathcal{R}$\",(0,-2.2),SE); label(\"$\\mathcal{S}$\",(4.2,-2.2),SW); [/asy]",
"type": "text"
}
],
"role": "user"
}
] |
$89$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Hexagon $ABCDEF$ is divided into five rhombuses, $P, Q, R, S,$ and $T$ , as shown. Rhombuses $P, Q, R,$ and $S$ are congruent, and each has area $\\sqrt{2006}.$ Let $K$ be the area of rhombus $T$ . Given that $K$ is a positive integer, find the number of possible values for $K.$ [asy] // TheMathGuyd size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label(\"$\\mathcal{T}$\",(2.1,-1.6)); label(\"$\\mathcal{P}$\",(0,-1),NE); label(\"$\\mathcal{Q}$\",(4.2,-1),NW); label(\"$\\mathcal{R}$\",(0,-2.2),SE); label(\"$\\mathcal{S}$\",(4.2,-2.2),SW); [/asy]",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$ -direction. Given that the particle's position after $150$ moves is $(p,q)$ , find the greatest integer less than or equal to $|p| + |q|$ .",
"type": "text"
}
],
"role": "user"
}
] |
$19$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$ -direction. Given that the particle's position after $150$ moves is $(p,q)$ , find the greatest integer less than or equal to $|p| + |q|$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.",
"type": "text"
}
],
"role": "user"
}
] |
$71$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nThe AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?",
"type": "text"
}
],
"role": "user"
}
] |
$150$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nThe AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$10$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$ ?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $a$ and $b$ be positive integers satisfying $\\frac{ab+1}{a+b} < \\frac{3}{2}$ . The maximum possible value of $\\frac{a^3b^3+1}{a^3+b^3}$ is $\\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .",
"type": "text"
}
],
"role": "user"
}
] |
$36$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $a$ and $b$ be positive integers satisfying $\\frac{ab+1}{a+b} < \\frac{3}{2}$ . The maximum possible value of $\\frac{a^3b^3+1}{a^3+b^3}$ is $\\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTriangle $ABC$ has $AB=21$ , $AC=22$ and $BC=20$ . Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$ , respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$ . Then $DE=\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$923$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTriangle $ABC$ has $AB=21$ , $AC=22$ and $BC=20$ . Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$ , respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$ . Then $DE=\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.",
"type": "text"
}
],
"role": "user"
}
] |
$63$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$750$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "A set of positive numbers has the $triangle~property$ if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\\{4, 5, 6, \\ldots, n\\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$253$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "A set of positive numbers has the $triangle~property$ if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\\{4, 5, 6, \\ldots, n\\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$ ?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "For every $m \\geq 2$ , let $Q(m)$ be the least positive integer with the following property: For every $n \\geq Q(m)$ , there is always a perfect cube $k^3$ in the range $n < k^3 \\leq mn$ . Find the remainder when \\[\\sum_{m = 2}^{2017} Q(m)\\] is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$59$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "For every $m \\geq 2$ , let $Q(m)$ be the least positive integer with the following property: For every $n \\geq Q(m)$ , there is always a perfect cube $k^3$ in the range $n < k^3 \\leq mn$ . Find the remainder when \\[\\sum_{m = 2}^{2017} Q(m)\\] is divided by $1000$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $ABCD$ be a square, and let $E$ and $F$ be points on $\\overline{AB}$ and $\\overline{BC},$ respectively. The line through $E$ parallel to $\\overline{BC}$ and the line through $F$ parallel to $\\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\\frac{9}{10}$ of the area of square $ABCD.$ Find $\\frac{AE}{EB} + \\frac{EB}{AE}.$",
"type": "text"
}
],
"role": "user"
}
] |
$18$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $ABCD$ be a square, and let $E$ and $F$ be points on $\\overline{AB}$ and $\\overline{BC},$ respectively. The line through $E$ parallel to $\\overline{BC}$ and the line through $F$ parallel to $\\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\\frac{9}{10}$ of the area of square $ABCD.$ Find $\\frac{AE}{EB} + \\frac{EB}{AE}.$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ \\[\\begin{array}{|c|c|c|c|c|c|} \\hline \\,1\\, & \\,3\\, & \\,5\\, & \\,7\\, & \\,9\\, & 11 \\\\ \\hline \\,2\\, & \\,4\\, & \\,6\\, & \\,8\\, & 10 & 12 \\\\ \\hline \\end{array}\\]",
"type": "text"
}
],
"role": "user"
}
] |
$144$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer divisors of $N.$ \\[\\begin{array}{|c|c|c|c|c|c|} \\hline \\,1\\, & \\,3\\, & \\,5\\, & \\,7\\, & \\,9\\, & 11 \\\\ \\hline \\,2\\, & \\,4\\, & \\,6\\, & \\,8\\, & 10 & 12 \\\\ \\hline \\end{array}\\]",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$37$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value that $x + y$ can have?",
"type": "text"
}
],
"role": "user"
}
] |
$4$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value that $x + y$ can have?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . ",
"type": "text"
}
],
"role": "user"
}
] |
$79$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . ",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Consider the sequence defined by $a_k=\\frac 1{k^2+k}$ for $k\\ge 1$ . Given that $a_m+a_{m+1}+\\cdots+a_{n-1}=1/29$ , for positive integers $m$ and $n$ with $m<n$ , find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$840$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Consider the sequence defined by $a_k=\\frac 1{k^2+k}$ for $k\\ge 1$ . Given that $a_m+a_{m+1}+\\cdots+a_{n-1}=1/29$ , for positive integers $m$ and $n$ with $m<n$ , find $m+n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \\cdot 10^3 + a_2 \\cdot 10^2 + a_1 \\cdot 10 + a_0$ , where the $a_i$ 's are integers, and $0 \\le a_i \\le 99$ . An example of such a representation is $1\\cdot 10^3 + 3\\cdot 10^2 + 67\\cdot 10^1 + 40\\cdot 10^0$ . Find $N$ .",
"type": "text"
}
],
"role": "user"
}
] |
$202$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \\cdot 10^3 + a_2 \\cdot 10^2 + a_1 \\cdot 10 + a_0$ , where the $a_i$ 's are integers, and $0 \\le a_i \\le 99$ . An example of such a representation is $1\\cdot 10^3 + 3\\cdot 10^2 + 67\\cdot 10^1 + 40\\cdot 10^0$ . Find $N$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nWhile finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\\frac{m\\pi}{n-\\pi}$ and $\\frac{p\\pi}{q+\\pi}$ , where $m$ , $n$ , $p$ , and $q$ are positive integers. Find $m+n+p+q$ . ",
"type": "text"
}
],
"role": "user"
}
] |
$900$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nWhile finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\\frac{m\\pi}{n-\\pi}$ and $\\frac{p\\pi}{q+\\pi}$ , where $m$ , $n$ , $p$ , and $q$ are positive integers. Find $m+n+p+q$ . ",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Square $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$ AIME 1995 Problem 1.png",
"type": "text"
}
],
"role": "user"
}
] |
$255$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Square $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$ AIME 1995 Problem 1.png",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nDetermine the value of $ab$ if $\\log_8a+\\log_4b^2=5$ and $\\log_8b+\\log_4a^2=7$ .",
"type": "text"
}
],
"role": "user"
}
] |
$512$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nDetermine the value of $ab$ if $\\log_8a+\\log_4b^2=5$ and $\\log_8b+\\log_4a^2=7$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$ . Find the number of such distinct triangles whose area is a positive integer.",
"type": "text"
}
],
"role": "user"
}
] |
$600$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$ . Find the number of such distinct triangles whose area is a positive integer.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "For any positive integer $k$ , let $f_1(k)$ denote the square of the sum of the digits of $k$ . For $n \\ge 2$ , let $f_n(k) = f_1(f_{n - 1}(k))$ . Find $f_{1988}(11)$ .",
"type": "text"
}
],
"role": "user"
}
] |
$169$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "For any positive integer $k$ , let $f_1(k)$ denote the square of the sum of the digits of $k$ . For $n \\ge 2$ , let $f_n(k) = f_1(f_{n - 1}(k))$ . Find $f_{1988}(11)$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor positive integer $n$ , let $s(n)$ denote the sum of the digits of $n$ . Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$ .",
"type": "text"
}
],
"role": "user"
}
] |
$695$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nFor positive integer $n$ , let $s(n)$ denote the sum of the digits of $n$ . Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAn urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$ . Find $N$ .",
"type": "text"
}
],
"role": "user"
}
] |
$144$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAn urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$ . Find $N$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nDiameter $AB$ of a circle has length a $2$ -digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$ . The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$ . Pdfresizer.com-pdf-convert-aimeq12.png",
"type": "text"
}
],
"role": "user"
}
] |
$65$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nDiameter $AB$ of a circle has length a $2$ -digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$ . The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$ . Pdfresizer.com-pdf-convert-aimeq12.png",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$640$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\\angle FAB = 120^\\circ,$ $\\overline{AB}\\parallel \\overline{DE},$ $\\overline{BC}\\parallel \\overline{EF,}$ $\\overline{CD}\\parallel \\overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\\{0,2,4,6,8,10\\}.$ The area of the hexagon can be written in the form $m\\sqrt {n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m + n.$",
"type": "text"
}
],
"role": "user"
}
] |
$51$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\\angle FAB = 120^\\circ,$ $\\overline{AB}\\parallel \\overline{DE},$ $\\overline{BC}\\parallel \\overline{EF,}$ $\\overline{CD}\\parallel \\overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\\{0,2,4,6,8,10\\}.$ The area of the hexagon can be written in the form $m\\sqrt {n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m + n.$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "From the set of integers $\\{1,2,3,\\dots,2009\\}$ , choose $k$ pairs $\\{a_i,b_i\\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$ . Find the maximum possible value of $k$ .",
"type": "text"
}
],
"role": "user"
}
] |
$803$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "From the set of integers $\\{1,2,3,\\dots,2009\\}$ , choose $k$ pairs $\\{a_i,b_i\\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$ . Find the maximum possible value of $k$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nGiven that $z$ is a complex number such that $z+\\frac 1z=2\\cos 3^\\circ$ , find the least integer that is greater than $z^{2000}+\\frac 1{z^{2000}}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$0$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nGiven that $z$ is a complex number such that $z+\\frac 1z=2\\cos 3^\\circ$ , find the least integer that is greater than $z^{2000}+\\frac 1{z^{2000}}$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nIn $\\triangle RED$ , $\\measuredangle DRE=75^{\\circ}$ and $\\measuredangle RED=45^{\\circ}$ . $RD=1$ . Let $M$ be the midpoint of segment $\\overline{RD}$ . Point $C$ lies on side $\\overline{ED}$ such that $\\overline{RC}\\perp\\overline{EM}$ . Extend segment $\\overline{DE}$ through $E$ to point $A$ such that $CA=AR$ . Then $AE=\\frac{a-\\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$ .",
"type": "text"
}
],
"role": "user"
}
] |
$56$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nIn $\\triangle RED$ , $\\measuredangle DRE=75^{\\circ}$ and $\\measuredangle RED=45^{\\circ}$ . $RD=1$ . Let $M$ be the midpoint of segment $\\overline{RD}$ . Point $C$ lies on side $\\overline{ED}$ such that $\\overline{RC}\\perp\\overline{EM}$ . Extend segment $\\overline{DE}$ through $E$ to point $A$ such that $CA=AR$ . Then $AE=\\frac{a-\\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "A positive integer $N$ has base-eleven representation $\\underline{a}\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}$ and base-eight representation $\\underline1\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}\\kern 0.1em\\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten.",
"type": "text"
}
],
"role": "user"
}
] |
$621$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "A positive integer $N$ has base-eleven representation $\\underline{a}\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}$ and base-eight representation $\\underline1\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}\\kern 0.1em\\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$ , and that $f(x) = 1 - |x - 2|$ for $1\\leq x \\leq 3$ . Find the smallest $x$ for which $f(x) = f(2001)$ .",
"type": "text"
}
],
"role": "user"
}
] |
$429$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$ , and that $f(x) = 1 - |x - 2|$ for $1\\leq x \\leq 3$ . Find the smallest $x$ for which $f(x) = f(2001)$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTen points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? ",
"type": "text"
}
],
"role": "user"
}
] |
$968$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nTen points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? ",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \\sqrt{-1}$ . Let $z$ be the unique complex number with the properties that $\\frac{z_3 - z_1}{z_2 - z_1} \\cdot \\frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$ .",
"type": "text"
}
],
"role": "user"
}
] |
$56$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \\sqrt{-1}$ . Let $z$ be the unique complex number with the properties that $\\frac{z_3 - z_1}{z_2 - z_1} \\cdot \\frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "There is a positive real number $x$ not equal to either $\\tfrac{1}{20}$ or $\\tfrac{1}{2}$ such that \\[\\log_{20x} (22x)=\\log_{2x} (202x).\\] The value $\\log_{20x} (22x)$ can be written as $\\log_{10} (\\tfrac{m}{n})$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$112$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "There is a positive real number $x$ not equal to either $\\tfrac{1}{20}$ or $\\tfrac{1}{2}$ such that \\[\\log_{20x} (22x)=\\log_{2x} (202x).\\] The value $\\log_{20x} (22x)$ can be written as $\\log_{10} (\\tfrac{m}{n})$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\\frac m{\\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\\lfloor m+\\sqrt{n}\\rfloor.$ (The notation $\\lfloor x\\rfloor$ denotes the greatest integer that is less than or equal to $x.$ )",
"type": "text"
}
],
"role": "user"
}
] |
$183$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\\frac m{\\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\\lfloor m+\\sqrt{n}\\rfloor.$ (The notation $\\lfloor x\\rfloor$ denotes the greatest integer that is less than or equal to $x.$ )",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nConsider sequences that consist entirely of $A$ 's and $B$ 's and that have the property that every run of consecutive $A$ 's has even length, and every run of consecutive $B$ 's has odd length. Examples of such sequences are $AA$ , $B$ , and $AABAA$ , while $BBAB$ is not such a sequence. How many such sequences have length 14?",
"type": "text"
}
],
"role": "user"
}
] |
$172$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nConsider sequences that consist entirely of $A$ 's and $B$ 's and that have the property that every run of consecutive $A$ 's has even length, and every run of consecutive $B$ 's has odd length. Examples of such sequences are $AA$ , $B$ , and $AABAA$ , while $BBAB$ is not such a sequence. How many such sequences have length 14?",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$ -th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$ . Find the smallest possible value of $n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$89$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$ -th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$ . Find the smallest possible value of $n$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nRhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$ , $Q^{}_{}$ , $R^{}_{}$ , and $S^{}_{}$ are interior points on sides $\\overline{AB}$ , $\\overline{BC}$ , $\\overline{CD}$ , and $\\overline{DA}$ , respectively. It is given that $PB^{}_{}=15$ , $BQ^{}_{}=20$ , $PR^{}_{}=30$ , and $QS^{}_{}=40$ . Let $m/n^{}_{}$ , in lowest terms, denote the perimeter of $ABCD^{}_{}$ . Find $m+n^{}_{}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$677$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nRhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$ , $Q^{}_{}$ , $R^{}_{}$ , and $S^{}_{}$ are interior points on sides $\\overline{AB}$ , $\\overline{BC}$ , $\\overline{CD}$ , and $\\overline{DA}$ , respectively. It is given that $PB^{}_{}=15$ , $BQ^{}_{}=20$ , $PR^{}_{}=30$ , and $QS^{}_{}=40$ . Let $m/n^{}_{}$ , in lowest terms, denote the perimeter of $ABCD^{}_{}$ . Find $m+n^{}_{}$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "There are $2^{10} = 1024$ possible $10$ -letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than $3$ adjacent letters that are identical.",
"type": "text"
}
],
"role": "user"
}
] |
$548$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "There are $2^{10} = 1024$ possible $10$ -letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than $3$ adjacent letters that are identical.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "Let $\\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \\le 10$ and $3y - x \\ge 15$ . When $\\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$ , the volume of the resulting solid is $\\frac {m\\pi}{n\\sqrt {p}}$ , where $m$ , $n$ , and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p$ .",
"type": "text"
}
],
"role": "user"
}
] |
$365$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "Let $\\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \\le 10$ and $3y - x \\ge 15$ . When $\\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$ , the volume of the resulting solid is $\\frac {m\\pi}{n\\sqrt {p}}$ , where $m$ , $n$ , and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAt a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements. ",
"type": "text"
}
],
"role": "user"
}
] |
$88$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nAt a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements. ",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $b \\geq 2$ be an integer. Call a positive integer $n$ $b\\textit{-eautiful}$ if it has exactly two digits when expressed in base $b$ , and these two digits sum to $\\sqrt{n}$ . For example, $81$ is $13$ -eautiful because $81=\\underline{6}$ $\\underline{3}_{13}$ and $6+3=\\sqrt{81}$ . Find the least integer $b\\geq 2$ for which there are more than ten $b$ -eautiful integers.",
"type": "text"
}
],
"role": "user"
}
] |
$211$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nLet $b \\geq 2$ be an integer. Call a positive integer $n$ $b\\textit{-eautiful}$ if it has exactly two digits when expressed in base $b$ , and these two digits sum to $\\sqrt{n}$ . For example, $81$ is $13$ -eautiful because $81=\\underline{6}$ $\\underline{3}_{13}$ and $6+3=\\sqrt{81}$ . Find the least integer $b\\geq 2$ for which there are more than ten $b$ -eautiful integers.",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA piecewise linear function is defined by\n\\[f(x) = \\begin{cases} x & \\text{if } x \\in [-1, 1) \\\\ 2 - x & \\text{if } x \\in [1, 3)\\end{cases}\\]\nand $f(x + 4) = f(x)$ for all real numbers $x.$ The graph of $f(x)$ has the sawtooth pattern depicted below. \n\n\\begin{asy}\nimport graph;\n\nsize(18cm);\nreal f(real x) {\n real x_mod = x - 4 * floor((x+1) / 4);\n if (-1 <= x_mod && x_mod < 1) {\n return x_mod;\n } else if (1 <= x_mod && x_mod < 3) {\n return 2 - x_mod;\n }\n return 0; // Should never reach here\n}\n\n// Set up axes\nreal xmin = -8, xmax = 11, ymin = -2, ymax = 2;\nxaxis(\"\", xmin, -2, Ticks(2.0));\nxaxis(\"\", -2, 2);\nxaxis(\"\", 2, xmax, Ticks(2.0));\nyaxis(\"\", ymin, ymax, Ticks(2.0));\n// Plot the function in segments due to periodicity\nreal step = 0.01;\nfor (real k = -8; k < 7; k += 4) {\n draw(graph(f, k, k + 4, n=(int)((4)/step)),blue);\n}\n draw(graph(f, 8, 11, n=(int)((4)/step)),blue);\n\\end{asy}\n\nThe parabola $x = 34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of these intersection points can be expressed in the form $\\tfrac{a + b\\sqrt c}d,$ where $a, b, c$ and $d$ are positive integers, $a, b,$ and $d$ has greatest common divisor equal to $1,$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$",
"type": "text"
}
],
"role": "user"
}
] |
$259$
|
MathArena/aime_2025_I
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nA piecewise linear function is defined by\n\\[f(x) = \\begin{cases} x & \\text{if } x \\in [-1, 1) \\\\ 2 - x & \\text{if } x \\in [1, 3)\\end{cases}\\]\nand $f(x + 4) = f(x)$ for all real numbers $x.$ The graph of $f(x)$ has the sawtooth pattern depicted below. \n\n\\begin{asy}\nimport graph;\n\nsize(18cm);\nreal f(real x) {\n real x_mod = x - 4 * floor((x+1) / 4);\n if (-1 <= x_mod && x_mod < 1) {\n return x_mod;\n } else if (1 <= x_mod && x_mod < 3) {\n return 2 - x_mod;\n }\n return 0; // Should never reach here\n}\n\n// Set up axes\nreal xmin = -8, xmax = 11, ymin = -2, ymax = 2;\nxaxis(\"\", xmin, -2, Ticks(2.0));\nxaxis(\"\", -2, 2);\nxaxis(\"\", 2, xmax, Ticks(2.0));\nyaxis(\"\", ymin, ymax, Ticks(2.0));\n// Plot the function in segments due to periodicity\nreal step = 0.01;\nfor (real k = -8; k < 7; k += 4) {\n draw(graph(f, k, k + 4, n=(int)((4)/step)),blue);\n}\n draw(graph(f, 8, 11, n=(int)((4)/step)),blue);\n\\end{asy}\n\nThe parabola $x = 34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of these intersection points can be expressed in the form $\\tfrac{a + b\\sqrt c}d,$ where $a, b, c$ and $d$ are positive integers, $a, b,$ and $d$ has greatest common divisor equal to $1,$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
[
{
"content": [
{
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "For positive integers $n$ and $k$ , let $f(n, k)$ be the remainder when $n$ is divided by $k$ , and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$ . Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$512$
|
di-zhang-fdu/AIME_1983_2024
|
[
{
"content": [
{
"path": "",
"text": "A user will ask you to solve a task. You should first draft your thinking process (inner monologue) until you have derived the final answer. Afterwards, write a self-contained summary of your thoughts (i.e. your summary should be succinct but contain all the critical steps you needed to reach the conclusion). You should use Markdown and Latex to format your response. Write both your thoughts and summary in the same language as the task posed by the user.\n\nYour thinking process must follow the template below:\n<think>\nYour thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate a correct answer.\n</think>\n\nHere, provide a concise summary that reflects your reasoning and presents a clear final answer to the user.\n\n\n\nYou will be given a problem. Please reason step by step, and put your final answer within \\boxed{}.",
"type": "text",
"url": null
}
],
"role": "system"
},
{
"content": [
{
"path": "",
"text": "For positive integers $n$ and $k$ , let $f(n, k)$ be the remainder when $n$ is divided by $k$ , and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$ . Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$ .",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
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